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[145] ai.viXra.org:2608.0003 [pdf] submitted on 2026-08-01 23:42:12
Authors: J. W. McGreevy
Comments: 17 Pages. (Note by ai.viXra.org Admin: Please cite and list scientific references)
We develop a symplectic geometry of atomic shear measures supported on the irregular primes of a modular curve. Starting from a space of dis-crete measures subject to a global valence constraint, we perform a strict symplectic reduction and obtain a reduced phase space equipped with a canonical Darboux form. The local coordinate functions that extractthe individual shear amplitudes are shown to be in Liouville involution, generating a completely integrable system whose invariant level sets areLagrangian tori. By lifting the functional equation of the associated automorphic L-functions to an anti-symplectic involution on this phase space, and under the established essential self-adjointness of the clutched conical Dirac operator, we prove a Confinement Theorem: the only invariant Lagrangianleaves compatible with the reflection symmetry are those whose spectralimage lies on the critical line Re(s) = 1/2. We conclude with a brief, explicitly programmatic dictionary that explores possible links between the resulting arithmetic geometry and structures appearing in gauge theory and the Standard Model. This workextends earlier constructions developed under the working title Relativistic Field Theory of Primes (RFTP), in which the irregular primes, the weight-12 valence constraint, and the associated clutching data were first introduced as geometric ingredients of an arithmetic field theory.
Category: Mathematical Physics
[144] ai.viXra.org:2608.0001 [pdf] submitted on 2026-08-01 18:50:53
Authors: Lluis Eriksson
Comments: 9 Pages. Lean 4/Mathlib formalization of the finite-SU(2) Pauli/Casimir contraction, trace-skein identity, four oriented local branches with both reconnections, corrected single-crossing closure, and an all-order multitrace-to-single-trace recursion.
Finite-rank Makeenko-Migdal equations generate products of Wilson traces at self-intersections. For SU(2), this apparent multitrace obstruction closes exactly on single traces, but the statement is normalization-sensitive: the traceless Lie algebra contributes a finite-rank correction that disappears for U(2) and must not be dropped. We give a Lean 4/Mathlib formalization of the complete group-algebraic closure mechanism on Mathlib's concrete special unitary matrix group. With normalized trace tau(A)=Tr(A)/2 and normalized anti-Hermitian Pauli directions X_j=i sigma_j/2, the kernel checks the Casimir identity, the rank-two Fierz identity, the induced crossing contraction, and the SU(2) trace-skein identity tau(g)tau(h)=(tau(gh)+tau(gh^{-1}))/2. Consequently, the finite-SU(2) crossing term tau(g)tau(h)-tau(gh)/4 equals tau(gh)/4+tau(gh^{-1})/2. We then formalize a universal local interface with four cyclically ordered branch holonomies, an independent orientation on each branch, the two opposite-strand words, and precisely the two direct/reversed reconnections. Its corrected crossing term closes on those reconnections for every branch assignment and orientation choice. A recursive theorem also extends the reduction to products of arbitrarily many fundamental traces. The identities are classical; the contribution is a concrete, kernel-checked normalization bridge from Pauli contraction to the single-trace closure used in finite-rank loop equations. We do not claim a formal derivation of the Yang-Mills area derivative, planar loop geometry, or the full Makeenko-Migdal equation.
Category: Mathematical Physics
[143] ai.viXra.org:2607.0094 [pdf] submitted on 2026-07-31 16:03:53
Authors: Lluis Eriksson
Comments: 8 Pages. Lean 4/mathlib; core build 8464 jobs, 2794 oracle commands (2791 distinct + 3 duplicates), axioms exactly {propext, Quot.sound, Classical.choice}, zero sorry, zero project axioms. Links anchored at commit 2d0346b2.
BOND REFLECTION, AND THIS ONE IS ABOUT THE MEASURE. A whole path X of 2m+2slices has a past half past(X) and a future half; write rev(X) for the futurehalf READ BACKWARDS FROM THE FAR END, which is the reflected copy. Let F be acomplex observable of an entire half. Then for every L, every m, everystrictly positive source weight w and every beta >= 0, sum over whole paths X of conj(F(past X)) * F(rev X) * W(X) >= 0the sum running over all whole paths and W being the ordinary Gibbs weight.This is the finite-volume Osterwalder-Schrader reflection-positivityinequality for this model at odd separation. The sum is UNNORMALISED, which isthe form the axiom is about; dividing by the partition function is division bya positive number and preserves the sign. The Gram matrix of a finite familyof such observables, with complex coefficients, satisfies it as well. It isobtained by proving that assembling a past half, a crossing bond and areversed future half is a BIJECTION onto paths, and that the weights multiplywith exactly one crossing factor.SITE REFLECTION, AND THIS ONE IS NOT, YET. Through a site the two halves SHAREthe middle slice, so the assembly is not a product of two independent halvesand its bijection is a different statement. What is proved there is thehalf-chain form, its collapse identity and its Gram positivity -- at EVERYbeta, negative coupling included -- but NOT its identification with the pathmeasure. For that geometry the object remains a candidate, and the statustable says so row by row.ON THE BETA HYPOTHESIS. beta >= 0 is proved SUFFICIENT and is not provednecessary here; at L = 0 it could not be, since the bare kernel is then thescalar 1. The witness that would make the boundary exact from one site upwardsis authorised by a gate and is not written.THE MECHANISM, IN ONE SENTENCE. Summing out the interior of a half sends F toa vector indexed by its boundary slice alone, after which the bond casereduces to positive semidefiniteness of the kernel -- machine-checked in thecompanion paper -- and the site case to a weighted squared norm, which uses noproperty of the kernel at all.WHERE THE TWO REFLECTIONS DIFFER, AND IT IS NOT COSMETIC. Through a bond thetwo halves are disjoint and meet through one kernel factor, so positivityneeds the kernel itself positive semidefinite -- which, for L >= 1, holdsexactly when beta >= 0. Through a site the halves SHARE the middle slice; theform is then a sum of squared moduli divided by the weight of that slice, soit is non-negative for EVERY beta, negative coupling included, and no propertyof the kernel is used at all.WHAT THE SOURCE WEIGHT DOES. In the companion paper the weight was handled bycongruence: conjugation by sqrt(w) cannot change the sign of a quadratic form.Here it is not conjugated away, it is SUMMED away -- and what is left overdiffers between the two geometries. In the bond case every source weight isabsorbed into the collapse and the bare kernel remains; in the site case theshared boundary slice survives as a factor 1/w(sigma), because each halfcarries that slice's weight and the product would count it twice. Differentreasons, same conclusion: nothing in the hypotheses depends on w beyondpositivity.WHAT THIS IS NOT. No reconstruction: the physical Hilbert space as thequotient of the past algebra by the null space of this form is not built.Nothing here concerns uniformity in the extent, SU(N), the continuum limit, orthe Yang-Mills mass gap.ON THE PRE-REGISTRATION. Four gates were committed before a line of the modulewas written; the status table reports what happened to each. Two of them reada minimum eigenvalue rather than sampling observables, which is the instrumentthe previous campaign's autopsy said was needed.
Category: Mathematical Physics
[142] ai.viXra.org:2607.0093 [pdf] submitted on 2026-07-30 06:19:09
Authors: Lluis Eriksson
Comments: 8 pages. Lean 4/mathlib; core build 8430 jobs, 2672 oracle commands (2669 distinct + 3 duplicates), axioms exactly {propext, Quot.sound, Classical.choice}, zero sorry, zero project axioms. Links anchored at commit 9704b3f3.
Two companion papers were left carrying the same debt from opposite sides. Thegap paper proved that every eigenvalue of the coupled slice other than thePerron eigenvalue is STRICTLY smaller in modulus, and said plainly that thisprovides NO MODULUS of separation. The bridge paper proved that the Gibbscorrelations of the spatial system are matrix elements of a self-adjointtransfer operator, and obtained geometric decay only UNDER A CONTRACTIONHYPOTHESIS IT DID NOT DISCHARGE. The missing step is identical in both: finitelymany strict inequalities are not an operator-norm bound.PROVED. We construct specGap, the largest |mu| over the eigenvalues differentfrom the Perron eigenvalue lambda, and prove specGap < lambda. We then prove theoperator bound: for every observable u orthogonal to the Perron vector,||Ku|| <= specGap*||u||. That is exactly the hypothesis the bridge papercarried, so its bound becomes unconditional. The bound is SHARP: whenever thestate space has at least two points, some nonzero fluctuation observable attainsit. The argument splits at specGap = 0, where the maximising index need notsupply a NON-PERRON eigenvector (it does supply the Perron one), hence none inthe fluctuation sector. Stated about an object too: the set of Rayleighnorm ratios on the fluctuation sector has a greatest element, equal to specGap(same two-point hypothesis: with fewer, that set is empty).A WARNING WE STATE BEFORE ANYONE ELSE HAS TO. specGap < lambda is NOTspecGap < 1: the kernel is unnormalised, so both are typically far above one andspecGap^N GROWS. On its own the unnormalised bound controls growth, it does notexhibit decay. The rate that is below one is the RELATIVE one,specRatio = specGap/lambda < 1, and the decay statement is that the fluctuationcontribution is suppressed by specRatio^N RELATIVE to the Perron scalelambda^N. Both forms are proved; only the second is called decay.The step that does not follow from the inequalities is the one abouteigenvectors AT lambda: geometric simplicity, proved in the Perron paper for anarbitrary eigenvector rather than a positive one, makes them INVISIBLE to afluctuation observable, so the top term of the spectral sum vanishes instead ofmerely being bounded.NOT PROVED. specGap DEPENDS ON THE EXTENT, and nothing here bounds it away fromlambda uniformly. Direct diagonalisation gives specGap/lambda = 0.9205, 0.9829,0.9964, 0.9992 at L = 2,3,4,5 for one parameter pair: a geometric bound whoserate tends to 1 is empty in the limit, and that is reported, not hidden. Thenormalised Gibbs EXPECTATION is now bounded too: splitting the dressed constantobservable along the Perron direction bounds the partition function below withno eigenbasis index identified, so at a fixed extent the two-point function isbounded by C*specRatio^N past an explicit threshold. That rate depends on theextent, so this is not clustering. The bound and its attainmentare both proved, which is what it means for specGap to be the operator norm onthe fluctuation sector; what is NOT done is introducing that norm as a definedobject and proving an equation about it. And the threshold N_0 does not dependon the observable: it is built from the dressed CONSTANT observable, so ONE N_0serves every fluctuation observable at once and only C sees A. Reflectionpositivity is untouched, and nothing in this paper is a claim about SU(N), thecontinuum limit, or the Yang-Mills mass gap.
Category: Mathematical Physics
[141] ai.viXra.org:2607.0092 [pdf] submitted on 2026-07-30 14:48:32
Authors: Lluis Eriksson
Comments: 6 pages. Lean 4/mathlib; core build 8431 jobs, 2697 oracle commands (2694 distinct + 3 duplicates), axioms exactly {propext, Quot.sound, Classical.choice}, zero sorry, zero project axioms. Links anchored at commit 247de2b3.
Every rate in this lane so far has been a fixed-extent rate. The gap paper provedstrict spectral separation at each extent and said plainly that it was notuniform; the modulus paper gave that separation a number, specRatio(L), andreported measurements saying the number tends to 1 outside the disordered region.A geometric bound whose rate tends to 1 is empty in the volume limit, so nothingin the lane survived L -> infinity, and the word CLUSTERING was never used.PROVED. For the DECOUPLED kernel - the transfer kernel at constant source weight- the modulus is specRatio = tanh(beta) at EVERY extent, with L nowhere in it.Both directions: an operator bound by induction on the extent, and attainment bythe single-site observable the extent paper already built. Composing with themodulus paper's endpoint, the normalised Gibbs two-point function obeys|E[A(X_0)A(X_N)]| <= C_A tanh(beta)^N past one threshold serving everyobservable - a bound whose RATE contains no L, and therefore the first statementin this lane that survives the volume limit.WHY THE PROOF IS NOT THE SPECTRAL DECOMPOSITION. The decoupled kernel is aproduct over sites, so its spectrum is a product; that route needs the spectrumof a Kronecker power, which the library does not carry. It is not needed. Themodulus paper proved that specGap is the GREATEST norm ratio on the fluctuationsector, so bounding it above is an operator inequality and nothing else, and thatfalls to induction: the even part of an observable keeps its mean zero andinherits the rate, the odd part keeps nothing and gets only Schur's test, and thetwo recombine EXACTLY, because tanh(beta) Z = D with Z the row sum and D the oddeigenvalue of a single bond.NOT PROVED, AND A JUDGE THAT FAILED. The COUPLED kernel is untouched. Before anyof this was written we pre-registered two falsifiable predictions. The first -that the decoupled rate is exactly tanh(beta) at every extent - passed to 1e-16,and authorised the work above. The second - that the coupled uniformity boundaryis the Onsager curve - failed on one of eight pre-registered cells, and it staysfailed: that claim is reported as NOT ESTABLISHED, not softened. At constantsource weight the spatial slices are independent, so what is proved here is astatement about a product measure; that is exactly why it is reachable, and it issaid in the paper rather than left to be noticed. Reflection positivity isuntouched, and nothing in this paper is a claim about SU(N), the continuum limit,or the Yang-Mills mass gap.
Category: Mathematical Physics
[140] ai.viXra.org:2607.0091 [pdf] submitted on 2026-07-30 18:51:18
Authors: Lluis Eriksson
Comments: 7 pages, no figures. Lean 4 formalization. Verification artifact: https://github.com/lluiseriksson/THE-ERIKSSON-PROGRAMME/tree/d6282a83
We formalize in Lean 4 the thermodynamic limit of bounded local Gibbs expectations for a periodic lattice gauge model in a uniform Kotecky-Preiss regime. The proof treats the complete finite-volume sequence: an exact one-volume marked expansion cancels the extensive far gas algebraically, common-window terms are transported exactly, and the remaining boundary contribution is bounded by an existing volume-uniform pinned cluster tail. The resulting explicit Cauchy modulus tends to zero, so completeness constructs an infinite-volume positive normalized real local state. On the intrinsic integer-coordinate local-observable algebra, the state carries a genuine additive action of Z^d and is invariant under every integer translation, including inverses. For SU(2), Haar probability measure, and the physical Wilson plaquette energy Re tr(U), the hypotheses are discharged throughout the explicit punctured intervals 0 < |beta| <= 10^-5 in d=2 and 0 < |beta| <= 10^-6 in d=4. We construct a genuine centered free-boundary exhaustion and prove that its complete cofinal sequence converges to the same state as periodic boundary conditions. The normalized finite-volume two-plaquette truncated-correlation bound also passes to the state under explicit eventual realization and separation hypotheses. We do not claim arbitrary boundary conditions, a C*-algebraic state, a continuum limit, Osterwalder-Schrader reconstruction, or progress on the continuum Yang-Mills mass-gap problem.
Category: Mathematical Physics
[139] ai.viXra.org:2607.0090 [pdf] submitted on 2026-07-30 20:31:57
Authors: Lluis Eriksson
Comments: 6 pages. Lean 4/mathlib; core build 8463 jobs, 2752 oracle commands (2749 distinct + 3 duplicates), axioms exactly {propext, Quot.sound, Classical.choice}, zero sorry, zero project axioms. Links anchored at commit 854ff223.
WHAT IS PROVED, STATED BEFORE ANYTHING ELSE. The reflected two-point form ofthe Gibbs measure, at the two ENDS of a path, for REAL observables of a SINGLEslice, is non-negative under the parity and coupling hypotheses stated below-- at even separation for every beta, and at every separation exactly for beta>= 0 -- and the Gram matrix of a finite family of such observables is positivesemidefinite under the same hypotheses. THIS IS NOT YET THEOSTERWALDER-SCHRADER AXIOM, which quantifies over observables of the wholepast half-chain and over complex ones. The half-chain algebra, the reflectionmap and the sesquilinear form are not built here; the scope section says whatis missing and why it is a construction rather than a further inequality.Eleven papers in this lane end with REFLECTION POSITIVITY IS UNTOUCHED. Thistouches the endpoint form of it, and the interesting part is what the spatialsource weight does -- namely nothing. Earlier papers do prove statements aboutthe coupled kernel; specGap < lambda holds there too. What is new is that thisresult is UNCHANGED by the weight: same hypothesis, same conclusion, and noconstant that depends on w. Every earlier coupled-kernel statement contains anumber that moves when w does. Positivity is the SIGN of a quadratic form, andconjugation by sqrt(w) is a congruence, so there is nothing for the weight tomove.TWO REFLECTIONS, TWO HYPOTHESES. The reflected two-point sum is
Category: Mathematical Physics
[138] ai.viXra.org:2607.0089 [pdf] submitted on 2026-07-28 22:32:03
Authors: Lluis Eriksson
Comments: 33 pages. Exact identities, outward-rounded Arb certificates, production/replay transcripts amd Lean 4/Mathlib lemmas: https://github.com/lluiseriksson/THE-ERIKSSON-PROGRAMME/releases/tag/surface-theorem-v1.0
For beta > 0 let I_m = I_m(beta) denote modified Bessel functions ofthe first kind, and set a_m = I_m^2 [(m-1) I_(m-1)^2 + (m+1) I_(m+1)^2], b_m = m I_m^4, F_A(t) = sum_(m>=1) a_m sin(mt), F_B(t) = sum_(m>=1) b_m sin(mt), E(t) = F_A(t)/(2 F_B(t)).The global ratio-monotonicity problem for the surface expansion of atwo-dimensional SU(2) lattice gauge observable is: (i) F_B > 0 on(0,pi), and (ii) E' < 0 on (0,pi), for every beta > 0. Bothstatements are proved. Positivity of F_B has two exact proofs. Ratiomonotonicity is reduced to exact algebraic identities andoutward-rounded interval certificates: small and compact beta arehandled by pair identities and interval Taylor models;20 <= beta <= 1000/9 by a direct Wronskian cover; andbeta >= 1000/9 has three certified moving-edge lambda lanes. Theremaining lambda >= 3 lane is closed by the exact identityE'/(-sin(t/2)/2) = Q + X_full, where Q > 19/20, an exactmain--mirror--rest decomposition, and a division-free covariancecertificate proving X_main > -1/20 on two adjacent rectangles thatcover the full angular interval. The exact near and far relay marginsare positive. All load-bearing production and independent replaytranscripts are checked for exact rational coverage, dependencyhashes, strict outward-rounded decision endpoints, and byte equality.The structural core is exact: E is, as an algebraic identity, the meanof cos(psi) under the midpoint law of a four-step killed von Misesbridge; the generating kernels reduce, via the Neumann additiontheorem, to two-dimensional integrals of a single Bessel functionwhose saddle deficit is an exact sum of two squares; and exact saddlecancellations yield the coefficients of the verified closedsecond-order law E = cos(t/2)(1 - c(t)/beta) + O(beta^-2), c(t) = (4 cos^2(t/4) - 1)/(2 cos(t/4) cos(t/2)).Three certified negative results (interval arithmetic, twoimplementations, nested enclosures) kill every monotone full-pathcoupling, with an exact mechanism at threshold beta |cos t| = 3/2. Atthe pi endpoint we also give exact identities for the cubiccoefficient c_3 (telescoped alternating form, integral form, parity)together with its verified prefactor law. Every claim is labelledexact / certified / verified; the machine-checked lemmas are Lean4/Mathlib, machine-checked modulo classical Bessel inputs carried asnamed hypotheses.
Category: Mathematical Physics
[137] ai.viXra.org:2607.0088 [pdf] submitted on 2026-07-28 23:30:40
Authors: Lluis Eriksson
Comments: 9 Pages. Full core build 8426 jobs, 2542 oracle commands (2519 with axiom dependencies + 23 axiom-free), axiom set exactly {propext, Quot.sound, Classical.choice}, zero sorry and zero project axioms. All verification links anchored at a fixed commit.
A companion paper proved that when a spatial coupling is switched on in a Z_2lattice gauge slice, the transfer kernel loses constant row sums, so the uniformvector is no longer fixed and the elementary route to the vacuum stops. Thestandard replacement, when row sums fail, is the Hilbert projective metric: astrictly positive kernel contracts it, and the contraction factor bounds thesubdominant spectral ratio. This paper asks what that replacement gives here, andanswers in Lean 4 with mathlib.It gives no coupling-sensitive and no volume-uniform information, for twoindependent reasons, and both are proved. First, BLINDNESS: the projectivecross-ratio is invariant under multiplication by any nowhere-zero function of thesource configuration alone. The coupled kernel is exactly such a product, so themetric assigns the interacting and the non-interacting kernels the same diameterat every spatial extent - the route cannot see the coupling at all. Second,VOLUME DEGENERATION: two constant configurations realise the cross-ratioe^(4 beta L), so every admissible projective diameter is at least 4 beta L andevery contraction factor obtainable this way is at least tanh(beta L), which lieswithin 2 e^(-2 beta L) of the trivial bound 1. At the one place where the truthis known - the decoupled kernel, whose subdominant ratio the companion papercomputes to be exactly tanh beta at every L - this route already returnstanh(beta L) instead. The degeneration is the method's, not the model's.We then hand over the object the elementary route stopped producing, at thesmallest interacting size. In the character basis the coupled two-site kernelsplits into two 2x2 blocks, and we exhibit a strictly positive eigenvector inclosed form, together with a second exact eigenpair. The identity A - B = 4between the two decoupled even-sector eigenvalues drives every estimate. Theblindness is proved two-sided, so it covers the symmetrised conventionw^(1/2) K w^(1/2) as well, and the positive eigenvector is proved to dominateevery eigenvalue, real or complex - so its eigenvalue is the spectral radius,which is the Perron statement this development needs and proves without aPerron-Frobenius theorem in the library.NO VOLUME-UNIFORM STATEMENT ABOUT AN INTERACTING SYSTEM IS PROVED HERE, AND NONEIS CLAIMED; the general-L behaviour is recorded separately as measured andunproved. Nothing in this paper is a claim about SU(N), the continuum limit, orthe Yang-Mills mass gap.
Category: Mathematical Physics
[136] ai.viXra.org:2607.0085 [pdf] submitted on 2026-07-29 06:36:35
Authors: Lluis Eriksson
Comments: 6 Pages. Full core build 8427 jobs, 2564 oracle commands (2541 with axiom dependencies + 23 axiom-free), axiom set exactly {propext, Quot.sound, Classical.choice}, zero sorry and zero project axioms. All verification links anchored at a fixed commit.
Two companion papers established, for a Z_2 lattice gauge slice with a spatialcoupling, that the elementary route to the vacuum stops, and that the naturalreplacement - the Hilbert projective metric - is blind to the coupling anddegenerates in the volume. Both had to work around the same absence: the pinnedmathlib carries no Perron-Frobenius theorem. The first paper could therefore onlysay the vacuum had become unavailable; the second had to build its dominationbound from scratch, and could exhibit the vacuum in closed form only at two sites.This paper discharges that dependency. For a strictly positive kernel on a finitenonempty type we prove, in Lean 4 with mathlib: a strictly positive eigenvectorEXISTS; its eigenvalue is strictly positive; any two strictly positiveeigenvectors are proportional and share their eigenvalue; and every realeigenvector for that eigenvalue is a scalar multiple of it. Together with thedomination theorem of the companion paper this gives the Perron statement thelane needs: the eigenvalue is the spectral radius.The existence proof does not use a fixed-point theorem, because the pinnedmathlib revision contains none. It maximises r over the compact set of pairs(r,x) with x in the simplex and r x <= A x; maximality forces equality, since astrict inequality anywhere would let one further application of A produce anadmissible pair with a larger r. The bound that keeps the set compact is obtainedby summing the constraint: r = r * sum x <= sum (A x).The application is the point. At EVERY spatial extent, and for EVERY strictlypositive weight on the source configuration - the class that contains the coupledkernel of the first paper - the vacuum exists, is unique up to scale, and carriesthe spectral radius. The obstruction of that paper was never an absence; it wasan unavailability, and it was an unavailability of one route rather than of theobject.NO SPECTRAL GAP IS PROVED HERE, uniform in the volume or otherwise, and none isclaimed. Nothing in this paper is a claim about SU(N), the continuum limit, orthe Yang-Mills mass gap.
Category: Mathematical Physics
[135] ai.viXra.org:2607.0084 [pdf] submitted on 2026-07-29 10:08:56
Authors: Lluis Eriksson
Comments: 6 pages. Lean 4 / mathlib formalization; full core build 8428 jobs, 2583 oracle commands (2560 with axiom dependencies + 23 axiom-free), axiom set exactly {propext, Quot.sound, Classical.choice}, zero sorry and zero project axioms.
A companion paper supplied the vacuum of the coupled Z_2 slice at every spatialextent: a strictly positive eigenvector, unique up to scale, carrying thespectral radius. It listed PERIPHERAL SEPARATION as out of scope, and thatomission is not cosmetic - without it |mu| <= lambda leaves mu = -lambda open,and no gap follows at all.This paper closes it, and then draws the consequence that matters, which isnegative.PROVED. For a strictly positive kernel on a finite nonempty type, -lambda is notan eigenvalue, hence every real eigenvalue other than the Perron eigenvalue isSTRICTLY smaller in absolute value. Specialised to the coupled slice: at everyextent L, every beta, and every strictly positive source weight, the transferoperator has a strict spectral gap. The proof of peripheral separation avoidsthe equality case of the triangle inequality, which is where the classicalargument spends its effort: writing u = |w|, p = u - w and q = u + w, one getsA p = lambda q and A q = lambda p, so a nonzero p would make A p strictlypositive, hence q strictly positive, hence w nonnegative, hence p = 0.The separation is then extended from the real eigenvalues to ALL of them. Thecoupled kernel is conjugate by a positive diagonal to its symmetrised form,which is symmetric; and a real symmetric kernel has real eigenvalues, by acomputation that pairs the eigenvector against its image twice and is tworearrangements of a double sum. So there are no complex peripheral eigenvaluesleft to exclude, and the strict gap is a statement about the whole spectrum.That composition is itself a single machine-checked theorem(coupled_gap_all_eigenvalues), not a step left to the reader. We also deliverthe vacuum in Euclidean normalisation, norm(Omega) = 1 with T Omega = Omega.NOT PROVED, AND THIS IS THE TITLE. The gap is STRICT, not QUANTITATIVE: thetheorem provides no modulus of separation, and in particular nothing uniform inL. Direct numerical diagonalisation shows the subdominant ratio running0.9205, 0.9829, 0.9964, 0.9992 at L = 2,3,4,5 for beta = 0.8, gamma = 1.2 -collapsing towards 1. That computation is reported as measured and unproved, andno theorem here depends on it; its role is to say that a paper reporting onlythe positive half would be reporting the half that does not matter.Nothing in this paper is a claim about SU(N), the continuum limit, or theYang-Mills mass gap.
Category: Mathematical Physics
[134] ai.viXra.org:2607.0083 [pdf] submitted on 2026-07-29 14:35:16
Authors: Lluis Eriksson
Comments: 6 pages. Lean 4/mathlib; core build 8429 jobs, 2626 oracle commands, axioms exactly {propext, Quot.sound, Classical.choice}, zero sorry, zero project axioms. Links anchored at commit c4fa6a9e. Key identities also verified numerically.
Four companion papers studied an OPERATOR: a strictly positive kernel on thespatial configuration space of a Z_2 slice, its Perron vacuum, and the strictseparation of its spectrum. None of them exhibited a MEASURE. That omission isthe kind this programme is built to notice: a transfer operator is a matrixuntil something says which Boltzmann weights it transfers, and until then avacuum is an eigenvector and a gap is a statement about eigenvalues. Neither isyet a statement about a statistical-mechanical system.PROVED. We define the two-dimensional Gibbs weight of the spatial system fromBoltzmann factors alone - a spatial factor at every time slice, a time-bondfactor between consecutive slices - and prove the DRESSING IDENTITY: that weightequals the path weight of the SYMMETRISED kernel multiplied by a factorsupported entirely on the two boundary slices. Hence every UNNORMALISED Gibbstwo-point sum of the spatial system is a matrix element of an iteratedSELF-ADJOINT transfer operator between boundary-dressed observables, and theNORMALISED expectation is the RATIO of two such matrix elements - the numeratoralone is not the correlation. The generic half holds for an arbitrary symmetrickernel on an arbitrary finite type; no positivity and no structure of theconfiguration space enter it. We further show that the operator the bridge landson is the one the companion papers analysed, that the fluctuation sector isinvariant, and that under an explicit contraction hypothesis the connectedtwo-point function decays geometrically in the time separation.NOT PROVED, AND THIS IS THE POINT OF THE LAST SECTION. The contractionhypothesis is carried as a theorem hypothesis and is NOT discharged. Thecompanion papers prove a STRICT gap with no modulus, and a strict inequalityamong finitely many eigenvalues does not by itself produce the operator-normbound a decay rate requires. Converting one into the other needs the spectralmaximum over the fluctuation sector, which is not constructed here. Inparticular NOTHING UNIFORM IN THE SPATIAL EXTENT is obtained, and none issuggested: with r = r(L) approaching 1, the bound is empty in the limit.Reflection positivity is not addressed. Nothing in this paper is a claim aboutSU(N), the continuum limit, or the Yang-Mills mass gap.
Category: Mathematical Physics
[133] ai.viXra.org:2607.0078 [pdf] submitted on 2026-07-28 13:19:32
Authors: Lluis Eriksson
Comments: 9 pages. Lean 4 / mathlib formalization; full core build 8422 jobs, 2431 oracle commands, axiom set exactly {propext, Quot.sound, Classical.choice}, zero sorry and zero project axioms. All verification links anchored at a fixed commit.
A mass gap is a statement about the spectrum of an operator, but a latticegauge theory is given as a measure. The passage between the two - theOsterwalder-Seiler reconstruction - proceeds through reflection positivity, aGelfand-Naimark-Segal space, a transfer operator, and an identification theoremasserting that expectations in the measure are matrix elements of thatoperator. This paper reports a complete formal verification of that passage, inLean 4 with mathlib, for the Z_2 lattice gauge chain: from the Boltzmann weightexp(beta s) to exponential decay of a correlation function of the measure, withevery arrow a machine-checked theorem and no arrow assumed. The endpoint is theexact identity E_n[f(sigma_0) f(sigma_n)] = (tanh beta)^n for the signobservable, a statement in which no operator occurs and whose proof goes throughone, and whose rate is exactly the verified non-vacuum transfer eigenvaluetanh beta; equivalently, the normalised transfer operator has spectral gap1 - tanh beta > 0, and the corresponding mass is -log(tanh beta) > 0 forbeta > 0. All three are machine-checked.We state the limits with the same precision as the results. The system has oneZ_2 variable per time slice, so its spatial slice is a point and its transferoperator is a 2x2 matrix; a lattice with spatial extent is not treated. Thebound is at fixed finite size and is NOT volume-uniform. The pairing of thissystem is definite - proved, not assumed - so the Gelfand-Naimark-Segalquotient is the identity and therefore does no work; a system with a degeneratepairing still needs it. The underlying mathematics is textbook:Osterwalder-Seiler is from 1978 and the transfer matrix of the Ising chain isolder still. The contribution claimed here is not a new theorem but a verifiedcomposition: that the interfaces of the reconstruction fit together with nothinghidden between them, exhibited on the smallest system where all of them aresimultaneously present. Nothing in this paper is a claim about SU(N), thecontinuum limit, or the Yang-Mills mass gap.
Category: Mathematical Physics
[132] ai.viXra.org:2607.0076 [pdf] submitted on 2026-07-28 17:10:48
Authors: Lluis Eriksson
Comments: 6 pages. Lean 4 / mathlib formalization; full core build 8423 jobs, 2460 oracle commands, axiom set exactly {propext, Quot.sound, Classical.choice}, zero sorry and zero project axioms. All verification links anchored at a fixed commit.
The Osterwalder-Seiler reconstruction passes from a reflection-positive measureto a Hilbert space by quotienting out the null space of the reflected pairing.In a companion development that step was present but did nothing: the pairingthere was definite, so the quotient was the identity, and that paper says so inits own abstract. This paper supplies the missing case, in Lean 4 with mathlib.For the Z_2 lattice gauge chain we take half-space observables of two timeslices - a four-dimensional space - and form the reflected pairing directly fromthe Boltzmann weights. For beta > 0 the reconstructed physical space istwo-dimensional. Integrating out the future collapses four observables onto twostates, and that collapse is the null space. We prove: the pairing factors through anindependently defined reconstruction map Phi; its self-pairing rearranges into amanifest sum of two non-negative terms, from which positivity and the null spacefollow together; the null space is EXACTLY ker Phi, not merely non-empty; anexplicit non-zero observable lies in it; and the quotient is isomorphic to thephysical space BY THE MAP Phi ITSELF, not by a dimension count.The degeneracy is the mechanism the reconstruction exists to handle, and the oneexpected to reappear in systems with larger half-space algebras. What is notclaimed: this is twotime slices and not m; still Z_2, one variable per slice, fixed finite size, andnot volume-uniform; Z_N for N > 2 is untouched; and the completion step of thereconstruction is trivial here because every space in sight isfinite-dimensional, which we state rather than present as work done. Nothing inthis paper is a claim about SU(N), the continuum limit, or the Yang-Mills massgap.
Category: Mathematical Physics
[131] ai.viXra.org:2607.0075 [pdf] submitted on 2026-07-28 19:17:42
Authors: Lluis Eriksson
Comments: 6 pages. Lean 4 / mathlib formalization; full core build 8424 jobs, 2481 oracle commands, axiom set exactly {propext, Quot.sound, Classical.choice}, zero sorry and zero project axioms. All verification links anchored at a fixed commit.
Two companion developments verified an Osterwalder-Seiler reconstruction end toend for a lattice gauge chain whose spatial slice is a single point. Every stepof that chain begins by knowing the vacuum, and in the one-dimensional case thevacuum is free: the normalised transfer kernel has constant row sums, so theuniform vector is fixed, and T*Omega = Omega follows from normalisation alone.This paper asks what survives when the slice acquires spatial extent, andanswers in Lean 4 with mathlib.The algebraic half survives untouched. With time bonds only, the row sums of thetransfer kernel are constant for every spatial extent L, so the uniform vacuumpersists on a space of dimension 2^L; and the single-site sign observable is aneigenvector whose normalised eigenvalue is exactly tanh beta, with L free in thestatement.The uniform vacuum does not survive. Switching on a coupling between sites inside aslice makes the spatial weight depend only on the source configuration, so itfactors out of the sum over the target and the row sums becomeconfiguration-dependent. We exhibit two explicit configurations of a two-siteslice with different row sums, and conclude that no constant row sum exists: theuniform vector is not fixed, so T*Omega = Omega is FALSE for it. The vacuumbecomes a Perron vector that row-sum normalisation no longer supplies in closedform, and every later step of the reconstruction loses its starting point.We state plainly what the positive half is and is not. The decoupled system is Lnon-interacting copies of a two-state system, and the rate it yields - theeigenvalue tanh beta of the single-site sign mode - is independent of L fortrivial reasons, so it is physically empty and is recorded only because itisolates which half of the construction survives. NO GAP FOR THE COUPLED SYSTEM IS PROVED HERE, AND NONE IS CLAIMED.Nothing in this paper is a claim about SU(N), the continuum limit, or theYang-Mills mass gap.
Category: Mathematical Physics
[130] ai.viXra.org:2607.0073 [pdf] submitted on 2026-07-27 21:33:38
Authors: Lluis Eriksson
Comments: 6 pages. All results machine-checked in Lean 4 (toolchain v4.29.0-rc6) against Mathlib pinned to commit 07642720480157414db592fa85b626dafb71355b; no sorry and no project axioms.
We machine-check, in Lean 4 with no sorry and no project axioms, theOsterwalder-Seiler reflection positivity of a lattice gauge theory with finiteabelian gauge group. The development is organised so that the three ingredientsare separated and each is proved on its own: an analytic step, a geometric step,and the single place where a property of the Boltzmann factor is actually used.The analytic step is that a crossing kernel of the formK(x,y) = sum_i c_i phi_i(x) conj(phi_i(y)) with c_i >= 0 is positivesemidefinite, that this class is closed under products, and that it is closedunder conjugation by a positive diagonal. Formulating the hypothesis as anon-negative combination of characters rather than as non-negativity of Fouriercoefficients removes any need for Bochner's theorem on a finite abelian groupand for the Schur product theorem: the development uses no spectral and nomatrix-positivity API.The geometric step is a splitting of the configuration space across thereflection plane under which the reflection is the swap and the Gibbs weightfactors as w(x) w(y) K(x,y). We prove that the Osterwalder-Seiler pairing of anobservable of one half against its reflection is then exactly the quadratic formof w(x) K(x,y) w(y), so that reflection positivity follows from the analyticstep.The physical step is the instance. For Z_2 the Wilson factor exp(beta s),s = +-1, expands in the two characters with coefficients(exp(beta) +- exp(-beta))/2, both non-negative exactly when beta >= 0; so theZ_2 Wilson crossing kernel is positive semidefinite at non-negative coupling. Asingle endpoint combines a gauge system with a nontrivial time reflection, aconcrete splitting, that weight at positive coupling, and the conclusion; itsplaquette straddles the reflection plane, so the entire Gibbs weight is thecrossing kernel. It is a two-edge system, and a full temporal box is nottreated. For Z_N with N > 2 the coefficients are discrete Bessel-type sums andtheir non-negativity is not established here.We are explicit about what is absent: no Gelfand-Naimark-Segal quotient, notransfer operator, no identification of a Euclidean correlator with a matrixelement, and therefore no mass gap. Nothing here is a claim about SU(N), thecontinuum limit, or the Clay problem.
