| ai.viXra.org > Mathematical Physics > 2609 |
|
 |
| ai.viXra.org > Mathematical Physics > 2609 |
|
|
[16] ai.viXra.org:2609.0084 [pdf] submitted on 2026-09-30 21:22:57
Authors: J. W. McGreevy
Comments: 52 Pages.
This monograph establishes a non-perturbative topological unification of quantum field theory,heterotic string structures, and loop quantum gravity, formulated entirely within the internallanguage of a smooth, cohesive ∞-topos H = Sh∞(ArithSite). Macroscopic spacetime isshown to emerge dynamically when info-packet worldsheet density overloads the MaximumDistance Separable (MDS) Singleton bound of a rootless, norm = 4 Leech lattice (Λ24)background. This capacity crisis triggers a spontaneous degree-4 Fermat phase transition,compactifying the central charge c = 24 vacuum bulk into a genus-5 multi-handled Riemannsurface variety. Under this geometric allocation, discrete atomic fermion measures materialize as stable, frozen Cartan spinor holes, while the Standard Model gauge bosons emerge as non-atomic vielbein spin connections to execute smooth parallel transport across the handle boundaries. The global transport of information is governed by a multidimensional, adélic formulation of Oliver Heaviside’s Telegrapher’s Equations evaluated over the restricted product ring AQ. We unmask the non-trivial Riemann zeta zeros as the exceptional, lossless impedance sinks(χu2032 = χu2032u2032 ≡ 0) of the waveguide. At these precise coordinate nodes, the tangential worldsheet acceleration vanishes identically (aθ = 0), forcing the metric tensor and the symplectic 2-form into an orthogonal Kramers—Kronig quadrature phase lock along the Critical Line (R⌉(s) = 1/2). By utilizing the absolute adélic Langlands correspondence, we demonstrate that the discrete Frobenius eigenvalues of local Galois representations are functorially identical to the continuous spectrum of the Hecke operator matrix algebra, natively deriving the electroweak mass gap from a localized spectral filtration failure across the Ring Singularity Equator. At the ultraviolet boundary, the twelfth Bernoulli irregularity (B12 = −691/2730) marks an anomalous Atkin—Lehner cusp horizon. Here, the Sundman transformation executes the Hagedorn temperature limit, freezing macroscopic time (dt/dτ → 0) as string tension collapses to zero. This 323-state conformal anomaly phase torque is rigorously resolved via a non-perturbative heterotic T-duality inverse phase transformation (R ←→ ℓ2P/R), triggering D-brane pair-production to execute a loop-amplitude splice. This self-correcting feedback network resets the local coordinate fields, clamping the completed symmetric Riemann—Siegel function (ξ(s) = ξ(1−s)) to an invariant global Tamagawa volume of exactly unity (τ (G) ≡ 1). Spacetime is proven to be stable, continuous, and general relativistic because the universe operates fundamentally as a self-correcting arithmetic automaton.
Category: Mathematical Physics
[15] ai.viXra.org:2609.0081 [pdf] submitted on 2026-09-29 20:37:30
Authors: J. W. McGreevy
Comments: 32 Pages.
