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[37] 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
[36] 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
[35] 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
[34] 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
[33] 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
[32] 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
[31] 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
[30] 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
[29] 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
[28] 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
[27] 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
[26] 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
[25] 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
[24] 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
[23] 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
[22] 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
[21] 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
[20] 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
[19] 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
[18] 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
[17] 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
[16] 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
[15] 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
[14] 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
[13] 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
[12] 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
[11] 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
[10] 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
[9] 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
[8] 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
[7] 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
[6] 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
[5] 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
[4] 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
[3] 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
[2] 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
[1] 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