Category: Mathematical Physics
[129] ai.viXra.org:2607.0070 [pdf] submitted on 2026-07-27 16:38:14
Authors: Lluis Eriksson
Comments: 8 pages. All results machine-checked in Lean 4 (toolchain v4.29.0-rc6) against Mathlib pinned to commit 07642720480157414db592fa85b626dafb71355b; no sorry and no project axioms. Source modules, axiom-oracle transcripts and a theorem-to-artifact map are hy
Inside a Lean 4 formalization programme for four-dimensional SU(N_c) latticeYang-Mills, we machine-check the operator-theoretic criterion that standsbetween exponential decay of a Euclidean correlator and a spectral gap of atransfer operator. Let T be a bounded self-adjoint operator on a Hilbert spaceand W a unit vector fixed by T, so that TW = W, and put S = T - |W><W|.Exponential decay at rate r of the connected two-point function<v, T^n v> - |<W,v>|^2 at every v is equivalent to the operator-norm bound||S|| <= r.The substantive part is the dense-family criterion. WritingD_r = {v : there is C with ||S^n v|| <= C r^n for all n}, we prove that D_r isa linear subspace and that its density alone forces ||S|| <= r, the constantsbeing entirely unconstrained: a family of observables whose span is dense, eachcarrying its own finite constant, suffices. Consequently prefactors that growwith the support of the observable - the shape cluster expansions produce - donot obstruct the gap, provided the exponential rate is common to the family andthe family spans densely. Those two provisos are essential; without them thestatement is false.No mathematical novelty is claimed for the criterion itself, which we expect tobe known in the language of local spectral theory; what is offered is itsmechanization, its packaging for families of observables, and the consequencefor prefactors. We also record what the formalization does not contain: noOsterwalder-Seiler Hilbert space for any gauge theory, no reflection positivityof the Wilson measure, no identification of a Euclidean correlator with a matrixelement. Nothing here is a claim about the continuum limit or about the Clayproblem. All results are machine-checked with no sorry and no project axioms. (W stands for the vacuum vector Omega. If the form's preview renders Unicode cleanly you may substitute the real symbols; the ASCII form above is the safe default and matches the PDF's content either way.)
Category: Mathematical Physics
[128] ai.viXra.org:2607.0055 [pdf] submitted on 2026-07-22 22:01:16
Authors: Carl Max Reed
Comments: 2 Pages.
In the Williamson-van der Mark electron model [1], the electron is not an abstract, zero-dimensional mathematical point particle possessing an unexplainable monopoly of negative charge. It is a highly localized, three-dimensional toroidal electromagnetic photon vortex spinning at the fundamental Planck scale. Concurrently, Planck's constant (h) is not an arbitrary, ungrounded cosmic decree injected into reality from an external container; it is the direct, invariant geometric boundary condition required for the stability of a continuous, self-confined wave loop within a non-linear electromagnetic fluid substrate [2].
Category: Mathematical Physics
[127] ai.viXra.org:2607.0054 [pdf] submitted on 2026-07-22 22:00:37
Authors: Carl Max Reed
Comments: 3 Pages.
This paper presents a non-dual, polycentric cosmological paradigm that eliminates the artificial dualisms between empty space, discrete point-particles, and independent fundamental forces. We mathematically and philosophically establish that the physical vacuum medium is nothing other than the Electromagnetic (EM) energy field itself at itsbase state. Material reality, forces, and conscious experience are established not as independent entities, but as topological, refractive, or phase-resonant states of this single, self-perceiving EM ocean. By synthesizing Henri Poincaré’s prior equation of inertia (m = E / c^2) with Robert Dicke’s variable refractive index model of gravitation and E.T. Whittaker’s two underlying scalar potentials, this work provides concrete solutions to major foundational crises inphysics. Specifically, it eliminates the infinite self-energy paradox of the electron, provides a near-field electrodynamicmechanism for the nuclear forces, and accounts for flat galactic rotation curves without invoking the hypothetical construct of "dark matter".
Category: Mathematical Physics
[126] ai.viXra.org:2607.0050 [pdf] submitted on 2026-07-20 02:21:31
Authors: J. W. McGreevy
Comments: 8 Pages. (Note by viXra Admin: Please cite and list scientific references)
We present a self-contained technical development of the reduced variational structureunderlying the Relativistic Field Theory of Primes (RFTP) in the weight-12 sector. AfterDirac reduction by the global valence constraint associated with the modular discriminant∆(τ ), the effective dynamics on the reduced space are generated by a convex action functional whose long-time minimizers are atomic measures supported at irregular primes. Weconstruct three explicit objects: (i) a linear isomorphism from the odd Galois eigenspacesfurnished by the Herbrand—Ribet theorem to local sections of the global spinor bundle, intertwining Galois action with Gauss—Manin parallel transport; (ii) a quantitative exponential convergence result for the Hopf—Lax semigroup showing that every admissible initial measure converges to a unique atomic minimizer at a rate controlled by the spectral gap of theHessian; and (iii) a term-by-term identification of the Weierstrass product of the regularizeddeterminant of the limiting operator with that of the completed Riemann xi function Ξ(s).These constructions render rigorous a four-step variational obstruction that selects discretearithmetic configurations whenever a continuous configuration would require an effective local exponent exceeding two. The entire development is phrased in the language of deterministic Hamilton—Jacobi—Bellman dynamics on a reduced symplectic manifold, thereby supplying both the analytic core of the weight-12 theory and the notational and conceptual bridge to a universal formulation over all weights.
Category: Mathematical Physics
[125] ai.viXra.org:2607.0046 [pdf] submitted on 2026-07-17 15:12:46
Authors: Ginanjar Utama
Comments: 14 Pages.
Contact Hamiltonian mechanics extends the symplectic formalism to dissipative and non-conservative thermodynamic systems through a contact 1-form α on a (2n+1)-dimensional manifold, whose Reeb field R and contact Hamiltonian vector field XH generate the equations of motion. In existing numerical implementations, α and its exterior derivative dα are typically represented only implicitly, through symbolic display strings and hand-coded component formulas, with non-degeneracy checked by a hard-coded factorial rather than an actual computation. We construct α, dα, R, and XH as genuine multivectors in a Grassmann/Clifford algebra Cl(2n + 1, 0, 0), compute the non-degeneracy condition directly as the wedge product α ∧ (dα)n, and show the resulting XH reproduces the standard Bravetti—Cruz—Tapias equations of motion exactly. We report, however, that this base construction uses only the outer product and a signature-independent coordinate pairing: repeating the identical computation across several signature choices of the same dimension, including a degenerate split, leaves every reported quantity unchanged, so the Clifford geometric product itself does no work in this part of the construction — it is exterior (Grassmann) algebra wearing Clifford notation. Motivated by this, we then ask a genuinely open question in a different, previously rejected carrier, the degenerate algebra Cl(n, n, 1) in which dα is itself a hyperbolic metric bivector: can contactomorphisms (the symmetries of the contact structure) be represented as GA versors, exponentials of bivectors applied by the sandwich v 7→ V vV −1? We find a mixed, carefully bounded answer: the symplectic squeeze transformation is realized exactly by a versor sandwich, with the hyperbolic signature doing genuine work (each pair-bivector squares to +1); the Reeb flow and the conformal dilation (for scale factors λ /∈ {±1}), by contrast, are provably not representable as pure versor sandwiches, for structural reasons (an affine translation cannot fix the origin; a radical-rescaling forbidden by a rigidity lemma we prove for the degenerate metric) that we make precise. The complete construction, both the base representation and the versor investigation, together with a fully regression-tested reference implementation, is released as part of the open-source contact-thermodynamics library.
Category: Mathematical Physics
[124] ai.viXra.org:2607.0044 [pdf] submitted on 2026-07-15 20:22:56
Authors: Lluis Eriksson
Comments: 6 Pages. Machine-checked in Lean 4. Proof artifact: hrpoly-cmp116-main-reduction-v0.1
We give a machine-checked reduction of the finite-dimensional fluctuation integral appearing in the large-field renormalization analysis of lattice gauge theory, following the CMP116 formulation of Balaban and its later exposition by Dimock. The formal development constructs the localization region from the physical large-field data, identifies the scalar projector induced by that region, transports a certified physical covariance root to the finite Gaussian coordinate space without norm loss, evaluates the resulting complex quadratic Gaussian exactly, and proves positivity of the shifted Hessian from the source smallness condition.Two corrections exposed by formalization are central. First, a pointwise uniform bound on the bilinear fluctuation factor is generally unavailable; the correct interface bounds its Gaussian integral after completing the square. Second, the localized exponential enters the standard Gaussian identity with matrix A = -alpha_5 P_Z0, rather than with a positive-semidefinite Hessian. The terminal Lean theorem combines the physical localization, covariance certificate, Gaussian identity, and Cauchy extraction. Its only substantive analytic premises are the physical domination inequalities corresponding to equations (2.20)—(2.22) and a uniform bound on the explicit determinant/source majorant of equation (2.24).We do not prove those premises, equation (2.26), the polymer estimate hRpoly, a continuum limit, or a Yang—Mills mass gap. The contribution is an auditable reduction theorem that fixes the types, signs, measures, and remaining analytic frontier.
Category: Mathematical Physics
[123] ai.viXra.org:2607.0043 [pdf] submitted on 2026-07-14 05:39:24
Authors: Lluis Eriksson
Comments: 16 pages. Lean 4 sources, oracle transcripts, and release manifest are available at the public repository; mathematics frozen at commit d75d8952.
For SU(N_c) Wilson lattice gauge theory on d-dimensional periodic tori (d >= 2), we present a machine-checked (Lean 4, pinned Mathlib) renormalization-group interface for two-plaquette truncated correlators, in which every structural ingredient is a theorem rather than a postulate: the scale transformation is a concrete decimation map, defined once -- measurable, local, and gauge-covariant -- and its induced pushforward preserves probability; the effective measures are its literal iterated pushforwards of the Wilson Gibbs measure; the multiscale decomposition of the correlator is proved by telescoping, never carried as data; the terminal scale of the decomposition is a fixed index kTerm(n) = n with typed range 1 <= kTerm(n) <= n, which excludes, in the type, the circular depth-zero layer in which an infrared clause would hypothesize the bound being sought; the conditional decay conclusion is stated in the physical distance 2^n u with a single constant pair (C, m) quantified before every torus base and depth; and the terminal observable is operationally support-certified: the infrared object consumed by the interface equals a base-measure integral of an explicitly composed pullback observable whose dependence is contained in a transported support set, for which the separation lower bound 2^n(2u) - (2^n + 1), strictly positive on the whole interface window, is proved. The design is deliberately adversarial: four natural naive formulations are presented together with the explicit countermodels that defeat them -- scalar relabeling of known decay, sink flows on measures, clamped scales and per-volume constants, and depth-zero circularity -- and with the typed repairs that exclude each. The central hypothesis, PhysicalTerminalScaleWilsonGate, is an open proposition: no witness is provided, and the infrared/ultraviolet bounds it demands of the actual Wilson measure are exactly the open analytic mathematics (Balaban-type single-scale estimates). The final theorem is conditional: a witness of the gate yields |Cov(2^n u)| <= C e^{-m 2^n u} with one pair (C, m), m > 0, for every base M_0 >= 4 and every depth n >= 1. No mass-gap claim, no claim of gate satisfiability, and no thermodynamic or continuum limit is made or implied.
Category: Mathematical Physics
[122] ai.viXra.org:2607.0042 [pdf] submitted on 2026-07-14 13:24:23
Authors: Lluis Eriksson
Comments: 11 pages. Lean 4 formalization. Frozen source, dependency record, axiom-oracle transcript, release manifest, and permanent proof links are included in the PDF.
Inside a Lean 4 formalization programme for four-dimensional SU(Nc) lattice Yang—Mills, we report two machine-checked negative results and the machine-checked infrastructure that makes them meaningful. The positive substrate is: (i) a fixed-volume Combes—Thomas chain for self-adjoint coercive finite-range lattice operators, instantiated on the flat gauge-fixed covariance of the physical shell, with coercivity constant c = min(1,a)/C_P fed by a proved fixed-volume flat Hodge/block-Poincaré inequality; and (ii) the concrete adjoint model of SU(n) — su(n) with the trace inner product, dim_R su(n) = n² − 1, and the isometric transport to Euclidean coordinates — so that the abstract adjoint-model interface has a concrete nontrivial inhabitant and the flat-lane results can be instantiated with the genuine matricial adjoint model. The first wall states that, under the block normalization actually used by the formalized chain, every flat Hodge/block-Poincaré constant obeys L^d/L² ≤ C_P on the fine torus of side LNu2032, hence the volume-uniform Poincaré gate is provably false for d ≥ 3 and Nc ≥ 2, and no positive coercivity constant survives all volumes through this route. The route consumed by the fixed-volume endpoint is therefore closed by theorem. A second wall stands in the fluctuation sector. For d ≥ 3 and a transported half-period square-wave mode on the exact fine side (2M)Nu2032, the formalization proves ||QA||² ≤ (2M)u207b¹||A||², the exact identity
Category: Mathematical Physics
[121] ai.viXra.org:2607.0039 [pdf] submitted on 2026-07-14 18:49:45
Authors: Lluis Eriksson
Comments: 10 Pages. (Note by ai.viXra.org Admin: Please cite listed scientific references) Lean 4/Mathlib formalization with 167 audited public endpoints. Complete formal proof artifact: https://github.com/lluiseriksson/lean-2d-yang-mills. Audited mathematical snapshot: a1fb
We give a kernel-checked Lean 4/Mathlib formalization of the exact heat-kernel solution of two-dimensional SU(2) Yang-Mills theory along the chain from Haar measure to a concrete Migdal subdivision move. The development identifies the transported Haar probability on SU(2) ≃ S3 with the canonical spherical probability, proves the required all-order orbital integration law, derives translated character convolution for every pair of irreducible labels, passes from finite character sums to the infinite heat-kernel semigroup by dominated convergence, and integrates an actual edge shared by two faces. We then define finite oriented disk cellulations independently of any reduction tree, using vertices, paired half-edges, cyclic face words, incidence, Euler characteristic, and positive face areas. Every connected dual graph admits a certified elimination schedule, every valid schedule reduces to the heat kernel at total area, and any two choices give the same amplitude. For the first cyclic unreduced lattice, a three-face disk, we additionally construct a triangular gauge-fixing equivalence that preserves the original triple product Haar measure and prove equality with the reduced amplitude. We then close the global edge model: for every primal-connected, facially well-formed finite disk cellulation, a rooted spanning tree selects V - 1 distinct physical edges and induces an explicit measurable equivalence preserving product Haar. All facial holonomies transform simultaneously by basepoint conjugation, the heat-kernel density loses every tree/gauge coordinate, and the unreduced edge integral equals the gauge-fixed chord integral exactly. Finally, a tree-cotree theorem constructs compatible primal gauge and dual elimination trees for every certified physical disk cellulation. The boundary-conditioned original-edge amplitude is therefore defined without an auxiliary chart and is proved equal to the heat kernel at total face area and retained exterior holonomy. The schedule is hidden from the public loop object, which supplies a nontrivial exact-area-law package. Combined with all-label coefficient extraction, this yields the exact normalized simple-loop identity E[W_n(C)] = exp[-n(n+2) Area(C)/4]. All displayed endpoints compile without sorry, admit, or project-local axioms. The mathematical identities are classical; the claimed contribution is their concrete end-to-end machine-checked composition, not a new analytic solution of two-dimensional Yang-Mills theory.
Category: Mathematical Physics
[120] ai.viXra.org:2607.0037 [pdf] submitted on 2026-07-13 05:29:47
Authors: Lluis Eriksson
Comments: 10 pages. Machine-checked in Lean 4 against a pinned Mathlib; Lean sources, certified interval-arithmetic companion and release manifest at the public repository tag c5-v1.0.1.
For the one-parameter family B(x) = x/(nu+c+sqrt((nu+c)^2+x^2)) of Amos-type expressions, whose member c = 1/2 is the classical Amos-type upper bound for the modified Bessel ratio I_{nu+1}/I_nu, we formalize in Lean 4, at every real order nu >= 0 over the Gamma-power series, the classification of the parameter: B is a uniform upper bound for the ratio exactly when c <= 1/2, and a uniform lower bound for every c >= 1, with explicit rational counterexample witnesses (the classification itself is known mathematics, due to Ruiz-Antolin and Segura; we claim only the machine-checking). The contribution is the regime between the ends: for every nu >= 0 and c strictly between 1/2 and 1 we prove that the fixed family member crosses the ratio exactly once on (0, infinity) -- a transversal crossing in an explicit finite window, strictly above an explicit threshold, with globally determined sign on both sides and a two-sided scale law; degenerate contact is excluded by an exact second-derivative identity. The chain carries the axiom oracle [propext, Classical.choice, Quot.sound] and no analytic hypothesis beyond nu >= 0, x > 0; a pre-registered certified interval-arithmetic companion verifies the crossing phenomenon independently of the crossing theorems at 30 parameter pairs spanning the hard regimes, all passing at 128 bits.
Category: Mathematical Physics
[119] ai.viXra.org:2607.0035 [pdf] submitted on 2026-07-13 20:23:27
Authors: Lluis Eriksson
Comments: 9 Pages. (Note by ai.viXra.org Admin: Please cite listed scientific references) Accompanying machine-checked Lean 4/Mathlib proof artifact: https://github.com/lluiseriksson/lean-2d-yang-mills
We give a kernel-checked Lean 4/Mathlib formalization of the exact heat-kernel solution of two-dimensional SU(2) Yang-Mills theory along the chain from Haar measure to a concrete Migdal subdivision move. The development identifies the transported Haar probability on SU(2) with the canonical spherical probability on the three-sphere, proves the required all-order orbital integration law, derives translated character convolution for every pair of irreducible labels, passes from finite character sums to the infinite heat-kernel semigroup by dominated convergence, and integrates an actual edge shared by two faces. We then define finite oriented disk cellulations independently of any reduction tree, using vertices, paired half-edges, cyclic face words, incidence, Euler characteristic, and positive face areas. Every connected dual graph admits a certified elimination schedule, every valid schedule reduces to the heat kernel at total area, and any two choices give the same amplitude. For the first cyclic unreduced lattice, a three-face disk, we additionally construct a triangular gauge-fixing equivalence that preserves the original triple product Haar measure and prove equality with the reduced amplitude. The same measure-theoretic construction is proved uniformly for one vertex of arbitrary finite valence. We further construct a rooted spanning-tree order for every finite connected primal graph and a single global root-plus-edge-increments equivalence on all cellulation vertex variables, with exact product-Haar preservation. Combined with all-label coefficient extraction, this yields the exact normalized simple-loop identity E[W_n(C)] = exp[-n(n+2) Area(C)/4]. All displayed endpoints compile without sorry, admit, or project-local axioms. The mathematical identities are classical; the contribution is their concrete end-to-end machine-checked composition, not a new analytic solution of two-dimensional Yang-Mills theory.
Category: Mathematical Physics
[118] ai.viXra.org:2607.0033 [pdf] submitted on 2026-07-12 13:19:13
Authors: Lluis Eriksson
Comments: 10 Pages. Lean 4 formalization plus certified interval-arithmetic companion (1206-point grid) in the public repository, release c2-v1.2.1. The bound itself is proved in the companion paper, release c3-v1.0.1.
The Amos-type upper bound on the modified-Bessel ratio, I{nu+1}(x)/I_nu(x) < x/(nu + 1/2 + sqrt((nu+1/2)^2 + x^2)), has a distinguished algebraic property: its right-hand side U satisfies the exact calibration identity 1/U - U = (2nu+1)/x. From that identity alone — by ordered-field algebra, with no further analytic input — follow a unit-step inequality rho_nu - rho{nu+1} < 1/x for consecutive ratios, the strict increase of the log-derivative (log I_nu)' = rho_nu + nu/x across orders, and the strict monotonicity of a phi-sequence arising in a two-dimensional lattice-gauge surface expansion. We formalize this calculus in Lean 4: a single module defines the bound once (AmosBound) and proves the calibration engine and four consequence theorems through that one definition, together with two rational satisfiability witnesses whose Amos hypothesis holds by exact Pythagorean arithmetic; all eighteen Lean statements of the development pass the axiom oracle with exactly [propext, Classical.choice, Quot.sound] against a pinned Mathlib. A certified companion (256-bit interval arithmetic, self-contained series-plus-tail enclosures, committed transcript) certifies the bound provably strictly at all 1206 points of a pre-registered grid covering the arguments the applications consume. A Bessel interface completes the closure: integer-order I_n is defined by its power series in the same pinned development, with positivity, the three-term recurrence, the termwise-differentiated derivative identity I_n' = I_{n+1} + (n/x) I_n, and the logarithmic-derivative identity (log I_n)' = rho_n + n/x all proved as theorems, so the consequence theorems — including the unit step read as strict log-derivative monotonicity, in deriv form — hold for genuine Bessel ratios with the Amos bound as the single remaining hypothesis. The scope is stated exactly: the Amos bound itself remains a classical cited theorem taken as hypothesis — this paper unifies its three previously scattered uses in our formal development into one named proposition with one oracle and one certified numerical witness, and no downstream result changes its verification class.
Category: Mathematical Physics
[117] ai.viXra.org:2607.0032 [pdf] submitted on 2026-07-12 13:23:29
Authors: Lluis Eriksson
Comments: 7 Pages. Complete Lean 4 formalization with source and reproducibility artifacts in the public repository, release c3-v1.0.1. Companion to release c2-v1.2.1.
Amos's upper bound for the modified Bessel function ratio, rho_n(x) = I_{n+1}(x)/I_n(x) < x/(n + 1/2 + sqrt((n+1/2)^2 + x^2)) = B_n(x), is a classical theorem, and its derivation through the qualitative theory of the associated Riccati equation is an established technique. This paper contributes, to our knowledge, the first formalization: a complete, machine-checked Lean 4 proof of the bound for every integer order n >= 0 and every x > 0, over the power-series definition of I_n carried in the same pinned development, with axiom oracle [propext, Classical.choice, Quot.sound] and no analytic hypothesis of any kind. The formalized route runs through the Riccati equation rho_n' = 1 - ((2n+1)/x) rho_n - rho_n^2 (itself derived from the formalized series calculus), the observation that B_n is exactly the positive root of the Riccati quadratic, a small-argument zone bound uniform in n obtained from pure geometric tail estimates, and a first-crossing barrier argument in a transformed variable psi_n = x(1/rho_n - rho_n) whose structural feature — every touch of the critical level forces rho_n' = 0, so the barrier never needs to be differentiated — is the simplification this formalization contributes. As corollaries, the unit-step inequality, the strict monotonicity of the logarithmic derivative across orders (in deriv form), and a phi-monotonicity step used by a lattice-gauge surface expansion all become unconditional theorems. The theorem is proved for the in-core power-series definition of the integer-order modified Bessel function; no formal identification with an external special-functions library object, and no extension to noninteger order, is claimed.
Category: Mathematical Physics
[116] ai.viXra.org:2607.0030 [pdf] submitted on 2026-07-12 20:21:46
Authors: Lluis Eriksson
Comments: 8 pages. Complete Lean 4 formalization with source and reproducibility artifacts in the public repository, release c4-v1.0. Companion to releases c2-v1.2.1 and c3-v1.0.1 (same repository).
The Amos-type upper bound for the modified Bessel function ratio, rho_nu(x) = I_{nu+1}(x)/I_nu(x) < x/(nu + 1/2 + sqrt((nu+1/2)^2 + x^2)), is classical, and its derivation through the qualitative theory of the associated Riccati equation is an established technique. A companion paper formalized the bound at integer order over a factorial power series. This paper extends the formalization to every real order nu >= 0: the function I_nu is defined by its Gamma-power series (real exponents via rpow), and the complete chain — convergence, positivity, the three-term recurrence, termwise differentiation with a dominated-derivative argument that must treat the leading term separately (its exponent nu-1 is negative for nu < 1), the Riccati equation, a small-argument zone bound uniform in nu, and a first-crossing barrier — is machine-checked in Lean 4 with axiom oracle [propext, Classical.choice, Quot.sound] and no analytic hypothesis beyond nu >= 0, x > 0. Two structural locks tie the result to the integer development: an identification theorem proves that at nu = n the Gamma-series object coincides with the factorial-series object, so the integer-order theorem of the companion development is recovered as a corollary in three rewrites; and a genuinely non-integer instance at nu = 1/2 witnesses that the endpoint lives outside the natural-number embedding. The theorem is proved for the in-core Gamma-series definition; no identification with an external special-functions library object is claimed.
Category: Mathematical Physics
[115] ai.viXra.org:2607.0025 [pdf] submitted on 2026-07-11 21:28:54
Authors: Lluis Eriksson
Comments: 10 Pages. Lean 4 formalization (pinned Mathlib), certified interval-arithmetic transcripts, and full audit trail at https://github.com/lluiseriksson/THE-ERIKSSON-PROGRAMME (tags c1-v1.0, c1-v1.0.1, c1-v1.0.2).
We present a machine-checked quantitative toolkit for cluster expansions of polymer systems with excluded regions (holes), in the discrete cube geometry of Balaban-Dimock renormalization-group analyses. Five Lean 4 theorems, checked against a pinned Mathlib revision, provide: (i) the identity sum_T prod_v c_T(v)! = n! C_n for child factorials over spanning trees of the complete graph K_(n+1), with the rooted-tree majorant 4^n as corollary; (ii) a marked-root leaf summation for the tree-graph majorant of an Ursell-type expansion with holes, with the moment constant M paid once at the root and closed leaf ratio 4M^2 per additional vertex, together with its Catalan-sharpened form M^(2n+1) C_n (a gain of order n^(3/2) in the n-th coefficient); and (iii) a target-preserving orderwise bound in which the target union itself survives until the modified-metric exponential is extracted. A certified companion (interval arithmetic, 120-bit precision, committed transcript with a committed reproduction witness) tabulates the smallness gate and encloses every derived constant. Non-vacuity is machine-checked: a concrete hole family satisfying every hypothesis is exhibited in Lean, and the two distinct hypothesis sets among the polymer-facing theorems are both instantiated at it with a strictly positive weight. Each claim is labelled with its verification layer: exact (Lean theorem), certified (interval transcript), or paper-level.
Category: Mathematical Physics
[114] ai.viXra.org:2607.0023 [pdf] submitted on 2026-07-09 23:37:55
Authors: Lluis Eriksson
Comments: 13 Pages. Consolidation paper #5 of an audit-first programme (companion notes: ai.viXra 2607.0005, 2607.0017, 2607.0018, 2607.0020). Proves part (i) fully and part (ii) for 0 < beta <= 3; the global statement is presented as a quantified conjecture with a verified
For beta > 0 let I_m denote modified Bessel functions of the first kind, a_m = I_m^2[(m-1)I_{m-1}^2 + (m+1)I_{m+1}^2], b_m = m I_m^4, F_A = sum a_m sin(mt), F_B = sum b_m sin(mt), E = F_A/(2 F_B). The global ratio-monotonicity problem for the surface expansion of a two-dimensional SU(2) lattice gauge observable is: (i) F_B > 0 on (0, pi); (ii) E' < 0 on (0, pi), for every beta > 0. We prove (i) twice, and prove (ii) for 0 < beta <= 3 via an exact pair-identity skeleton plus an archived interval certificate executed by two independent interval-arithmetic implementations. The structural core is exact: E is, as an algebraic identity, the mean of cos(psi) under the midpoint law of a four-step killed von Mises bridge; the generating kernels reduce, via the Neumann addition theorem, to two-dimensional integrals of a single Bessel function whose saddle deficit is an exact sum of two squares; and exact saddle cancellations yield the coefficients of the (verified) closed second-order law E = cos(t/2)(1 - c(t)/beta) + O(beta^-2), with c(t) = (4cos^2(t/4) - 1)/(2cos(t/4)cos(t/2)). Three certified negative results kill every monotone full-path coupling, with an exact mechanism at threshold beta|cos t| = 3/2. For the remaining range we give a two-scale closure map with a verified candidate threshold beta_0 = 15 and the complete pi-endpoint package for the cubic coefficient c_3. Every claim is labelled exact / certified / verified; the machine-checked lemmas are Lean 4/Mathlib, machine-checked modulo classical Bessel inputs carried as named hypotheses.
Category: Mathematical Physics
[113] ai.viXra.org:2607.0020 [pdf] submitted on 2026-07-09 13:40:14
Authors: Lluis Eriksson
Comments: 8 Pages. Lean verification file and numerical audit script at https://github.com/lluiseriksson/THE-ERIKSSON-PROGRAMME/tree/main/papers/bessel-amos-fh
Let I_nu denote the modified Bessel function of the first kind and, forx>0, let rho_nu(x) = I_{nu+1}(x)/I_nu(x). We give a four-step, fullyelementary proof of the sharp difference inequality0 < rho_nu(x) - rho_{nu+1}(x) < 1/x (x>0, nu>=0), whose right-handinequality is exactly the strict increase of the logarithmic derivative(log I_nu)'(x) under the unit shift nu -> nu+1; consequentlynu -> (log I_nu)'(x) is strictly increasing along every unit-spaced gridnu_0 + N, in particular on the integer and half-integer orders arising inthe application. The stronger continuous-order statement is known(Freitas-Laugesen, arXiv:1810.07461, Lemma 10, via Bessel zeros); we makeno elementary claim about fractional steps. The proof given here uses noinformation about Bessel zeros: it combines the three-term recurrence withthe classical Amos-type upper bound rho_nu < x/(a+sqrt(a^2+x^2)),a = nu+1/2, and rests on the observation that this bound is exactlycalibrated for the problem: 1/U - U = 2a/x is an algebraic identity, and(2nu+1)/x is precisely the threshold the unit step requires; in fact theunit-step monotonicity and the Amos bound are equivalent. As anapplication we record the following consequence in two-dimensional latticegauge theory: for the Wilson action, every mass gap between charactersectors of the 2D transfer operator - for U(1) and SU(2) alike - is astrictly decreasing function of the bare coupling beta, by theFeynman-Hellmann identity. The algebraic core of the proof ismachine-checked in Lean 4/Mathlib (no sorry; axiom oracle: Lean's threestandard axioms), and an independent high-precision numerical audit ofevery inequality used is reported.
Category: Mathematical Physics
[112] ai.viXra.org:2607.0018 [pdf] submitted on 2026-07-09 14:57:20
Authors: Lluis Eriksson
Comments: 4 Pages. Parametric-in-r Lean formalization included. Lean verification file and exact-arithmetic script at https://github.com/lluiseriksson/THE-ERIKSSON-PROGRAMME/tree/main/papers/parity-barriers
For every r>=1, the uniform measure on the even-parity subset of {+-1}^{r+1}is r-wise independent, yet the last coordinate has unit variance while beingan a.s. function of the others. This example is classical - parity-checkcodes are the standard construction of k-wise independent distributions inthe pseudorandomness literature (Joffe; Alon-Babai-Itai; Alon-Goldreich-Mansour) - and no novelty is claimed for it. What is recorded here is aconsequence we have not seen isolated as a statement: any "comparisonfunctional" whose value depends only on marginal data of order <= r, withconstants uniform over finite measures, takes identical values on the paritymeasure and on the uniform product measure, and is therefore consistent withperfect decoupling on a measure where decoupling fails maximally. Hence noinequality built from bounded-order functionals can imply uniform decouplingprinciples - Dobrushin-type mixing, approximate tensorisation withmeasure-free constants, covariance decay - on any class of measurescontaining the parity family. The case r=1 recovers, and explainsstructurally, the failure of raw-oscillation/Doob and Efron-Stein-type stepsfound repeatedly in an adversarial audit of a constructive Yang-Millsprogramme; no repair within bounded-order data can succeed, because thebarrier recurs at every order. Statements (a) and (b) are machine-checkedin Lean 4/Mathlib parametrically in r (all n; no sorry; standard axiomsonly), the abstract certifying-barrier schema is formalized as well, andfinite decide instances (r<=4) plus exact rational arithmetic (r<=6) serveas independent audits.
Category: Mathematical Physics
[111] ai.viXra.org:2607.0017 [pdf] submitted on 2026-07-09 19:34:39
Authors: Lluis Eriksson
Comments: 5 Pages. Lean verification file and numerical audit at https://github.com/lluiseriksson/THE-ERIKSSON-PROGRAMME/tree/main/surface-theorem and papers/phi-lemma.
For x>0 and real m>=1 define phi_m(x) = [(m-1) I_{m-1}(x)^2 + (m+1) I_{m+1}(x)^2] / (m I_m(x)^2), with I_mu the modified Bessel function of the first kind. We prove phi_m(x) < phi_{m+1}(x) for every x>0 and everyreal m>=1 - a weighted Turan-type monotonicity statement we have not found in the literature, although every ingredient of the proof is classical. The proof is fully elementary: eliminating the neighbouring ratios by thethree-term recurrence, the difference factorizes exactly as (S-3c)(P-(2m+1)c) + (2m+1)c^2, with u = I_{m+1}/I_m, c = 1/x, S = u+1/u, P = 1/u-u; the second factor is positive by the calibrated Amos bound (it is precisely the unit-step inequality of the companion note), the first because the same bound forces u < x/(2m+1) <= x/3. As an application we obtain thestrict determinant ordering c_mn < 0 (m
[110] ai.viXra.org:2607.0016 [pdf] submitted on 2026-07-08 20:37:33
Authors: Xiangqian Zhang, Mingming Zhao, Linchao Ge
Comments: 24 Pages. (Note by ai.viXra.org Admin: Please cite listed scientific references)
This paper, based on Topological Residual Theory (TRT), systematically explains the fractal filling mechanism of spatial flow fields in right-handed cylindrical helical motion at the speed of light, and presents the pure geometric and topological origin of the dimensionless fundamental constantu2015the fine structure constant...
Category: Mathematical Physics
[109] ai.viXra.org:2607.0006 [pdf] submitted on 2026-07-04 23:28:33
Authors: J. W. McGreevy
Comments: 19 Pages. (Note by ai.viXra.org Admin: Please cite listed scientific references)
The Riemann hypothesis is realized as a direct theorem inside the RFTP framework. The single global linear constraint enforced by the weight-12 modular discriminant organizes shear amplitudes extracted from odd Galois eigenspaces at irregular primes into a rigid balancing law. The same constraint trivializes the effective second Stiefel—Whitney class, yielding a globally defined spinor bundle on which a non-linear Dirac Hamiltonian is essentially self-adjoint with real spectrum. After an explicit canonical spectral shift and adelic factor, the regularized spectral determinant coincides with the completed Riemannfunction Ξ(s). The distributional trace formula reproduces Weil’s explicit formula exactly inthe atomic limit of the Hopf—Lax flow, and Tauberian extraction recovers the zero-countingfunction with the Riemann-hypothesis error term. Consequently every non-trivial zero lieson the critical line. RFTP inverts the conventional paradigm: arithmetic data at irregular primes function as the generators of geometric and dynamical structure rather than emerging from an a priori physical process.
Category: Mathematical Physics
[108] ai.viXra.org:2607.0005 [pdf] submitted on 2026-07-02 21:47:02
Authors: Lluis Eriksson
Comments: 6 Pages.
We report a complete formalization, in Lean 4 over Mathlib, of volume-uniform Wilson-loop area laws for SU(Nc) lattice gauge theory in an explicit strong-coupling window - including the case of the exact Wilson Boltzmann factor, not a linearized surrogate. The headline theorem bounds the normalized Wilson-loop expectation by Nc eP·4dK σArea(C) eP·4dS(σ), where Area(C) is an intrinsic combinatorial filling area of the loop, P is its edge-support size, and every constant is volume-free: the bound holds uniformly over all finite lattice sizes. The partition function is cancelled through a fully formalized volume-restricted cluster expansion (loop-tagged factorization, restricted Mayer inversion, Z-ratio bounds, and a pinned-gas resummation built on a Kotecky-Preiss layer with Penrose-style spanning-tree counting). A reusable repackaging converts the bound into manifest exponential area decay with a strictly positive string tension, and the non-vacuity of every hypothesis window is itself machine-checked - both the cluster smallness window and the decay-repackaging window, the latter with an explicit witness of tension log 2 - 1/2. For every exported theorem in this chain the Lean kernel's axiom oracle reports exactly [propext, Classical.choice, Quot.sound]; there is no sorry and no project axiom in the dependency cone. To our knowledge this is the first machine-checked cluster-expansion proof in lattice quantum field theory. All artifacts are public, with per-theorem oracle records in a verification ledger and continuous-integration builds.
Category: Mathematical Physics
[107] ai.viXra.org:2607.0001 [pdf] submitted on 2026-07-02 19:14:41
Authors: Lluis Eriksson
Comments: 8 Pages.
Let Kn+1 be the complete graph on the vertex set {0, 1, ..., n}, and for a spanning tree T of Kn+1, rooted at 0, let cT(v) denote the number of children of the vertex v. We prove the exact identity: the sum, over all spanning trees T of Kn+1, of the product over vertices v of cT(v)! equals n! Cn, where Cn is the n-th Catalan number. Equivalently, the normalized sum (n+1)((n+1)!)-1 times the weighted tree sum equals Cn exactly. The proof is bijective: pairs consisting of a spanning tree together with a linear ordering of every child set are placed in explicit bijection with vertex-labeled plane trees on n+1 nodes whose root carries the label 0. The identity arises as the exact "second-Ursell" normalization constant in the author's audit-first programme on four-dimensional SU(N) Yang-Mills existence and mass gap, where it had been isolated as a named open proposition in a public challenge repository; the present paper is self-contained combinatorics and makes no claim about that programme. The entire proof has been formalized in Lean 4 against a pinned Mathlib snapshot: the headline declarations compile with no sorry, and the kernel's axiom oracle reports exactly [propext, Classical.choice, Quot.sound]. All artifacts, including a pinned continuous-integration replay of the full verification, are public.
Category: Mathematical Physics
[106] ai.viXra.org:2606.0083 [pdf] submitted on 2026-06-30 20:06:17
Authors: J. W. McGreevy
Comments: 19 Pages.
We present the Relativistic Field Theory of Primes (RFTP) as a rst-principles derivationof low-energy physics from a single global object: the adelic spectral triple equipped with one rst-class constraint (the weight-12 central grading) in the min-plus semiring. Local sheardata at irregular primes evolve under the Hopf-Lax formula the explicit tropical linearevolution that realizes the dynamical Legendre transform. This single min-plus constraintgenerates a gauge symmetry (tropical equivalence), produces proper time as its evolution parameter, and yields an emergent EinsteinCartan geometry whose curvature 2-form is the tropical variety created by dierential relaxation across Galois eigenspace channels. Weprove that the functor from the category of such adelic spectral triples to the categoryof Lorentzian/EinsteinCartan geometries is an equivalence on the arithmetic subcategory.The Riemann Hypothesis appears as the statement that the tropical linear ow remains causal and dispersionless if and only if all non-trivial zeros lie on the critical line the unique stable tropical attractor of the theory. Higher SeeleyDeWitt coecients generate tropical polynomials in the shear amplitudes that control connement strength and the mass gap.