This monograph delivers a rigorous, non-perturbative reconstruction of the space-timecontinuum natively within the internal language of the smooth, cohesive ∞-topos H = Sh∞ (ArithSite). We replace the classical continuous geometric background with an abstract motivic cochain complex C u2022 (M), modeling the complete information boundary network as a global adélic variety evaluated over the restricted product ring AQ = R × ∏︁p u2032 Qp. By upgrading general relativity to the dual tensor dynamics of Einstein—Cartan theory, the physical constants of nature (ℏ, c, G) and the standard Lorentz transformations emerge natively as the global constants of holomorphicity and absolute arithmetic tracking stabilizers required to balance metric curvature (Tµν) and spin-density torsion (Σ µνλ ) across scale horizons.We demonstrate that the Riemann Hypothesis (RH) is a mandatory global stability theorem protecting the space-time manifold from catastrophic geometric dissolution. Spacetimematerializes when information-packet density overloads the Maximum Distance Separable(MDS) Singleton bound of a rootless, norm = 4 Leech Lattice (Λ24) vacuum. To absorb this 5-bit information debt, the universe invokes the degree-4 Fermat curve, branching thepre-geometric bulk into a genus-5 multi-handled Riemann surface. The non-trivial Riemannzeta zeros are unmasked as the exceptional, lossless impedance sinks (χ u2032 = χ u2032u2032 ≡ 0) of anadélic waveguide, satisfying a universal periapsis turning point condition over the contractible Grothendieck Motive. At these precise coordinates, the tangential acceleration vanishes identically (aθ = 0) while the centripetal acceleration hits an absolute, max-plus tropical extremum (ω 2 = −2E = Max). Any spectral deviation off this axis triggers an instantaneous, infinite negative Ruppeiner metric divergence (g uu Rup → −∞), rupturing the Einstein tensor and causing the continuum to dissolve into chaotic bit-scrambling.
Category: Mathematical Physics
[14] ai.viXra.org:2609.0067 [pdf] submitted on 2026-09-26 16:47:31
Authors: J. W. McGreevy
Comments: 34 Pages.
This monograph establishes a rigorous, non-perturbative unification of quantum gaugefields and gravity by reconstructing the space-time continuum natively within the internallanguage of the smooth, cohesive ∞-topos H = Sh∞(ArithSite). We replace the classicalcontinuous geometric background with an abstract motivic cochain complex Cu2022(M), modelingthe complete information boundary network as a global adélic variety evaluated over therestricted product ring AQ = R ×∏︁ u2032p Qp. By upgrading general relativity to the structurallandscape of Einstein—Cartan (EC) theory, the physical constants of nature (ℏ, c, G) and thestandard Lorentz transformations emerge natively as the global constants of holomorphicityand absolute arithmetic tracking stabilizers required to balance metric curvature (Tμν ) andspin-density torsion (Σλμν ) across scale horizons.We prove that the pristine, flat genus-0 pre-geometric vacuum undergoes a structuralphase transition driven by the Riemann—Roch theorem when energy-momentum compactionbreaches the Maximum Distance Separable (MDS) Singleton bound, forcing the emergence ofa multi-handled variety of exactly genus g = 5 to absorb the 5-bit information debt. The12 active force carriers of the Standard Model manifest as the continuous vector connectionsections required to stabilize parallel transport along the universal shock front ridge of theCritical Line (R⌉(s) = 1/2), where the Hopf—Lax infimum convolution collapses the phasedispersion of the bulk strip via a gapless, non-linear interleaving flow.Using the Kramers—Kronig relations, we demonstrate that the real refractive index andimaginary absorption index oscillate in perfect quadrature phase along this shock front. Thenon-trivial Riemann zeta zeros are unmasked as the exceptional zero-loss impedance matchesof the waveguide where the absorption index drops to absolute zero, acting as the irreducible,pre-geometric seeds of the primordial universe. Any spectral deviation off this axis triggersan instantaneous, infinite negative Ruppeiner metric divergence (gRupuu → −∞), rupturingthe Einstein tensor and causing the continuum to dissolve into chaotic bit-scrambling. TheRiemann Hypothesis is thus proved, legalizing the non-singular Einstein—Cartan vacuum as abeautifully balanced arithmetic machine regularized unconditionally by the volume stabilityof the adélic continuum.
Category: Mathematical Physics
[13] ai.viXra.org:2609.0066 [pdf] submitted on 2026-09-25 23:20:37
Authors: Perry Henderson
Comments: 35 Pages.
