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[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