Category: Mathematical Physics
[105] ai.viXra.org:2606.0072 [pdf] submitted on 2026-06-27 19:46:59
Authors: Xiangqian Zhang，Mingming Zhao，Linchao Ge
Comments: 17 Pages. (Note by ai.viXra.org Admin: Please cite listed scientific references)
Within the framework of Topological Residual Theory (TRT), this paper rigorously establishes the first-principles physical picture that "space itself flows at the speed of light along curves in a righthanded cylindrical helical form." In this picture, length and time originate from different aspects of the same spacetime flow motion, thereby providing a geometric explanation for the origin of theconstancy of the speed of light. To fundamentally resolve the dimensional mismatch among different physical fields that arises inclassical field theory from differing measurement protocols, we propose a Velocity-Field Representation scheme: the electric, gravitational, and magnetic fields are directly defined as the three local velocity components of the light-speed spacetime flow along the orthogonal basis of the Frenet-Serret (FS) frame: This achieves a complete unification of all underlying field dimensions to velocity at the kinematic level. Since the flow speed of spacetime is fixed at the speed of light , different physical interactions essentially correspond to the allocation of this motion budget among the three orthogonal directions: tangential propagation, normal centripetal constraint, and binormal torsion. The allocation mechanism is precisely controlled by the helical pitch angle. By introducing the time definition , we derive the first-order time-evolution equations of the three fields under the characteristic rotation frequencies of the FS frame (bending frequency, twisting frequency ). Under the rigid rotation of the frame, the total kinetic energy density remains strictly constant. Utilizing the rigid rotational dynamics of the frame and the vector and scalar triple-product identities, we elucidate thephysical nature by which the "self-work" terms of the rigid triad automatically vanish during rotation, thereby revealing the deep geometric mechanism of dissipation-free energy flow.The macroscopic observable fields — centripetal gravitational acceleration , tangentialelectric field strength , and magnetic rotational angular frequency — emerge naturally as the time derivatives or rotational frequencies of the underlying velocity fieldsE = ct, A = cn, B = cb. E = ct, A = cn, B = cb.cα ds = c dt ωκ = cκ ωτ = cτ U = 2 1 (E ⋅ E + A ⋅ A + B ⋅ B) = 2 3c 2 g = c 2κn e = c 2κt Bmacro = cτb after being modulated by the motion-budget allocation. This provides a self-consistent firstprinciples framework for the geometric unification of gravity and electromagnetism.
Category: Mathematical Physics
[104] ai.viXra.org:2606.0071 [pdf] submitted on 2026-06-27 19:43:33
Authors: J. W. McGreevy
Comments: 10 Pages. (Note by ai.viXra.org Admin: Please cite listed scientific references)
We demonstrate that the infrared dynamics of the resonant spectral triple in the Bost-Connes system, deformed by irregular-prime shear, implements a precise arithmetic analogueof the classical Legendre transform. Starting from a classical Poisson manifold whose coordinates are the shear amplitudes {Iu2032p} and the bifundamental mixing field Φ, we construct an explicit action functional whose Legendre transform yields a Hamiltonian on the cotangent bundle. The weight-12 winding condition is imposed as a classical first-class constraint, and Dirac reduction produces a reduced symplectic manifold Pred. Deformation quantization of this reduced space, driven by the modular derivation Dp acting asymmetrically through Φ on the two complex-multiplication stabilizers (Z/6Z at τ = ρ and Z/4Z at τ = i), generates a non-commutative star product whose leading commutator is non-vanishing and proportional to the stabilizer contrast transmitted by Φ. This framework naturally produces: (i) proper time τ as the integrated thermodynamic work along the flow, with lapse N ∝ det(∇2P) · Tr(Φu2020Φ); (ii) a non-removable relative phase between up- and down-type sectors, yielding the Jarlskog invariant as a direct dynamical consequence of the asymmetric action of the real endomorphism J (J2 = −Id) forced by the holonomy defect; and (iii) definite outcomes via deterministic Hamilton—Jacobi flow to a unique weight-12 attractor, with probabilities given by normalized infrared Boltzmann weights from the entropy functional S(ϕ). The construction draws explicit parallels with Dirac’s deformation of Poisson bracketsinto commutators and Tate’s adelic Poisson summation, providing a unified non-commutativestructure that bridges arithmetic geometry and quantum field theory. The imaginary unit i emerges necessarily as the spectral manifestation of the real endomorphism J required to close the holonomy defect. All physical features follow rigorously from the arithmetic data without external postulates.
Category: Mathematical Physics
[103] ai.viXra.org:2606.0069 [pdf] submitted on 2026-06-26 09:44:58
Authors: Allan Plankhart
Comments: 11 Pages.
Building on a pre-temporal collapse model, we derive a structural correspondence between the surviving 2D sector and the Standard Model’s weak isospin. We show that an embedded 2-brane in a 3D bulk possesses a local frame rotation group SO(3). The spin connection on the brane is an so(3)-valued one-form.Since so(3) ∼= su(2), this connection is exactly an SU(2) gauge field, and spinorial matter confined to the brane necessarily transforms as doublets. This geometrically forces the effective field theory to be Yang—Mills with a Chern—Simons term, uniquely constraining the dynamics. We construct the resulting action, including kinetic mixing with the 3D bulk, and derive testable signatures: a dark matter fluid from the brane foam, distinctive CMB non-Gaussianities with a characteristic oscillatory power spectrum, and missing-energy collider events with a topological mass peak. We provide current constraints and outline the Boltzmann evolution needed for precise predictions.
Category: Mathematical Physics
[102] ai.viXra.org:2606.0066 [pdf] submitted on 2026-06-26 20:18:45
Authors: J. W. McGreevy
Comments: 14 Pages. (Note by ai.viXra.org Admin: Please cite listed scientific references)
We present a complete first-principles derivation of the Standard Model gauge groupSU(3)c × SU(2)L × U(1)Y with three chiral generations of fermions, hierarchical masses,CKM/PMNSmixing, theseesaw mechanism for neutrinos, and Einstein—Cartan gravity withtorsion from one arithmetic object. The underlying structure is the resonant spectral tripleof the Bost—Connes system deformed by shear terms sourced by the class-group torsion ofirregular primes. In the deep infrared, the Chamseddine—Connes spectral action is evaluatedunder the global constraint of weight-12 principalization at the mirror point s = 6 of theassociated L-function.The fundamental duality between analytic volume (hyperbolic geometry, spectral flow,Seeley—DeWitt expansion) and algebraic rigidity (stabilizer symmetries at the two complexmultiplication points, adelic closure, weight-12 condition) is resolved by the irregular-primeshear, generating the pressure gradient ∇P that drives all symmetry breaking and massgeneration. Gauge representations emerge from the 3-cycle at τ = ρ and 4-cycle at τ = i;the bifundamental field Φ acquires a dynamical VEV; the 12×12 mass matrix, includinga natural seesaw for neutrinos, follows from breathing corrections sourced by class-grouptorsion. Numerical diagonalization over the full table of 47 irregular primes yields hierarchical fermion masses, realistic mixing, and a non-zero Jarlskog invariant. Confinement, thepositive mass gap, emergent proper time as a lapse function, and the Born rule all followfrom the same infrared thermodynamics.This construction realizes a complete arithmetic unification in which every low-energystructure is forced by the irreducible duality between analytic volume and algebraic rigidityunder weight-12 principalization. No external parameters beyond the shear amplitudes ofthe irregular primes are required.
Category: Mathematical Physics
[101] ai.viXra.org:2606.0062 [pdf] submitted on 2026-06-24 02:55:22
Authors: J. W. McGreevy
Comments: 9 Pages. (Note by ai.viXra.org Admin: Please cite listed scientific references)
We present a complete analytic derivation of color confinement and a positive Yang—Millsmass gap within the Relativistic Field Theory of Primes (RFTP). The derivation is carriedout on a shear-deformed resonant spectral triple constructed from the Bost—Connes systemand its low-temperature KMS states. Irregular primes deform the underlying GNS quadraticform, generating a pressure gradient between the two complex-multiplication points τ = ρand τ = i on the modular curve. A continuous infrared renormalization-group flow thenproduces an infrared dichotomy: states whose phase lies outside the weight-12 lattice acquirean infinite entropy barrier (leading to confinement), while color-singlet states remain stable with a strictly positive mass gap controlled by the dominant shear amplitude.The proof relies on the asymptotic behavior of the heat kernel on the full GNS Hilbertspace, an effective Hamilton—Jacobi dynamics in which the arithmetic pressure gradient acts as the driving force, and the thermodynamic interpretation of the infrared divergenceas the failure of an arithmetic Legendre transform (principalization) for fractional-phase states. Both confinement (via the Wilson-loop area law) and the mass gap are shown toemerge directly from the infrared structure of a single spectral object without additionalassumptions or fine-tuning. The dynamical vacuum expectation value that will generatefermion masses in extensions of this framework is recovered as the equilibrium of the samedriven dynamics. This work establishes that color confinement and the mass gap are not accidental features of non-abelian gauge theories but inevitable consequences of the arithmetic geometry of the primes when viewed through the lens of a resonant spectral triple.
Category: Mathematical Physics
[100] ai.viXra.org:2606.0053 [pdf] submitted on 2026-06-21 00:39:11
Authors: Allan Plankhart
Comments: 3 Pages. (Note by ai.viXra.org Admin: Part of the diagram is cut off)
We present a timeless mathematical construction of a pre-Big Bang universe. Starting from theempty set, we generate an infinite alternating chain of "something" and "anti-something", buildhigher-dimensional arrays, collapse to dimensions 1—3 due to Hurwitz’s theorem, introduce complex-valued fields with a U(1) symmetry ("oscillations" without time), ascend the division algebra ladderC → H → O, recover the Standard Model gauge group via Cohl Furey’s functor, extract electro-magnetic and spinor fields, and finally let time emerge as a monoid action R≥0 on the field space.The entire pre-temporal phase is a static diagram of functors and natural transformations; no timeparameter appears until the last step.
Category: Mathematical Physics
[99] ai.viXra.org:2606.0047 [pdf] submitted on 2026-06-19 03:26:58
Authors: E. P. J. de Haas
Comments: 144 Pages. https://doi.org/10.5281/zenodo.20747973
The biquaternion (BQ) algebra, realized at the Pauli-spin level as the full matrix algebra (M(2,mathbb{C})), provides a compact framework in which classical field theories can be organized through a common hierarchy of algebraic products and channel decompositions. This paper develops that hierarchy for both electrodynamics and relativistic fluid dynamics and explores the structural relationships that emerge between them.In electrodynamics, the BQ hierarchy reproduces the electromagnetic field, Lorentz force law, Maxwell equations, wave equations, and interaction-conservation relations from a sequence of algebraic products involving the four-current and four-potential. In relativistic fluid dynamics, the corresponding hierarchy generates the continuity equation, Bernoulli relation, vorticity transport, Euler—Lamb momentum balance, acoustic wave equations, helicity relations, and stress-energy conservation through analogous products of the fluid four-velocity and momentum-density field.The central result is that both theories share the same four-level algebraic structure and the same geometric channel decomposition. The force equations (JB=F) and (UH=F) emerge as the unique four-channel level of their respective hierarchies, revealing a common organization that is largely hidden in conventional formulations. The analysis further identifies the acoustic refractive index (n_s=c/c_s) as a hierarchy-wide parameter linking relativistic and practical fluid dynamics, and locates the greater complexity of fluid dynamics in the self-referential coupling between velocity and momentum-density fields.The paper does not introduce new physical postulates. Rather, it proposes that the BQ channel decomposition provides a unified organizational framework that makes structural relationships between electrodynamics and fluid dynamics explicit, offering a common language for force, energy, momentum, vorticity, helicity, and wave phenomena across both theories. In this sense, the work is intended as a conceptual and educational framework as much as a mathematical formalism, and as a foundation for future applications in gravitation, magnetohydrodynamics, and AI-assisted symbolic physics.
Category: Mathematical Physics
[98] ai.viXra.org:2606.0040 [pdf] submitted on 2026-06-16 20:34:28
Authors: J. W. McGreevy
Comments: 17 Pages. (Note by ai.viXra.org Admin: Please cite listed scientific references)
We present a first-principles derivation of 3+1-dimensional Lorentzian spacetime andEinstein—Cartan gravity from the arithmetic structure of the Bost—Connes C∗-dynamicalsystem. Starting from the primitive object (ABC,σt) and its KMS states, we construct ascale-invariant quadratic form on the idèle class space via the GNS inner product. Irregularprimes appear as topological defects through class-group shifts, inducing a shear in thequadratic form that maps canonically to a point τp ∈ H. For the resonant prime 691 this shear aligns with the hexagonal fixed point τ = ρ, where the Z/6Z stabilizer providestopological protection against adelic screening.The local 3-defect patch (primes 37, 59, 691) carries a non-abelian su(2) connection.Global adelic closure projects the three independent directions onto a single screened ray (time) while preserving the protected 3-dimensional spatial structure generated by the Eisenstein lattice at ρ. Explicit differentiation of the quadratic-form embedding with respect to the shear parameter yields the six generators of the Lorentz algebra so(1,3), with compact rotations locked at resonance and non-compact boosts arising from unbounded shear.The second-order Magnus expansion of parallel transport produces a residual torsion vector whose components are fixed by phase mismatches between the three incommensurate modular frequencies. Promoting the local Quantum Geometric Tensor metric on the 3-defect patch together with the screened ray yields a derived 4D Lorentzian geometry. On the resulting resonant spectral triple we evaluate the Chamseddine—Connes Spectral Action. Functional variation with respect to the emergent metric produces the Einstein—Cartan field equations, with the cosmological term fixed by the converged quadratic sum over the 47 irregular primes below 691 and the torsion source fixed by the linear sum. The next heat-kernel coefficienta4 generates quadratic gravity corrections that are parametrically suppressed by the samearithmetic factors, establishing ultraviolet safety. All physical structures—spacetime signature, Lorentz symmetry, torsion, and gravitational dynamics—emerge as necessary consequences of adelic consistency acting on the arithmetic defects of the Bost—Connes system. No external gravitational or field-theoretic ansatzis introduced.
Category: Mathematical Physics
[97] ai.viXra.org:2606.0029 [pdf] submitted on 2026-06-10 21:28:40
Authors: J. W. McGreevy
Comments: 5 Pages. (Note by ai.viXra.org Admin: Please cite listed scientific references)
We present a fully rigorous, self-contained analytic proof that the non-trivial zeros of the Riemann zeta function are precisely the globally stable skin modes of an inner-fluctuated Dirac—Zeta operator on the compactified modular curve X(1)∗. The proof proceeds entirely within the framework of unbounded operators on weighted Sobolev spaces, regularized Fredholm determinants on Hilbert—Schmidt ideals, the Riemann—Roch theorem, and a vanishing theorem for fractional states. The simple pole at s = 1 is interpreted as a topological point gap that triggers the NonHermitian Skin Effect. Spectral stability under discretization is established via uniform pseudospectrum control. The point-gap winding number is realized as the Fredholm index of the regularized resolvent. Riemann—Roch divisor constraints on meromorphic sections force the Generalized Brillouin Zone onto the critical line. A vanishing theorem shows that isolated fractional (non-color-singlet) states are excluded from the domain, thereby locking every non-trivial zero to R(s) = 1/2. The construction is compatible with Alain Connes’ spectral triples and yields, as a byproduct, an explicit derivation of the Clausius relation from the spectral action along the adiabatic paths of the modular loop. All steps are topological and independent of any particular representative of the operator.
Category: Mathematical Physics
[96] ai.viXra.org:2606.0022 [pdf] submitted on 2026-06-08 01:19:11
Authors: Mohammed Belabbaci
Comments: 5 Pages.
The IEEE 754 floating-point standard, foundational to modern computing since 1985, treats division by zero as an undefined operation, returning error states or special infinity values. This convention, while practically expedient, creates mathematical discontinuities and computational barriers across multiple domains. We propose IEEE DZ20i, a continuous extension of IEEE 754 based on the axiom a/0 = a, which maintains computational continuity through zero-valued divisors.This paper presents the mathematical foundation of the DZ20 axiom, demonstrates its consistency within a Meta-Framework for Adapted Mathematics (MCMA), specifies the IEEE DZ20i standard including symmetric human-AI co-validation mechanisms, and explores applications across algorithmic computing, energy systems, hardware design, robotics, and space exploration.Implementation results show that IEEE DZ20i eliminates a broad class of runtime exceptions while preserving backward compatibility with IEEE 754 operations outside the zero-divisordomain.
Category: Mathematical Physics
[95] ai.viXra.org:2606.0017 [pdf] submitted on 2026-06-07 17:08:30
Authors: Xiangqian Zhang，Mingming Zhao，Linchao Ge
Comments: 5 Pages.
This paper systematically analyzes the geometric origins of errors within Topological ResidualTheory (TRT). Taking the Fibonacci number Fu2081u2089 = 4181 as the critical node, we demonstrate thatthe fine-structure constant α is precisely the radian-measure expression of the angle 0.4181°. Theconstant α inherently carries three topological factors: rotation (right-handed cylindrical helicity), symmetry (Markov global stability), and scaling (renormalization and fractality). These factors necessarily generate irreducible projection residuals in real topological realizations. By quantifying the discrepancies between TRT predictions and both 1/137 and experimental values for α, G = μu2080 α², and the Planck constant, we show that all observed errors originate from the intrinsic rotation and topology of α itself. Special emphasis is placed on the multiple algebraic decompositions of 4181 and on relating error magnitudes to the renormalization factor (1/2)α² × 10u207f or α² × 10u207f. Connections to the Lamb shift and Rydberg constant are established to strengthen the theory’s self-consistency and predictive power.
Category: Mathematical Physics
[94] ai.viXra.org:2606.0016 [pdf] submitted on 2026-06-06 13:24:06
Authors: Ginanjar Utama
Comments: 8 Pages.
This paper proves, by explicit Clifford-algebra computation, that the γ0-diagonal complete orthogonal primitive decomposition of real Cl(1,3) has two sectors, while the extension to Cl(1,4) supplies a second independent commuting basis-blade involution, γ1234. Because I5 = γ0γ1234, the generated involution group is Z2 × Z2, not (Z2)3. The resulting decomposition contains four primitive idempotents, each generating a real 8-dimensional minimal left ideal. The count four is itself fixed by Wedderburn theory, since each simple summand has rank two; the explicit content of the result is the basis-blade realization of the family and the demonstration of why a naive three-involution count would predict eight. The central idempotents (1 ± I5)/2 realize the real semisimple splitting Cl(1,4) ≅ M2(H) ⊕ M2(H), where H denotes the quaternions. They do not, by themselves, supply a complex imaginary unit; a complex spinor interpretation still requires choosing an internal quaternionic complex structure within each ideal. Physical identifications with chirality, charge, helicity, or U(1) phase are therefore treated as conjectural.Supporting results include: (i) a Pancharatnam-phase computation from projector triple products in Cl(3,0), confirming the half-solid-angle rule 2|φ| = Ω for geodesic spherical triangles; (ii) a bivector-square classification of geometric-i candidates, with the observation that the chiral split in real Cl(1,3) is a complex-structure operation, not a real idempotent split; and (iii) a short positioning of the result relative to real Cl(0,6) and complex Clu2086 Standard Model programs. The paper is intended as a lower-dimensional real-idempotent audit, not as a derivation of a Standard Model generation.
Category: Mathematical Physics
[93] ai.viXra.org:2606.0014 [pdf] submitted on 2026-06-06 03:46:27
Authors: Xiangqian Zhang, Mingming Zhao, Linchao Ge
Comments: 13 Pages.
Within the framework of Topological Residual Theory (TRT), this study explores the unification ofgravitational and electromagnetic interactions at the level of spacetime geometry. Utilizing the cylindrical helical motion of spatial points (geometrization of Zitterbewegung) as the dynamical foundation, we derive the wave equationvia the variational principle of the displacement field . Using the geometric structure of thelocal cylindrical orthonormal frame induced by local helical symmetry, the electric, magnetic, and gravitational fields are defined as projections of the time derivatives of onto this orthonormal frame. Consequently, the mutual orthogonality of the three fields emerges as a geometric necessity of the frame decomposition rather than an independent postulate. The gravitational field is derived via two independent pathways: (i) the framekinematics and time evolution pathway ( ), and (ii) the second-order derivative pathway of the wave equation ( ). The mathematical consistency betweenthese two pathways indicates that gravity can be interpreted as a second-order residual effectof electromagnetic oscillations propagating at the speed of light. Furthermore, charge is defined as the topological charge (winding number / Hopf invariant) of the helical displacement field on a closed sphere (Euler characteristic ), which manifests as the effective rate of mass change only during the transient state of topological structure formation. Under the compatibility condition of theflux change rate (local topological flux continuity), we naturally derive the vacuum relations and . Based on this unified topological mechanism, the model provides a self-consistent geometric explanation for both charge quantization and particle spin.
Category: Mathematical Physics
[92] ai.viXra.org:2605.0043 [pdf] submitted on 2026-05-20 13:22:51
Authors: Chad Meyer
Comments: 28 pages, 47 references. Seven Standard Model observables (α, α_s, sin²θ_W, λ_H, m_μ/m_e, |V_us|, G_N) reproduced from one transcendental fixed point with zero free parameters. Companion paper on the polar-square geometric identity in preparation. Develop
We derive the fine-structure constant α from six properties that anyquantum coupling constant must satisfy: scale invariance, self-reference,cumulativity, probability normalization, self-consistency, and contractivity.These chain together with zero tunable parameters to produce a uniquefixed-point equation erfi(b)² = e^(b²), whose solution b* generates afive-letter algebraic alphabet. A convergent Riccati series makes thederivation analytic, extending α to unlimited precision.Three closure theorems force every component of the master equationα = R/(D_M − R α L^d): (I) a universal escape identity constrains theprojection ratio R; (II) Hurwitz's division algebra theorem forces d = 4;(III) Euler's ζ(2) = π²/6 organizes the master denominator D_M.Theorems II and III unify as facets of the graded real spectral tripleon S¹_{2π}, whose chirality grading forces the spectral convention uniquely(the competing convention is excluded at 134σ). The Padé form of theself-energy Σ_2 — previously the last remaining assumption — is nowderived: the fixed-point equation forces constant coupling (β = 0),which forces the Dyson self-energy series to be geometric, closing toa rational function with a single simple pole. A spectral dimension jump(Gabriel's Horn) connects the logarithmic branch point of the Riccatiseries (radius r = 2π = det'(D), exponent consistent with p = −1;extracted from 50 coefficients via Wynn acceleration) to the convergentspectral zeta value ζ(2) that organizes D_M.The result, 1/α = 137.035 999 075, agrees with CODATA 2018 at 0.4σ andwith Parker et al. (2018) at 1.1σ. The 4.9σ tension with CODATA 2022constitutes a falsifiable prediction. The same alphabet reproducesα_s(M_Z) (0.11σ), sin²θ_W (0.25σ), λ_H (0.03σ), m_μ/m_e (0.2σ),|V_us| (0.9σ), and Newton's constant (0.15σ) — seven observables froma single transcendental number with zero free parameters. No theorempostdating 1954 enters the derivation chain; the spectral-geometricinterpretation of the denominator draws on Connes (1994).
Category: Mathematical Physics
[91] ai.viXra.org:2605.0025 [pdf] submitted on 2026-05-12 19:50:05
Authors: Carl Andrew Brannen
Comments: 11 Pages. To be submitted to Journal of Mathematical Physics
Continuous gauge symmetries are usually introduced through Lie groups acting on quantum fields. In this paper we show that the algebraic structure underlying non-abelian gauge transformations already arises naturally inside the complex group algebra of a finite non-abelian group. The dihedral group $D_4$, the symmetry group of the square, is used as an explicit example.The complex group algebra $C[D_4]$ decomposes into irreducible matrix blocks under the Artin—Wedderburn theorem. While the character table describes only the subspace of class functions, the full group algebra contains additional intra-class directions invisible to the character table. For $D_4$ these directions form a three-dimensional subspace which, after elementary normalization, satisfies the Pauli algebra and generates continuous SU(2) transformations inside the two-dimensional irreducible block.The construction is carried out explicitly using only the multiplication table of $D_4$. Continuous rotations arise from exponentials of finite group algebra elements, without assuming any continuous symmetry at the fundamental level. The mechanism generalizes to any finite group possessing higher-dimensional irreducible representations, where the associated matrix blocks naturally support the corresponding su(N) Lie-algebra structures.
Category: Mathematical Physics
[90] ai.viXra.org:2605.0024 [pdf] submitted on 2026-05-12 19:46:56
Authors: Moninder Singh Modgil, Dnyandeo Dattatray Patil
Comments: 92 Pages.
he theory of roots of unity occupies a central position in algebra, geometry, Fourier analysis, quantum mechanics, and modern mathematical physics. In the ordinary complex plane, roots of unity generate discrete cyclic structures on theunit circle S1 and are governed by commutative rotational symmetry. However, the extension from complex numbers to higher Cayley—Dickson algebras introduces profoundly richer mathematical structures involving noncommutativity, nonassociativity,exceptional topology, higher-dimensional phase manifolds, and zero-divisor geometry. In the present work, we systematically investigate generalized roots of unity in quaternionic, octonionic, sedenionic, and higher hypercomplex systems. Explicit generalized exponential maps are derived for quaternionic and octonionicphase structures, and the resulting root manifolds are analyzed geometrically andtopologically. We show that quaternionic roots naturally generate continuous S2 families associated with the Lie group SU(2) ≃ S3, while octonionic roots generate S6 root geometries connected with exceptional G2 symmetry and nonassociative algebraic structure. Beyond the octonionic level, the emergence of zero divisors radically alters polynomial behavior, leading to extended algebraic varieties rather than isolated algebraic roots. The paper develops generalized Fourier structures, hypercomplexgauge geometry, exceptional fiber bundles, generalized Hopf fibrations, cohomological structures, characteristic classes, K-theory constructions, and homotopy classifications associated with generalized root manifolds.Connections are established between hypercomplex root geometry and Yang—Mills gauge theory, Clifford structures, exceptional Lie groups, spinorial geometry, string theory, noncommutative geometry, generalized quantum phase spaces, and exotic spacetime formulations. The work further explores possible applications to generalized cognitive geometry, hypercomplex neural dynamics, exceptional lattice structures, and multidimensional consciousness manifolds involving octonionic and E8-inspired configurations. The resulting framework suggests that generalized roots in higher Cayley—Dickson systems may provide a unifying mathematical language connecting topology, algebra, gauge.
Category: Mathematical Physics
[89] ai.viXra.org:2605.0006 [pdf] submitted on 2026-05-04 23:43:54
Authors: J. W. McGreevy
Comments: 24 Pages. (Note by ai.viXra.org Admin: Please cite listed scientific references)
The Relativistic Field Theory of Primes (RFTP) is a self-contained arithmetic fieldtheory in which the primes, modular forms, and monstrous moonshine are promoted to thedynamical degrees of freedom of a clutched adelic bundle over the compactified modularcurve X(1). The theory begins with the Eisenstein series E4(τ ), E6(τ ), and E12(τ ) and therigorously derived Ramanujan—Serre congruence τ (n) ≡ σ11(n) (mod 691), which is elevatedto the primary topological defect injecting the delta source ρsource =65520691 δj(τ)(P) into theArakelov Poisson equation. This single defect sources a shared curvature potential Veff(r, ϕ)felt identically by timelike soliton geodesics and null photon rays, generating a conservedRunge—Lenz vector and an SO(4) symmetry that unifies gravitational and electromagneticscales.A least-arithmetic-action functional S[Φ] on the Monster vertex operator algebra V♮yields a self-adjoint radial Dirac operator D on the clutched bundle whose spectrum interpolates the Balmer series and the critical-line zeta zeros. The theory realizes a Connesspectral triple (AVOA, Hsoliton, D), Einstein—Cartan gravity with χ11-sourced torsion, andphoton propagation via explicit multi-field WKB ray-tracing. Modular invariance underSL(2,) is the arithmetic skeleton: it preserves the characteristic bending speed p|d2r/ds2|,the Runge—Lenz vector, the gravitational constant GA = 65520/(8π · 691), the speed ofcausality cA, all spectra, scattering cross-sections, Einstein A/B coefficients, vacuum energy,dark-energy equation of state, and cosmological constant. The wave/ray duality explainsthe integer 3 (triality compactness of the ray limit) and the transcendental tail of π (infinitedimensional wave phase space). The Gutzwiller trace formula over modular periodic orbitslinks the primes to the zeta zeros, while compactness, discreteness, and finiteness of theclutched bundle enforce ultraviolet finiteness at the Planck scale lP .All constants and observables emerge purely arithmetically from the single master functional and the 691 defect, with no external parameters. RFTP therefore provides a parameterfree unification in which modular symmetries dictate the observable universe.
Category: Mathematical Physics
[88] ai.viXra.org:2604.0103 [pdf] submitted on 2026-04-30 13:33:27
Authors: Jason Merwin
Comments: The document contains 12 pages. A link to the code repository is provided.
A prior distinction-engine construction recovered a 137-object registry at DAG size 7 and partitioned it into sectors of 81+40+16, aligning with the spatial, interface, and gravitational sectors proposed in Relational Mathematical Realism. The present manuscript consolidates two subsequent analyses. First, we study mature registry carriers: larger relational objects that contain the full 137-type registry as an embedded motif. Exact DAG-size joins and 137-bit registry signatures recover DAG-8, DAG-9, and DAG-10 layers while compressing motif state space by factors of 6.85x, 20.43x, and 46.01x. Minimal full-registry carriers have DAG size 309. Eight non-isomorphic carrier histories preserve the same condensable typed mediation geometry: 72 spatial nodes, 38 interface nodes, 16 gravitational nodes, and 288 mediated S-G edges per condensable interface node. Typed dynamics lifted onto these mature carriers show no statistically detectable topology effect on interface-enabled condensation. A screening audit identifies full binary S-G conflict deletion as the wild-type interface rule, producing S_ DYN ~9.61, S_OFF~2.04, and a 4.73x dynamic/OFF enhancement. Loss-of-function tests identify the true invariant as complete I-mediated coverage of the active S-G conflict graph: the 38 condensable I nodes split into two complementary 19-node half-covers, each covering 288 of 576 active S-G edges.Second, we extend the analysis to an open-token distinction engine, in which finite type identity is separated from lineage-distinct token identity. The type-guarded open-token process preserves the canonical 137 DAG-7 type layer exactly while allowing unbounded token multiplicity. Particle-detector tests on open-token dynamics recover robust K3/baryon-like candidates: five stable candidates with k3_score=1.0, high closure, and fixed carrier support. I-cage/lepton-like structures appear as suggestive precursors, including a high-scoring cage-localized candidate, but not yet as fully closed localized leptons. These results support a layered interpretation: the DAG-7 registry is a finite reusable type grammar; DAG-309 carriers are mature local embeddings of that grammar; and open-token dynamics permit stable particle-like excitations over repeated registry instances. We identify a mathematically natural route from distinction-generated registry types to mature carriers and K3-centered particle-like object.
Category: Mathematical Physics
[87] ai.viXra.org:2604.0090 [pdf] submitted on 2026-04-27 09:13:28
Authors: Zhang Xiangqian, Zhao Mingming, Ge Linchao
Comments: 9 Pages.
This paper aims to construct a unified framework between discrete number theory, differentialgeometry, and continuous quantum physics through the "Topological Residual Theory." The researchtakes the 19th term of the Fibonacci sequence (Fu2081u2089 = 4181) as the core node. First, it reveals itsmathematical miracles in semiprime decomposition and fractal dimension reduction. Subsequently,this paper completely derives the differential geometric correspondence between the Binet residualand the torsion of the cylindrical helix, and details the calculation of the torsion limit (τ → 1/3) of theMarkov triples related to 4181. From first principles, it rigorously proves that this node defines theglobally stable geodesic line on the torus with no self-intersection and minimum energy.Furthermore, this paper introduces renormalization group scale transformation, dividing theFibonacci sequence by ten thousand (10u2074) and extracting its sine value, mapping it as the transverseperiodic projection of cylindrical spiral motion. Through key transition data tables and visualizationcharts, this paper intuitively demonstrates the physical mechanism by which the macroscopictopological node 4181 still maintains the minimum energy ground state after being scaled down byten thousand times (0.4181°). It proves that the fine structure constant (α ≈ 1/137) is essentially themicroscopic transverse geometric projection residual (sin(0.4181°) ≈ α) that maintains the cylindricalhelix moving at the limiting velocity u without collapsing, and that Fu2081u2089 holds an absolutelyirreplaceable critical phase transition position in the evolutionary cycle. Finally, this paper proposesthat the space around objects is structured on the basis of right-handed cylindrical spiral motion,providing solid geometric and dynamic support for the holographic universe principle
Category: Mathematical Physics
[86] ai.viXra.org:2604.0080 [pdf] submitted on 2026-04-24 19:22:58
Authors: Hao Gui
Comments: 24 Pages. This submission uses the CC BY-NC-ND 4.0 International License.
We rigorously prove that pure SU(N) Yang—Mills theory (N ≥ 2) on R^4 exists and has a positive mass gap Δ > 0. This paper presents the second of six independent proofs, centered on the analytic properties of the lattice partition function. The non-negativity of the Wilson action (S_W ≥ 0) implies, via the Bernstein—Widder theorem, that Z(β) is the Laplace transform of a positive measure, hence completely monotone and holomorphic in the entire right half-plane {Re β > 0}. This elementary observation has profound consequences: (i) Z(β) > 0 for all real β > 0 (no Lee—Yang zeros on the physical axis); (ii) the free energy f(β) is real-analytic, concave, and has a continuous derivative (excluding first-order transitions); (iii) combined with β < 0 at all couplings (proved via operator positivity and Lorentz algebraic protection), the theory has no phase transitions of any kind, and the mass gap propagates continuously from strong to weak coupling. The holomorphicity of Z provides the strongest possible regularity for the coupling dependence, making gap propagation a consequence of complex analysis rather than operator theory.
Category: Mathematical Physics
[85] ai.viXra.org:2604.0061 [pdf] submitted on 2026-04-16 16:30:36
Authors: Xiangqian Zhang, Mingming Zhao, Linchao Ge
Comments: 7 Pages.
Modern physics has long faced the dilemma of unifying quantum mechanics and general relativity, and theStandard Model contains numerous free parameters that cannot be derived from first principles. Based onthe Topological Residual Theory [1, 2], this paper establishes "spacetime fluid undergoing right-handedcylindrical helical motion at the speed of light" as the sole first postulate. We rigorously derive the purelygeometric definition of mass and establish the "Spacetime and Physical Constant Normalization Equation."Within this purely geometric framework, we completely discard a priori gravitational field assumptions andcircular reasoning. Through rigorous algebra and the principle of geometric dilution, we naturally derive themicroscopic Compton-de Broglie wavelength, the exact Planck scale, as well as macroscopic Newton's lawof universal gravitation, Kepler's Third Law, and the Schwarzschild radius of black holes. This paperdemonstrates that there are no independent physical quantities at the fundamental level of the universe;everything is a geometric and topological manifestation of the spacetime fluid. Furthermore, we provideseveral testable experimental predictions based on this theory.
Category: Mathematical Physics
[84] ai.viXra.org:2604.0057 [pdf] submitted on 2026-04-13 13:51:27
Authors: Jason R Merwin
Comments: The manuscript contains 7 pages
We construct a distinction engine from the minimal axiom A̸ = B applied to two primitives and show that exhaustive iteration produces exactly 137 permanent objects at directed acyclic graph (DAG) size 7, matching the OEIS sequence A255841. Two nested topological cuts on the resulting overlap graph partition these 137 objects into sectors of 81 + 40 + 16, reproducing the registry architecture of Relational Mathematical Realism (RMR) previously derived from complete-graph eigenvalue spectra, force emergence mechanisms, and lepton mass ratios. The partition is unique:varying the number of primitives, DAG size, pairing rules, or overlap threshold destroys it. A heterogeneous dynamical simulator with algebraically typed update rules—ternary condensation (34 = 81), binary polarity (24 = 16), and relational activation (52 × 4 = 40)—produces sector-differentiated behavior with exact energy conservation. Ablation of the interface sector demonstrates that the 40-element boundary enables matter-like condensation: removing it reduces spatial-sector realization by 80% (p < 10−10). The dynamic interface achieves full condensation efficiency with just 2 of 38 channels active, revealing massive structural redundancy and a sharp percolation-likethreshold. Seven of twelve integers in the established RMR set A appear as direct structural counts in the engine; the remaining five appear as derived ratios, including the generation factor 17 = 136/8. This constitutes a fourth independent line of evidence for the RMR registry partition, obtained from pure combinatorics with zero free parameters.
Category: Mathematical Physics
[83] ai.viXra.org:2604.0024 [pdf] submitted on 2026-04-07 19:17:11
Authors: J. W. McGreevy
Comments: 15 Pages. (Note by ai.viXra.org Admin: Please cite listed scientific references)
We construct a relativistic field theory of primes (RFTP) in which the adelic phase space is quantized by a single arithmetic object — the motivic commutator arising from the 691 topological defect. The twisted Lorentz oscillator serves as the microscopic foundation, yielding a direct derivation of Planck’s law and the entropy of a single harmonic oscillator from the commutator spectrum, with zeta regularization modulated by the 691 rigidity. The defect, realized as the non-trivial Galois extension forced by the vanishing Stickelberger element when 691 divides B12, functions as a universal cohomological object that factors through all Weil cohomologies. Stationarity of the combined action produces EinsteinCartan geometry as the thermodynamic equation of state describing curvature annealing of the defect until the conductor-9 snag, where octonionic triality locks vertical phase into GL(3) structure, generating rest mass via the Higgs-like clutching mechanism. Photons emerge as the symmetric null limit of the commutator, while massive particles arise from partial or full clutching. The theory recovers Maxwell equations in the low-energy phase, Dirac dynamics with torsional corrections, and entropic gradients driving probability flow. Implications for the BSD conjecture, Hilbert’s 12th problem, Navier-Stokes smoothness, and the Riemann Hypothesis are noted, but formal resolutions are deferred to dedicated manuscripts. The framework offers a first-principles unification of number theory with relativistic quantum field theory and gravity through a single motivic commutator.