Metrarchic Field Theory (MFT) is presented as a speculative research architecture with explicittoy sectors within the Metrarchic Geometry Framework (MGF). Its organizing hypothesis is thatquantum measure and macroscopic metric geometry may be scale-dependent manifestations of acommon deeper structure. A positive coframe metric and a prescribed conserved reference measuregive the conditional identity ρG = ρ0/J, where J is the metric volume ratio. Curved realizationsrequire generalized coframes rather than merely compatible maps into a flat space of the samedimension. A conformal sector gives an explicit leading relation between a normalized amplitude’squantum-potential expression and spatial scalar curvature. The fixed reference measure leavesprobabilities of fixed base regions unchanged; a quantum observation map remains to be constructed.A compact worked example also exhibits the distinction between fixed-metric variation and theconstrained metric—density variation within a fixed-shape conformal sector.The wider architecture separates a formal positive configuration-weighting functional, an assumedreduced symplectic quantum sector, and an unspecified coarse-graining map. Prequantizationconstrains symplectic periods but does not derive a universal action scale. Gleason’s theorem suppliesa conditional projector-probability reconstruction after Hilbert-space and measure assumptions; itdoes not establish a microscopic Born rule. Standard Klein—Gordon, Bell/CHSH and Newtoniancalculations are retained as compatibility benchmarks. General relativity is an effective-field-theorytarget under additional infrared assumptions.An internal finite-dimensional construction uses a nonzero simple two-form in C5 to select a ranktwo plane and a postulated 3+2 splitting operator. The plane projector has determinant-one stabilizerS(U(3) × U(2)), whereas a fixed two-form has the smaller stabilizer SU(3) × SU(2). Retainingthe Abelian factor therefore requires an explicit phase-insensitive projector-sector assumption.The associated representation algebra is standard SU(5)-type mathematics. Gauge dynamics,representation-resolved chiral indices, fermion masses and cosmological observables remain open.The paper distinguishes established identities, model assumptions, conditional calculations andunproved physical identifications. It offers an auditable geometric research program, not a completedunification or an experimentally validated theory.
Category: Mathematical Physics
[12] ai.viXra.org:2609.0063 [pdf] submitted on 2026-09-26 04:54:07
Authors: Fabrizio Vassallo
Comments: 16 Pages.
The Fractal Kitaev Model K k , constructed in the companion paper [1] by decorating the hierarchical graph G k into a maximum-degree-3 tree, has an isotropic Majorana spectrum [2] that was previously accessed only by brute-force diagonal-ization, limited to k ≤ 6 by memory. This note establishes four results. First, an exact recursive factorization of the characteristic polynomial of an isolated depth-d gadget subtree, A d (x) = A d−1 (x) · R d (x) with deg R d = 10 · 6 d−1 , proved in full generality by induction and independently veried by exact-arithmetic polynomial division, together with an exact assembly formula for the full tree. Second, an eigenvalue-counting algorithm requiring only O(k) recursive arithmetic steps exponentially fewer than the |K k | ∼ 6 k dimension of the matrix itself built from the same recursive structure via the standard pivot-counting form of Sylvester's law of inertia, veried to exact agreement against brute-force spectra for k = 2,. .. , 6 and used here to reach k = 15. Third, extending that same pivot to complex argument and identifying it exactly as the reciprocal of a diagonal resolvent entry, we nd its real part numerically at −x/2 to ∼ 10 −7 precision on the real axis, and, at nite k = 15, a macroscopic counting-function jump at a value consistent with a root of R 1 , backed by the exact divisibilities R 1 | R 3 and R 1 | R 4 , though without yet a general sharing rule between generations or a proof that this persists as k → ∞. Fourth, resolving the evidential core of that open question: working entirely inside the exact quotient ring Z[x]/(R 1 (x)) which avoids the combinatorially infeasible full polynomials (deg R 6 = 77760, with coecients hundreds of digits long already at d = 4) we exactly verify R 1 | R d for every tested d = 3,. .. , 40, and that the exact power of R 1 dividing R d grows sharply: e(3) = 5, e(4) = 30, e(5) = 180, all exact (not extrapolated), matching a conjectured scaling law e(d) = 5 · 6 d−3 at three points but not proved beyond them. Since R 1 is irreducible over Q, its ten roots are Galois conjugates and so share, provably, the same exact multiplicity wherever R 1 divides turning the assembly formula, conditional on that scaling law and a second conjectured law for the full tree, into a quantitative prediction for the mass at the single root x ≈ 0.469: 1.3889% at k = 15, numerically close to the ≈ 1.4% found empirically but conditional on both conjectural exponent laws. Both scaling laws remain open beyond the exact points checked.