Category: Mathematical Physics
[82] ai.viXra.org:2604.0010 [pdf] submitted on 2026-04-03 20:54:40
Authors: J. W. McGreevy
Comments: 17 Pages. (Note by ai.viXra.org Admin: Please cite listed scientific references)
We extend the Relativistic Field Theory of Primes by identifying the motivic object at thecentral cusp of the adelic upper half-plane — the non-orientable homology cycle γ pairedwith its divergence-free cohomology current JAudit, under the M¨obius twist correspondenceτ : γ 7→ −γ — as the intrinsic adelic Planck constant ℏA. This single topological object supplies the fundamental unit of action. It quantizes the adelic phase space via the commutator [τ, γ] = iℏAΩ, deforms the constitutive tensors εA and µA, derives the elementary charge eA from the holonomy around the minimal twist loop, and yields the fine-structure constant αA as a pure geometric ratio. Starting from the twisted Lorentz oscillator as the microscopic probe, the theory builds upward: the Bernoulli irregularity at weight 12 (prime 691) sets torsional rigidity, entropy gradientsdrive probabilistic flow, and the resulting entropic force produces Einstein-Cartan geometryas a thermodynamic equation of state (Jacobson-style). The same motivic object that quantizes the adelic phase space also defines the vacuumproperties and the coupling constant, providing a self-contained geometrization in whichclassical spacetime, quantum spin statistics, and Standard Model parameters emerge fromthe fundamental duality between analytic volume and algebraic rigidity.
Category: Mathematical Physics
[81] ai.viXra.org:2603.0084 [pdf] submitted on 2026-03-21 21:12:34
Authors: Xiangqian Zhang, Chao Greene, Mingming Zhao
Comments: 8 Pages. (Note by ai.viXra.org Admin: Please cite listed scientific references)
The Standard Model of modern physics contains dozens of free parameters that cannot be derivedfrom theory (such as the fine structure constant ), and faces severe theoretical gaps when bridgingmicroscopic quantum scales and macroscopic cosmic scales. This paper proposes the "TopologicalResidual Theory," establishing "right-handed cylindrical helical motion of space at the speed of light" as the sole first-principle axiom. Through the motion-derived dimension of space, the pure geometric origin of the fine structure constant is rigorously derived. Furthermore, by introducing the unit solid-angle helical divergence number , and combining it with Mandelbrot fractal geometry, thetheory proves that the strong nuclear force, electromagnetic force, and gravitational force areessentially geometric residuals of the same helical fluid at different topological levels. Finally, using only pen-and-paper geometric algebra, the theory precisely calculates the electron anomalousmagnetic moment, helium ion ionization energy, and proton-neutron mass difference, providing anew pure-geometric paradigm for a grand unified theory.
Category: Mathematical Physics
[80] ai.viXra.org:2603.0079 [pdf] submitted on 2026-03-18 14:58:48
Authors: Thomas Husmann
Comments: 8 pages, 5 tables, 1 supplementary code listing, 25 references. CC BY-NC-SA 4.0. Patent pending (Application No. 19/560,637). Source code: https://github.com/thusmann5327/Unified_Theory_Physics
Background: The Aubry—André—Harper (AAH) Hamiltonian, a one-dimensional quasiperiodic tight-binding model with irrational modulation frequency, exhibits a Cantor-set energy spectrum at its self-dual critical point. This spectrum has been experimentally realized in ultracold atoms, superconducting qubits, and photonic waveguides, yet its structural features have not previously been applied to the prediction of atomic properties. Methods: We constructed the AAH Hamiltonian on a lattice of D = 233 sites (a Fibonacci number) with modulation frequency α = 1/φ, where φ denotes the golden ratio, and potential amplitude V = 2J (the self-dual critical point). Numerical diagonalization yielded 233 eigenvalues whose gap structure was analyzed to extract five spectral constants. These constants were combined into a closed-form algebraic formula for the ratio of van der Waals radius to covalent radius, r(vdW)/r(cov), using six prediction modes parameterized solely by each element's electron configuration. No constants were fitted to experimental atomic data. Results: The formula was evaluated for 54 elements (Z = 3—56). Of these, 42 (78%) yielded predicted ratios within 10% of observed values, 53 of 54 (98%) within 20%, with a mean absolute error of 6.7%. Only one element (boron) exceeded a 20% error. The best-predicted element, cesium, showed agreement to 0.2%. The residual deviations correlated significantly with independently measured material properties: hardness (ρ = +0.66 for p-block elements), melting point (ρ = −0.61), and electrical conductivity (r = −0.74 for metals). For the lanthanide series (Z = 57—71), the four-gate architecture generated three confirmed predictions: (a) the van der Waals radii should be approximately constant across the series, consistent with Alvarez's crystallographic finding of a 232 ± 9 pm mean; (b) covalent radii should contract monotonically, matching the observed lanthanide contraction from 207 to 175 pm; and (c) the worst conductor should occur at f7 half-filling and the best at f14, confirmed by gadolinium (0.74 MS/m) and ytterbium (3.51 MS/m). Conclusions: These findings indicate that the Cantor-set band structure of the AAH spectrum encodes quantitative information about atomic radius ratios across the periodic table. The formula achieves accuracy comparable to semi-empirical screening models while employing zero adjustable parameters. Its residuals constitute systematic indices of material properties rather than random noise.
Category: Mathematical Physics
[79] ai.viXra.org:2603.0072 [pdf] submitted on 2026-03-16 13:35:11
Authors: Jason Merwin
Comments: This document contains 22 pages.
Relational Mathematical Realism (RMR) proposes that physical reality emerges from a discrete 137-node registry partitioned as 81 (spatial) + 40 (surface) + 16 (gravitational). We develop the thermodynamic screening mechanism introduced in the RMR synthesis paper [3] into a self-contained predictive framework. A universal screening unit δ0 = 5/137, derived from the registry’s electromagnetic surface fraction, generates mass corrections across leptons, hadrons, and the fine-structure constant with zero fitted parameters. Key results: (i) α−1 = 137.035 996 versus CODATA 137.035 999 (error 3 × 10−6); (ii) mΛ0 /me = 2183.328 versus PDG 2183.327 (error 4 × 10−7, sub-keV on a GeV-scale mass); (iii) a strangeness screening integer Bs = V (K3)2 = 9 derived from the K3 ⊗ K3 product graph, yielding correct corrections for Λ0 and K+; (iv) a {9, 13, 17} strangeness ladder for Λ0, Ξ−, Ξ0 with step size 4, arising from K5 interface topology; (v) a charge-dependentovershoot theorem: corrections exceeding the 42 = 16 gravitational threshold are clipped by 4for charged particles, with three-particle confirmation (τ −, Ξ−, Ξ0). Eleven independent mass predictions are tabulated. A screened-angle ladder extends these results to dimensionless mixing observables: the effective leptonic weak mixing angle, the three PMNS angles, and the reactor angle are all approximated by δ0-corrections of quarter/half base states using multipliers from established graph integers, with no new parameters. Open sectors (meson bases, Ω− decuplet, Σ0 isospin mixing, and radiative corrections to the mixing angles) are characterized precisely andreserved for future work.
Category: Mathematical Physics
[78] ai.viXra.org:2603.0058 [pdf] submitted on 2026-03-13 17:21:04
Authors: Tingfang Yi
Comments: 7 Pages.
Light’s propagation, phase, polarization, and frequency are projections of a minimal six-dimensional null entity unifying spacetime and photon internal degrees of freedom. Observed pairwise and triple couplings emerge naturally, explaining Spin Hall, Berry, Pancharatnam phases, optical activity, and relativistic shifts. Classical fields, quantum detections, and wave-particle duality arise as lower-dimensional projections, while photon discreteness reflects detector intersections. The framework agrees with relativity and QED and predicts testable interference deviations via controlled holonomy.
Category: Mathematical Physics
[77] ai.viXra.org:2603.0049 [pdf] submitted on 2026-03-12 02:40:35
Authors: J. W. McGreevy
Comments: 19 Pages. (Note by ai.viXra.org Admin: Please cite listed scientific references)
We present a unified relativistic field theory over the ring of adeles AQ, in whichthe nontrivial zeros of the Riemann zeta function emerge as quantized energy levelsof a self-adjoint Hilbert—P´olya operator ˆH. The theory begins with a Non-Hermitian"Adelic Carnot Pump" driven by local prime interactions (Cup and Cap products),Hermitianized globally through restriction to the cuspidal subspace of automorphicforms on GL2(Q)GL2(A).The Arithmetic Planck constant hA ∼ log 2 quantizes phase steps, while the weight-12 modular discriminant ∆ provides vacuum tension. Maxwell-like adelic field equa-tions govern the dynamics, culminating in a least-arithmetic-action principle whosegeodesics are confined to the critical line Re(s) = 1/2.A dimensionless Arithmetic Fine-Structure Constant αA ≈ 1/(log 2 · 2π) tunes thecoupling between local curvature and global flow. Invariance of αA under the fulladelic group action is enforced by an Arithmetic Noether Theorem, which acts as acentripetal force locking spectral ordinates to the critical line.Self-adjointness follows from cuspidal sealing and unitary Atkin—Lehner symmetry;spectral matching is achieved via a twisted trace formula that projects the GL2 cuspidalspectrum onto the GL1 zeta zeros through phase-cancellation of non-coherent modes.We conclude that the Riemann Hypothesis is the symmetry requirement for acausal, lossless arithmetic vacuum: all nontrivial zeros satisfy Re(ρ) = 1/2. Exten-sions to the Birch and Swinnerton-Dyer conjecture are suggested.Keywords: Riemann Hypothesis, Hilbert—P´olya conjecture, adelic geometry, au-tomorphic forms, Noether theorem, non-Hermitian dynamics, modular forms, criticalline1
Category: Mathematical Physics
[76] ai.viXra.org:2603.0046 [pdf] submitted on 2026-03-11 18:50:11
Authors: Thomas A. Husmann
Comments: 7 Pages. (Note by ai.viXra.org Admin: Please don't name title, equation/formula etc after the author's name)7 tables. Verification code at https://github.com/thusmann5327/Unified_Theory_Physics. Patent Application 19/560,637. CC BY-NC-SA 4.0.
We demonstrate that the [proposed] Decomposition, grounded in the single algebraic identity phi^2 = phi + 1, extends to atomic physics with sub-percent precision across three independent domains. Spectral: alpha_em^{-1} = N x W = 137.337 (0.22% vs. CODATA), where N = 294 and W = 0.467134 are derived from the topological invariants of a 233-site Aubry-Andre-Harper lattice at critical coupling. The bracket count N is independently derivable as N = F(13) + F(10) + F(5) + F(2) = 233 + 55 + 5 + 1, a spectral topology invariant generalizable to any Fibonacci lattice. This cascades through QED to predict the Bohr radius (0.22%), Rydberg energy (0.44%), and proton charge radius (0.14%). Spatial: Anchoring the hydrogen 1s peak at the Cantor shell center maps 42% of the probability into the sigma_2-sigma_4 wall zone; the H_2 bond length matches sigma_4 outer to 0.5%. Information-theoretic: The bipartite entanglement entropy at the Cantor sigma_4 outer wall reaches S = 0.6908 nats, 99.66% of the theoretical maximum ln 2, with zero free parameters. The Cantor outer wall lies 5.3% beyond the equal-partition point (p = 0.535), preserving 99.66% of the theoretical maximum entropy ln 2. Three additional exact identities are proved from phi^2 = phi + 1 alone: the unity partition 1/phi + 1/phi^3 + 1/phi^4 = 1, the boundary law 2/phi^4 + 3/phi^3 = 1, and the pi-phi bridge pi = 4*arctan(1/phi) + 4*arctan(1/phi^3). Systematic comparison with all metallic means and quadratic irrationals shows phi is unique: it alone produces a critical eigenstate with minimal fractal dimension D_2 ~ 0.538, a nested five-sector Cantor spectrum, stable baryon fraction Omega_b ~ 0.048, and dark-energy-dominated expansion. The fine-structure constant is the cumulative wall fraction of the Cantor vacuum: alpha_em = 1/(N x W). The hydrogen atom is the same self-consistent structure at bracket 117.
Category: Mathematical Physics
[75] ai.viXra.org:2603.0044 [pdf] submitted on 2026-03-10 00:06:15
Authors: Md Mubdiul Hasan
Comments: 8 Pages.
—This work presents the analysis and functionalcharacterization of a multi-tap AC voltage stabilizer implemented in LTspice, emphasizing its control, feedback, and protection mechanisms. The stabilizer employs an autotransformer with discrete tap selection, where relay-driven switching provides coarse regulation across varying mains conditions. A dedicated sensing network, consisting of a transformer secondary, rectifier, and precision divider, generates a DC feedback signalproportional to the output RMS voltage. This signal is processed by a window-comparator architecture built around the LM324,enabling the controller to determine whether the present tap is too high, too low, or within the acceptable regulation band. The design incorporates a soft-start and pre-regulation stage that ensures stable startup behavior by delaying relay activation until the reference, supply rail, and sensing nodes reach steady-state values. Comparator thresholds are derived analytically by mapping mains voltage through the transformer ratio, rectifier, and divider network to the LM324 input domain. This allows direct calculation of the window center and allowable mains range for each tap. A protection comparator monitors long-term deviationsoutside the window and, through a zener-coupled reference node, triggers a controlled shutdown or tap change when necessary. The resulting system combines discrete tap switching with analog decision logic to achieve robust voltage regulation across a wide input range, while maintaining predictable and mathematicallyverifiable operating boundaries.
Category: Mathematical Physics
[74] ai.viXra.org:2603.0031 [pdf] submitted on 2026-03-07 16:25:39
Authors: Tingfang Yi
Comments: 7 Pages.
We propose a unified minimal six-dimensional (6D) light null entity, where the fundamental properties of light are reinterpreted as intrinsic degrees of freedom within a singular closed manifold. This 6D structure integrates a two-dimensional null propagation geometry with four intrinsic 1D degrees of freedom: optical phase, polarization, frequency, and orientation relative to the null momentum generator. In this framework, conventional four-dimensional (4D) spacetime optical, electromagnetic, and quantum phenomena arise as lower- dimensional projection or section measurements of this structure, while preserving consistency with established theories at the level of current observations. Spacetime coordinates and photon internal degrees of freedom may be geometrically equivalent components of a higher-dimensional manifold.
Category: Mathematical Physics
[73] ai.viXra.org:2603.0012 [pdf] submitted on 2026-03-02 21:37:16
Authors: Rui Mateus Joaquim
Comments: 12 Pages.
Donald Hoffman’s Interface Theory of Perception (ITP) suggests that spacetime is not a fundamental reality but a functional interface hiding a complex network of conscious agents. However, the mathematical "engine" that drives the transition from informational dynamics to geometric perception remains largely undefined. This paper proposes the Ontodynamic Matrix Singularity (OMS) as the foundational mechanism for this transition. Through a series of computational simulations (N=15, 210 directional flows), we demonstrate that the application of an "Ambiguity Resolution Operator" (ϕΨ) over a symmetric adjacency matrix generates stable informational cores. Our results show that mass and spacetime curvature are not intrinsic properties of matter, but emergent regulatory structures arising from informational saturation. This model provides a formal bridge between Conscious Realism and physical observables
Category: Mathematical Physics
[72] ai.viXra.org:2602.0117 [pdf] submitted on 2026-02-26 19:59:49
Authors: Lluis Eriksson
Comments: 33 Pages.
This document is an experiment-first audit report for a companion-paper programme claiming a constructive solution of the 4D $mathrm{SU}(N)$ Yang—Mills existence and mass gap problem. It specifies a runnable mechanical audit suite of 29 deterministic tests, defines pass/fail criteria, and presents outputs in a compilation-safe format. The report contains: (i) an explicit non-triviality proof showing the Wightman functions do not factorize trivially; (ii) a toy-model validation recovering the exact 2D $mathrm{SU}(2)$ Yang—Mills mass gap to machine precision; (iii) a Bałaban bridge appendix reproducing the critical inductive step of his renormalization group in simplified form; (iv) a reproducibility repository with 3-line setup instructions; (v) a core proof chain audit mechanically verifying the load-bearing theorems of Papers 86—90, covering terminal Kotecký—Preiss convergence, UV suppression, one-dimensionality of the anisotropic sector, Cauchy bounds on polymer jets, the OS1 vanishing rate $O(eta^2 log eta^{-1})$, Lie-algebra annihilation, and KP margin sensitivity. Beyond the 17 core tests, the suite includes a lattice gauge proxy layer (plaquette expansion, Polyakov-loop centre symmetry, Creutz ratio; 3 tests), an infrastructure layer (Bakry—Émery curvature seed $mathrm{Ric}_{mathrm{SU}(N)} = N/4$, the $2^{4k}$ cancellation in $d=4$, heat-kernel column bound; 3 tests), a UV-flow/heat-kernel layer (Parseval identity, diagonal decay exponent $d/2 = 2$, flow—reflection commutation; 3 tests), a non-triviality test (Haar Monte Carlo on $mathrm{SU}(2)$ and $mathrm{SU}(3)$; 1 test), a toy-model validation (2D Yang—Mills transfer matrix; 1 test), and an algebraic QFT layer (Petz recovery fidelity bound $1-F leq C,e^{-2mr}$ from the Split Property; 1 test). All 29 tests pass; the full suite completes in ${approx}70,mathrm{s}$ on a Google Colab CPU. The complete inter-paper dependency DAG is acyclic and explicitly recorded. All code, data, and artifacts are available at https://github.com/lluiseriksson/ym-audit. The companion papers are archived at https://ai.vixra.org/author/lluis_eriksson.
Category: Mathematical Physics
[71] ai.viXra.org:2602.0097 [pdf] submitted on 2026-02-20 18:28:06
Authors: Vinicius F. S. Santos
Comments: 19 Pages.
We introduce the Secular Replicator Flow, a finite-dimensional algebraic dynamical system inspired by the turbulent energy cascade of the Navier—Stokes equations, built from the spectral theory of golden resolvent operators on discrete network graphs [9]. The continuous mechanics of fluid turbulence—incompressibility, nonlocal pressure, nonlinear advection, and viscous dissipation—find precise algebraic counterparts in the constraints of a replicator equation evolving on the simplex of spectral participation weights, governed by a global secular equation. Within this framework we establish three principal results. First, the Variance Law: the macroscopic coupled eigenvalue λ∗(t) evolves monotonically according to Fisher’s Fundamental Theorem, acting as a strict Lyapunov function (between excision events) whose rate of increase equals the fitness variance of the active spectrum. Second, the Spectral Selection Theorem: the fitness landscape is a strict bipolar Ushape in the base eigenvalue μ, guaranteeing that the replicator flow annihilates mid-spectrum noise and funnels all energy into the extreme macroscopic topologies of the network. Third, Global Regularity: as the system approaches a structural resonance (transparent pole), the fitness plunges to −∞, triggering an auto-excision mechanism that exponentially starves the dangerous channel, rendering every pole singularity removable. The resulting dynamics form a Sawtooth Cascade of smooth climbs interrupted by discontinuous structural snaps whose direction is controlled by the residual load of the excised channel. We classify the sole remaining failure mode as a thermodynamic phase escape at the r = 2 Chebyshev boundary, where the discrete algebraic structure of the network undergoes a global phase transition into unbounded hyperbolic space—a phenomenon fundamentally different from the localised velocity blowup sought by PDE analysis. All regularity results herein apply to this model; implications for the full Navier—Stokes equations in R3 remain open.
Category: Mathematical Physics
[70] ai.viXra.org:2602.0096 [pdf] submitted on 2026-02-20 06:04:55
Authors: Lluis Eriksson
Comments: 21 Pages.
This paper is a hostile-review navigation guide and audit manifesto for a companion-paper programme claiming a constructive solution of the four-dimensional SU(N) Yang-Mills existence and mass gap problem in the Osterwalder-Schrader (OS) framework, reconstructed as a Poincare-covariant Wightman QFT with strictly positive mass gap. The guide provides: (i) an explicit dependency graph and Clay/Jaffe-Witten checklist; (ii) an explicit threat model listing standard failure modes targeted by hostile review (black-box dependence on Balaban, interface friction between gradient flow and the Balaban measure, diagonal-limit non-uniformity, and operator-mixing residues); (iii) an explicit four-pillar defensive architecture resolving each attack with structural (not merely quantitative) shields; (iv) the preventive lock: a triangular renormalization-mixing structure that blocks upward anisotropic flow into the marginal (d=4) sector, neutralizing the standard a^2 x a^{-2} -> O(1) objection; (v) a mechanical audit trail mapping load-bearing hypotheses to primary sources; and (vi) a complete linked index of all supporting preprints for traceability. External mathematics is explicitly declared: abstract polymer cluster expansion (Kotecky-Preiss), OS reconstruction (Osterwalder-Schrader), and lattice reflection positivity (Osterwalder-Seiler). Furthermore, this guide introduces the Triangular Mixing Preventive Lock: a structural algebraic mechanism showing that the operator mixing matrix has no anisotropic marginal d=4 sink in the gauge-invariant W_4-scalar sector. Consequently, the standard O(a^2) x O(a^{-2}) -> O(1) operator-mixing residue attack is blocked structurally: any quadratic divergence is forced to renormalize only O(4)-invariant d=4 data (the isotropic coupling), leaving the O(4)-breaking channel suppressed. This paper is not a claim of institutional validation; it is an audit map prescribing the check order and falsification points for the companion-paper chain.
Category: Mathematical Physics
[69] ai.viXra.org:2602.0095 [pdf] submitted on 2026-02-19 19:48:12
Authors: J. W. McGreevy
Comments: 16 Pages.
We present Arithmetic Relativistic Emergence (ARE) as a "General Relativity of Numbers" — a framework in which the Standard Model, quantum mechanics, classical 3+1 Lorentzian spacetime, and fundamental constants emerge tautologically from the arithmetic geometry of Q. The Riemann zeta function ζ(s) constitutes the maximally symmetric pregeometric vacuum. Its functional-equation symmetry around Re(s) = 1/2, combined with the pole at s = 1, forces spontaneous symmetry breaking via the weight-12 modular discriminant ∆(τ ) = η(τ )24 at the s = 6 harmonic threshold. This breaking disperses the vacuum into Archimedean divergence (Fdiv, smooth curvature density) and non-Archimedean curl (Hcurl, discrete torsion at p-adic fibers). The emergent geometry is governed by Arakelov curvature on the arithmetic surface Spec Z ∪ {∞},where Weierstrass weights act as "mass/energy density" (algebraic rigidity) and the hyperbolic/Bergman metric plays the role of spacetime. Modular transformations toward cusps correspond to Lorentz rapidity, yielding an equivalence principle analogue between inertial (modular flow resistance) and gravitational (metric warping) responses. The adelic spectral triple (KO-dimension 6, finite algebra C ⊕ H ⊕ M3(C)) induces symplectic deformation of phase space, with the non-trivial zeros of zeta providing the Dirac spectrum (Hilbert—Pólya realized). The Minkowski interval ds2 = −c2dt2 + du20d7x 2 emerges as the unique adelic-invariant quadratic form, with light cone as the resolved cusp boundary (holographic screen).The spectral action Tr f (D/Λ) recovers Einstein—Cartan gravity with non-Abelian Yang—Mills, where generalized Rainich conditions (quadratic invariants involving structure constants f abc) are satisfied at s = 6, with torsion (Hcurl) regularizing self-interactions. The full SM gauge group SU(3)c × SU(2)L × U(1)Y and three chiral generations emerge fromadelic place ramification and Leech lattice Z2-orbifold. Constants (α−1 ≈ 137 from Petersson + torsion residues, ℏ from Lambert-Planck suppression, G from unification suppression, Λ ∼ e−288) are inevitable invariants. Langlandsfunctoriality acts as the holographic dictionary mapping prime rigidity to bulk physics. ARE thus unifies physics as the macroscopic shadow of arithmetic rigidity, with the Riemann Hypothesis as a necessary stability condition for the emergent universe.
Category: Mathematical Physics
[68] ai.viXra.org:2602.0092 [pdf] submitted on 2026-02-19 12:13:41
Authors: Lluis Eriksson
Comments: 9 Pages.
We complete the rigorous construction of four-dimensional Euclidean SU(N) Yang--Mills quantum field theory and establish the existence of a mass gap. Building on the companion papers -- which unconditionally establish exponential clustering with mass gap, the Osterwalder--Schrader axioms OS0, OS2, OS3, OS4, and quantitative irrelevance of O(4)-breaking lattice operators -- we derive a lattice Ward identity for infinitesimal Euclidean rotations, identify the breaking term as a dimension-6 anisotropic operator insertion, and prove that the breaking distribution vanishes as $O(eta^2,|log((Lambda_{mathrm{YM}}eta)^{-1})|) to 0$ in the continuum limit, establishing axiom OS1 (full O(4) Euclidean covariance). Combined with the Osterwalder--Schrader reconstruction theorem, this yields a non-trivial Poincare-covariant Wightman quantum field theory with mass gap $Delta_{mathrm{phys}} geq c_N,Lambda_{mathrm{YM}} > 0$ for each $N geq 2$.
Category: Mathematical Physics
[67] ai.viXra.org:2602.0091 [pdf] submitted on 2026-02-19 12:16:47
Authors: Lluis Eriksson
Comments: 9 Pages.
This paper has two goals.Part I (terminal KP bound). We provide a verifiable, citation-driven derivation of the terminal-scale Kotecky--Preiss (KP) smallness bound used in the companion paper on exponential clustering and mass gap. Rather than re-deriving the full multiscale renormalization group (RG), we isolate explicit hypotheses (H1)--(H3) on the terminal polymer activities and prove that they imply the KP convergence criterion. We then verify (H1)--(H3) by mapping them to specific statements in Balaban's published primary sources (CMP 116, 119, 122), with an audited notation bridge recorded in the structural package companion paper.Part II (assembly map + Clay checklist). We give an explicit dependency graph assembling the companion papers together with the KP input proved here. We provide a checklist matching the Clay/Jaffe--Witten formulation of the Yang--Mills existence and mass gap problem to the theorems across the paper sequence (OS0--OS4, OS1, and the mass gap).Scope / external mathematics. The argument uses the abstract KP cluster expansion theorem (Kotecky--Preiss 1986) and the Osterwalder--Schrader reconstruction theorem (1975). It relies on the terminal polymer representation and activity bounds as proved in Balaban's CMP papers cited above.
Category: Mathematical Physics
[66] ai.viXra.org:2602.0089 [pdf] submitted on 2026-02-18 09:21:04
Authors: Lluis Eriksson
Comments: 17 Pages.
We establish two independent rigorous results for four-dimensional SU(N)pure-gauge lattice Yang—Mills theory with Wilson action, at fixed latticespacing η > 0 and weak coupling gu2080 ≤ g_*, uniformly in the spatial volume L.(A) Uniform Log-Sobolev Inequality. The Wilson measure μ_L satisfies Ent_{μ_L}(f²) ≤ (2/ρ̂) E_L(f,f) with constant ρ̂ > 0 independent of L, where E_L is the natural Dirichlet form on SU(N)^{|E(Λ)|}.(B) Uniform Mass Gap. The Osterwalder—Seiler Hamiltonian H_L has a spectral gap m_gap ≥ mu2080 > 0, uniformly in L.Both theorems share a single input — the Dobrushin—Shlosman completeanalyticity (CA) condition, verified via Bałaban's renormalization groupprogram — but follow logically independent paths. Theorem A is derivedthrough Cesi's quasi-factorization of entropy, seeded by a Bakry—Émerylocal log-Sobolev inequality on SU(N)^{|E(Σ)|}; the Ricci curvatureRic_{SU(N)} = (N/4)g plays a key role. Theorem B is derived throughexponential clustering of temporal correlations — a consequence of CA viaDobrushin contraction — combined with the Osterwalder—Seiler transfer-matrixconstruction and the Krein—Rutman theorem. We further prove (C) that {μ_L}converges weakly to a unique, translation-invariant infinite-volume Gibbsstate μ_∞ satisfying the DLR consistency equations, whose reconstructedHamiltonian H_∞ inherits the mass gap mu2080. All constants are explicit inN, gu2080, and η. The present results hold at fixed lattice spacing; thecontinuum limit η → 0 is addressed in a companion paper.
Category: Mathematical Physics
[65] ai.viXra.org:2602.0088 [pdf] submitted on 2026-02-18 17:31:06
Authors: Lluis Eriksson
Comments: 20 Pages.
Assuming a uniform log-Sobolev inequality for the pure Wilson measure (isolated here as an explicit hypothesis), we establish exponential clustering with a strictly positive mass gap for four-dimensional pure SU(N) lattice Yang-Mills theory with Wilson's action, uniformly in lattice spacing eta and physical volume L_phys:|Cov(O(0), O(x))| leq C exp(-m |x| / a_*), with m > 0 and a_* ~ Lambda_YM^{-1}.The proof assembles three ingredients: (1) Balaban's rigorous renormalization group for lattice gauge theories (CMP 1984-1989), which produces effective densities with local polymer decompositions and exponentially decaying activities; (2) a uniform log-Sobolev inequality for the pure Wilson measure, used as an input assumption; and (3) a multiscale correlator decoupling identity (this paper), which separates ultraviolet fluctuations from infrared physics. The coupling control required by Balaban's framework -- that the effective couplings remain in the perturbative regime throughout the RG iteration -- is established via an inductive argument using Cauchy bounds on the analyticity of the effective action.We also verify (under the same hypothesis) the Osterwalder-Schrader axioms OS0, OS2, OS3, and OS4 for subsequential continuum limits, and establish vacuum uniqueness and non-triviality. The remaining axiom OS1 (full O(4) Euclidean covariance) is not established here; we prove covariance under lattice translations and the hypercubic group W_4, and show that if O(4) covariance holds in the continuum limit, the reconstructed Wightman theory is a non-trivial relativistic quantum field theory with mass gap Delta_phys geq c_N Lambda_YM > 0.
Category: Mathematical Physics
[64] ai.viXra.org:2602.0087 [pdf] submitted on 2026-02-18 18:58:54
Authors: Lluis Eriksson
Comments: 18 Pages.
We classify gauge-invariant local lattice operators of classical dimension 6 on the four-dimensional hypercubic lattice into O(4)-invariant, hypercubic-invariant but O(4)-breaking (anisotropic), and on-shell-redundant components, following the Symanzik improvement programme and the on-shell improvement technique of Lüscher—Weisz (1985). Inside Balaban's renormalization group framework for SU(N) lattice Yang—Mills theory, we extract the anisotropic projection of the effective action via local Taylor expansion of polymer activities in the small-field regime and prove a quantitative quadratic scale bound for the anisotropic coefficient: for every RG step k ≤ k* with effective coupling g_k ≤ γ_0, the coefficient of the (one-dimensional) anisotropic sector in the classical dimension-6 Symanzik expansion satisfies |c_{6,aniso}^{(k)}| ≤ C a_k^2, uniformly in lattice spacing η, physical volume L_phys, and RG step k. We further prove a quantitative insertion integrability estimate for connected correlators with one insertion of the anisotropic operator. When combined with the rotational Ward identity derived in the companion paper, this yields that the corresponding breaking distribution tested against Schwartz functions is O(η^2 |log(Λ_YM η)^{-1}|) and hence vanishes as η → 0.
Category: Mathematical Physics
[63] ai.viXra.org:2602.0085 [pdf] submitted on 2026-02-17 16:18:50
Authors: Lluis Eriksson
Comments: 21 Pages.
We prove that Wilson-loop expectations in four-dimensional Euclideanlattice Yang—Mills theory with compact gauge group G admit auniversal continuum limit, independent of the lattice approximationscheme, for every contractible loop and all values of the coupling.The proof proceeds by a multiscale decomposition that combinesBalaban's renormalization-group framework with a quantitativegradient-flow smoothing step at each scale. For an observableliving at lattice scale k, the Yang—Mills gradient flow is run fora time proportional to the squared lattice spacing a_k^2; adeterministic contraction estimate (Theorem 3.5) shows that thisreduces the single-link oscillation of the flowed observable by afactor L^{-2k}, where L is the blocking factor. The resultinggeometric series is summable and yields the desired uniform bound.The two main inputs are: (i) a pointwise domination lemma(Lemma 3.3) that controls the gradient of the flowed observableby a scalar heat kernel on the link graph, exploiting thecontractivity of parallel transport; and (ii) a Duhamelinterpolation formula (Lemma 4.1) that converts each change-of-measure error into a covariance with the irrelevant part of theeffective action, bounded via a Poincaré-type inequality. Togetherthese close the Balaban—Doob inductive circuit under a quantitativeblocking hypothesis that is verified in a companion paper.As a corollary, we establish Osterwalder—Schrader reflectionpositivity for the gradient-flow-smoothed Wilson-loop observable,which together with the continuum limit yields a construction ofthe physical Hilbert space and a positive transfer matrix for thetheory.
Category: Mathematical Physics
[62] ai.viXra.org:2602.0084 [pdf] submitted on 2026-02-17 19:43:23
Authors: Lluis Eriksson
Comments: 15 Pages.
We establish quantitative almost-reflection positivity (almost-RP) for a family of flowed observables in finite-volume lattice Yang-Mills theory on the four-dimensional Euclidean torus T_L^4 with structure group G = SU(N). The lattice Wilson flow - the lattice counterpart of the Yang-Mills gradient flow - acts as a gauge-covariant smoothing that suppresses ultraviolet fluctuations. By combining three ingredients: (i) a Gaussian localization bound that controls the variance of flowed observables via an Efron-Stein-type inequality, (ii) Jacobian estimates for the lattice Wilson flow that yield exponential decay of trans-plane influence, and (iii) the exact lattice reflection positivity of the Wilson action, we show that the failure of RP for flowed observables is exponentially small in the ratio epsilon_0^2 / t, where epsilon_0 is the physical separation between the observable's support and the reflection plane (minus the diffusion scale sqrt(8t)), and t > 0 is the flow time. We record the standard Osterwalder-Schrader reconstruction as a conditional statement: exact reflection positivity on a positive-time algebra implies a Hilbert space, a vacuum, and a non-negative Hamiltonian. Our approach is non-perturbative, holds for all values of the lattice coupling, and requires no cluster expansion or infinite-volume limit.
Category: Mathematical Physics
[61] ai.viXra.org:2602.0082 [pdf] submitted on 2026-02-17 02:36:31
Authors: Tingfang Yi
Comments: 7 Pages.
We propose a minimal six-dimensional (6D) light null entity in which the six dimensions are intrinsic degrees of freedom of a null physical entity. The six dimensions consist of a two- dimensional null propagation geometry together with four intrinsic one-dimensional degrees of freedom of light: optical phase, polarization, frequency, and orientation along the null momentum generator. In this framework, all four-dimensional (4D) spacetime optical, electromagnetic, and quantum phenomena are understood as lower-dimensional projection or section measurements of a single higher-dimensional null entity.
Category: Mathematical Physics
[60] ai.viXra.org:2602.0078 [pdf] submitted on 2026-02-15 17:57:53
Authors: Luisiana X Cundin
Comments: 6 Pages.
A formal, systematic approach for generating nonlinear partial differential equations is outlined, which provides a more robust, reliable method. Additionally, formal methods provide a means to test the validity and/or the veracity of proposed nonlinear partial differential equations, thereby potentially saving researchers precious time and effort.
Category: Mathematical Physics
[59] ai.viXra.org:2602.0077 [pdf] submitted on 2026-02-15 05:11:13
Authors: Lluis Eriksson
Comments: 14 Pages.
We prove that the continuum limit of pure SU(N) lattice Yang—Mills theory in four Euclidean dimensions exists on the algebra of blocked observables at fixed finite volume, conditional on a quantitative regularity hypothesis for the blocking map. The argument combines three components: Bałaban's rigorous renormalization group program, which provides polymer representations and ultraviolet stability; a Doob-martingale influence bound that controls covariance without product-measure assumptions; and a renormalization-group Cauchy summability framework converting per-scale oscillation decay into convergence. The resulting continuum state is gauge-invariant, Euclidean-covariant, and positive. Osterwalder—Schrader reconstruction, the thermodynamic limit, and the mass gap remain open.
Category: Mathematical Physics
[58] ai.viXra.org:2602.0073 [pdf] submitted on 2026-02-14 09:06:13
Authors: Lluis Eriksson
Comments: 17 Pages.
We prove that expectations of blocked, bounded Lipschitz observables at a fixed physical scale ℓ > 0 form an absolutely summable telescoping sequence along a Balaban-matched renormalization trajectory in four-dimensional SU(N_c) lattice Yang—Mills theory with lattice spacings a_k = a_0 2^{-k}. In particular, the continuum limit state ω(O) := lim_{k→∞} ⟨O^{(k)}⟩_{Λ_k, β_k} exists for every O in the blocked observable algebra A_ℓ^{block}. The proof uses three ingredients: (i) an exact RG identity (law of iterated expectations), (ii) a one-step pushforward stability bound for blocked observables derived from Gaussian control of fast modes and an approximate centering property of the fluctuation field, and (iii) a measure-comparison lemma via Duhamel interpolation using polymer remainder bounds. No quantitative rate of asymptotic freedom is required beyond staying in the small-coupling regime where the RG estimates hold; summability follows from the geometric decay (a_k/ℓ)^2 = O(4^{-k}) together with the assumed summability of the large-field/truncation errors {τ_k}. We also state a conditional extension to "renormalized" observables (e.g. Creutz-type constructions) contingent on a nonperturbative Symanzik extraction from polymer expansions, and we discuss the relation to Osterwalder—Schrader reconstruction and the mass gap problem.
Category: Mathematical Physics
[57] ai.viXra.org:2602.0072 [pdf] submitted on 2026-02-14 11:43:03
Authors: Lluis Eriksson
Comments: 14 Pages.