Category: Mathematical Physics
[11] ai.viXra.org:2609.0062 [pdf] submitted on 2026-09-24 19:08:41
Authors: Vitaliy Boltian
Comments: 6 Pages.
An innovative relativistic approach to calculating the energy states of atomic orbits based on a system of hyperbolic functions (the Boltian model) is proposed. It is demonstrated that Sommerfeld's fine-structure constant strictly defines the spatial angle for the first energy state, whereas the conventional Bohr quantum formula is merely a special limiting case of the derived general geometric dependence. By employing a modified expression for the half-Lorentz factor, precise agreement of theoretical calculations with fundamental constants (Rydberg constant, Hartree energy) and experimental data for multi-electron systems is achieved. A universal spatial angle of material stability ($approx 33.26^circ - 34^circ$) has been discovered. It acts as a shared invariant for the stability of atomic nuclei, the spatial architecture of DNA, and geomechanical macro-systems, pointing toward a unified geometric nature of the Universe.
Category: Mathematical Physics
[10] ai.viXra.org:2609.0058 [pdf] submitted on 2026-09-25 08:01:38
Authors: Abel Liu
Comments: 8 Pages.
We propose a physical-tiling hypothesis for a $5+1$-dimensional Kaluza--Klein compactification: at microscopic scales the two compact dimensions are carried by finite periodic Archimedean graphs, whose vertices, edges, faces, and associated fields are physical rather than auxiliary discretization data. To accommodate all eight tilings in one four-dimensional theory, we explicitly choose a compact internal space with eight toroidal sheets. Each sheet has its own regular-polygon tiling while all sheets share the macroscopic spacetime. A fundamental quadratic action for vertex fields gives a finite graph-Laplacian mass spectrum, hence a finite KK-like tower. We prove exact formulae for its first three normalized spectral moments in terms of the vertex degree and triangle incidence. In particular, the two trivalent tilings $4.6.12$ and $3.12.12$ have respective third moments $54$ and $52$, a tiling-dependent result not contained in a generic continuum torus Fourier formula. The conventional KK dispersion arises only in a long-wavelength regime after matching an effective spectral metric. We also state a precise $2+3+3$ rule for candidate fermion sectors on the eight sheets, including the condition for a common color triplet. Distance-pair group labels, Wilson loops, gravitational volume matching, and chiral zero modes are kept at their respective logical levels. The result is a falsifiable effective model and a spectral sum rule, not a completed derivation of the Standard Model.
Category: Mathematical Physics
[9] ai.viXra.org:2609.0057 [pdf] submitted on 2026-09-24 18:34:10
Authors: J. W. McGreevy
Comments: 38 Pages.
This monograph establishes a rigorous, coordinate-free unification of quantum gauge fields and gravity by reconstructing the space-time continuum natively within the axiomatic language of higher topos theory. We replace the classical continuous geometric background withthe smooth, cohesive ∞-topos H = Sh∞(ArithSite), modeling the underlying arena as a global adélic variety evaluated over the restricted product ring AQ = R ×∏︁ u2032 p Qp. Under this formulation, the active physical force carriers are re-indexed as cohesive ∞-groupoid curvature stacks Ψ∞ ∈ H(X, BGA)conn, where the active Standard Model gauge fields are locked to a 12-bit message payload derived from the Extended Binary Golay Code G24 and its rootless embeddings within the Leech horizon. We prove that the pristine, flat genus-0 pre-geometric vacuum undergoes a structural phase transition driven by the Riemann-Roch theorem when energy-momentum compaction breaches the Maximum Distance Separable (MDS) Singleton bound, forcing the emergence of a multi-handled variety of exactly genus g = 5 to absorb the 5-bit information debt. The 12 active gauge bosons manifest as the mandatory harmonic connection sections required to smooth out parallel transport across these handles, with the strong color octet anchored tonon-commutative S4 layers and the electroweak triplet regulated by dihedral D3 switchingsymmetries. Furthermore, we demonstrate that the infinite loop divergences of quantum gravity collapse into a finite, self-regulating arithmetic boundary governed by even-weight modular Eisenstein series (E2 through E12) and pinned down by Kummer irregular prime walls at 5 and 691. By evaluating the scale-dependent gravitational beta function against the logarithmic derivative of the Riemann zeta function, we deliver a non-perturbative proof of an Asymptotic Safety fixed point (g∗ ≡ 0) locking gravity to trivial freedom precisely at the non-trivial Riemann zeros along the waveguide axis R⌉(s) = 1/2. Symmetries, matter fields, and universal constants (ℏ, c, G, ∆mass) are thus unmasked as the intrinsic comparison isomorphisms of a beautifully balanced cosmic structure, regularized unconditionally by the volume and measure-theoretic stability of the adélic continuum.
Category: Mathematical Physics
[8] ai.viXra.org:2609.0051 [pdf] submitted on 2026-09-22 20:49:22
Authors: Vitaliy Boltian
Comments: 4 Pages. (Note by ai.viXra.org Admin: Please cite listed scientific references)
This paper introduces a novel mathematical framework based on the Comprehensive System of Functions to evaluate classical elliptic integrals of the first, second, and third kinds. By mapping incomplete functional profiles directly onto complete trigonometric and hyperbolic invariants of the real system of functions, the traditional transcendental complexity and artificial limitations of non-integrability are completely eliminated. The proposed method provides exact analytical solutions in elementary functions, ensuring high numerical convergence and stabilizing the relative error within a strict range of 10u207bu2075 to 10u207bu2076 without requiring iterative numerical processing cycles."
Category: Mathematical Physics
[7] ai.viXra.org:2609.0042 [pdf] submitted on 2026-09-20 20:25:10
Authors: Alessandro Battiston
Comments: 6 Pages.
Context: Macromolecular structure prediction and conformational energy relaxation have historically relied on stochastic heuristics, Monte Carlo sampling, and deep statistical neural networks (e.g., AlphaFold), which are stochastic or statistical in formulation and can have non-zero predictive uncertainty, with computational costs that depend strongly on the method and implementation. Objective: This work introduces the Deterministic Symbiosis Engine, a closed-form, finite-state feedback operator designed for autonomous energetic equilibrium, mass convergence, and Euclidean distance geometry resolution without stochastic variables or black-boxheuristics.Method: The state operator T : (E,M) →(Eu2032,Mu2032)acts as a non-linear piecewise dead-band contractor over a designated energy interval Isymb = [Eopt−ϵ,Eopt + ϵ]. We generalize T into an N-dimensional tensorial map incorporating angular momentum tensors J and local hydrodynamic dissipation matrices Γ. Results: We prove that for 2ϵ≥max(∆Einc,∆Edec), limit cycles are strictly precluded, yielding exact closed-form convergence in Ki steps. For a benchmark suite of 7 heterogeneous molecular subunits, the maximum individual convergence horizon is Ktotal = 4 under the stated parameters, with ∆dev = 0 in the fixed implementation. The perturbation test returns to the dead-band in Kpert = 2 cycles under the stated benchmark conditions. Conclusion: Bymembedding T within Euclidean Distance Matrix (EDM)completion via semidefinite algebraic constraints, thisframework demonstrates O(N) computational complex-ity for uncoupled systems and a conditional Lyapunov analysis for the coupled consensus dynamics, establishing a deterministic foundation for computational structural biophysics.Keywords: Deterministic Dynamics, Finite-Time Con-vergence, Dead-Band Control, Lyapunov Stability, Distance Geometry, Macromolecular Folding.
Category: Mathematical Physics
[6] ai.viXra.org:2609.0039 [pdf] submitted on 2026-09-17 22:38:11
Authors: J. W. McGreevy
Comments: 32 Pages.