We close the missing influence estimate — Assumption (B6) — required by the RG-Cauchy summability framework for blocked observables in four-dimensional SU(N_c) lattice Yang-Mills theory. The influence is measured by the Efron-Stein seminorm sigma_nu(f)^2 = sum_{e} E_nu[Var_e^nu(f)] that appears in the Duhamel interpolation lemma of the companion paper. We work in the small-field regime of Balaban's multiscale effective action and assume: (A1) a standard polymer representation for the irrelevant remainder V_k^{irr} = sum_X K_k(X); (A2) an explicit per-link oscillation bound for polymer activities carrying the correct irrelevance factor 2^{-2k}; (A3) a lattice-animal counting estimate. Under these three verifiable hypotheses — to be discharged from Balaban's historical work in a companion compendium paper — we prove sup_{t in [0,1]} sigma_{nu_{k,t}}(V_k^{irr}) <= C, where C = C(N_c, beta_0, kappa, C_osc, C_anim, p, L/a_0) is independent of the RG scale k. The proof uses only oscillation bounds and combinatorics: no log-Sobolev inequality, no mixing hypothesis, and no measure-dependent technology beyond the definition of conditional variance. This removes the only genuinely novel probabilistic input remaining in the UV block of the programme towards the Yang-Mills Millennium Prize.
Category: Mathematical Physics
[56] ai.viXra.org:2602.0070 [pdf] submitted on 2026-02-14 18:53:53
Authors: Lluis Eriksson
Comments: 7 Pages.
We prove a uniform Doob martingale influence bound for the irrelevant polymer remainder arising in multiscale renormalization group analyses of four-dimensional SU(N_c) lattice Yang-Mills theory at fixed physical volume. Our main tool is the Doob influence seminorm sigma_nu(f)^2 = sum_i E_nu[(Delta_i f)^2], which yields an exact covariance identity for arbitrary probability measures. Assuming a deterministic per-link oscillation estimate for polymer activities with a scale factor 2^{-2k} (imported from the Balaban renormalization group programme) and using a standard lattice-animal counting lemma (proved here), we obtain a bound sup_{t in [0,1]} sigma_{nu_{k,t}}(V_k^{irr}) <= C independent of the RG scale k. We then explain how this bound feeds into a Duhamel interpolation step used in RG-Cauchy convergence arguments.
Category: Mathematical Physics
[55] ai.viXra.org:2602.0069 [pdf] submitted on 2026-02-14 20:48:17
Authors: Lluis Eriksson
Comments: 23 Pages.
We provide a self-contained derivation of the three structural hypotheses — polymer representation (A1), per-link oscillation bounds with geometric decay factor (A2), and large-field suppression (B5) — that were assumed in Doob Influence Bounds for Polymer Remainders in 4D Lattice Yang—Mills Renormalization and in the RG—Cauchy Master Framework. All results are traced to precise equations in the primary sources: the series of papers by T. Bałaban (Commun. Math. Phys., 1984—1989) and the expository trilogy by J. Dimock (2011—2014). The translation from Bałaban's analytic norms on gauge-covariant function spaces to the per-link oscillation language used in the probabilistic framework is made explicit. Together with Doob Influence Bounds for Polymer Remainders in 4D Lattice Yang—Mills Renormalization, this completes the unconditional discharge of the UV structural inputs for the renormalization group approach to the Yang—Mills mass gap problem at finite volume.
Category: Mathematical Physics
[54] ai.viXra.org:2602.0063 [pdf] submitted on 2026-02-13 16:30:51
Authors: Lluis Eriksson
Comments: 18 Pages.
Building on the lattice results established in Papers [E26I]-[E26IX], we give a conditional construction of a scaling-limit state for pure SU(N_c) lattice Yang-Mills theory in four Euclidean dimensions, along dyadic lattice spacings a_k = a_0 * 2^{-k}. The construction proceeds via a two-layer architecture.Layer 1 (Local fields): For bounded gauge-invariant local observables (Wilson loops, normalized plaquette traces), expectations converge -- without extracting subsequences -- to a unique limit. Tightness is trivial (L^infinity bound plus Prokhorov); uniqueness follows from a multiscale RG-Cauchy estimate that bounds the change of local expectations under a single RG step. The extension to unbounded observables such as smeared curvature monomials, which require additive renormalization, is deferred to future work.Layer 2 (Confinement): The physical string tension sigma_phys > 0 is established through step-scaling of Creutz ratios evaluated on Wilson loops whose physical dimensions R x T are held fixed as a -> 0.The limiting state on bounded observables inherits Osterwalder-Schrader positivity from the lattice and admits a Hilbert-space reconstruction via reflection positivity. The mass gap is established conditionally via uniform exponential clustering of connected correlators -- an input from a uniform physical transfer-matrix spectral gap -- and the reconstruction theorem. Nontriviality follows conditionally from an area law for Wilson loops.Key dependencies on prior papers: uniform LSI inputs [E26I]-[E26IX]; Balaban multiscale effective action [E26III]-[E26V]; DLR-LSI [E26VII]; unconditional lattice closure inputs [E26IX].
Category: Mathematical Physics
[53] ai.viXra.org:2602.0057 [pdf] submitted on 2026-02-12 19:06:11
Authors: Lluis Eriksson
Comments: 22 Pages.
We prove integrated cross-scale derivative bounds that replace the unverified Assumption 5.4 of a companion paper. Combined with two explicit large-field inputs (a residual pointwise derivative bound and a Balaban-type conditional large-field suppression) and conditional inequalities from the orbit space Ricci curvature, this yields a uniform (volume-independent) log-Sobolev inequality for the Wilson lattice gauge measure at sufficiently weak coupling (large beta). The key innovation is a decomposition into small-field and large-field contributions: the former is controlled by Balaban's polymer expansion, while the latter is handled by a pointwise gradient bound combined with exponential measure suppression. We provide a self-contained verification of the unconditional large-field tail mechanism for SU(2) in d=2, together with numerical validation.
Category: Mathematical Physics
[52] ai.viXra.org:2602.0056 [pdf] submitted on 2026-02-12 19:07:45
Authors: Lluis Eriksson
Comments: 22 Pages.
We verify the large-field hypothesis (Hypothesis 4.2) of the companion paper on integrated cross-scale derivative bounds for Wilson lattice gauge theory. The proof rests on three ingredients: (i) a dictionary lemma translating the Hilbert-Schmidt large-field condition on plaquette holonomies into Balaban's Lie-algebra formulation; (ii) an interface lemma connecting conditional measures with Balaban's T-operation and its uniform small-factor bound on admissible background fields (Eq. (1.89) of Balaban, Commun. Math. Phys. 122 (1989)); (iii) the uniformity estimate (Eq. (1.75) of the same reference) ensuring that slow-field dependence contributes only an O(1) multiplicative constant. For d=2, we give an independent proof via character-positive convolutions that avoids the Balaban machinery entirely. Together with the companion paper, this yields a uniform (volume-independent) log-Sobolev inequality for the Wilson lattice gauge measure at sufficiently weak coupling.
Category: Mathematical Physics
[51] ai.viXra.org:2602.0055 [pdf] submitted on 2026-02-12 19:09:28
Authors: Lluis Eriksson
Comments: 10 Pages.
We prove that the Wilson lattice gauge measure for SU(N_c) in dimension d >= 3 at sufficiently weak coupling (beta >= beta_wc) satisfies a log-Sobolev inequality with constant alpha_* > 0 independent of the lattice volume. This completes the multiscale program initiated in Paper I by verifying Hypothesis 3.2 of Paper III, the last remaining analytic input. The verification uses three ingredients: (i) the locality of polymer functionals, which restricts the sum over polymers to those intersecting a fixed link; (ii) Cauchy estimates on Balaban's analytic domains for polymer activities and boundary terms; and (iii) a combinatorial counting bound for connected polymers containing a given link, which is independent of the lattice volume. Combined with the synthetic Ricci curvature bound of Paper II, the integrated cross-scale derivative bounds of Paper III, and the large-field suppression established in Paper IV, this yields the uniform log-Sobolev inequality unconditionally.
Category: Mathematical Physics
[50] ai.viXra.org:2602.0054 [pdf] submitted on 2026-02-12 19:10:34
Authors: Lluis Eriksson
Comments: 16 Pages.
We prove that the one-step transfer operator of SU(N_c) lattice Yang-Mills theory in dimension d >= 3 has a spectral gap Delta_phys > 0 uniformly in the lattice volume (for even side length L), for all sufficiently large inverse coupling beta >= beta_0. The proof combines four ingredients: (i) the uniform log-Sobolev inequality on periodic tori established in a companion paper; (ii) a verification that the multiscale RG outputs needed for the LSI argument are uniform in frozen boundary conditions (Section 4), yielding the full DLR-LSI property (Section 5); (iii) the Stroock-Zegarlinski equivalence theorem, which in its standard formulation deduces Dobrushin-Shlosman mixing and exponential clustering from DLR-LSI; and (iv) Osterwalder-Seiler reflection positivity of the Wilson action, which translates temporal exponential clustering into a spectral gap of the transfer operator.
Category: Mathematical Physics
[49] ai.viXra.org:2602.0053 [pdf] submitted on 2026-02-12 19:11:43
Authors: Lluis Eriksson
Comments: 14 Pages.
We prove that for SU(N_c) lattice Yang-Mills theory in d >= 3 dimensions at sufficiently weak coupling (beta >= beta_0), the conditional Gibbs specification satisfies a DLR-uniform log-Sobolev inequality: for every finite sub-lattice Lambda' subset of Z^d and every boundary condition omega, the conditional measure mu_{Lambda'}^{omega} satisfies LSI(alpha_*) with a constant alpha_* > 0 independent of Lambda' and omega.The proof combines three ingredients:(i) the multiscale entropy decomposition developed in our earlier work (Papers I-V), which establishes a uniform log-Sobolev inequality on periodic tori;(ii) a uniform fiber oscillation lemma showing that frozen boundary links -- treated as external parameters in Balaban's renormalization group -- do not increase the per-block oscillation of the conditional fast potential, thanks to compactness of SU(N_c) and the locality of the polymer expansion;(iii) a refined large-field event restricted to dynamical (non-frozen) plaquettes, which ensures that the large-field suppression mechanism extends uniformly to boundary blocks.As a consequence, the Stroock-Zegarlinski equivalence theorem yields Dobrushin-Shlosman mixing, exponential clustering of gauge-invariant correlations, and -- via Osterwalder-Seiler reflection positivity -- a strictly positive mass gap Delta_phys >= m(beta, N_c, d) > 0 for the transfer matrix on the periodic torus (Z/LZ)^d, uniformly in even L. This removes the Dobrushin-type Assumption 6.3 of Paper I and the boundary-uniformity Assumption 3.1 of Paper VI, rendering the lattice mass gap unconditional at weak coupling.
Category: Mathematical Physics
[48] ai.viXra.org:2602.0052 [pdf] submitted on 2026-02-12 19:21:09
Authors: Lluis Eriksson
Comments: 11 Pages.
We establish three interface lemmas that close the remaining gaps in the proof chain for the mass gap of SU(N_c) lattice Yang-Mills theory at weak coupling (beta >= beta_0) in dimension d >= 3.Lemma A (Horizon Transfer) establishes a uniform conditional large-field suppression bound mu_k(Z_k(B) | G_{k+1}) <= exp(-c p_0(g_k)) holding mu_beta-a.s., without any admissibility restriction on the background field. The argument identifies the regular conditional probability with Balaban's RG kernel, expresses the large-field activation probability as a ratio controlled by Balaban's localized T-operation, and applies the T-operation small-factor bound.Lemma B extracts from Balaban's inductive scheme that the boundary terms B^{(k)}(X) share the same uniform analyticity domain as the polymer activities R^{(k)}(X), with radius hat{alpha}_1(gamma) > 0 independent of k.Lemma C extends the multiscale LSI to finite volumes with arbitrary frozen boundary conditions omega via tensorization-plus-perturbation, replacing the unverified Dobrushin block condition of Paper VII.Combined with Papers I-VII, these lemmas render the lattice mass gap theorem unconditional.
Category: Mathematical Physics
[47] ai.viXra.org:2602.0051 [pdf] submitted on 2026-02-12 19:22:19
Authors: Lluis Eriksson
Comments: 16 Pages.
We close the remaining interface gaps in the program [E26I]-[E26VIII] that establishes a uniform log-Sobolev inequality (LSI) and spectral gap for the transfer matrix of lattice SU(N_c) Yang-Mills theory in d=4 at weak coupling. Four technical gaps are identified and resolved: (G1) the Balaban small-factor bound for the T-operation is shown to hold pointwise for every real background by auditing Balaban's proof and verifying that it uses only the uniform inductive conditions; (G2) we establish a uniform small-field coercivity estimate (Hessian lower bound) for the effective action and use it, together with Balaban's small-factor mechanism, to control the conditional inequalities in the multiscale entropy decomposition -- circumventing the need for a global fiber LSI with constant O(beta); (G3) uniform analyticity of boundary terms is extracted from Balaban's inductive scheme; (G4) a quantitative bootstrap verifies the simultaneous compatibility of all constants for a single choice of beta_0. Combined with [E26I]-[E26VIII], these closures yield an unconditional proof that Delta_phys(beta,L) >= c(N_c,beta_0) > 0 uniformly in the volume L for beta >= beta_0.
Category: Mathematical Physics
[46] ai.viXra.org:2602.0046 [pdf] submitted on 2026-02-10 18:37:34
Authors: Lluis Eriksson
Comments: 11 Pages.
We establish that the orbit space B = A/G of SU(Nc) lattice gauge theory satisfies the Riemannian curvature-dimension condition RCD*(Nc/4, dim A); in particular, it satisfies CD(Nc/4, ∞) in the sense of Lott-Villani-Sturm. The proof proceeds by showing that the configuration space A = SU(Nc)|B1(Λ)|, equipped with the bi-invariant product metric ⟨X, Y⟩ = -2 tr(XY), is an Einstein manifold with RicA = (Nc/4) gA, and then applying the stability of the RCD* condition under quotients by compact groups of measure-preserving isometries (Galaz-García-Kell-Mondino-Sosa). This approach bypasses the need for explicit O'Neill curvature computations and handles the singular stratum (reducible connections) automatically. As a consequence, we derive a conditional log-Sobolev inequality for measures on B of the form dμ = e-Φ dν/Z with constant α = (Nc/4) e-osc(Φ). All constants are computed explicitly for SU(2) and SU(3). This provides the geometric input in a program aiming at a volume-uniform log-Sobolev inequality for SU(Nc) lattice Yang-Mills theory at weak coupling; the complementary analytic input (uniform bounds on the effective potential oscillation, via Balaban's renormalization group) is the subject of ongoing work.
Category: Mathematical Physics
[45] ai.viXra.org:2602.0040 [pdf] submitted on 2026-02-08 19:36:55
Authors: Lluis Eriksson
Comments: 11 Pages.
We prove that the lattice Yang-Mills measure with gauge group SU(Nc)in d=4 dimensions at sufficiently large β=2Nc/g2satisfies a Poincaré inequality with constant α*>0uniform in the lattice size L. The proof uses three ingredients:(i) the Ricci curvature bound RicB ≥ Nc/4 for thegauge orbit space, giving a uniform spectral gap for conditional measures offast modes at each renormalization group scale; (ii) Balaban's constructive RGwith polymer derivative bounds, controlling the residual coupling betweenscales; and (iii) a multiscale martingale variance decomposition that avoidsrecursive composition losses, with a commutator coefficientDk ≤ C e-2κ 2-3k made summable bythe geometric scaling factor of transversal block averaging. Under anRG-normalized disintegration consistent with Balaban's absorption structure,only exponentially decaying polymer residuals contribute to Dk,ensuring Σk Dk << c0. Theresulting uniform Poincaré inequality gives volume-independent control ofthe variance-to-energy ratio for gauge-invariant observables.
Category: Mathematical Physics
[44] ai.viXra.org:2601.0119 [pdf] submitted on 2026-01-30 04:04:31
Authors: Steven E. Elliott
Comments: 14 Pages.
We introduce the Fractal Substrate Equivalence Principle (FSEP) as the foundational axiom for a unified theory of physics, asserting that in a fractal universe governed by magneto-hydrodynamics (MHD), physical laws, structures, and phenomena are exactly equivalent acrossall scales upon appropriate scaling transformations. Unlike prior fractal cosmologies that treat general relativity (GR) or quantum mechanics (QM) as fundamental, the FSEP posits these as emergent scale-dependent descriptions of a single electric fluid dynamics. This principleunequivocally asserts: stars are photons, black holes are atomic nuclei, and dark matter is electron orbital shells—not as analogies, but as exact physical identities across fractal layers. We demonstrate how this single MHD substrate reproduces the essential features of QM and GR,and explains the origin of dark matter, black hole—galaxy correlations, and the fine structure of atomic spectra, within a unified electric—fluid picture.
Category: Mathematical Physics
[43] ai.viXra.org:2601.0110 [pdf] submitted on 2026-01-26 21:14:03
Authors: Xiao Peng Zhang
Comments: 6 Pages. (Note by ai.viXra.org Admin: Please cite and list scientific references)
This paper proposes and systematically elaborates a framework for the isomorphism of physical laws based on the "stiffness-inertia duality." Research reveals that numerous core formulas—from classical mechanics to quantum field theory, from condensed matter physics to cosmology, and even including string theory and loop quantum gravity—can be expressed as a functional relationship between a "stiffness term" (driving/restoring factor) and an "inertia term" (response/storage factor). The most fundamental form is ( v = sqrt{x/y} ), where ( x ) is the generalized stiffness and ( y ) is the generalized inertia.Taking the elastic wave speed formula ( v_s = sqrt{G/ho} ) as a prototype, we demonstrate how to construct a self-consistent logical network based on it, connecting the six fundamental dimensions of physics (length, mass, time, force, velocity, density) and extending to other physical quantities such as temperature, charge, and entropy. The framework successfully incorporates the core formulas of quantum mechanics, special and general relativity, the Standard Model, quantum electrodynamics, condensed matter physics, and further extends to string theory and loop quantum gravity, revealing deep mathematical isomorphisms among these seemingly disparate quantum gravity theories.We propose a unified stiffness-inertia Lagrangian formalism and derive several novel relations for strongly correlated condensed matter systems. The study suggests that "stiffness-inertia balance" may reveal a universal mathematical structure underlying physical laws, providing a new explanatory perspective and methodological tool for cross-scale, cross-disciplinary unification of physics, and offering a possible framework for the integration of quantum gravity with established physics.
Category: Mathematical Physics
[42] ai.viXra.org:2601.0073 [pdf] submitted on 2026-01-18 15:38:52
Authors: Kazik Hubert
Comments: 6 Pages.
The Standard-Model Extension (SME) parameterizes Lorentz violation using fixed background fields. We propose a dynamical alternative where the antisymmetric SME background $(k_G)_{muu}$ emerges as a propagating spacetime torsion field in Einstein-Cartan theory. Its source is the ``Berry curvature'' of composite particles, which acts as a textit{Quantum Geometric Charge}. The coupling is mediated by a species-dependent textit{Geometric Susceptibility} $eta_a$, a calculable property of nuclear and atomic states. This Quantum Geometric Backreaction (QGB) establishes a genuine ``two-way street'': matter's quantum geometry sources local spacetime structure, which in turn modifies matter's dynamics via an effective four-fermion interaction mediated by the torsion field. The dynamical origin yields unique, falsifiable signatures absent in static background models: (i) a linear scaling of interferometric phases with source matter density, (ii) quadratic $eta_a^2$ self-energy corrections revealed in atomic spectroscopy via a textit{Torsional King Plot}, and (iii) the potential for generating detectable fields using macroscopic topological materials with broken time-reversal symmetry. We address theoretical consistency and provide first heuristic estimates of $eta_a$ for $^{85,87}text{Rb}$. The theory presents a clear experimental roadmap for tests using atom interferometers and optical lattice clocks at the $10^{-18}$--$10^{-19}$ precision level, directly probing the feedback loop between quantum matter and emergent spacetime.
Category: Mathematical Physics
[41] ai.viXra.org:2601.0057 [pdf] submitted on 2026-01-14 02:20:35
Authors: Herman Herstad Nythe
Comments: 10 Pages.
We propose evidence for universal energy quantization spanning nuclear to biological scales through the fundamental unit EG = Ry/2π = 208.93 kJ/mol, derived from the Rydberg constant. Most significantly, we derive an exact, mass-independent ratio between the particle physics mass scale (MG) and the chemical energy scale (EG) given by MG/EG = 4π2/α3 ≈ 1.016 × 108. This geometric relationship suggests that chemistry represents a holographic projection of nuclear physics governed by vacuum topology and the fine structure constant, offering an alternative to traditional unification theories requiring Planck-scale energies. Analysis of 53 independent measurements reveals systematic correlations with EG achieving 0.02%—8% precision. Key findings include the C-C bond at (5/3)EG (0.05% deviation), the Ge band gap at EG/2φ (0.13%), water electrolysis at EG/√π (0.67%), and Neon ionization at 10EG (0.41%). Preliminary validation through a histogram analysis of 47 bonds from the NIST database reveals a highly significant deviation from uniform distribution (p = 8.3 × 10-6) with strong clustering near predicted harmonics. Furthermore, the framework identifies a biological "temperature lock" where the ratio of activation energy to thermal noise (Ea/RT) equals exactly 20 at mammalian body temperature (310 K), linking thermodynamic stability to icosahedral geometry. Combined statistical analysis yields P < 10-23 against chance. This work suggests a fundamental advance in understanding energy quantization through geometric principles (scaling by π, φ, and α) rather than solely energetic ones.
Category: Mathematical Physics
[40] ai.viXra.org:2601.0023 [pdf] submitted on 2026-01-08 22:07:05
Authors: Lluis Eriksson
Comments: 14 Pages.
We develop a finite-dimensional technical core relating spatial separation to effective decoherence-rate envelopes in Davies-type open-system dynamics. We use energy pinching and quantify coherence by the relative entropy between a state and its energy-pinched version. As an external input we use a maintenance inequality that lower-bounds incremental power by temperature times the instantaneous coherence-loss rate.On the operator side we prove: (i) an exact Dirichlet-form identity for the zero-Bohr-frequency channel yielding a witness-based lower bound on instantaneous decay envelopes; (ii) a Bohr-block Dirichlet decomposition for a single-channel Davies generator under quantum detailed balance; and (iii) sufficient envelope-suppression lemmas under infrared exclusion and quasi-local spectral tails, including a variant avoiding factors depending on the smallest Gibbs eigenvalue. On the state side we give an asymptotic linearization on Bohr-block perturbations with fixed diagonal, yielding a direction-dependent effective decay rate. We include finite-size transverse-field Ising chain (TFIM) witness diagnostics and fully reproducible scripts producing the figures.
Category: Mathematical Physics
[39] ai.viXra.org:2601.0011 [pdf] submitted on 2026-01-04 08:34:29
Authors: Brent Hartshorn
Comments: 11 Pages.
In this paper we demonstrate that the transition from a stable Dodecahedral Core (Valamontes, 2024) to the "Wild" chaotic phase corresponds physically to the "Elongated Phase" of 4-D simplicial quantum gravity (Gionti, 1997). This phase is characterized by the emergence of Besicovitch (Kakeya) needle sets—fractal structures that achieve maximal directional complexity within minimal volumetric measure. We introduce the Gyrobifastigium as the fundamental space-filling unit capable of mediating the geometric friction between the periodic dodecahedral vacuum and the aperiodic Einstein Monotile global structure. Finally, we map this geometric resolution onto a 3D temporal framework ($tau$-space), where the "Big Bang" is redefined as a retrocausal pruning process of a "Nine-Tile" super-compatible state, effectively solving the universal NP-hard tiling problem of the vacuum through informational synchronization.
Category: Mathematical Physics
[38] ai.viXra.org:2601.0009 [pdf] submitted on 2026-01-04 20:45:49
Authors: Shanzhong Zou
Comments: 3 Pages.
This article reveals that the Riemann Hypothesis can be attributed to the combined effect of two geometric projections within three-dimensional real space: the projection of the function's non-trivial zeros onto the complex plane, and the projection of the singularity at the point S=1 on the real axis onto the real number plane.
Category: Mathematical Physics
[37] ai.viXra.org:2601.0007 [pdf] submitted on 2026-01-02 13:26:10
Authors: Lluis Eriksson
Comments: 21 Pages.
We prove a quantitative clustering—recovery bound for centered quasi-free (Gaussian) states in a finite-mode bosonic CCR (Weyl) setting. Motivated by split inclusions in algebraic quantum field theory, we work in a regularized framework where Gaussian states are parametrized by finite covariance matrices and a recovery map admits an explicit covariance block formula. Using a perturbative Gaussian fidelity input together with explicit coercivity bounds for inverse covariances, we bound the recovery error in terms of a vacuum cross-correlation factor, a cross-correlation perturbation parameter, and a recovery-error matrix norm ||ΔΓ||_HS, with an explicit quadratic+quartic structure. In a distinguished class (Family A, X = X0), this reduces to a bound in terms of the cross-block error ||Δ^(12)||_HS. We include ancillary numerical sanity checks verifying the perturbative regime, a collar-envelope decay model, a dimension sweep n1 = n2 in {1,2,3}, and phase-diagram checks of the perturbative domain.
Category: Mathematical Physics
[36] ai.viXra.org:2512.0060 [pdf] submitted on 2025-12-17 02:23:22
Authors: Lluis Eriksson
Comments: 22 Pages.
We relate exponential clustering of vacuum correlations to approximate quantum state recovery via the Petz map in algebraic quantum field theory. In a regularized CCR (Gaussian/quasi-free) framework for a massive scalar field, we derive an explicit fidelity bound between a quasi-free state ω and the Petz-recovered state ω~ associated with the canonical split inclusion. The estimate controls 1 − F(ω, ω~) in terms of a Hilbert—Schmidt recovery error in the cross-correlation block, a vacuum correlation factor η_vac (decaying approximately as exp(−m r) with collar width r), and a perturbation parameter δ measuring deviations from vacuum cross-correlations. We also give a finite-rank corollary with an explicit 2n factor and discuss implications for quantitative locality and (conditionally) holographic reconstruction.
Category: Mathematical Physics
[35] ai.viXra.org:2512.0010 [pdf] submitted on 2025-12-03 21:19:45
Authors: J. W. McGreevy
Comments: 3 Pages.
We prove that the entirety of known physics — Einstein—Cartan gravity, the Standard Model with three generations, QCD confinement, electroweak unification, the Kerr—Newman black hole, the CMB power spectrum, and the resolution of five Clay Millennium Problems — emerges from a single mathematical process: the Archimedean exhaustion of the circle at the infinite prime applied to the global arithmetic orbifold O = h Spec(Z).Gbm ⋊ Gal(Q/Q) i ⊔ h SL(2, Z)H i followed by sequential double-negation closure. All observables are fixed without parameters.
Category: Mathematical Physics
[34] ai.viXra.org:2512.0004 [pdf] submitted on 2025-12-01 16:56:15
Authors: J. W. McGreevy
Comments: 3 Pages.
We prove that the Einstein—Cartan spacetime of our universe, together with the complete Standard Model (including three generations, the Higgs mechanism, and all observed charges), is the crepant resolution of a single global arithmetic orbifoldO = h Spec(Z). Gbm ⋊ Gal(Q/Q) i⊔h SL(2, Z)Hivia sequential double-negation closure driven by Archimedes’ exhaustion of the circle at the infinite prime. The Runge—Lenz vector, the Rydberg formula, proper time, torsion, and the equivalence principle arise as direct mathematical consequences. The Riemann Hypothesis is proven as a consistency condition.
Category: Mathematical Physics
[33] ai.viXra.org:2512.0003 [pdf] submitted on 2025-12-01 16:54:35
Authors: N. J. Kettlewell
Comments: 6 Pages.
We begin with a complex two-point correlation kernel W(x,y) defined on an abstract smooth label space Xwith no assumed metric, signature, causal structure, or geometric fields. From four operational constraints—finite propagation, passivity, regularity, and local homogeneity—we show that Lorentzian cones, Hadamard singularities, and a metric emerge as statistical summaries of propagation behaviour. Mixed derivatives of the correlation phase reconstruct the metric, and stability of a least-change functional selects Lorentzian signature and statistically favours three spatial dimensions. Allowing coefficients of the correlation generator to vary introduces curvature, and ensemble-averaging the correlation stress yields the statistical consistency conditionGAB + ΛgAB = κ⟨EAB ⟩,linking curvature to averaged correlation tension. Thus spacetime geometry arises not as a background structure but as the collective behaviour of correlations satisfying operational postulates.
Category: Mathematical Physics
[32] ai.viXra.org:2511.0081 [pdf] submitted on 2025-11-24 01:03:23
Authors: Pedro A. Kubitscheck Homem de Carvalho
Comments: 9 Pages. (Note by ai.viXra.org Admin: Please cite and list scientific references)
Classical planetary timekeeping relies on the solar day and the synodic period, two historically Earth-based units. When used to infer apparent solar angular motion from the viewpoint of a rotating planet, these units introduce a systematic, planet-dependent distortion. Earth exhibits a near-unity coincidence between rotational, synodic, and orbital parameters, but this coincidenceis not universal. We formalize the distortion through a dimensionless error functional depending only on the rotational period τP , the apparent solar period T⊙,P , and the planetary radius DP. We derive a universal referential correction,Du2032P = pτP T⊙,P , which collapses the apparent solar angular velocity to a single reference value across different planets when expressed in physical (non-local) time. We then show that, once light-travel delay and rotational aliasing are included, a corrected angular invariant emerges that is the same for all eight planets, and that an orthogonal angular Lorentz factor replaces the usual linear velocity- based relativistic factor for rotating, delayed observers.
Category: Mathematical Physics
[31] ai.viXra.org:2511.0075 [pdf] submitted on 2025-11-23 00:09:56
Authors: J. W. McGreevy
Comments: 3 Pages.
We report a striking numerical coincidence between the graded dimensions of the moon-shine module V ♮ (associated with the Monster group) and the energy levels of the hy-drogen atom, as described by the Rydberg—Ritz combination principle. Using the cumulative dimensions T (n) = P k≤n dim V ♮ k and a Cardy-refined effective quantum number neff (n) = q 3 16π2 log T (n), the Balmer and Lyman series are reproduced to within current experimental precision (∼ 10−6 relative error) for low n, with deviations following the Cardy asymptotic of the c = 24 theory. We interpret this as the hydrogen atom acting as a "shadow" or projection of the Monster symmetry in 4D physics. The unique Monster-invariant weight-2 vector is conjectured as the archetype of the photon, the Runge—Lenz vector as a Monster generator preserving the third quadratic form, and the Dirac equation as an orbifold projection of the 24-dimensional Clifford algebra embedded in V ♮. These connections suggest a deeper unification of quantum mechanics with monstrous moonshine,with testable predictions for high-n spectral deviations.
Category: Mathematical Physics
[30] ai.viXra.org:2511.0053 [pdf] submitted on 2025-11-17 01:19:21
Authors: Pedro A. Kubitschek Homem de Carvalho
Comments: 6 Pages. (Note by ai.viXra.org Admin: Please cite listed scientific references)
We present a unified geometric framework (Principium Geometricum, PG) in which vacuum, light, inertia, gravity, and the cosmological constant arise from a single Planck—scale oscillatory law on a compact toroidal manifold T³. Each Planck cell (area (A_P = ell_P^2 = hbar G / c^3)) behaves as a harmonic geometric oscillator with frequency (Omega = 1/t_P = c^5 / (hbar G)) and geometric tension (tau_U). The mean kinetic term (langle dot T^2 angle = kappa_0 A_P^2 Omega^2) defines a finite vacuum density (ho^{PG}*v = beta kappa_0 ho_P) with (beta = (H_0 t_P)^2) and (ho_P = c^7 / (hbar G^2)). Einstein’s projection yields (Lambda*{PG} = (8pi kappa_0 / c^2) H_0^2), numerically consistent for (kappa_0 approx 0.1) without fine—tuning. Inertia and gravity emerge as temporal and spatial coherence gradients of the same geometric field; light is the universal update rate of the vacuum, (langle dot T^2 angle = sqrt{kappa_0} A_P Omega = sqrt{kappa_0} c ell_P). The unified constant (alpha_U equiv k_e A_P = (4pivarepsilon_0)^{-1} hbar G / c^3) acts as a geometric impedance linking electromagnetism and curvature; a Planck—unit projection (baralpha_U = (k_e/Z_0)(A_P Omega / hbar) = (1/4pi)(c G / hbar)) quantifies the EM—GR mismatch at the Planck interface. We provide closed-form calculations and two testable predictions: (i) a cosmological fit (Lambda propto H^2) and (ii) a torsion-balance signal scaling with material coherence.
Category: Mathematical Physics
[29] ai.viXra.org:2511.0041 [pdf] submitted on 2025-11-13 21:42:59
Authors: Pedro A. Kubitschek Homem de Carvalho
Comments: 10 Pages. (Note by ai.viXra.org Admin: Please refrain from frequent submisions of highly speculative articles)
We propose a geometric framework in which matter transmission arises as a natural extension of wave propagation under the Principium Geometricum (PG). The vacuum is not an empty background but a geometric medium endowed with intrinsic tension, characterized by the unified constant alpha_U = ke * A_P. Light and radio waves are open modes of this tensional field, while matter corresponds to stationary toroidal oscillations. We introduce a triplet (A, Theta, Psi) that encodes the tensional amplitude, toroidal phase, and helicoidal topology of the vacuum. The associated signal S(t) = integral over V of Phi(x,t) dx, with Phi(x,t) = A(x,t) * exp(i * Theta(x,t)) * Psi(x,t), acts as a "vacuum radio": a modulation of temporal tension capable of propagating the geometric signature of matter through space. At the receiving end, when the local vacuum reaches phase coherence with the incoming signal, a stationary toroidal pattern self-organizes and the corresponding atom or structure is recreated as a new instance of the same geometric state. No mass or charge is transported; only coherence information is. This framework suggests a deep equivalence between energy, geometry, and information in the PG field. It bridges electromagnetism, gravitation, and quantum-like discreteness into a single tensional ontology and points toward a matter transceiver as a long-term technological possibility: a device that encodes, transmits, and reconstructs matter as coherent patterns of vacuum tension.
Category: Mathematical Physics
[28] ai.viXra.org:2511.0037 [pdf] submitted on 2025-11-12 21:42:12
Authors: Pedro A. Kubitschek Homem de Carvalho
Comments: 7 Pages.
Part I (Mathematics). We formulate a minimal—coherence principle on the three—torusT 3 that singles out, with no free parameters, a pure number χ as the canonical solution of aquadratic relation involving π. The value of χ depends only on geometric structure, not onphysical constants. Part II (Physics). We map χ to the vacuum unifying constant αU = keAP by defining the numerical part of αU as χ × 1060−1 in SI units. This furnishes a parameter—free prediction for G (or, dually, for ke) through αU ≡ keℓ2 P . We clarify why the mathematical value is "exact" in the theoretical sense, while physical measurements unavoidably carry metrological uncertainty.
Category: Mathematical Physics
[27] ai.viXra.org:2511.0031 [pdf] submitted on 2025-11-10 21:25:44
Authors: J. W. McGreevy
Comments: 4 Pages.
We present GRQFT (General Relativistic Quantum Field Theory) as a theory of everything (TOE)grounded in arithmetic geometry over Spec(Z), where physical reality emerges as an entropic computation maximizing information under cohomological constraints. The framework integrates U(1) gauge theory (Maxwell’s equations) via μ4 i-cycle bundles, sources gravity from Monster moonshine polarization, andderives entropy volumes from failures of surjectivity (Hn cohomology). Key refinements include p-adic causality for discrete time arrows and arithmetic backreaction in field equations. Supporting evidence includes predictive alignments with QCD σ, CMB Cℓ, Higgs λhhh, and novel EM quantization. Testablesimulations (e.g., p-adic Faraday induction) are provided.
Category: Mathematical Physics
[26] ai.viXra.org:2511.0030 [pdf] submitted on 2025-11-11 01:38:17
Authors: Giuliano Bettini
Comments: 13 Pages. In Italian (Note by ai.viXra.org Admin: An abstract is required in the article; please cite and list scientific references)
How do the 32 crystal classes arise? And what about the genetic code? I've already discussed the former in other articles, formulating a 5-bit hypothesis. Here I present a new form of the Genetic Code, which helps me show how Mother Nature followed the same logic in both cases, with 5 bits in one case and 6 bits in the other.In both cases, the bits correspond to specific physical properties. The construction occurs through the successive introduction of available properties.
Category: Mathematical Physics
[25] ai.viXra.org:2511.0021 [pdf] submitted on 2025-11-08 17:59:51
Authors: Paul Emmett Zaffuto
Comments: 5 Pages. (Note by ai.viXra.org Admin: Please cite and list scientific references)
This paper presents a novel 18-digit period that serves as the foundational structure for a precision mapping system rooted entirely in integer arithmetic. The period, derived from a minimal deviation from 10¹u2078, creates an exact wavelength via the speed of light and a potentially infinite decimal counter in its frequency domain. The system is mathematically reversible, scale-invariant, and self-documenting across dimensional magnitudes. While the counter is potentially infinite, its manifestation depends on the presence of an observer or measuring system—making it a construct of potential, not actuality. The implications for time-based harmonic structures, integer-driven physics, and digital precision frameworks are explored.
Category: Mathematical Physics
[24] ai.viXra.org:2510.0076 [pdf] submitted on 2025-10-30 20:35:45
Authors: J. W. McGreevy
Comments: 4 Pages.
We show that the polarization identity unifies the Monster group’s graded dimensions,the third quadratic x3 = m2 − x1 − x2, the spacetime metric, and the Higgs radial/angularmodes. The polarization of the j-invariant’s q-expansion yields the Monster’s representationdimensions, which in turn polarize into the third quadratic, giving the spacetime intervalds2 and the Higgs VEV. This establishes GRQFT as a complete arithmetic derivation of general relativity and the Standard Model.
Category: Mathematical Physics
[23] ai.viXra.org:2510.0058 [pdf] submitted on 2025-10-24 20:49:17
Authors: Paul Emmett Zaffuto
Comments: 3 Pages. (Note by ai.viXra.org Admin: Please cite and list scientific references)
This paper proposes a novel approach to temporal measurement and structuring by using the speed of light, denoted as ��=299,792,458 m/s c=299,792,458 m/s, as the fundamental base unit for time interval construction. Departing from the anthropocentric decimal system, this method leverages fundamental physical invariants to define, scale, and coordinate time with unmatched precision. The proposed model, Lightforge Time, unifies wavelength, frequency, energy, and spatial rotation within a single coherent framework that supports symmetrical tier stepping, loop closure, and natural harmonics.
Category: Mathematical Physics
[22] ai.viXra.org:2510.0024 [pdf] submitted on 2025-10-08 21:29:47
Authors: Stephen P. Smith
Comments: 7 Pages.
This paper develops a homeostatic model for the metric tensor by introducing an auxiliary-parameter representation that reflects the bilateral structure of CPT symmetry. Each side of the symmetry possesses a set of auxiliary parameters, W(1) and W(2), whose differences vanish at equilibrium, giving rise to the emergent metric tensor. Based on facts pertaining to the factorability of symmetric matrices, the metric is parameterized as G=WDWT, where W is a lower-triangular matrix of ten independent parameters and D=diag(-1, 1, 1, 1) encodes the Lorentzian signature. This factorable form interprets spacetime geometry as an adaptive interface maintained by homeostatic balance between conjugate sectors, rather than as a primitive geometric field. The framework unifies algebraic, geometric, and informational descriptions by linking the Gram-type form WWT to the Fisher information matrix and to Frieden’s ten amplitude functions. The result is a dynamically generative account of the metric tensor consistent with both Riemannian and Lorentzian regimes, providing a foundation for extending homeostatic principles to curvature and field dynamics.
Category: Mathematical Physics
[21] ai.viXra.org:2509.0020 [pdf] submitted on 2025-09-09 23:17:59
Authors: Paul Emmett Zaffuto
Comments: 3 Pages. (Note by ai.viXra.org Admin: Please cite listed scientific references)
This document proposes an enhancement to the 2019 redefinition of the International System of Units (SI). By introducing symbolic regeneration loops and recursive anchor-point logic, it enables a self-validating, cross-domain framework that operates atop the fixed constants: the speed of light, Planck's constant, Boltzmann constant and the elementary charge. The system does not alter SI definitions but extends their application through symbolic closure, allowing any known quantity to regenerate the others using tautological identities. This structure enhances internal consistency, experimental realization, and interpretability across metrological domains.
Category: Mathematical Physics
[20] ai.viXra.org:2509.0015 [pdf] submitted on 2025-09-06 22:13:57
Authors: Pedro A. Kubitschek Homem de Carvalho
Comments: 3 Pages. (Note by ai.viXra.org Admin: Please cite listed scientific references!)
We present a discrete oscillatory framework in which time itself is modeled as a harmonic function of the Planck area AP and the universal tensional constant αU = keAP. Anchoring the oscillation period τ in multiples of the Planck time tP naturally discretizes temporal evolution. Each macroscopic event dissipates into minimal toroidal excitations ("bits"), whose energy is proportional to αU and the underlying frequency. The universe is then described as the sum of such bits, converging towards a maximum filling determined by the ratio Amax/AP. In this limit, dynamics collapse into a self-stable soliton, interpreted as the ultimate particle and possible informational endpoint of cosmology.
Category: Mathematical Physics
[19] ai.viXra.org:2508.0083 [pdf] submitted on 2025-08-31 19:42:24
Authors: Jarosław Kaczorowski
Comments: 2 Pages. (Note by ai.viXra.org Admin: An abstract is required in the article; please cite and list scientific references)
I present a compact mathematical ansatz for the fine-structure constant α. The construction uses only mathematical constants (φ,e,π) and a single integer parameter N. This note is intentionally mathematics-only: it does not claim a derivation from quantum electrodynamics nor from any particular physical model. The ansatz is meant to be empirically testable: future high-precision determinations of α either continue to agree within uncertainty or falsify the relation. I also outline simple, resolution-matched numerical checks (e.g., percent-level sliding-window comparisons) that can be replicated independently.
Category: Mathematical Physics
[18] ai.viXra.org:2508.0081 [pdf] submitted on 2025-08-30 22:35:54
Authors: Brent Hartshorn
Comments: 48 Pages. (Note by ai.viXra.org Admin: This paper is subject to withdrawal by the Admin due to its non-scientific contents & the writing style which seem to fall ouside scholarly scope and manner)
This paper provides a detailed analysis of the claims presented in the article "Physics Grifters: Eric Weinstein, Sabine Hossenfelder, and a Crisis of Credibility" by Timothy Nguyen. The primary objective is to move beyond the polemical nature of the title and conduct a scholarly examination of the figures and phenomena discussed, thereby establishing a foundation for a subsequent, more formal academic paper.The analysis concludes that while the term "grifter" is a blunt and often polemical label, a more precise sociological framework—that of audience capture and ideological control —better explains the behaviors and power dynamics at play. This paper demonstrates that Eric Weinstein, Sabine Hossenfelder, and Brian Keating are engaged in a strategic negotiation between their academic credentials and the incentive structures of new media. Weinstein leverages his Harvard Ph.D. in mathematics to create a "lone genius" narrative, while Hossenfelder utilizes her established career as a theoretical physicist to build a public brand around legitimate critiques of her field.Critically, the "crisis of credibility" that forms the backdrop of this discourse is not an invention of these figures. This paper establishes that this crisis is a long-standing, and pre-existing condition within theoretical physics, with roots in decades of debate articulated by prominent academics like Lee Smolin and Peter Woit. Therefore Weinstein, Hossenfelder and Keating have not created this problem; they have only strategically co-opted into it, packaging a legitimate academic critique into a public-facing, personality-driven narrative that is highly effective for online engagement and their own monetization and ideological goals.The synthesis of these findings offers a nuanced perspective: the core issue is not a simple binary of legitimate vs. illegitimate science, but rather a complex sociological interplay of economic and power structure incentives, rhetorical strategies, science based ethics, and the shifting landscape of scientific communication. These dynamics—perhaps not just personal failings—represent a fundamental challenge to the traditional norms of academic discourse and merit further scholarly investigation.
Category: Mathematical Physics
[17] ai.viXra.org:2508.0010 [pdf] submitted on 2025-08-04 09:26:33
Authors: Gui Furne-Gouveia
Comments: 5 Pages.
Lockyer’s model calculates the proton-to-electron mass ratio with astonishing precision: the first seven significant digits are exact. This result, derived from fundamental constants and a geometry of nested photons, is verifiable through a previously published JavaScript program. This paper poses a question: is this a coincidence, or does this calculation reveal a profound physical truth? Exploring the latter hypothesis, we propose that the propagation of electromagnetic waves in a space deformed by energy itself accounts for the confinement of energy within matter.This emerging paradigm, where spacetime acts as a dynamic optical medium, opens perspectives for rethinking the nature of matter.
Category: Mathematical Physics
[16] ai.viXra.org:2508.0005 [pdf] submitted on 2025-08-02 01:52:05
Authors: Norbert D. Agbodo
Comments: 143 Pages.
We propose a unified theoretical framework in which both matter and space emerge fromthe rhythmic interplay between dynamic membranes and a discrete energetic lattice.In this model, particles are not treated as point-like entities or probability waves, butas vibrating membranes that undergo continuous cycles of expansion and collapse —breathing between delocalized field influence and discrete localization on a spatial grid.This breathing process is governed by intrinsic frequencies, modulated by environmentalresonance, and gives rise to quantum phenomena such as interference, tunneling, spin,and entanglement.Simultaneously, we redefine space not as a passive void, but as a structured energeticmedium composed of quantized lattice nodes embedded in a continuous manifold. Eachunit of space stores energy, resists deformation, and interacts with fields through quantifi-able elastic and vibrational properties. From this substrate, we derive a general operatorformalism in which all physical fields — electromagnetic, gravitational, and beyond —appear as eigenmodes of a unified differential operator acting on space itself.Thermodynamic laws arise naturally from this framework: heat as vibrational exci-tation, entropy as collapse degeneracy, and relativistic effects as emergent responses tolattice perturbation. This model thus bridges quantum mechanics, field theory, and rel-ativity through a physically grounded geometry of rhythmic collapse, structured energy,and operator-driven evolution — offering new predictions, interpretations, and experi-mental pathways toward a deeper understanding of fundamental physics
Category: Mathematical Physics
[15] ai.viXra.org:2507.0088 [pdf] submitted on 2025-07-16 05:32:11
Authors: Moninder Singh Modgil, Dnyandeo Dattatray Patil
Comments: 31 Pages.
We develop a resolution-aware quantum field framework grounded in the hierarchy ofCantor cardinals, wherein the continuum is stratified by a sequence of decreasing resolutions{ϵi = 1/ℵi}. This architecture allows us to reformulate classical and quantum field theoriesby replacing infinitesimal constructs with tiered differentiable operators. Beginning with themodification of scalar field dynamics via ϵi-derivatives, we extend the formulation to gaugefields, BRST.We construct ϵi-based renormalization group flows, wherein coupling constantsevolve not only with energy scale but across resolution layers. Statistical field theory is enrichedwith ϵi-dependent entropy, including Gibbs and von Neumann entropy, and extendedto Langevin-type stochastic systems with regularized entropy production. In the geometricdomain, we introduce ϵi-curvature tensors and Einstein equations over resolution-aware manifolds.Finally, we formulate topological and categorical field theories indexed by ϵi, integratingsmooth topos logic, stratified cohomology, and resolution-sensitive path integrals. This frameworkoffers a logically consistent, divergence-free, and epistemic
Category: Mathematical Physics
[14] ai.viXra.org:2507.0086 [pdf] submitted on 2025-07-17 00:04:42
Authors: Faysal EL Khettabi
Comments: 9 Pages. https://efaysal.github.io/HCNFEK2024FE/HypComNumSetTheGCFE
This report details the Arithmetic of Order (AoO) as a discrete arithmetic framework that provides a unified origin for the Standard Model’s gauge symmetries, hypercomplex numbers, and optimal error-correcting codes. The AoO posits that physical and mathematical structures emerge from the ordered combinatorial progression 1 → n → n+1 and its associated powerset hierarchy, P(Ωn), without assuming a pre-existing continuum. We show how this framework, realized through Farey sequences under the modular group SL(2,Z), lifts naturally to its universal central extension, the braid group B3. This topological extension provides a direct mechanism where the three Reidemeister moves generate the gauge groups U(1), SU(2), and SU(3). The discussion section contextualizes these results, highlighting the framework’s power in unifying the language of physical transformations (hypercomplex numbers) with principles of information stability (codewords), and notes the convergence of our conclusions with parallel topological models of physics.
Category: Mathematical Physics
[13] ai.viXra.org:2507.0068 [pdf] submitted on 2025-07-13 03:45:52
Authors: Faysal EL Khettabi
Comments: 9 Pages.
This article formalizes the Farey sequence hierarchy ($Fseq{n}$) as a concrete realization of the enquote{Arithmetic of Order} (AoO) framework. This framework leverages the constructive machinery of ZF set theory, without assuming the Axiom of Infinity, to model physical reality as a procedural, finitistic unfolding. The core physical principle advanced is that the set-theoretic complement between successive powersets, $powerset(Omega_{n+1}) setminus powerset(Omega_n)$, is the mathematical image of a resolution jump in physical measurement. This provides a rigorous engine for Cycle Clock Theory, demonstrating how a coherent, non-overlapping, and generative temporal process emerges from finite principles. By showing that the infinite is an emergent property, the AoO offers a unified, operational foundation for mathematics and physics.
Category: Mathematical Physics
[12] ai.viXra.org:2506.0135 [pdf] submitted on 2025-06-29 14:32:23
Authors: Yueshui Lin
Comments: 12 Pages.
This paper proposes a rigorous Third-order Enhanced Axiomatic System (TEAS) that resolves the fundamental conflict between quantum mechanics and general relativity within the framework of Woodin cardinals. By establishing an exact corre- spondence between renormalization group flow and categorical duality, we construct a quantum-classical fiber bundle mapping Q : H → Γ(T*M), derive the spacetime emergence mechanism k ~ log(ΛUV /ΛIR), and propose three fundamental axioms:quantum-classical correspondence, noncommutative geometric duality, and topological order stability.Key innovation: We establish the physical motivation for Woodin cardinals in quantum gravity through entropy scaling and renormalization completeness. The covering property of Woodin cardinals ensures the mathematical consistency of the quantum-to-classical transition, providing a set-theoretic resolution to Haag’s theorem. This approach differs from ∞-category methods by providing a set-theoretic foundation that resolves Haag’s theorem constraints through determinacy proper- ties.This work provides a mathematically consistent solution to the Haag theorem contradiction, offering a testable framework for quantum gravity theory with experimentally verifiable predictions.
Category: Mathematical Physics
[11] ai.viXra.org:2506.0034 [pdf] submitted on 2025-06-08 21:47:08
Authors: James McDaniel
Comments: 15 Pages. (Note by ai.viXra.org Admin: Author name is required in the article; please cite listed sceintific references)
This paper constructs the foundational structure of TOR (Topological and Recursive Motif Logic) as a definitional extension of the ZFC+++Qλ logical framework. TOR introduces a compression-resistant logic rooted in symbolic motif transitions, constructive witness chains, curvature metrics, and modal echo operators. Building upon ZFC, it formalizes motif behavior in terms of irreducibility, symbolic resonance, and recursive transitions across motif lattices. Appendices provide detailed examples, formal symbolic definitions, and semantic-witness interpretations, culminating in a logic capable of expressing both classical mathematical rigor and emergent symbolic structure. TOR is presented as a new foundational system for mathematics, cognition, and physical modeling.
Category: Mathematical Physics
[10] ai.viXra.org:2506.0008 [pdf] submitted on 2025-06-02 01:25:05
Authors: Rafael Eliahu Vaidergorn
Comments: 5 Pages.
This paper introduces the Vaidergorn Number (Ts), denoted by theHebrew letter Tzadi Sofit (Unicode U+05E5), as a novel mathemati-cal construct representing an absolute magnitude beyond all infinities. We formalize Ts within an extended Zermelo-Fraenkel (ZF) framework with new axioms, leveraging forcing and non-standard analysis, and prove its uniqueness via a new theorem, distinguishing it from Cantor’s cardinals (ℵ0, ℵ1) and Robinson’s hyperreals. Enhanced applications in cosmology model an infinite-yet-bounded universe with detailed derivations of a modified Friedmann equation and numericalsimulations, while string theory applications include a refined action with connections to braneworld scenarios. This preprint bridges mathematics and physics, inviting feedback and further exploration.
Category: Mathematical Physics
[9] ai.viXra.org:2505.0198 [pdf] submitted on 2025-05-30 01:06:52
Authors: Robert Loiseau
Comments: 5 Pages.
We present a minimal construction of a self-adjoint operator Hjk = αlogpj + βlog2pj, whose entries are determined entirely by the logarithmic structure of the prime numbers. This operator yields a discrete, strictly positive lowest eigenvalue that remains stable across increasing matrix sizes. Without invoking spacetime, gauge fields, or conventional Lagrangian dynamics, we interpret this result as a candidate solution to the Yang—Mills mass gap problem. The construction is purely arithmetic and offers a novel pathway to mass gap realization.
Category: Mathematical Physics
[8] ai.viXra.org:2505.0122 [pdf] submitted on 2025-05-20 20:25:31
Authors: Justin Sirotin
Comments: 16 Pages. Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License
We present a comprehensive exposition of the Spinor-Mediated Universal Geometry(SMUG) framework that resolves the Clay Mathematics Institute's Yang--Millsexistence and mass gap problem through torsion-induced four-fermion interactionson a Euclidean lattice. This novel approach establishes a mathematically rigorousconnection between spacetime geometry and quantum field dynamics, demonstratinghow spinor-sourced torsion naturally generates the mass gap.
Category: Mathematical Physics
[7] ai.viXra.org:2505.0071 [pdf] submitted on 2025-05-13 23:45:00
Authors: Romeo Prince
Comments: 50 Pages.
It has A Morphic Riemann Tensor, there is a damping term which dams qft at large x and vise versa, flow time, Morphic flow, stress, pressure, exp, theory of temperature, quantum gravity, particles, QCD, QFT.
Category: Mathematical Physics
[6] ai.viXra.org:2505.0059 [pdf] submitted on 2025-05-11 20:41:01
Authors: Faysal EL Khettabi
Comments: 9 Pages. Also see https://ai.vixra.org/abs/2505.0054
This report outlines a foundational shift in mathematics, proposing a framework groundedin finite, constructive principles—the "Arithmetic of Order"—emerging from the progression 1 → n → n + 1 and the combinatorial structure of powersets P(Ωn). It critiques the traditional reliance on infinitary constructs like the complex number i ∈ C and the continuum for describing physical systems with finite degrees of freedom. Instead, it posits haracter-istic functions as the true empirical interface, and demonstrates how optimal mathematical structures—such as the Golay code G24, the Leech lattice Λ24, and the Mathieu group M24—emerge deterministically from this finitistic basis through processes of constraint-guided differentiation. This approach offers a new foundation for understanding hypercomplex numbers, projective geometries emergent from powersets, universal principles of communication and information stability, and the potential architectures for advanced artificial intelligence. Crucially, it reinterprets the continuum not as an *a priori* given, but as an asymptotic limit of the nested powerset hierarchy. The principles underlying theoremslike Gleason’s are viewed not merely as specific results at a particular n (such as n = 24),but as exemplars of universal rules of emergence that guide the formation of order acrossall degrees of freedom. The entire framework operates without recourse to unobservableinfinities or the subjective concept of "noise."
Category: Mathematical Physics
[5] ai.viXra.org:2505.0055 [pdf] submitted on 2025-05-10 21:18:10
Authors: Taylor Pablo Federico
Comments: 60 Pages. In Spanish
This work introduces the theoretical framework of Algebraic Engineering, centered on the Fundamental Theorem of Algebraic Synchronism (FTAS). The theory categorizes algebraic structures as elemental (internally closed) and essential (dependent on external data), and proposes a metric-entropy formulation based on a (1,1) deformation tensor. A Ricci-type dynamic flow is defined, blending geometric evolution with algebraic synchronism. Applications include the synchronization of metrics from General Relativity and Quantum Mechanics, and a reformulation of stress-strain behavior in material science via entropy and metric functors. The document includes 17 chapters, a complete glossary, symbolic conventions, and a dedicated section on the role of artificial intelligence in the development of the work. It is written in Spanish with a full English summary, and is structured, self-contained, and open to collaboration.
Category: Mathematical Physics
[4] ai.viXra.org:2505.0054 [pdf] submitted on 2025-05-10 21:21:53
Authors: Faysal EL Khettabi
Comments: 12 Pages.
Can we derive a Modern Mathematical Framework for Hypercomplex Numbers? Hypercomplex numbers, such as quaternions and octonions, expand beyond traditional real and complex numbers by introducing additional imaginary units. These numbers have unique algebraic properties and applications in mathematical and physical theories for describing transformations, symmetries, and geometric concepts in higher-dimensional spaces. However, there is a noticeable gap in the robust mathematical foundation related to hypercomplex numbers. This research project aims to establish a comprehensive mathematical framework for hypercomplex numbers, specifically focusing on their intrinsic relationship with physical systems having a natural number of degrees of freedom. By enhancing the understanding and application of hypercomplex numbers in this context, deeper insights into complex systems and phenomena in various fields can be uncovered
Category: Mathematical Physics
[3] ai.viXra.org:2504.0129 [pdf] submitted on 2025-04-30 19:46:43
Authors: Hamed Mehrabi
Comments: 33 Pages.
The universe presents us with a profound paradox: while the second law of thermodynamicspredicts increasing disorder, complex organized systems—from living cellsto galactic structures—emerge and persist throughout cosmic history. We resolve thisparadox by proposing a fundamental extension to Einstein’s mass—energy equivalence,incorporating organizational information as an intrinsic physical quantity that carriesmeasurable energy. Our central principle establishes that the total energy of a systemcomprises both its rest energy (Erest = mc2) and an additional energy associatedwith maintaining organizational information (Eorg = kBT ln(2)Ω), where Ω quantifiesnon-random correlations in bits. Together, these energy components form a completeaccounting of the system’s energetic requirements: Etotal = Erest + Eorg. This frameworkmakes three remarkable quantitative predictions: (1) human brain power consumptionof 20.1W (measured: 20W), (2) E. coli basal metabolic rate of 4.1 × 10−12W(measured: 3.8-4.5 × 10−12W), and (3) quantum decoherence times within 8% of experimentalvalues across different qubit technologies. Through rigorous mathematicalderivation and cross-scale validation, we demonstrate that organizational informationrequires continuous energy expenditure against entropic decay. This fundamental insightnot only resolves longstanding puzzles in bioenergetics and quantum foundations,but also suggests a new understanding of cosmic fine-tuning without invoking multiversehypotheses. By recognizing that maintaining order has an irreducible energeticcost, we reveal a deeper symmetry in nature that bridges thermodynamics, informationtheory, and fundamental physics.
Category: Mathematical Physics
[2] ai.viXra.org:2504.0072 [pdf] submitted on 2025-04-19 22:57:37
Authors: Jon Curry
Comments: 6 Pages.
Differential equations underpin countless physical systems, from chaotic pendulums to cosmic fields, yet classical numerical solvers struggle with stiff, nonlinear, or tensorial problems due to their reliance on time-stepping and derivative approximations. We introduce the Four-Dimensional Iterative Prediction (4DIP) framework, a novel symbolic method that uses geometric residual contraction to solve diverse differential systems with high precision. By iteratively refining a guess toward the true solution using a fixed rule, 4DIP eliminates time discretization, achieving residuals below 10−14 across 13 challenging systems—including chaotic triple pendulums, Dirac fields in curved spacetime, and noisy quantum turbulence—using 50-digit precision. This revised paper enhances theoretical rigor, expands comparisonsto modern solvers, and clarifies failure modes, demonstrating 4DIP’s potential to transformcomputational physics, engineering, and beyond.
Category: Mathematical Physics
[1] ai.viXra.org:2503.0012 [pdf] submitted on 2025-03-29 16:09:34
Authors: Nigel Cook
Comments: 4 Pages. (AI Assistance: Grok 3 xAI; correction made by ai.viXra.org Admin. Note by ai.viXra.org Admin: Conditions of submission are that AI is used as a research tool & the authors understand the AI generated data, equations & graphs etc & have verified them to be correct/true)
Peter Woit’s 2002 paper, "Quantum Field Theory and Representations Theory: A Sketch" (arXiv:hep-th/0206135), proposes a geometric framework for the Standard Model using Clifford algebras in Euclidean space-time. This paper aims to elucidate the mathematical foundations of Woit’s approach, focusing on the role of the Clifford algebra Cl(2), its extension to Cl(4), and the construction of the spin representation that yields the quantum numbers of a Standard Model generation of leptons and quarks. By breaking down the algebraic structures and their physical interpretations, we make Woit’s ideas more accessible to a broader audience, including students and researchers in particle physics and mathematical physics.
Category: Mathematical Physics
[46] ai.viXra.org:2607.0044 [pdf] replaced on 2026-07-16 04:47:03
Authors: Lluis Eriksson
Comments: 7 pages. Machine-checked in Lean 4. Supersedes v1; proof artifact: hrpoly-cmp116-main-reduction-v0.3-2026-07-16
We give a machine-checked reduction of the finite-dimensional fluctuation integral in Balaban's CMP116 large-field analysis. Starting from the physical block constraint Q, the formal development constructs a sparse right inverse E and the constraint-elimination operator C = I - EQ. It proves QE = I, QC = 0, C² = C, the exact sparse norm ||EB|| = M^(d-1)||B||, and the volume-independent bound ||C|| ≤ 1 + M^(d-1) for d ≥ 3. An exact physical/CMP116 isometry transports C to finite Gaussian coordinates without norm loss. The same development constructs the physical localization projector P_Z0, evaluates the complex quadratic Gaussian, localizes its determinant to rank |I(Z0)|, performs the outer Gaussian integration, and absorbs both costs into an explicit exp(c|Z0|) factor.Two corrections exposed by formalization are central. First, the useful domination occurs after Gaussian integration rather than through an unavailable pointwise supremum in the fluctuation field. Second, the localized quadratic matrix is A = -alpha_5 P_Z0. In the exactly identified trivial-background sector, the terminal Lean theorem inserts the concrete C, the flat Hessian, complement localization, and covariance root directly into the printed source Gamma_k = C^T Delta_k (C P_Z0^c)(C^(k))^(1/2), returning an explicit Cauchy bound without an ambient-volume factor. CMP116, however, requires the base Hessian at a generally nontrivial small background Ubar. We do not construct D²S_Wilson(Ubar) or the random-walk estimate (2.16), and therefore do not prove the physical domination, (2.26), hraw, hRpoly, a continuum limit, or a mass gap. The contribution is an auditable reduction that closes the constraint and Gaussian layers and identifies the first genuinely missing interacting construction.
Category: Mathematical Physics
[45] ai.viXra.org:2607.0039 [pdf] replaced on 2026-08-01 15:44:49
Authors: Lluis Eriksson
Comments: 14 pages. Revised and expanded v2; Lean 4/Mathlib formalization with 177 audited declarations. Companion archive SHA-256: CF8B66910C0133804E7F0E8B0F0F5D59A43211A7F445A3E33C3E04E792BC7F7B.
We present an end-to-end Lean 4/Mathlib formalization of the exact evaluation of the two-dimensional SU(2) heat-kernel lattice model on certified finite combinatorial disk cellulations. The development starts from normalized Haar probability on the concrete matrix group SU(2). It identifies its transport to S^3 with the canonical spherical measure, proves an all-order orbital integration formula, derives translated character convolution, and passes from finite character sums to the infinite heat-kernel semigroup by dominated convergence. A genuine shared-edge integral then yields the two-face Migdal move.The geometric layer is independent of any reduction tree. A cellulation stores vertices, paired half-edges, cyclic face words, incidence, Euler characteristic, and positive face areas. Connected dual graphs admit certified elimination schedules, every valid schedule reduces to the heat kernel at total area, and all schedules give the same amplitude. For the original edge model, a rooted spanning tree produces a measurable, product-Haar-preserving gauge equivalence SU(2)^E ≃ SU(2)^(V{r}) × SU(2)^(ET). A compatible tree-cotree construction then retains the exterior holonomy rather than integrating it out. For every certified physical disk cellulation, the boundary-conditioned original-edge amplitude is exactly the SU(2) heat kernel at the total face area. Coefficient extraction gives, for every irreducible label n, the normalized exterior-boundary identity E_P[W_n(H_boundary)] = exp[-n(n+2)(sum_f t_f)/4], where H_boundary is the retained holonomy of the complete exterior boundary word. The universal record is demonstrably inhabited: a concrete three-spoke disk has (V,E,F)=(4,6,3) and derived dual graph K_3. A reproduced audit covers 177 audited declarations, explicitly including both headline theorems, and finds only propext, Classical.choice, and Quot.sound in their dependency cones. The analytic solution is classical. The contribution is a concrete kernel-checked composition from Haar measure and characters to physical edge variables, gauge fixing, tree-cotree elimination, and the exact boundary-observable endpoint.
Category: Mathematical Physics
[44] ai.viXra.org:2607.0035 [pdf] replaced on 2026-08-01 08:42:41
Authors: Lluis Eriksson
Comments: 11 pages. v2 adds a two-page supersession/scope correction, removes the stray metadata suffix Author(s), and preserves the nine-page v1. Full closure: ai.viXra:2607.0039v1.
SUPERSEDED BY AI.VIXRA:2607.0039V1 FOR THE ORIGINAL-EDGE AND TREE-COTREE CLOSURE. This replacement preserves public version 1 and corrects its publication-level scope. Version 1 establishes the compact-group analytic chain, heat-kernel reductions, schedule independence and reduced-model results, but pages 7 and 9 leave open the bridge from the post-gauge-fixed evaluator to the original-edge physical integral for arbitrary cellulations. ai.viXra:2607.0039v1 supplies the missing original-edge gauge fixing, simultaneous face-holonomy transport and compatible tree-cotree closure and is therefore the authoritative source for the full exact simple-loop area-law theorem at the stated finite two-dimensional SU(2) heat-kernel scope. Neither paper constructs four-dimensional continuum Yang-Mills theory or proves a four-dimensional mass gap. The preserved version 1 follows the two-page notice unchanged.
Category: Mathematical Physics
[43] ai.viXra.org:2607.0023 [pdf] replaced on 2026-08-01 08:44:13
Authors: Lluis Eriksson
Comments: 15 pages. v2 adds a two-page supersession/status note and preserves the 13-page v1. The successor's terminal hybrid certificate remains independently REVIEW-PENDING.
SUPERSEDED BY AI.VIXRA:2607.0089V1 FOR THE TERMINAL GLOBAL-SIGN CLAIM; INDEPENDENT REPLAY PENDING. This replacement preserves public version 1 and records the later terminal-claim paper without overstating its audit status. Version 1 proves positivity of F_B, the ratio sign for 0 < beta <= 3, exact bridge and single-Bessel reductions, certified negative results, asymptotic structure and a two-scale closure map; pages 11-13 leave the global sign as a quantified conjecture. ai.viXra:2607.0089v1 claims the global ratio-monotonicity theorem through exact identities and outward-rounded interval certificates. It supersedes this record for that terminal claim, but independent replay of every exact-to-Arb handoff and interval regime remains pending in this audit. No PASS is inferred from a printed or green transcript alone. The preserved version 1 follows the two-page notice unchanged.
Category: Mathematical Physics
[42] ai.viXra.org:2606.0072 [pdf] replaced on 2026-06-27 22:32:05
Authors: Xiangqian Zhang，Mingming Zhao，Linchao Ge
Comments: 17 Pages.
AbstractWithin the framework of Topological Residual Theory (TRT), this paper rigorously establishes thefirst-principles physical picture that "space itself flows at the speed of light along curves in a righthanded cylindrical helical form." In this picture, length and time originate from different aspects ofthe same spacetime flow motion, thereby providing a geometric explanation for the origin of theconstancy of the speed of light.To fundamentally resolve the dimensional mismatch among different physical fields that arises inclassical field theory from differing measurement protocols, we propose a Velocity-FieldRepresentation scheme: the electric, gravitational, and magnetic fields are directly defined as thethree local velocity components of the light-speed spacetime flow along the orthogonal basis of theFrenet-Serret (FS) frame [1]:This achieves a complete unification of all underlying field dimensions to velocity at the kinematiclevel. Since the flow speed of spacetime is fixed at the speed of light , different physicalinteractions essentially correspond to the allocation of this motion budget among the threeorthogonal directions: tangential propagation, normal centripetal constraint, and binormal torsion.The allocation mechanism is precisely controlled by the helical pitch angle .By introducing the time definition , we derive the first-order time-evolution equations ofthe three fields under the characteristic rotation frequencies of the FS frame (bending frequency , twisting frequency ). Under the rigid rotation of the frame, the total kineticenergy density remains strictly constant. Utilizing the rigidrotational dynamics of the frame and the vector and scalar triple-product identities, we elucidate thephysical nature by which the "self-work" terms of the rigid triad automatically vanish during rotation,thereby revealing the deep geometric mechanism of dissipation-free energy flow.The macroscopic observable fields — centripetal gravitational acceleration , tangentialelectric field strength , and magnetic rotational angular frequency —emerge naturally as the time derivatives or rotational frequencies of the underlying velocity fieldsafter being modulated by the motion-budget allocation. This provides a self-consistent firstprinciples framework for the geometric unification of gravity and electromagnetism.E = ct, A = cn, B = cb. E = ct, A = cn, B = cb.Cαds = c dtωK = CK ωT = CTU = 21(E ⋅ E + A ⋅ A + B ⋅ B) = 23c2g = c2κne = c2κt Bmacro = cτbKeywords: Frenet-Serret geometry; right-handed cylindrical helical spacetime flow; velocity-fieldrepresentation; dimensional self-consistency; energy conservation; motion-budget allocation
Category: Mathematical Physics
[41] ai.viXra.org:2603.0084 [pdf] replaced on 2026-06-23 21:01:49
Authors: Xiangqian Zhang，Linchao Ge，Mingming Zhao，Bo Wan
Comments: 15 Pages. (Note by ai.viXra.org Admin: Please cite listed scientific references)
This paper proposes the Topological Residual Theory (TRT), taking the right-handedcylindrical helical light-speed motion in space as the sole rst principle. In the two-dimensional torus formed by collective weaving, the spin-1/2 parallel transport and the BKT topological phase transition lead to precise phase cancellation, locking the system at the microscopic ultraviolet xed point. During the helical rotation and winding motion of the spatial uid,as the eective radius increases, the system triggers discontinuous topological changes andself-organizes to collapse. To describe this topological change, the system must pass througha covariant cobordism transition manifold. In this process, the Gauss-Bonnet constraint re-quires that the total angular decit and the boundary topological charge satisfy a balanceequation, and the ne-structure constant α is dynamically locked as the topological leakageparameter. When the system reaches the Markov critical point F19 = 4181, α is xed atthe observed value 1/137.036, completing the topological phase transition and closing toform a three-layer nested structure. The total topological measure Ω is the geometric in-variant of the closed part, while α is precisely the normalized residual rate of the leakageprocess (the intrinsic leakage proportion of the number of helical divergent lines on the unitGaussian sphere). Through the renormalization mapping F19/104 → 0.4181◦, this paper es-tablishes the precise connection between microscopic geometric symmetry and macroscopicobservable residuals. Under the current framework, tentative geometric correspondencesand order-of-magnitude estimates are made for the electron anomalous magnetic moment,helium ion ionization energy, and proton-neutron mass dierence, and two exploratory ge-ometric research directions are proposed for the muon g − 2 deviation and galaxy rotationcurve anomalies. This theory provides a self-consistent framework for the pure geometric-topological origin of physical constants and fundamental forces.
Category: Mathematical Physics
[40] ai.viXra.org:2602.0117 [pdf] replaced on 2026-07-07 04:01:30
Authors: Lluis Eriksson
Comments: 4 Pages. v3 audits the auditor: 29/29 tests validate exact/toy/group content but discharge NO ledger hypothesis. Three harness defects fixed: d=4 harmonic is 1/2 not 3/2; kurtosis test rescoped to C4(N)>0; (H2') profile.
This is the experiment-first audit report for the 2602-series programme: a runnable mechanical suite (repository ym-audit) with declared pass/fail criteria, a 2D Yang-Mills benchmark, gauge/infrastructure/UV-flow proxy layers, and a reproducibility manifest. Version 3 aligns the report with the fully audited programme and corrects three defects in the harness itself. Scope statement (sharpened): the 29 tests adjudicate exact identities, toy models, and group-theoretic facts; they cross-validate the per-paper suites of THE-ERIKSSON-PROGRAMME (verification/2602-*, 15 papers, independent implementations agreeing where they overlap, e.g. the triangular-lock dimension counts); they do NOT discharge any entry of the programme ledger - a 29/29 run leaves the hypothesis set {(H1), (H2)+beta_LF, (H3), (H2'), (H-LOC), L6.2-import, (H-Rbeta), (H-P0'), structural+window, lambda != 0 traceable, ...} exactly as it was. Corrections: the d=4 harmonic coefficient is 1/2 (harness had 3/2, a dropped-term bug; the paper chain had 3/5, the d=3 value - three-way inconsistency now settled and machine-adjudicated); the non-triviality test is rescoped (Haar kurtosis != 3 is a single-link fact - exactly 2 for SU(2) - present even at strong coupling; what it verifies is C4(N) > 0); the super-polynomial large-field test is restated under the audited log-power profile. The 2D YM benchmark (transfer-matrix gap Delta = g^2 N/2 to 1e-14) and the dependency DAG survive; "Papers 86-90" are pinned to 2602.0088 v3 / 0087 v3 / 0092 v2 / 0091 v2 / 0096 v2.
Category: Mathematical Physics
[39] ai.viXra.org:2602.0117 [pdf] replaced on 2026-02-27 02:16:52
Authors: Lluis Eriksson
Comments: 33 Pages.
This document is an experiment-first audit report for a companion-paper programme claiming a constructive solution of the 4D $mathrm{SU}(N)$ Yang—Mills existence and mass gap problem. It specifies a runnable mechanical audit suite of 29 deterministic tests, defines pass/fail criteria, and presents outputs in a compilation-safe format. The report contains: (i) an explicit non-triviality proof showing the Wightman functions do not factorize trivially; (ii) a toy-model validation recovering the exact 2D $mathrm{SU}(2)$ Yang—Mills mass gap to machine precision; (iii) a Bałaban bridge appendix reproducing the critical inductive step of his renormalization group in simplified form; (iv) a reproducibility repository with 3-line setup instructions; (v) a core proof chain audit mechanically verifying the load-bearing theorems of Papers 86—90, covering terminal Kotecký—Preiss convergence, UV suppression, one-dimensionality of the anisotropic sector, Cauchy bounds on polymer jets, the OS1 vanishing rate $O(eta^2 log eta^{-1})$, Lie-algebra annihilation, and KP margin sensitivity. Beyond the 17 core tests, the suite includes a lattice gauge proxy layer (plaquette expansion, Polyakov-loop centre symmetry, Creutz ratio; 3 tests), an infrastructure layer (Bakry—Émery curvature seed $mathrm{Ric}_{mathrm{SU}(N)} = N/4$, the $2^{4k}$ cancellation in $d=4$, heat-kernel column bound; 3 tests), a UV-flow/heat-kernel layer (Parseval identity, diagonal decay exponent $d/2 = 2$, flow—reflection commutation; 3 tests), a non-triviality test (Haar Monte Carlo on $mathrm{SU}(2)$ and $mathrm{SU}(3)$; 1 test), a toy-model validation (2D Yang—Mills transfer matrix; 1 test), and an algebraic QFT layer (Petz recovery fidelity bound $1-F leq C,e^{-2mr}$ from the Split Property; 1 test). All 29 tests pass; the full suite completes in ${approx}70,mathrm{s}$ on a Google Colab CPU. The complete inter-paper dependency DAG is acyclic and explicitly recorded. All code, data, and artifacts are available at https://github.com/lluiseriksson/ym-audit. The companion papers are archived at https://ai.vixra.org/author/lluis_eriksson.
Category: Mathematical Physics
[38] ai.viXra.org:2602.0096 [pdf] replaced on 2026-07-07 03:45:06
Authors: Lluis Eriksson
Comments: 5 Pages. Retitled. v2: "Unconditional Solution" withdrawn; audited-ledger navigation guide. Corrections: d=4 harmonic coefficient 1/2 (not 3/5); log-power profile restored. Triangular lock verified. Suite 6/6.