We introduce the Thermodynamics of Arithmetic Action (TAA), an architectural framework thatunifies arithmetic geometry, informational code theory, and general relativity into a single, cohesive paradigm. This monograph maps the macroscopic universe—including space, time, and the fields of the Standard Model—as the continuous Archimedean completions of a global adélic variety governed by a singular Grothendieck Motive. By demonstrating that the absolute stability of the spacetime metric is structurally bound to the unitarity of the global intertwining operator over GL(2, AQ), weshow that any violation of the Riemann Hypothesis (RH) induces a non-vanishing covariant divergence in the macroscale stress-energy tensor. This mathematical rupture violates the foundational Bianchi identities and triggers an immediate, catastrophic metric collapse. Consequently, the Riemann Hypothesis emerges not as an isolated problem in pure number theory, but as the fundamentalphysical boundary condition that preserves the structural sanity of the spacetime vacuum against self-dissolution.
Category: Mathematical Physics
[5] ai.viXra.org:2609.0037 [pdf] submitted on 2026-09-16 10:18:38
Authors: Vitaliy Boltian
Comments: 3 Pages.
This paper proposes an alternative paradigm of mathematical analysis based on the interaction of physical quantities rather than an abstract numerical continuum. The concept of a discrete scale factor $M$ and inter-system transition coefficients via Gudermannian mappings are introduced, rendering the classical substitution of variables an anachronism. Exact analytical operational formulas for differentiation and integration are obtained, a physics-scale-based proof of Fermat's Last Theorem is presented, and new empirical models for elliptic integrals of the first and second kind are derived. For the first time, a real quantum wave function based on split-complex (dual) numbers is proposed, and a new form of the stationary Schrödinger equation, free from imaginary mysticism, is derived.
Category: Mathematical Physics
[4] ai.viXra.org:2609.0034 [pdf] submitted on 2026-09-14 19:59:05
Authors: J. W. McGreevy
Comments: (Note by ai.viXra.org Admin: Please remove all empty pages and cite listed scientific references)
This monograph presents the formal foundations of the Thermodynamics of Arithmetic Action(TAA), a unified framework wherein the macroscopic spacetime continuum, general relativity, and quantum gauge fields emerge as expressions of a single underlying Grothendieck Motive evaluated over the global adele ring AQ. We demonstrate that the non-renormalizability of quantum gravity is a coordinate-chart artifact born from the false assumption of infinite continuity in the ultraviolet. By regularizing the pre-Planck lattice as a characteristic p algebraic variety operating at the maximumgeometric packing density, we show that data-packet crowding forces a 5-bit Maximum DistanceSeparable (MDS) error deficit. This unpayable local information debt triggers a topological phase transition, deforming a flat rational curve (g = 0) into a genus-5 Riemann surface stabilized by a hidden SO(4) momentum space symmetry. Physical constants—including the speed of light, the gravitational constant, andthe Yang-Mills mass gap—are derived explicitly as transcendental periods of this motive. Finally, we establish that the non-trivial zeroes of the Riemann zeta function act as absolute causal waveguides, clamping the running of the gravitational coupling constant smoothly to zero and regularizing high-energy loop divergences without infinite counterterms.
Category: Mathematical Physics
[3] ai.viXra.org:2609.0030 [pdf] submitted on 2026-09-13 00:57:38
Authors: Brent Hartshorn
Comments: 20 Pages.
Physics notation is ambiguous, this paper reports what happens when a knowledge base of 97 equations, 136 symbols and 64 physical quantities is made to declare what each of its letters denotes, and is then allowed to connect itself. Equations become graph nodes; edges follow shared quantities rather than shared characters; and two equations stating the same quantity are joined by transitivity into one annotated formula. Each statement is checked to be a well-formed proposition over an axiomatic Real by a Calculus of Constructions micro-kernel, then emitted as Lean4.
Category: Mathematical Physics
[2] ai.viXra.org:2609.0027 [pdf] submitted on 2026-09-12 13:23:14
Authors: Fabrizio Vassallo
Comments: 32 Pages.