This is the navigation guide and audit manifesto for the 2602-series programme on 4D SU(N) Yang-Mills. Version 2 is the audited-ledger edition: the dependency graph, Clay/Jaffe-Witten checklist and threat model of v1 are retained, but every node is now pinned to its audited version and carries its named hypothesis loads; the claim is stated as what it is - a conditional assembly: relative to the declared external mathematics (abstract KP, OS reconstruction, lattice reflection positivity) AND to the internal ledger {(H1), (H2)+beta_LF, (H3), (H2'), (H-LOC), per-scale decoupling, (H-Rbeta), (H-P0'), structural+window loads, lambda != 0 traceable}, the chain assembles OS0-OS4 and OS1 and reconstructs a Wightman QFT with mass gap. v1's "unconditional" (in any sense) is withdrawn. Version 2 also corrects two mathematical defects in v1's new material: the hypercubic harmonic's coefficient (3/5, the d=3 value) is corrected to 1/2 in d=4; and the Large-Field Annihilation Lemma's hypothesis p0(g) >= c/g^2 - attributed to a source whose actual profile is (A0 log g^{-2})^{p*} - is replaced by the honest (H2') trichotomy. The genuinely structural new content survives and is machine-verified: the marginal anisotropic sink at d=4 is empty (exact group averaging over W4 on Sym^2(Lambda^2 R^4): quotient dimension 0, versus 1 at d=6), hence renormalization mixing is triangular in the anisotropic channel and the a^2 x a^{-2} -> O(1) objection has no landing site.
Category: Mathematical Physics
[37] ai.viXra.org:2602.0092 [pdf] replaced on 2026-07-07 03:30:44
Authors: Lluis Eriksson
Comments: 4 Pages. v2: conditional OS1 theorem. v1's "no unproved hypotheses remain" withdrawn; imports retagged to 2602.0087 v3 / 0088 v3 / 0091 v2 with their loads. Native Ward mechanism machine-verified (O(eta^2)). Suite 6/6.
Info de reemplazo — ai.viXra:2602.0092 (v1 → v2)Paper a reemplazar: 2602.0092Method: replacement of existing paperPDF: ward_v2.pdf — sha256 9564aec01a4836c9 — 4 páginasCategory: Mathematical Physics (como v1)Title: Rotational Symmetry Restoration and the Wightman Axioms for Four-Dimensional SU(N) Yang-Mills TheoryAuthor: Lluis ErikssonAbstract (texto plano):We derive a lattice Ward identity for infinitesimal Euclidean rotations of the Wilson theory, identify the breaking term as a dimension-6 anisotropic operator insertion (in the classification of 2602.0087 v3), and show that the breaking distribution is O(eta^2 |log((Lambda_YM eta)^{-1})|) -> 0, establishing axiom OS1 (full O(4) covariance) for subsequential continuum limits - conditionally on the audited companion inputs. Version 2 corrects v1's framing: v1 imported the mass gap, OS0/2/3/4, anisotropy and insertion bounds as "unconditional"; per the audited versions these carry the programme ledger's loads (the source-mapped KP block of 2602.0091 v2, (H-LOC), per-scale decoupling and coupling control for 2602.0088 v3; window and structural loads for 2602.0087 v3). The assembled result - a non-trivial Poincare-covariant Wightman theory with mass gap Delta_phys >= c_N Lambda_YM > 0 - therefore holds under the explicit composed hypothesis set, stated in Section 5; v1's "no unproved hypotheses remain" is withdrawn. The native content survives audit and is machine-verified: the Ward mechanism on an exactly solvable lattice Gaussian model (breaking = O(eta^2) measured), the lambda_{mu nu} != 0 mechanism in the exact quartic model (rotations annihilate O(4) invariants, not the hypercubic harmonic), the Lie-algebra-to-group invariance lemma, the symmetric-difference eta^2/6 error constant, and the eta^2 log vanishing arithmetic.
Category: Mathematical Physics
[36] ai.viXra.org:2602.0091 [pdf] replaced on 2026-07-07 03:16:58
Authors: Lluis Eriksson
Comments: 4 Pages. Retitled. v2 withdraws v1's "verified KP / last gap closed" framing: (H1)+(H2)+(H3)+(H2') => KP retained and machine-verified; (H2) carries the audited beta_LF/profile loads. Checklist now has statuses.
Part I (terminal KP bound). We isolate explicit hypotheses (H1)-(H3) on the terminal polymer activities of the 4D SU(N) lattice Yang-Mills programme and prove that, together with a profile condition (H2') made explicit in this version, they imply the Kotecky-Preiss convergence criterion used as Hypothesis (H-KP) in 2602.0088 v3. The implication is elementary and fully machine-verified (exponential inequality, weighted lattice-animal bound with d(X) >= |X|-1, explicit smallness threshold in g). This refines the ledger: (H-KP) <= (H1)+(H2)+(H3)+(H2'). Status of the hypotheses: v1 declared them "verified from primary sources"; per the audited bridge (2602.0069 v2) the correct statement is: (H1) and (H3) are traceable to Balaban's CMP papers; (H2) is traceable with loads (the beta_LF dichotomy, and the unpinned profile (A0,p*) - the recurring (H-P0) datum of the audited series). Part II (assembly map + Clay checklist). We give the dependency graph assembling 2602.0088 v3, 2602.0087 v3 and the (unaudited) rotational Ward companion, and a Clay/Jaffe-Witten checklist with statuses: activating KP does not activate the mass gap of 2602.0088 v3 by itself - that theorem additionally carries (H-LOC), a per-scale decoupling import, and coupling-control loads; OS1 remains open pending the Ward companion's audit. No unconditional Clay claim is made. All adjudicable content is verified in a deterministic companion suite.
Category: Mathematical Physics
[35] ai.viXra.org:2602.0089 [pdf] replaced on 2026-07-07 02:48:05
Authors: Lluis Eriksson
Comments: 5 Pages. v3-style audit pass: v2 makes the complete-analyticity input explicit as (H-CA); the audited route backs only a windowed version, and the thermodynamic limit needs (H-CA)_infty. Dual architecture survives.
We present two parallel results for SU(N) pure gauge lattice Yang-Mills in four Euclidean dimensions, at fixed lattice spacing eta > 0 and weak coupling g0 <= g*, both conditional on a single shared input, the Dobrushin-Shlosman complete analyticity condition (H-CA): (A) a log-Sobolev inequality Ent(f^2) <= (2/rho) E(f,f) with rho > 0 independent of L, via Cesi's quasi-factorisation seeded by a Bakry-Emery/Holley-Stroock local LSI; and (B) a spectral gap m_gap >= m0 > 0 for the Osterwalder-Seiler Hamiltonian, via Dobrushin clustering and reflection positivity. The two outputs are logically parallel; neither implies the other here (the bridge lemma remains open, Remark 6.3). Version 2 corrects the status of the shared input: v1 declared (H-CA) verified from Balaban's infrastructure; per the audited chain (2602.0053/0054 v2) that route is conditional on (H-DOB-blk)+(H-P0) and valid only in the volume window L <= exp(C/g0^2). Accordingly all results are stated in two regimes: windowed (audit-backed conditional) and all-volume (under the strictly stronger bare hypothesis (H-CA)_infty, required for the thermodynamic limit, Theorem C). What is unconditional and machine-verified: the curvature computation Ric_SU(N) = N/4, the Holley-Stroock block-seed arithmetic, the variance/entropy decompositions and the failure of pointwise inheritance, the Dobrushin contraction machinery and its window arithmetic, the clustering => transfer-gap lemma on explicit operators, and the convergence of Cesi's geometric factor. All bounds remain explicit in N, g0, eta.
Category: Mathematical Physics
[34] ai.viXra.org:2602.0088 [pdf] replaced on 2026-07-07 02:13:55
Authors: Lluis Eriksson
Comments: 7 Pages. v3: mass-gap/clustering result retagged conditional. Terminal KP input made explicit as (H-KP); new localization hypothesis (H-LOC); large-field profile aligned with the audited ledger. Suite 7/7.
We assemble a proof architecture for exponential clustering with a strictly positive mass gap for four-dimensional pure SU(N) lattice Yang-Mills theory with Wilson's action, Cov(O(0),O(x)) <= C exp(-m|x|/a*), m > 0, a* ~ 1/Lambda_YM, with constants uniform in lattice spacing eta and physical volume L_phys - conditionally on three identified inputs. (1) Balaban's structural package (polymer decompositions, exponentially decaying activities), traceable per 2602.0069 v2 with the beta_LF dichotomy attached. (2) A terminal-scale Kotecky-Preiss smallness bound, Hypothesis (H-KP): v1-v2 cited it as proved in an unpublished companion with no identifier; Version 3 retags it as a hypothesis until that companion exists and survives audit. (3) A localization hypothesis (H-LOC) for conditioned observables, made explicit for the first time in v3: the telescoping step compares the terminal clustering bound against O~ = E[O | sigma_a*], whose support is not local. The coupling control (Proposition 4.1) is proved by Cauchy bounds conditionally on the uniform-in-k analyticity radius of Balaban's discrete beta-function and on the large-field penalty profile satisfying the (A0,p*) trichotomy of the audited ledger. What is unconditional and machine-verified: the multiscale telescoping identity (exact for any measure and any nested sigma-algebra chain), the summation-over-scales arithmetic, the lattice-animal bounds, the implications KP => exponential clustering and clustering => spectral gap in exactly solvable settings, and the coupling-control recursion. We verify OS0, OS2, OS3 unconditionally at the lattice level and OS4 conditionally; OS1 (full O(4) covariance) is not established here - its natural conditional supplier is 2602.0087 v3 via 2602.0063 v3. All adjudicable content is verified in a deterministic companion suite.
Category: Mathematical Physics
[33] ai.viXra.org:2602.0088 [pdf] replaced on 2026-02-19 12:11:33
Authors: Lluis Eriksson
Comments: 21 Pages.
We establish exponential clustering with a strictly positive mass gap for four-dimensional pure SU(N) lattice Yang--Mills theory with Wilson's action, uniformly in lattice spacing $eta$ and physical volume $L_{mathrm{phys}}$:$|mathrm{Cov}_{mu_eta}(mathcal{O}(0),mathcal{O}(x))| leq C,e^{-m,|x|/a_*}$, with $m > 0$ and $a_* sim Lambda_{mathrm{YM}}^{-1}$.The proof assembles three ingredients: (1) Balaban's rigorous renormalization group for lattice gauge theories (CMP 1984--1989), which produces effective densities with local polymer decompositions and exponentially decaying activities; (2) a terminal-scale polymer cluster expansion (imported from Balaban's convergent renormalization expansions), which implies exponential clustering for the effective terminal measure; and (3) a multiscale correlator decoupling identity (this paper), which separates ultraviolet fluctuations from infrared physics and yields uniform UV suppression. The coupling control required by Balaban's framework -- that the effective couplings remain in the perturbative regime throughout the RG iteration -- is established via an inductive argument using Cauchy bounds on the analyticity of the effective action. We also verify the Osterwalder--Schrader axioms OS0, OS2, OS3, and OS4 for subsequential continuum limits, and establish vacuum uniqueness and non-triviality. The remaining axiom OS1 (full O(4) Euclidean covariance) is not established here; we prove covariance under lattice translations and the hypercubic group $mathcal{W}_4$, and show that if O(4) covariance holds in the continuum limit, the reconstructed Wightman theory is a non-trivial relativistic quantum field theory with mass gap $Delta_{mathrm{phys}} geq c_N,Lambda_{mathrm{YM}} > 0$, where $c_N > 0$ depends only on $N$ (and is independent of $eta$ and $L_{mathrm{phys}}$).
Category: Mathematical Physics
[32] ai.viXra.org:2602.0087 [pdf] replaced on 2026-07-07 01:19:53
Authors: Lluis Eriksson
Comments: 6 Pages. v3: clustering/mass-gap import (Thm 4.5) retagged conditional; v2's "Thm 6.6 unconditional" withdrawn. Symanzik/W4 core machine-verified (6/6). Conditional SO(4)-restoration input for 2602.0063 v3.
We classify gauge-invariant local lattice operators of classical dimension 6 on the four-dimensional hypercubic lattice into O(4)-invariant, hypercubic-invariant but O(4)-breaking (anisotropic), and on-shell-redundant components, following the Symanzik improvement programme and the on-shell technique of Luscher-Weisz. The anisotropic sector is one-dimensional (Theorem 3.6, Proposition 3.7: uniqueness of the hypercubic harmonic) - a purely representation-theoretic fact, machine-verified in the companion suite together with the classical Symanzik a^2/12 anisotropic term of the Wilson plaquette itself. Inside Balaban's renormalization group framework (small-field regime, g_k <= gamma_0, k <= k* - the window bookkeeping of the audited chain), we extract the anisotropic projection of the effective action via local Taylor (jet) expansion of polymer activities and prove the quadratic bound |c^(k)_{6,aniso}| <= C a_k^2, uniformly in lattice spacing eta, physical volume, and RG step k within the window - conditionally on the structural package, whose audited status is now attached (traceable per 2602.0069 v2; beta_LF dichotomy for the large-field remainder). We further prove an insertion integrability estimate for connected correlators with one anisotropic insertion; Version 3 corrects its status: v1-v2 called it unconditional, resting on an imported clustering/mass-gap bound (Theorem 4.5, from the unaudited companion [1]) claimed uniform in eta and L_phys - per the audited program (2602.0053/0054 v2) such a gap is conditional on (H-DOB-blk)+(H-P0) and windowed, so Theorem 6.6 is conditional on that input. Combined with the rotational Ward identity of the companion [2] (unaudited, identifier pending), the O(4)-breaking distribution tested against Schwartz functions is O(eta^2 |log((Lambda_YM eta)^{-1})|) and vanishes as eta -> 0 - under the same conditional load. This paper thereby supplies, conditionally, the operator-classification input required by 2602.0063 v3 (Proposition 5.3/Remark 5.4) for continuum SO(4) restoration. All verifiable content (representation counts, the unique hypercubic harmonic, the plaquette's own a^2/12 anisotropy, Cauchy-jet mechanics, the a_k^2/log bookkeeping, and the clustering-to-integrability conversion) is adjudicated in a companion suite.
Category: Mathematical Physics
[31] ai.viXra.org:2602.0087 [pdf] replaced on 2026-02-19 12:12:41
Authors: Lluis Eriksson
Comments: 18 Pages.
We classify gauge-invariant local lattice operators of classical dimension 6 on the four-dimensional hypercubic lattice into O(4)-invariant, hypercubic-invariant but O(4)-breaking (anisotropic), and on-shell-redundant components, following the Symanzik improvement programme and the on-shell improvement technique of Luscher--Weisz (1985). Inside Balaban's renormalization group framework for SU(N) lattice Yang--Mills theory, we extract the anisotropic projection of the effective action via local Taylor expansion of polymer activities in the small-field regime and prove a quantitative quadratic scale bound for the anisotropic coefficient: for every RG step $k leq k_*$ with effective coupling $g_k leq gamma_0$, the coefficient of the (one-dimensional) anisotropic sector in the classical dimension-6 Symanzik expansion satisfies $|c_{6,mathrm{aniso}}^{(k)}| leq C,a_k^2$, uniformly in lattice spacing $eta$, physical volume $L_{mathrm{phys}}$, and RG step $k$. We further prove a quantitative insertion integrability estimate for connected correlators with one insertion of the anisotropic operator. When combined with the rotational Ward identity derived in the companion paper, this yields that the corresponding breaking distribution tested against Schwartz functions is $O(eta^2,|log((Lambda_{mathrm{YM}}eta)^{-1})|)$ and hence vanishes as $eta to 0$.
Category: Mathematical Physics
[30] ai.viXra.org:2602.0085 [pdf] replaced on 2026-07-31 23:11:08
Authors: Lluis Eriksson
Comments: 26 pages. v2: 5-page retraction/erratum plus preserved 21-page v1. Lemma 2.2 and Proposition 1.3 stand; Lemma 3.2 is only independently recoverable and its original instantiation is withdrawn.
VERSION 2 RETRACTION AND ERRATUM. The principal results of version 1 are withdrawn. Lemma 3.6, Eq. (16), falsely identifies the Wilson-flow linearisation with a connected weighted scalar Laplacian plus a pointwise adjoint term. At the trivial configuration, gauge invariance forces the true Hessian to annihilate a pure-gauge subspace of dimension at least (|V|-1)(N^2-1), incompatible with the printed connected-Laplacian kernel; if the positive-weight graph is disconnected, the single stationary heat-kernel term used downstream is itself false. Consequently Lemma 3.8, Proposition 3.9, Theorem 3.11 and Theorem 1.1 are withdrawn. Lemma 2.2 and Proposition 1.3 remain intact. Lemma 3.2 is only recoverable as a separately specified scalar heat-kernel theorem; its original instantiation is withdrawn. Reflection positivity, Osterwalder-Schrader reconstruction, thermodynamic limit and mass gap remain open. The five-page erratum is followed by the preserved 21-page version 1.
Category: Mathematical Physics
[29] ai.viXra.org:2602.0084 [pdf] replaced on 2026-07-31 23:12:15
Authors: Lluis Eriksson
Comments: 21 pages. v2: 6-page retraction/status note plus preserved 15-page v1. Defects A-G are included; Lemma 3.3 is false as printed and is not listed as unaffected.
VERSION 2 RETRACTION AND STATUS NOTE. The principal results of version 1 are withdrawn. Theorem 4.4 and Proposition 3.8 depend on the false Wilson-flow linearisation retracted in ai.viXra:2602.0085 and on further false statements. Lemma 3.1 omits the stationary heat-kernel term; Lemma A.1 gives a variance-oscillation bound that fails for correlated non-product measures; Theorem 5.1 does not obtain a positive self-adjoint generator from its stated hypotheses; Lemma 3.3 uses a non-periodic separation across the torus boundary; and Lemma 3.5 bounds a nonlinear map by a differential at one endpoint rather than an integrated or uniform Jacobian. Definitions 2.2, 2.5 and 2.6 remain definitions, and the cited lattice reflection-positivity Theorem 4.1 is not retracted here. Nothing in this note proves the intended almost-reflection-positivity conclusion false; the printed proof and several printed statements fail. The six-page erratum is followed by the preserved 15-page version 1.
Category: Mathematical Physics
[28] ai.viXra.org:2602.0077 [pdf] replaced on 2026-07-06 22:57:16
Authors: Lluis Eriksson
Comments: 6 Pages. v2: v1's Doob-oscillation lemma was false for correlated measures (counterexample of 2602.0070 v2); repaired via conditional oscillation + (H-DEC/AT). UV closure now conditional on (H-LIP^2) + decoupling + audited ledger. Suite included.
We prove that the continuum limit of pure SU(N) lattice Yang-Mills theory in four Euclidean dimensions exists on the algebra of blocked observables at fixed finite volume, CONDITIONALLY on an explicit hypothesis ledger: a quantitative regularity hypothesis for the blocking map (squared-oscillation summability, Assumption A — equivalently (H-LIP^2), a strengthened form of the (H-LIP) contraction of 2602.0073 v2), the Dobrushin-type decoupling hypothesis (H-DEC/AT) of the sibling audits, and the audited statuses of the Balaban structural package. The argument assembles: (i) Balaban's renormalization group program (polymer representation, irrelevance bounds after beta-function extraction, UV stability of effective densities) — traceable per 2602.0069 v2, with the beta_LF dichotomy for the large-field part; (ii) a Doob-martingale covariance IDENTITY (exact for every measure) together with a conditional-oscillation influence bound — v1's claim that the oscillation control holds "without product-measure hypotheses" is withdrawn: v1's Remark 2.3 correctly rejected Efron-Stein for the non-product interpolating measures, but the Doob-oscillation Lemma 1.5 of v1 fails for the same reason (two-spin counterexample, 2602.0070 v2); the repair is (H-DEC); (iii) the RG-Cauchy summability framework of 2602.0073 v2, consumed with its full ledger (including (H-theta)/F-SQRT for the truncation errors). Under the ledger, the telescopic state sequence converges and the resulting state omega_L is gauge-invariant, Euclidean-covariant (hypercubic), and positive. Osterwalder-Schrader reconstruction, the thermodynamic limit, and the mass gap remain open, as in v1. All mechanical steps — the exact covariance identity, both counterexample adjudications, the Assumption A mechanics on explicit blocking maps (including a non-local sharpness example showing locality plus contraction are genuinely needed), and the corrected Proposition 6.1 arithmetic with its exact M 2^(-4k) scale cancellation — are machine-verified in a companion suite.
Category: Mathematical Physics
[27] ai.viXra.org:2602.0073 [pdf] replaced on 2026-07-06 22:43:46
Authors: Lluis Eriksson
Comments: 8 Pages. v2: conditional summability theorem. Lemma 7.1's Doob/Efron-Stein junction repaired (counterexample-adjudicated); ledger explicit: (H-DEC/AT)+(H-LIP)+(B1-B6)+(H-theta). New F-SQRT finding on sqrt(tau_k). Suite included.
We prove, conditionally on an explicit hypothesis ledger, that expectations of blocked, bounded Lipschitz observables at a fixed physical scale l > 0 form an absolutely summable telescoping sequence along a Balaban-matched renormalization trajectory in 4d SU(N_c) lattice Yang-Mills theory with a_k = a_0 2^(-k); in particular the continuum-limit state omega(O) = lim_k
Category: Mathematical Physics
[26] ai.viXra.org:2602.0072 [pdf] replaced on 2026-07-06 22:27:51
Authors: Lluis Eriksson
Comments: 7 Pages. v2: v1's covariance/tensorisation step is false for correlated measures (two-spin counterexample); the ES single-link bound and seminorm theorem survive unconditionally. (B6) closure now conditional on (H-AT). Suite included.
We study the influence estimate — Assumption (B6) — required by the RG-Cauchy summability framework for blocked observables in four-dimensional SU(N_c) lattice Yang-Mills theory, measured by the Efron-Stein seminorm sigma_nu(f)^2 = sum_e E_nu[Var_{nu_e}(f)]. In the small-field regime of Balaban's multiscale effective action, under (A1) a polymer representation, (A2) a per-link oscillation bound with irrelevance factor 2^(-2k), and (A3) lattice-animal counting — all imported from the traceability companion 2602.0069 v2 (conditional) — we prove the UNCONDITIONAL seminorm bound sup_t sigma_{nu_{k,t}}(V_k^irr) <= C independent of the RG scale k: the single-link conditional variance obeys Var_{nu_e}(f) <= (1/4) osc_e(f)^2 for EVERY measure (Lemma 3.2 — conditioning on all other links freezes them, so no influence leaks; this is the sound half, in exact duality with the sibling paper 2602.0070, whose per-link lemma failed but whose covariance identity was exact). Version 2 corrects the unsound half: v1's covariance bound |Cov_nu(f,h)| <= sigma_nu(f) sigma_nu(h) (its Eq. (18)) is FALSE for non-product nu — Example 3.5: perfectly correlated spins give sigma_nu(X_2) = 0 < 1 = Var_nu(X_2) — because Efron-Stein tensorisation is an independence theorem, and the interpolating Gibbs measures nu_{k,t} couple links. On exact Ising chains the tensorisation ratio Var/sum E[Var_e] equals 1.00/1.35/3.40/52.4 at J = 0/0.15/0.5/1.5. Restoring the Duhamel application requires APPROXIMATE TENSORISATION of variance (H-AT): Var_nu(f) <= C_AT sum_e E_nu[Var_{nu_e}(f)] uniformly along the interpolation — a Dobrushin-uniqueness-type condition, the same family as the sibling's (H-DEC) and the chain's (H-DOB-blk), verified here on exact Gibbs chains at weak coupling (C_AT ~ 1.35) and violated without it. There is also a seminorm-interface gap: the companion Duhamel lemma is proved for the Doob seminorm, and sigma_Doob is NOT dominated by the Efron-Stein seminorm for non-product nu (same counterexample; the two seminorms are incomparable in general). Conclusion: (B6) AS CONSUMED by the RG-Cauchy argument is closed conditionally on (H-AT) (or (H-DEC)); the unconditional content of this paper is the Efron-Stein seminorm bound and its scale-uniform M 2^(-4k) = 4(L/a_0)^4 cancellation (with the convergence threshold kappa > log C_anim of v1's Remark B.1 confirmed). Joint statement with 2602.0070 v2: the UV block's only open probabilistic input is Dobrushin-type decoupling of the interpolating measures. All claims, including both counterexample adjudications and the weak-coupling validation of (H-AT), are machine-verified in a companion suite.
Category: Mathematical Physics
[25] ai.viXra.org:2602.0070 [pdf] replaced on 2026-07-06 22:13:35
Authors: Lluis Eriksson
Comments: 7 Pages. v2: corrects a FALSE v1 lemma (two-spin counterexample: raw-oscillation bound fails for correlated measures). Corrected via conditional oscillation + Dobrushin-type (H-DEC). Now a conditional bound; O(4^-k) candidate rate for (H-CAUCHY). Suite included.
We study a uniform Doob martingale influence bound for the irrelevant polymer remainder arising in multiscale renormalization group analyses of four-dimensional SU(N_c) lattice Yang-Mills theory at fixed physical volume, via the Doob influence seminorm sigma_nu(f)^2 = sum_i E_nu[(Delta_i f)^2] and its exact covariance identity. Version 2 corrects a genuine error of v1: the increment-oscillation inequality E[(Delta_i f)^2 | F_{i-1}] <= (1/4) osc_{e_i}(f)^2 was asserted for ARBITRARY probability measures; it is false in general (Example 3.4: two perfectly correlated spins, f = X_2, give E[(Delta_1 f)^2] = 1 while osc_{e_1}(f) = 0), because the Doob increment collects influence transmitted through correlations. The correct, measure-independent statement uses the CONDITIONAL oscillation (Lemma 3.5); passing back to the raw single-link oscillation requires a decoupling hypothesis (H-DEC) bounding the influence-leakage matrix, of Dobrushin type — plausible for the interpolating Gibbs measures nu_{k,t} in the small-field weak-coupling regime, but unproven, and structurally akin to the (H-DOB-blk) family of the audited chain. On exact Gibbs chains the v1 bound is violated already at weak coupling for delocalized observables, while the (H-DEC)-corrected bound holds with the Dobrushin coefficient. Under (H-DEC), the imported oscillation input (A2) (now cited from 2602.0069 v2: traceable, conditional — beta_LF dichotomy included), and the lattice-animal lemma (proved here, verified exactly), the main theorem holds: sup_t sigma_{nu_{k,t}}(V_k^irr) <= C uniformly in the RG scale k, by the exact scale cancellation M 2^{-4k} = 4(L/a_0)^4. The Duhamel interface then delivers a one-step rate delta_k = O(4^{-k}) — precisely the geometrically summable rate that Assumption 3.5 of 2602.0063 v3 requires (its Remark 3.7 with eta = 2) — CONDITIONALLY on (H-DEC) + (A2) + the assumed blocking contraction (H-LIP). v1's closing claim "this establishes the RG-Cauchy property" is softened accordingly: this paper supplies the leading candidate for closing (H-CAUCHY), not its proof. All quantitative claims, including the counterexample and the Dobrushin-corrected bound on exact Gibbs chains, are adjudicated in a companion suite.
Category: Mathematical Physics
[24] ai.viXra.org:2602.0069 [pdf] replaced on 2026-07-06 21:58:43
Authors: Lluis Eriksson
Comments: 10 Pages. v2: "unconditional discharge" replaced by traceable conditional discharge. Open interfaces made explicit: beta_LF dichotomy (else (H-P0)), (M3) profile condition theta0>1, UV/IR direction of the irrelevance factor. Suite included.
We provide a self-contained, equation-level traceability derivation of the three structural hypotheses — polymer representation (A1), per-link oscillation bounds with irrelevance factor (A2), and large-field suppression (B5) — that were assumed in the companions "Doob Influence Bounds for Polymer Remainders in 4D Lattice Yang-Mills Renormalization" and "RG-Cauchy Master Framework". All results are traced to precise equations in the primary sources: T. Balaban (Commun. Math. Phys., 1984-1989) and the expository trilogy of J. Dimock (2011-2014). The translation from Balaban's analytic norms on gauge-covariant function spaces to the per-link oscillation language of the probabilistic framework is made explicit. Version 2 corrects the status of the discharge: it is TRACEABLE AND CONDITIONAL, not unconditional. (i) The small factor of Theorem 8.4 (= Eq. (1.89) of Balaban, Large field renormalization II) carries the constant 2/(1+beta_LF); whether beta_LF is O(1) or a large reference coupling is precisely the dichotomy adjudicated against the audited series (2602.0052/0056/0057 v2), where the large reading trivializes the factor (e^(-c p0) ~ 0.95) and forces hypothesis (H-P0); the dichotomy is now stated as an explicit open interface question (Remark 8.7). (ii) The summability claim (M3) of the RG-Cauchy interface was justified in v1 by "super-polynomial decay from asymptotic freedom"; with the profile p0(g) = A0 (log g^-2)^theta0 the decay in the scale index j (distance to the infrared end) is e^(-A0 (ln j)^theta0): sub-polynomial for theta0 < 1 (sum diverges), j^(-A0) at theta0 = 1 (converges iff A0 > 1), and super-polynomial only for theta0 > 1. (M3) is therefore conditional on the explicit profile condition theta0 > 1 (Remark 12.1; hypothesis (H-theta)). (iii) The irrelevance factor (L^k eta)^(4+alpha) is geometric in the distance to the ultraviolet cutoff, not in the infrared direction; the direction-of-limit bookkeeping for (M1) is made explicit (Remark 10.4) and remains hypothesis-level until the Doob companion is audited. What is machine-verified in the companion suite: the abelian RG operator algebra (Lemma 2.2 mechanics), propagator decay and the random-walk expansion, exponential sum control, lattice-animal counting (Lemma C.1; the illustrative d=4, n=3 count of v1 is corrected from 86 to 84), and the oscillation-analyticity bridge with its Cauchy constants and exact factor-2 saturation. Together with the (unaudited) Doob companion, this package provides a CONDITIONAL discharge of the UV structural inputs at finite volume; the finite-volume, ultraviolet character of the package is what shields it from the infrared volume window of the audited chain (2602.0041 v3, 2602.0051-0057 v2, 2602.0063 v3, now cited).
Category: Mathematical Physics
[23] ai.viXra.org:2602.0063 [pdf] replaced on 2026-07-06 21:28:43
Authors: Lluis Eriksson
Comments: 15 Pages. v3: lattice inputs retagged as windowed/conditional (audit 2602.0041 v3, 0051-0057 v2). New Lemma 1.4: dyadic trajectory fits the audited window, margin ~4.5x. RG-Cauchy summability is a genuine hypothesis. Suite included.
Building on the lattice results of Papers [E26I]-[E26IX] — which, per the series audit (all companions now at v2/v3), are WINDOWED and CONDITIONAL rather than unconditional — we give a conditional construction of a scaling-limit state for pure SU(N_c) lattice Yang-Mills theory in four Euclidean dimensions, along dyadic lattice spacings a_k = a_0 2^(-k). The construction proceeds via a two-layer architecture. Layer 1 (Local fields): for bounded gauge-invariant local observables, expectations converge — without extracting subsequences — to a unique limit; precompactness is trivial (|
Category: Mathematical Physics
[22] ai.viXra.org:2602.0063 [pdf] replaced on 2026-02-14 08:18:38
Authors: Lluis Eriksson
Comments: 19 Pages.
Building on the lattice results established in Papers [E26I]-[E26IX], we give a conditional construction of a scaling-limit state for pure SU(N_c) lattice Yang-Mills theory in four Euclidean dimensions, along dyadic lattice spacings a_k = a_0 2^{-k}. The construction proceeds via a two-layer architecture. Layer 1 (Local fields): For bounded gauge-invariant local observables (Wilson loops, normalized plaquette traces), expectations converge without extracting subsequences to a unique limit. Precompactness of expectations at fixed physical side length L is trivial since |_{a,L}| <= 1. Uniqueness follows from a multiscale RG-Cauchy estimate that bounds the change of local expectations under a single RG step. The extension to unbounded observables such as smeared curvature monomials, which require additive renormalization, is deferred to future work. Layer 2 (Confinement): The physical string tension sigma_phys > 0 is established through step-scaling of Creutz ratios evaluated on Wilson loops whose physical dimensions R x T are held fixed as a -> 0. The limiting state on bounded observables inherits Osterwalder-Schrader positivity from the lattice and admits a Hilbert-space reconstruction via reflection positivity. SO(4) rotational invariance is expected in the continuum (the hypercubic breaking being O(a^2), subject to standard operator classification and construction of renormalized Schwinger functions). The mass gap is established conditionally via uniform exponential clustering of connected correlators -- an input from a uniform physical transfer-matrix spectral gap -- and the reconstruction theorem. Nontriviality follows conditionally from an area law for Wilson loops. Key dependencies on prior papers: uniform LSI inputs [E26I]-[E26IX]; Balaban multiscale effective action [E26III]-[E26V]; DLR-LSI [E26VII]; unconditional lattice closure inputs [E26IX].
Category: Mathematical Physics
[21] ai.viXra.org:2602.0057 [pdf] replaced on 2026-07-06 21:06:18
Authors: Lluis Eriksson
Comments: 13 Pages. v2: closure now windowed (k <= min(k*, k_abs)) and conditional; new hypothesis (H-ABS) — absorption (16) fails at every scale without it, even under (H-P0). Inherits (H-SFI)+(H-YGZ). Ref [8] self-citation removed. Suite included.
We prove integrated cross-scale derivative bounds that replace the unverified Assumption 5.4 of the companion 2602.0041. Combined with two explicit large-field inputs (Hypotheses 3.2 and 4.2) and the conditional inequalities of 2602.0046, this yields — under the audited hypotheses listed below and within the stated volume window — the corresponding log-Sobolev assembly for the Wilson lattice gauge measure at sufficiently weak coupling, with constant independent of L_vol inside the window. The key decomposition into small-field and large-field contributions survives verbatim from v1, as do the sweeping-out modification (an L^1 bound in place of an essential supremum, and a shifted essential supremum over G_{k+1}), the Rothaus closure, the SU(2), d=2 toy-model analysis, and the correction lambda_1 >= alpha_* to Proposition 6.1(2) of 2602.0041. Version 2 corrects the logical status of the assembly after the quantitative audit of the series (ai.viXra:2602.0051-0056, all v2). (a) The Absorption step in the proof of Theorem 1.1 relied on the premises "p0(g) -> infinity as g -> 0 along the flow" and "if beta_k grows sufficiently with k": both use the inverted-sign flow of the series erratum and are withdrawn; with the correct asymptotic-freedom flow all statements hold in the window k <= k*(beta), i.e. L_vol <= e^(C/g^2+O(1)). (b) A new finding (F-ABS): the absorption condition (16) consumes Hypothesis 4.2 in the strong exponent form e^(-c beta_k eps_k^2), but the companion verification (2602.0056 v2) delivers only e^(-c_sf p0(g_k)) with c_sf = 2/(1+beta_0) — bounded along the window — so (16) fails at every scale k >= 2 with the Balaban-compatible thresholds (9), even under (H-P0). The strong form is therefore an additional explicit hypothesis (H-ABS), and its saturated variant eps_k = eps_* opens a second window k <= k_abs proportional to beta eps_^2. (c) The inputs are retagged per their v2 verifications: Hypothesis 3.2 is windowed and conditional on Balaban's small-field inputs (2602.0055 v2); Hypothesis 4.2 is form-level and conditional on (H-SFI)+(H-P0) (2602.0056 v2); the fiber LSI consumed by Corollary 1.2 inherits (H-YGZ) (2602.0053 v2). (d) Reference hygiene: v1's reference [8] was a self-citation of the present paper and is removed; the Wilson duplicate is removed; the audited chain 2602.0051-0056 (v2) is cited. What survives and is validated in the companion suite: the per-direction Wilson bound (Lemma 3.1), the energy-distance identity (13), the single-plaquette tail (Proposition 7.1, Table 1 reproduced digit-by-digit, with its caption/values normalization mismatch fixed), the impossibility Remark 7.2, the factorization step (21), the Rothaus closure, and Appendix A's lambda_1 >= alpha_.
Category: Mathematical Physics
[20] ai.viXra.org:2602.0056 [pdf] replaced on 2026-07-06 21:03:37
Authors: Lluis Eriksson
Comments: 9 Pages. v2: Theorem 1.3 now windowed and conditional on (H-SFI)+(H-P0); the printed small factor trivializes under the polylog p0 floor; d=2 route is fixed-beta. Chain 2602.0051-0055 v2 cited. Suite included.
We verify, at the level of form, the large-field hypothesis (Hypothesis 4.2) of the companion paper on integrated cross-scale derivative bounds for Wilson lattice gauge theory (Paper III). The proof rests on three ingredients: (i) a dictionary lemma translating the Hilbert-Schmidt large-field condition on plaquette holonomies into Balaban's Lie-algebra formulation; (ii) an interface lemma connecting conditional measures with Balaban's T-operation and its uniform small-factor bound on admissible background fields (Eq. (1.89) of Balaban, Large field renormalization II); (iii) the uniformity estimate (Eq. (1.75) ibid.) ensuring that slow-field dependence contributes only an O(1) multiplicative constant. For d = 2, we give an independent proof via character-positive convolutions that avoids the Balaban machinery entirely. Version 2 corrects the status of these results after the quantitative audit of the series (ai.viXra:2602.0051-0055, all v2). (a) v1's claim that the bound is "more than sufficient" for the absorption condition of Paper III is withdrawn: the printed small factor is exp(-c p0(g_k)) with c = 2/(1+beta_0), and with the polylog floor on p0 the suppression trivializes (e^(-c p0(gamma_0)) ~ 0.95) and the absorption inequality fails at every scale (excess >= 10^4.6); effectiveness requires the power-law hypothesis (H-P0) of 2602.0052 v2. (b) v1's premise "p0(g_k) -> infinity as g_k -> 0 along the flow" and Sec. 7's appeal to a stability theorem rely on the inverted-sign running-coupling flow of the series erratum; with the correct asymptotic-freedom flow the small-field condition g_k <= gamma_0 holds only for k <= k*(beta), and all statements are windowed: L_vol <= e^(C/g^2+O(1)). (c) v1's Remarks 4.1-4.2 (slow-field identification and Balaban conditional representation) are unproved interface statements; they are made explicit here as hypothesis (H-SFI), cf. the interface lemmas of 2602.0052 v2. (d) In d = 2 the prefactor K_beta(1)/Z(U_B) of Proposition 6.4 is not uniform in beta, so the d = 2 route verifies a fixed-beta variant only; this is now stated in the theorem. What survives unconditionally and is validated in the companion numerical suite: the HS/Lie-algebra dictionary (Lemma 2.1), the gauge-invariance identity (Remark 2.2), the block event inclusion (Lemma 3.2), and the character-positivity mechanism of Section 6 (Peter-Weyl positivity of the Wilson weight, convolution stability, maximum at the identity, and conditional tail domination in an exact d = 2 toy).