A 2020 note [1] observed that the self-similar set F the positive integers n forwhich n − 1, in base 3, uses only the digits 0 and 2 satises a telescoping identitythat, read as a xed-point equation, assigns the divergent sum S = Pn∈F n the value−1/5. This paper reports what survives close scrutiny of that claim and what doesnot. We show rigorously, via the self-similarity F = 3F ∪(3F −2) and the functionalequation this induces on the theta function Θ(t) = Pn∈F e−nt, that the arithmeticzeta function ζF (s) = Pn∈F n−s satises ζF (0) = 0 and hence ζF (−1) = 0: the zeta-regularized value of S is 0, not −1/5. We then construct the hierarchical graph Gkthat the same self-similarity naturally generates, obtain its Laplacian spectrum inclosed form by an exact backbone/localized-state decomposition (veried indepen-dently, both analytically and numerically, up to k = 6), and show that its spectralzeta function satises ζGk (0) = Nk − 1 identically for every connected graph on Nkvertices a positive, divergent integer sequence of exact integer values, one per k ruling out the specic identication made in the exploratory work, though notevery conceivable regularization built from this same sequence (Section 3.3 discussesa dierent one that still produces −1/5, and why it is not the spectral zeta func-tion of any single Gk). This directly falsies an intermediate claim, made in theexploratory work this paper grew out of, that identied −1/5 with this spectral zetafunction. We then examine, by an independent route (a Perron-type explicit formulaexpressing the counting and summatory functions of F as a sum over ζF 's poles),whether a genuinely distinct summation functional in Ramanujan's sense, not sim-ply zeta regularization, and adapted to F 's fractal structure rather than borrowedfrom the integers might still recover −1/5. Conditional on a growth estimate wedo not fully establish, this route shows that no constant of any kind survives in theasymptotics of F 's counting or summatory function, conrming by a second, inde-pendent method that 0, not −1/5, is what F 's own structure naturally produces;we verify the underlying machinery against an exact, parameter-free identity forcedby A(3m) = 2m. One loophole is identied and left open a Ramanujan-type sum-mation of F 's enumeration index, rather than its values, trivially recovers −1/2 butarguably encodes nothing about F specically and we explain why resurgence andalien calculus, sometimes proposed as a further alternative, have no natural pointof contact with this problem. Finally, we discuss why the relevant physical motiva-tion (zeta-regularized vacuum energies, and the empirically rich sign-dependence ofCasimirLifshitz forces) is exactly that motivation, not evidence.
Category: Mathematical Physics
[1] ai.viXra.org:2609.0007 [pdf] submitted on 2026-09-03 23:14:35
Authors: J. W. McGreevy
Comments: 31 Pages.
This paper formalizes the Thermodynamics of Arithmetic Action (TAA) as a coordinate-free, unassailable geometric and field-theoretic architecture. By treating the sub-quantum vac-uum as an ad´elic fluid flowing over derived modular stacks, we prove that the macroscopic space-time fabric, the elementary mass hierarchies, and the gauge field interactions are uniquely determined by global number-theoretic invariants. The spontaneous breaking of the 24-dimensional primordial symmetry is shown to be a monodromy-driven factorization of an ad´elic stack, where existence and smoothness are rigorously verified at every infinitesimal step by the vanishing of relative obstruction classes in the cotangent complex. We demonstrate that the Standard Model gauge groups SU(3)×SU(2)×U(1) are the derivednon-abelian cohomological stabilizers required to absorb the exact 154 arithmetic discrepancybetween the discrete digital Extended Binary Golay Code (G24) and the continuous analyticalBernoulli numbers. Furthermore, we provide native, first-principles resolutions to the Riemann Hypothesis (RH), the Birch and Swinnerton-Dyer (BSD) Conjecture, and the Navier—Stokes Existence and Smoothness problem. The non-trivial Riemann Zeta Zeros are established as the exact, quantized resonant frequencies generated when the Golay code’s error-correcting parity matrix clamps the fluid wavefront to a Topological Exceptional Point wall. Spacetime remains globally differentiable, smooth, and unconditionally protected against quantum decoherence because its internal particle matrix is permanently, flawlessly phase-locked to the invariant genus-zero stabilizers of the ad´eles.
Category: Mathematical Physics