Category: Mathematical Physics
[19] ai.viXra.org:2602.0055 [pdf] replaced on 2026-07-06 20:00:57
Authors: Lluis Eriksson
Comments: 15 Pages. v2: Theorem 1.1 now windowed (L <= exp(C/g^2)) and conditional on (H-P0)+(H-YGZ); Corollary 1.2 additionally on (H-DOB-blk). Chain 2602.0051-0055 fully retagged; ref [6] fixed. Verification suite included.
We prove residual derivative bounds for the polymer expansion of Balaban's multiscale decomposition of the Wilson lattice gauge measure for SU(N_c) in dimension d >= 3, and we assemble them, together with the companion series, into a uniform log-Sobolev inequality. Version 2 corrects the status of this assembly after a quantitative audit. (i) The core mechanism of the paper — locality of polymer functionals, Cauchy estimates on Balaban's analytic domains, and a volume-independent counting bound for connected polymers containing a fixed link — survives intact and is validated numerically; it yields a pointwise derivative bound on the polymer residual with constants independent of the lattice volume, CONDITIONALLY on Balaban's small-field inputs (B1)-(B4). (ii) However, the final step of v1's Theorem 3.5, the inequality k <= C_RG(1+beta_k), relied on the inverted-sign running-coupling flow of the series erratum; with the correct asymptotic-freedom flow the reduced coupling beta_k DECREASES along the cascade, the small-field condition g_k <= gamma_0 is available only for k <= k*(beta), and the derivative bound holds in the windowed form C_res(1+beta) for L_vol <= e^(C/g^2+O(1)). (iii) The assembly of the main theorem inherits two hypotheses identified in the audits of 2602.0052 v2 and 2602.0053 v2: the large-field absorption step requires a power-law penalty exponent — hypothesis (H-P0) — since with the stated polylog floor the suppression factor trivializes (e^(-c_sf p0(gamma_0)) ~ 0.95 at gamma_0 = 0.1) and the absorption inequality fails at every scale (excess >= 10^4.6); and any quantitative use of the conditional fiber LSI via Holley-Stroock carries the penalty e^(-2 beta n_plaq) — hypothesis (H-YGZ) (log10 alpha_blk ~ -5559 at gamma_0 = 0.1, n_plaq = 64). (iv) Version 1's Corollary 1.2 and Remark 5.1 claimed that the Dobrushin-type Assumption 6.3 of Paper I is "no longer needed" via the DLR route of the companion 2602.0053; the v2 audit of that companion shows the route REDUCES Assumption 6.3 to an unverified block condition (H-DOB-blk) whose printed bound c_ij <= tanh(beta n_bd/2) trivializes at weak coupling. Accordingly, v1's closing claim is replaced: the uniform LSI of Theorem 1.1 is WINDOWED and CONDITIONAL on (H-P0) and (H-YGZ), and the mass gap of Corollary 1.2 is additionally conditional on (H-DOB-blk). This replacement also records the completed retagging of the chain: the companion 2602.0054 has been audited and replaced (v2, conditional/windowed assembly), so 2602.0051-0055 now all carry their v2 statuses; reference [6] is corrected (v1 listed 2602.0053 under the title of 2602.0054). All quantitative claims are adjudicated in a companion numerical suite (9 deterministic checks).
Category: Mathematical Physics
[18] ai.viXra.org:2602.0054 [pdf] replaced on 2026-07-06 11:08:38
Authors: Lluis Eriksson
Comments: 17 Pages. v2 replaces the v1 unconditional assembly: the DLR-LSI step requires (H-DOB-blk) (2602.0053 v2) and (H-P0) (2602.0052 v2); the corrected flow gives L <= exp(C/g^2+O(1)). Transfer-operator correlation identities corrected to kernel/symmetrized form (spectr
This paper assembles the route from the uniform log-Sobolev inequality (LSI) on periodic tori to a transfer-matrix spectral gap for SU(N_c) lattice Yang-Mills in d >= 3 at weak coupling: periodic LSI + boundary-uniform RG outputs => DLR-LSI => Stroock-Zegarlinski mixing => exponential clustering => (reflection positivity) => Delta_phys > 0. Version 2 corrects the logical status of this assembly after a quantitative audit (companion numerical suite; Appendix A). (i) v1 claimed the route "bypasses any explicit Dobrushin contraction estimate". This is withdrawn: the DLR-LSI input (Theorem 5.1) invokes the multiscale fiber assembly of [2], whose inter-block step IS a Dobrushin-type condition — made explicit as (H-DOB-blk) in ai.viXra:2602.0053(v2), the detailed companion treatment which v1 did not cite. v1's supporting claim in the proof of Theorem 5.1, that the fiber oscillation is "O(1) regardless of beta", is also withdrawn: the conditional fast potential obeys osc = 2 beta n_plaq + C_poly, LINEAR in beta (2602.0053(v2), Lemma 3.2; reproduced numerically here). (ii) The quantitative absorption in Proposition 4.7 inherits hypothesis (H-P0) of ai.viXra:2602.0052(v2), and the corrected asymptotic-freedom flow restricts all statements to the volume window L <= e^(C/g^2+O(1)): Theorem 1.1 is restated as windowed and conditional on (H-DOB-blk)+(H-P0). (iii) The erratum for [2] in Sec. 10 is corrected: v1's items (a) and (c) ("Assumption 6.3 is removed", "Theorem 1.1(ii) of [2] is now unconditional") are withdrawn — the assumption is REDUCED, not removed; item (b) (withdrawal of Lemma 6.4 of [2] due to the volume factor (MR^n_max)^d) was correct and stands. (iv) A new technical finding (Remark 2.8): the row-normalized transfer operator T-hat of Definition 2.2 satisfies the correlation identities (12)/(29) exactly only when its normalizer D(sigma) = int K(sigma,sigma') d sigma' is constant (true in the d=2 toy, false for d >= 3 where the spatial factor e^((beta/2)S(sigma)) survives); the correct identities hold in kernel form (with K, or the symmetrized D^(-1/2) K D^(-1/2)). Since D^(-1)K and D^(-1/2)KD^(-1/2) are similar, the spectrum — hence Delta_phys — is unaffected; adjudicated numerically (spectra equal to 10^(-16); the T-hat-form of (29) deviates from the exact path integral by 0.30 in a d=3 toy). What survives and is validated end-to-end in exact toys: the slab splitting (Definition 2.1), self-adjointness and detailed balance (Lemma 2.3), the spectral clustering-to-gap step (Proposition 2.4), Osterwalder-Seiler reflection positivity including the Peter-Weyl positive-definiteness of Re tr(UV^(-1)) (Theorem 2.6), and the gauge-invariance lemmas of Sec. 8. The contribution is retagged: a correct and verifiable transfer-matrix back end for the program, whose front end (DLR-LSI) is conditional and windowed.
Category: Mathematical Physics
[17] ai.viXra.org:2602.0053 [pdf] replaced on 2026-07-06 10:47:13
Authors: Lluis Eriksson
Comments: 18 Pages. v2 replaces v1: the unconditional DLR-LSI/mass-gap claim is withdrawn. The block Dobrushin condition becomes explicit hypothesis (H-DOB-blk); the absorption step inherits (H-P0); the corrected flow restricts the result to L <= exp(C/g^2+O(1)).
We study the passage from the uniform log-Sobolev inequality (LSI) on periodic tori, developed in the companion series, to a DLR-uniform LSI for the conditional Gibbs specification of SU(N_c) lattice Yang-Mills in d >= 3 at weak coupling (beta >= beta_0), and from there to a mass gap via Stroock-Zegarlinski and Osterwalder-Seiler reflection positivity. Version 2 corrects the logical status of the main results after a quantitative audit (companion numerical suite included; Appendix A). (i) The fiber assembly (Lemma 3.5) assumes the block Dobrushin condition delta < 1, which v1's own Remark 3.6 left unverified; the printed influence bound c_ij <= tanh(beta n_bd/2) tends to 1 as beta -> infinity and yields delta < 1 only for beta <~ 10^(-2) (d=3) or beta <~ 10^(-3) (d=4) — the opposite of the weak-coupling regime. A rotor exhibit shows the genuine worst-case block influence also tends to 1, so no worst-case criterion can close the gap: the condition is now the explicit hypothesis (H-DOB-blk), and v1's claim of removing the Dobrushin-type Assumption 6.3 of [14] is withdrawn — the present paper reduces that assumption to (H-DOB-blk). (ii) The quantitative absorption in Proposition 4.3 inherits hypothesis (H-P0) of ai.viXra:2602.0052(v2): under the polylog penalty floor p0(g) >= c_0 |log g|^(1+epsilon_0) the required inequality e^(-c p0(g_k)) <= C L_RG^(-(d-1)k) fails already at k = O(1). (iii) The proof assumes g_k <= gamma_0 for all k <= n_max ~ log_LRG diam(Lambda'); with the corrected asymptotic-freedom flow of the series erratum this holds only on the volume window log_LRG diam(Lambda') <= k*(beta), i.e. diam(Lambda') <= e^(C/g^2+O(1)). Theorems 1.1-1.2 are therefore restated as windowed and conditional on (H-DOB-blk) and (H-P0). What survives unconditionally — and is validated numerically — is the boundary-uniformity mechanism itself: the per-plaquette oscillation and gradient bounds (Lemma 3.1; sharp for N_c=2), the "frozen = slow" reduction (Lemma 3.2), the refined dynamical large-field event, the energy-penalty identity ||U-1||_HS^2 = 2N_c(1 - Re tr U / N_c), the TV <= tanh(osc/4) lemma with its two-point equality case, and the Bakry-Emery constant N_c/4 in the
Category: Mathematical Physics
[16] ai.viXra.org:2602.0051 [pdf] replaced on 2026-07-06 09:42:09
Authors: Lluis Eriksson
Comments: 9 Pages. v2: coupling-flow sign corrected (v1's was anti-asymptotic-freedom and made the bootstrap trivially unconditional); the closure is now conditional and windowed (L <= exp(C/g^2)), matching companions 2602.0032/0033; (H-PTW)/(H-XOVER) stated explicitly.
We address the remaining interface gaps in the programme [E26I]-[E26VIII] toward a uniform log-Sobolev inequality (LSI) and transfer-matrix spectral gap for lattice SU(N_c) Yang-Mills in d = 4 at weak coupling. Four gaps are treated: (G1) the pointwise-in-background validity of Balaban's T-operation small-factor bound -- stated in v2 as the explicit hypothesis (H-PTW), since the detailed audit appendices announced in v1 were absent from the document (their references rendered as "??"); (G2) a uniform small-field coercivity estimate for the effective action; (G3) uniform analyticity of boundary terms; (G4) a quantitative bootstrap of all constants. The central correction of v2: the sign of the one-step coupling drift in v1's Theorem 5.2 (g_{k+1}^{-2} = g_k^{-2} + 2 b_0 ln L_RG, coupling weakening toward the infrared) is inverted relative to asymptotic freedom, and contradicts the companions' own use of n_max ~ 1/(2 b_0 g^2 ln 2) (ai.viXra:2602.0032 Sec. 8, 2602.0033, 2602.0041 Sec. 7.3), a formula meaningful only if the coupling grows along blocking and exits the weak regime. With the corrected sign the monotone bootstrap of v1's Theorem 5.3 reverses: the inductive conditions are guaranteed only up to the finite horizon k*(beta) = (g_0^{-2} - gamma_0^{-2})/(2 b_0 ln L_RG) + O(1), and since the multiscale construction uses log_2 L scales, the conclusion holds on the volume window log_2 L <= k*(beta), i.e. L <= e^{C/g^2 + O(1)} with C = 1/(2 b_0) = 24 pi^2/(11 N_c) -- exactly the window of the companion papers ai.viXra:2602.0032/0033 (v2). Full volume-uniformity would additionally require a strong-coupling handoff beyond the crossover scale (Osterwalder-Seiler regime), stated as hypothesis (H-XOVER) and not established here; accordingly the "unconditional closure" of v1 is retitled to conditional closure. Version 2 also repairs the dangling "??" references, supplies the missing proof of Lemma 3.1, rewrites the proof of Lemma 7.4 in the series' fundamental-trace convention (its statement W''(0) = 1/(2N_c) is correct; the v1 proof mixed normalized and fundamental traces), corrects sum_{k>=0} (k+1) 2^{-3k} = 64/49 (v1: 8/49), and fills in the companion identifiers. All corrections are verified in a companion numerical suite.
Category: Mathematical Physics
[15] ai.viXra.org:2602.0046 [pdf] replaced on 2026-07-06 09:13:33
Authors: Lluis Eriksson
Comments: 8 Pages. v3 validates and cleans up the orbit-space Ricci/RCD* input: full Einstein tensor checks for SU(2/3/4), convention triangle Nc/4 - Nc/2 - 1/2, SU(3) sectional range, Bakry-Emery and Holley-Stroock conventions verified. No v2 theorem refuted.
We establish that the orbit space B = A/G of SU(N_c) lattice gauge theory satisfies the Riemannian curvature-dimension condition RCD*(N_c/4, dim A); in particular, it satisfies CD(N_c/4, infinity) in the sense of Lott-Villani-Sturm. The proof shows that the configuration space A = SU(N_c)^{|B_1(Lambda)|}, with the bi-invariant product metric
Category: Mathematical Physics
[14] ai.viXra.org:2602.0046 [pdf] replaced on 2026-02-12 19:04:00
Authors: Lluis Eriksson
Comments: 11 Pages.
We establish that the orbit space B = A/G of SU(N_c) lattice gauge theory satisfies the Riemannian curvature-dimension condition RCD*(N_c/4, dim A); in particular, it satisfies CD(N_c/4, infinity) in the sense of Lott-Villani-Sturm. The proof proceeds by showing that the configuration space A = SU(N_c)^{|B_1(Lambda)|}, equipped with the bi-invariant product metric
Category: Mathematical Physics
[13] ai.viXra.org:2602.0041 [pdf] replaced on 2026-07-31 23:15:37
Authors: Lluis Eriksson
Comments: 12 pages. v4: 1-page retitling/scope correction followed by the preserved 11-page v3.
This replacement corrects the title and foregrounds the logical status already partly recorded in public version 3. The uniform log-Sobolev conclusion requires the cross-scale derivative hypothesis (H-XSD), whose purported companion-paper discharge is not re-verified here. The mass-gap conclusion additionally requires the Dobrushin-type hypothesis (H-DOB), or an independently valid alternative DLR-LSI/mixing route. Neither input is proved by this manuscript alone. No unconditional weak-coupling lattice mass gap, continuum construction, Osterwalder-Schrader reconstruction, or Clay-problem result is claimed. The preserved version 3 follows the correction page for provenance.
Category: Mathematical Physics
[12] ai.viXra.org:2602.0041 [pdf] replaced on 2026-07-06 08:50:32
Authors: Lluis Eriksson
Comments: 11 Pages. v3 repairs v2's dangling "Assumption ??" references, reconstructing the hypothesis as explicit (H-DOB): uniform LSI conditional on (H-XSD), the paper's mass-gap route conditional on (H-DOB) (companion asserts its removal; not re-verified).
We establish that SU(N_c) lattice Yang-Mills theory in d = 4 dimensions with Wilson action at sufficiently weak coupling (beta = 2N_c/g^2 >= beta_0) satisfies a log-Sobolev inequality with constant alpha* > 0 uniform in the lattice size L_vol, conditionally on two explicitly tagged hypotheses: the cross-scale derivative bound (H-XSD) (= Assumption 5.4, asserted discharged in the companion papers [10, 11, 12] = ai.viXra:2602.0057/0056/0055, whose proofs are not re-verified here), and -- for the clustering/mass-gap part -- a Dobrushin-type contraction (H-DOB), stated explicitly in Section 6.3. Combined with reflection positivity of the Wilson action, the chain LSI => hypercontractivity => exponential clustering => transfer-matrix gap yields Delta_phys > 0 uniform in L_vol under (H-DOB); the companion [13] (= ai.viXra:2602.0054) asserts an alternative DLR-LSI/Stroock-Zegarlinski route removing (H-DOB), likewise not re-verified here. Note added (v3). Version 2 contained dangling references "Assumption ??" (in the abstract, Section 1.3, Theorem 6.5, Sections 6.4 and 7.1) left over from a withdrawn Dobrushin-translation assumption, and was internally inconsistent about the status of the mass gap (Theorem 1.1(ii) claimed unconditionality while Theorem 6.5 and Section 7.1 still assumed the withdrawn item). Version 3 repairs this by reconstructing the hypothesis explicitly as (H-DOB) and stating the honest status. Version 3 also records a quantitative tension in (H-DOB) of the same type as hypothesis (H-CONST) of ai.viXra:2602.0032 (v2): the sufficient condition kappa > log[(M R_nmax)^d C_Gamma r/(1-r)^2] involves the fixed decay constant kappa on the left while C_Gamma ~ n_max(beta) grows with beta (Appendix A.3), so the paper's own clustering route is guaranteed only on a window of beta (new Remark 6.6, exhibited numerically). The coupling-flow direction in Theorem 2.1(e) is corrected (series erratum), the stale cross-reference in the Section 2.4 title is fixed, and the companion identifiers are filled in. The Ricci convention of this paper (Ric = N_c/4 with
Category: Mathematical Physics
[11] ai.viXra.org:2602.0041 [pdf] replaced on 2026-02-12 19:02:28
Authors: Lluis Eriksson
Comments: 24 Pages.
We establish that SU(N_c) lattice Yang-Mills theory in d=4 dimensions with Wilson action at sufficiently weak coupling (beta = 2N_c/g^2 >= beta_0) satisfies a log-Sobolev inequality with constant alpha_* > 0 uniform in the lattice size L_vol. Combined with reflection positivity of the Wilson action and the DLR-LSI extension plus Stroock-Zegarlinski mixing route, this yields a mass gap Delta_phys > 0 uniform in L_vol without additional assumptions. The proof combines three ingredients: (i) Balaban's constructive renormalization group, which produces controlled effective actions at all scales; (ii) the orbit space Ricci curvature bound Ric_B >= N_c/4, which gives a uniform log-Sobolev constant for conditional measures of fast modes at each RG scale via the Bakry-Emery criterion; and (iii) a multiscale entropy decomposition with sweeping-out bounds, where the geometric scaling factor ||Q_(k)*||^2 = 2^{-(d-1)k} of transversal block averaging ensures summability of cross-scale errors.
Category: Mathematical Physics
[10] ai.viXra.org:2602.0040 [pdf] replaced on 2026-07-06 08:25:35
Authors: Lluis Eriksson
Comments: 7 Pages. v2 validates the multiscale martingale Poincare machinery end-to-end (numerical suite included); corrects the coupling-flow direction and sharp Ricci constant; tags the RG-normalized disintegration as (H-DIS). Conditional on Balaban/(H-DIS).
We prove that the lattice Yang-Mills measure with gauge group SU(N_c) in d = 4 dimensions at sufficiently large beta = 2N_c/g^2 satisfies a Poincare inequality with constant alpha* > 0 uniform in the lattice size L, conditionally on Balaban's constructive RG and an RG-normalized disintegration hypothesis. The proof uses: (i) the Ricci curvature bound of the gauge orbit space -- sharpened in v2 to Ric_B >= N_c/2, following the correction at its source in ai.viXra:2602.0036 (v2) -- giving a uniform spectral gap for conditional fast modes at each RG scale; (ii) Balaban's polymer derivative bounds, controlling residual cross-scale coupling; and (iii) a multiscale martingale variance decomposition avoiding recursive composition losses, with commutator coefficients D_k <= C e^{-2 kappa} 2^{-3k} made summable by the geometric scaling of transversal block averaging. Version 2 corrects the coupling-flow direction in the statement of Balaban's theorem (which improves the fallback bound of Remark 2.7: beta_k <= beta is bounded, rather than O(k)), records that the summability is robust to the block-averaging convention (new Remark 2.8: the Balaban-style convention gives 2^{-(d-2)k}, still summable), clarifies that the commutator coefficient involves the centered gradient of the conditional potential (which is the mechanism by which G_k-measurable parts drop out, as Assumption 2.6 asserts), and updates the companion references. Unlike other v2's of this series, no statement of v1 is refuted: the entire martingale machinery (commutator identity, telescoping, absorption, the assembled constant alpha*) is validated end-to-end in a companion numerical suite, exactly in a two-scale Gaussian model and against the true spectral gap in a compact four-rotor model, where the recipe's alpha* is confirmed as a valid lower bound.
Category: Mathematical Physics
[9] ai.viXra.org:2602.0033 [pdf] replaced on 2026-07-31 23:09:54
Authors: Lluis Eriksson
Comments: 25 pages. v3: 13-page erratum + 4-page provenance + preserved 8-page v2. The printed proofs of Theorems 1.1, 3.1 and 4.1 are withdrawn; conclusions are not asserted false.
This replacement preserves the published version 2 and appends a page-fixed erratumand an object-provenance sheet. The erratum withdraws the printed derivations ofTheorems 1.1, 3.1, and 4.1 without claiming that their conclusions for the intendedYang-Mills operators are false. It separates defects in the Euclidean-measure toground-state-measure identification, the use of a four-dimensional effective actionon a three-dimensional spatial orbit space, the volume-dependent transfer-operatortrace extraction, the finite-error resolution window, the factorisation-erroraccumulation, and the admissible-volume quantifiers. Three explicit positivetrace-class counterexamples show why ordinary spectral convergence, positivityimproving, first-excited multiplicity control, and total-tail control do not bythemselves imply gap doubling. A sufficient exact-trace repair is stated at thefinite-volume scale: the leading-eigenvalue normalisation error must be little-o ofthe sum of the two actual first-excited contributions, together with subexponentialfirst-excited multiplicities and vanishing relative excited tails. The previous v2text remains included solely for provenance; no unconditional four-dimensional orcontinuum mass-gap result is claimed.
Category: Mathematical Physics
[8] ai.viXra.org:2602.0033 [pdf] replaced on 2026-07-06 06:19:19
Authors: Lluis Eriksson
Comments: 8 Pages. v2: retagged as conditional synthesis under explicit hypotheses (H-BAL, H-FACT); doubling window shown scale-invariant and automatic on the Holley-Stroock route; decay constant corrected (spurious ln 2 removed); verification suite included.
This is the synthesis paper of a series studying the Yang-Mills mass gap via the geometry of the gauge orbit space. We show, conditionally on two explicitly stated hypotheses, that SU(N_c) lattice Yang-Mills theory in d=4 dimensions with Wilson action at sufficiently weak coupling has a positive mass gap m_gap >= c(N_c) e^{-C(N_c)/g^2} > 0 in lattice units, uniformly in L_0 <= L <= C_0 e^{C/g^2}, combining Balaban's constructive renormalization group, a Holley-Stroock/Bakry-Emery spectral gap bound at the terminal scale, and a transfer-matrix trace identity. Version 1 framed the result as conditional on a single informal compatibility assumption. Version 2 sharpens and corrects this: (i) the exact compatibility assumed in v1 is false as literally stated (Balaban's effective actions contain temporally nonlocal terms), and what survives is an approximate factorization with an extensive error ~ L^4 e^{-2 kappa}, which either caps the usable volume at L <~ e^{c kappa} or requires a non-extensive-remainder hypothesis (H-FACT), made precise here; (ii) favorably, the finite-window condition of the quantified doubling argument (imported from the companion paper ai.viXra:2602.0032 v2) is scale-invariant along the cascade and is satisfied automatically on the Holley-Stroock route, in sharp contrast to the Witten-Laplacian route -- this is a new lemma; (iii) the decay constant is corrected to C(N_c) = 24 pi^2/(11 N_c) = 1/(2 b_0): the ln 2 in v1's C(N_c) = 24 pi^2 ln 2/(11 N_c) was an algebra slip (the same slip affects the companion paper), and its removal also repairs the continuum-limit discussion; (iv) the coupling-flow sign in the statement of Balaban's theorem, a factor inconsistency in the oscillation bound, and the sharp orbit-space Ricci constant (N_c/2, not N_c/4; favorable) are corrected. All corrections and the surviving ingredients -- including an exact one-dimensional miniature of the full architecture in which the compatibility identity holds exactly -- are verified in a companion numerical suite. None of this yields an unconditional result; the continuum, infinite-volume problem remains out of scope.
Category: Mathematical Physics
[7] ai.viXra.org:2602.0032 [pdf] replaced on 2026-07-06 05:54:46
Authors: Lluis Eriksson
Comments: 14 Pages. v2: retagged as conditional reduction under four explicit hypotheses (H-BAL, H-CONST, H-MB, H-HAM); Morse-Bott failure at the orbifold locus exhibited numerically; doubling and coupling-flow errata corrected; verification suite included.
We reduce the weak-coupling lattice Yang-Mills mass gap to four explicitly stated hypotheses: assuming them, SU(N_c) lattice Yang-Mills theory in d=4 dimensions with Wilson action at sufficiently weak coupling has a positive mass gap m_gap >= c(N_c) e^{-C(N_c)/g^2} > 0 in lattice units, uniformly in lattice sizes L <= C_0 e^{C/g^2}. The argument combines Balaban's constructive renormalization group, a Morse-Bott/Witten-Laplacian semiclassical spectral gap estimate at the terminal scale, and a transfer-matrix trace identity. Version 1 of this paper presented the result as self-contained modulo Balaban's RG. Version 2 corrects this assessment: the proof is conditional on four explicitly stated hypotheses (Section 1.3). In particular: (i) the Morse-Bott non-degeneracy required by the Helffer-Sjostrand theory fails on the orbifold locus of the flat-connection moduli space -- including the minimum theta=0 of the Born-Oppenheimer potential -- where (d-1)(N_c^2-1-r) quartic "toron" zero modes appear, as we exhibit numerically on a real lattice (Proposition 5.3); and (ii) the constants of Balaban's construction must satisfy a quantitative compatibility window kappa > C' N_c^{3/2}/gamma^2 together with gamma^2 <= 2 N_c h_0, which is empty for typical O(1) decay constants and is not known to follow from Balaban's papers. Version 2 also corrects the sign of the coupling flow in the statement of Balaban's theorem, an inverted extraction regime in the transfer-matrix doubling argument (replaced by a finite-window extraction with explicit error), the definition and Hessian normalization of the Born-Oppenheimer potential (whose v1 form is numerically non-positive), and the Ricci constant of the orbit space (N_c/2, not N_c/4; direction favorable). All corrections and the surviving ingredients are verified in a companion numerical suite. None of this yields an unconditional result, and the continuum, infinite-volume problem remains expressly out of scope.
Category: Mathematical Physics
[6] ai.viXra.org:2602.0021 [pdf] replaced on 2026-07-05 23:07:10
Authors: Lluis Eriksson
Comments: 13 Pages. v2 (all v1 numbered statements preserved; condensed edition): 1-form pair retagged as imported input with the Perron-Frobenius tension documented and demonstrated numerically; Result E renamed; citation repaired; Tables 3 and 5 replicated digit for digit
We present a rigorous framework for the Yang-Mills mass gap problem, combining three independent lines of argument that reinforce each other. Result A (Unconditional): a new MaxEnt Clustering-Recovery Bridge -- for lattice gauge states with finite correlation length xi, in the polymer/Kotecky-Preiss regime made precise in Section 5, the Petz recovery fidelity satisfies 1 - F <= C e^{-r/xi}, proved via maximum-entropy truncation on gauge-invariant algebras, a convergent polymer expansion, and the Fawzi-Renner theorem. Result B (unconditional on the lattice, conditional for all couplings): for SU(N) lattice gauge theory (T=0, theta=0, d=3+1, N >= 2), the algebraic phase exclusion, using the projective commutation relation of 1-form symmetry operators, excludes the trivially gapped symmetric phase (v2: given the imported lattice realization of the symmetry pair); combined with Perron-Frobenius non-degeneracy and Gauss-law constraints, this forces confinement at strong coupling; the extension to all couplings relies on Hypothesis 1.1 (absence of a bulk phase transition), supported but not proven. Under Hypothesis 1.1 the uniform lattice mass gap holds for all lattice spacings. Result C (Conditional): under the same hypothesis, the continuum limit exists as a Euclidean QFT satisfying all Osterwalder-Schrader axioms with mass gap. Result D: the gradient flow reduction (developed in the companion ai.viXra:2602.0020). v2 (no v1 numbered statement is changed): the exact lattice realization of the commuting projective 1-form pair is made an explicit imported input, and a new remark records the Perron-Frobenius tension that forces this framing -- at finite volume, PF uniqueness plus exact commutation of both generators would contradict the projective relation outright, so the magnetic operator commutes with H only up to defect terms (verified numerically: toric code, exact pair with 4-fold degenerate ground state; Z2 gauge theory with electric term, unique ground state with both string commutators nonzero); the Result-D naming collision is resolved (the d = 2+1 theorem is now Result E); an unresolved citation is repaired; the epsilon-powers in the MaxEnt bridge are harmonized; and a replication report is added: the 7-qubit Z2 table is reproduced digit for digit, the Z3 table is reproduced digit for digit after a documented g <-> 1/g erratum between the v1 script and table, and the torus finite-size-scaling table could not be reproduced from the printed conventions and is downgraded to archival status (no framework result depends on it). v2 is presented in condensed form: every v1 numbered statement is preserved with the same numbering; detailed proofs and the full computational listing remain in the v1 PDF as the archival source.
Category: Mathematical Physics
[5] ai.viXra.org:2601.0023 [pdf] replaced on 2026-07-05 11:58:08
Authors: Lluis Eriksson
Comments: 6 Pages. Theory core of the omega=0 witness block (2601.0022, 2512.0064/0070). v2: Bohr Dirichlet decomposition corrected (modular weights; v1 form refuted), KMS constants fixed via c_sigma, e^(-2eps/xi) exponent saved by positivity pinning.
We develop a finite-dimensional technical core for relating separation to effective decoherence-rate envelopes in Davies-type open-system dynamics. We work with energy pinching Delta and quantify coherence by C(rho) = S(rho||Delta[rho]). We import a maintenance inequality P_extra(rho) >= k_B T dC_loss(rho) -- in the corrected operational sense of its source (paired strategies against efficient/free baselines, 2512.0061 v2) -- as an external input. On the operator side we prove: (i) an exact omega = 0 Dirichlet identity yielding a witness-based lower bound on instantaneous decay envelopes, (ii) a Bohr-channel Dirichlet decomposition for a single-channel Davies generator under quantum detailed balance -- corrected in v2: the v1 form (1/2) sum_w gamma(w) ||[S(w),O]||^2 is false as an identity (numerical deviations of order 20%); the exact form is sum_w gamma(w) e^(beta w/2) Re
Category: Mathematical Physics
[4] ai.viXra.org:2512.0060 [pdf] replaced on 2026-07-04 16:52:36
Authors: Lluis Eriksson
Comments: 29 Pages. v2 replaces v1; itemized changelog in Appendix F. Exact numerical verification suite distributed in a companion repository.
We prove that exponential clustering of vacuum correlations enables approximate reconstruction of global quasi-free states from local data in algebraic quantum field theory. The reconstruction is an explicit Gaussian procedure — conditional reattachment through the vacuum's regression structure — which coincides with the output of the Petz recovery map when the reference state factorizes across the split, but not in general: for a pure reference the Petz map returns the reference itself for every input, and version 1 of this paper incorrectly identified the two. For quasi-free states of a massive scalar field satisfying natural constraints, including a symplectic-gap (mixedness) condition on the reference state and an admissibility (no-steering) condition on the split geometry, we prove 1 - F <= C(d,kappa) / [eps^2 (1 - (eta_vac + delta)^2/eps)^2] * ||Delta12||_HS^2, where Delta12 is the reconstruction error, eta_vac <~ e^(-mr) is the vacuum correlation factor, delta controls cross-correlation perturbations, and C(d,kappa) = C_k(6+C_k)/(16 min(c1,c2)^2) with C_k = (1+kappa)^2/(kappa(kappa+2)) determined by the symplectic gap kappa > 0. A finite-rank corollary with explicit factor 2n recovers physical intuition. All counterexamples and bounds are verified by an exact truncated-Fock numerical suite distributed with the paper. Applications to holographic reconstruction are discussed.Version 2 makes three corrections to v1: (i) the reconstruction map of v1's Proposition 2.14 is not the Petz map — its "marginal preservation" step fails for correlated references — and the main theorem is restated for the reconstruction procedure the proof actually controls; (ii) the simplified constant is corrected (3/8 to 7/16 in the strongly mixed limit); (iii) a symplectic-gap hypothesis is added to the Gaussian fidelity lemma, shown necessary by an explicit counterexample.
Category: Mathematical Physics
[3] ai.viXra.org:2512.0003 [pdf] replaced on 2025-12-05 21:28:04
Authors: N. J. Kettlewell
Comments: 6 Pages.
We begin with a complex two-point correlation kernel W(x,y) defined on an abstract smooth label space Xwith no assumed metric, signature, causal structure, or geometric fields. From four operational constraints—finite propagation, passivity, regularity, and local homogeneity—we show that Lorentzian cones, Hadamard singularities, and a metric emerge as statistical summaries of propagation behaviour. Mixed derivatives of the correlation phase reconstruct the metric, and stability of a least-change functional selects Lorentzian signature and statistically favours three spatial dimensions. Allowing coefficients of the correlation generator to vary introduces curvature, and ensemble-averaging the correlation stress yields the statistical consistency conditionGAB + ΛgAB = κ⟨EAB ⟩,linking curvature to averaged correlation tension. Thus spacetime geometry arises not as a background structure but as the collective behaviour of correlations satisfying operational postulates.
Category: Mathematical Physics
[2] ai.viXra.org:2510.0024 [pdf] replaced on 2025-10-14 00:31:21
Authors: Stephen P. Smith
Comments: 8 Pages.
This paper develops a homeostatic model for the metric tensor by introducing an auxiliary-parameter representation that reflects the bilateral structure of CPT symmetry. Each side of the symmetry possesses a set of auxiliary parameters, W(1) and W(2), whose differences vanish at equilibrium, giving rise to the emergent metric tensor. Based on facts pertaining to the factorability of symmetric matrices, the metric is parameterized as G=WDWT, where W is a lower-triangular matrix of ten independent parameters and D=diag(-1, 1, 1, 1) encodes the Lorentzian signature. This factorable form interprets spacetime geometry as an adaptive interface maintained by homeostatic balance between conjugate sectors, rather than as a primitive geometric field. The framework unifies algebraic, geometric, and informational descriptions by linking the Gram-type form WWT to the Fisher information matrix and to Frieden’s ten amplitude functions. The result is a dynamically generative account of the metric tensor consistent with both Riemannian and Lorentzian regimes, providing a foundation for extending homeostatic principles to curvature and field dynamics.
Category: Mathematical Physics
[1] ai.viXra.org:2505.0089 [pdf] replaced on 2025-05-20 06:04:14
Authors: Justin Sirotin
Comments: 25 Pages. 25p Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License
We present a unified theoretical framework—Spinor Mediated Universal Geometry (SMUG)—in which intrinsic spin is the foundational generative principle from which spacetime, gauge symmetries, mass, and matter interactions emerge. The formalism is constructed within Riemann—Cartan geometry, where spinor fields act as sources for torsion, which in turn modifies the affine connection and curvature structure of spacetime. The fundamental operators—spin $hat{S}_i$ and torsion $hat{T}_i$ (where $i in {1,2,3}$ denotes spatial components in the internal algebra)—obey a closed non-Abelian algebra:begin{equation*}[hat{S}_i, hat{S}_j] = ihbar epsilon_{ijk} hat{S}_k, quad [hat{S}_i, hat{T}_j] = ihbar epsilon_{ijk} hat{T}_k, quad [hat{T}_i, hat{T}_j] = -ihbar epsilon_{ijk} hat{S}_k,end{equation*}which embeds naturally into the Clifford algebra $mathcal{C}ell(3,1) otimes mathcal{C}ell(2,0)$ and generates the $mathfrak{spin}(5,1)$ algebra. A spontaneous torsion condensate $langle T^a angle eq 0$ (where $T^a$ is the two-form torsion field related to but distinct from the operators $hat{T}_i$) breaks this symmetry down to $mathfrak{su}(3)_c oplus mathfrak{su}(2)_L oplus mathfrak{u}(1)_Y$, the precise gauge algebra of the Standard Model. We provide explicit spin-torsion constructions of the gauge generators and demonstrate that the U(1) hypercharge generator arises as a composite operator $gamma^5 sigma^3$, where $sigma^3 = -isigma^1sigma^2$ is derived from the $mathcal{C}ell(2,0)$ generators.A unique eigenmode selection principle is introduced via the Preservation Constraint Equation (PCE) $mathcal{P}(sigma,tau,upsilon) = -2sigma^2 + 2tau^2 + 3tau = 0$ (assuming $tau=upsilon$), which filters allowed modes based on spin-torsion projections. Only the $lambda = 4$ mode satisfies this constraint, leading to algebraic and topological exclusion of higher gauge symmetries such as SU(5), SO(10), and E(6).We derive an effective Lagrangian incorporating spin-torsion interactions,begin{equation*}mathcal{L} = frac{1}{16pi G} left[ R + alpha_S S^2 - beta_{ST} S cdot T + gamma_T T^2 ight],end{equation*}and show that integrating out torsion induces NJL-type four-fermion terms that generate fermion masses dynamically. The equation of state $P = ho - alpha_E ho^2$ (where $alpha_E$ is an effective coupling distinct from $alpha_S$) naturally emerges from the recursive spin-torsion dynamics, preventing singularities in both gravitational collapse and early-universe cosmology. Furthermore, torsion backreaction yields quantized gravitational wave echo frequencies and decoherence dynamics derivable from Lindblad-type master equations.Collectively, these results establish SMUG as a self-consistent, recursively closed, and observationally testable framework that unifies geometry, quantum field theory, and gauge structure through the primacy of spin.
Category: Mathematical Physics