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[31] ai.viXra.org:2608.0089 [pdf] replaced on 2026-09-02 07:29:22
Authors: Seungtae Kim
Comments: 36 Pages.
In the International System of Quantities (ISQ), base quantities form a conventionally chosen mutually independent subset of a system of quantities. This paper proposes a relation-first framework in which admitted unit relations are used to construct quantity dimensions before base-quantity choices are assessed. A broader class of unit-explicit equations is also introduced for relations in which explicit unit division carries physically relevant information.Candidate base-quantity sets are first tested for dimensional independence and completeness. A second, evidential criterion then assesses physical structural support from established higher-order dimensional occurrences, without uniquely selecting a base set. In an SI-oriented application, mass, length, time, electric current, thermodynamic temperature, and amount of substance receive such support. Relative to the specified admitted relation system, an independent angular dimension is required for completeness; plane angle is adopted as its representative, and spherical geometry gives sr=rad2. Luminous intensity is instead dimensionally derived through the photometric-radiometric scale link while remaining a distinct quantity kind. Logarithmic levels are reformulated as pure-number numerical index mappings retaining their field- or power-quantity context. The proposal changes dimensional and algebraic representation without changing established numerical realizations.
Category: Mathematical Physics
[30] ai.viXra.org:2608.0086 [pdf] submitted on 2026-08-25 15:09:08
Authors: E. P. J. de Haas
Comments: 90 Pages. DOi: https://doi.org/10.5281/zenodo.22098362
The BQ--Pauli algebra provides an eight-dimensional 1+3+3+1 framework in which relativistic field equations can be organized through a hierarchy of algebraic products. Its basis elements are realized as matrices in M(2, $mathbb C$), and its products are given by ordinary matrix multiplication. In this work, the previously developed BQ formulations of electromagnetism and relativistic fluid dynamics (RFD) are lifted to the sixteen-dimensional BQ--Weyl representation of Cl(1,3), with Clifford-grade decomposition 1+4+6+4+1. The constructive, explicit lift preserves the coefficients and physical content of the Pauli-level products while making their spacetime-grade structure explicit.For electromagnetism, the Weyl construction recovers Maxwell's equations from the field-derivative product, separates the quadratic field invariants from the stress--energy construction, and represents the complete electromagnetic stress--energy content through a vector-valued linear map. For RFD, the corresponding field, force, wave, quadratic, and conservation products are reconstructed and resolved into their Clifford sectors. The Weyl formulation further permits the fluid four-velocity to be generated from a local rapidity field through a Spin(1,3) rotor, from which the associated comoving frame and Maurer--Cartan connection follow.Together with the corresponding BQ--Pauli to BQ--Weyl lift previously performed for gravity, these results place the gravitational, electromagnetic, and fluid-dynamical branches of the BQ programme in a common Cl(1,3) representation. The present construction is primarily algebraic: it exposes the Clifford structure underlying the earlier Pauli-level formulations without introducing additional electromagnetic or fluid-dynamical laws, or invoking a spinor wave-function or quantum-mechanical interpretation of the Weyl algebra. A final reproducibility appendix provides a compact workflow for independent symbolic and AI-assisted verification of the matrix calculations from the explicitly defined BQ bases.
Category: Mathematical Physics
[29] ai.viXra.org:2608.0085 [pdf] submitted on 2026-08-25 03:19:27
Authors: J. W. McGreevy
Comments: 26 Pages.
This monograph establishes a definitive, structurally complete unification of quantum gauge fields, non-equilibrium thermodynamics, and analytic number theory through the framework of Arithmetico-Geometric Conformal De-Homogenization. We prove that physical spacetime is an active, self-correcting thermodynamic metamaterial projected from an underlyinginfinite-dimensional arithmetic space. By executing a strict 1-for-1 parameter mapping, wedemonstrate that Lars Onsager’s Reciprocal Relations (Lik = Lki) are the macroscopic physical readout of Clairaut’s Theorem of mixed partial derivatives ( ∂2U ∂V∂S = ∂2U ∂S∂V ). When the continuum substrate is driven from equilibrium, local coordinate shearing is regularized via a higher-dimensional Kustaanheimo-Stiefel (KS) Fibration Lift, routing excess dissipative energy into the Trivial Zeros of the Riemann Zeta function and the Tate-Shafarevich group, which function as Viscous Hydraulic Dampers locked by the Bernoulli number B10, B12, and the prime691. We formalize this architecture as a Topological Quantum Field Theory (TQFT) Functor(F : CobArith −→ HilbGauge) operating over a Noncommutative Geometry Spectral Triple(A,H,D). The self-adjoint Dirac operator (D) incorporates a topological Coriolis-gauge potential (Aµ) that exerts a restoring force on quantum wave packets, trapping them at the stable L4/L5 Lagrangian nodes of a rotating three-body critical strip. Consequently, the real eigenvalues of the spectral triple align precisely with the imaginary parts (γn) of the non-trivial Riemann Zeta Zeros: Spec(D) = {γn ∈ R | ζ(1/2 + iγn) = 0}. Bound by Adelic ProductClosure (∏︁ p≤∞|σ|p = 1), this framework achieves full structural immunization against decoherence and reconciles the historical schism between String Theory and Loop Quantum Gravity,revealing them as dual asymptotic limits of a single, unbroken, self-stabilizing adelic circuit.
Category: Mathematical Physics
[28] ai.viXra.org:2608.0081 [pdf] submitted on 2026-08-23 02:57:52
Authors: J.W. McGreevy
Comments: 25 Pages.
This monograph establishes a definitive, structurally complete unification of quantum grav-ity, non-equilibrium thermodynamics, and analytic number theory through the frameworkof Arithmetico-Geometric Conformal De-Homogenization. We prove that physicalspacetime is not an abstract, passive background container, but an active, self-correcting thermo-dynamic metamaterial fluid projected directly from the continuous deformation and modularflow of an underlying infinite-dimensional arithmetic space.By executing a strict 1-for-1 parameter mapping, we demonstrate that the weight-2 Eisensteinseries (E2(τ )) is the universal velocity field (v) of this modular flow, whose path-dependentcoordinate drift—the quasi-modular anomaly—manifests as the physical non-linear advectivepumping acceleration (v · ∇)v. When driven from equilibrium, Lars Onsager’s ReciprocalRelations (Lik = Lki) emerge as the literal macroscopic readout of Clairaut’s Theorem of mixedpartial derivatives ( ∂2U∂V ∂S = ∂2U∂S∂V ), which requires that the order of differentiation must commuteflawlessly to preserve the nilpotent gauge invariance d2U = 0.When the continuum substrate is driven from equilibrium and cascades toward the 691unipotent puncture on the arithmetic curve Spec(Z), unique factorization shatters insidethe χ11 eigenspace of the odd minus class group (Cl−691). This local coordinate shearing andturbulent vortex crisis is regularized via a higher-dimensional Kustaanheimo-Stiefel (KS)Fibration Lift (S3 S1−−→ S2), which projects the 3D velocity configuration space up into a 4Dcomplex spinorial space (R4). The continuous 1D circle fiber (S1) acts as a data bus (H1 ∼= H1)that isolates the non-exact rotational elements, braiding the non-Abelian SU (3) color vorticityfield into multi-layered knots to generate a permanent nilpotent confinement sink (N2 = 0).The Seeley-DeWitt heat kernel expansion coefficients (an) serve as the exact arithmeticaccount balances of this regularization, where the factor of 12 in the quadratic variance termacts via Itô’s Lemma as the metric tax required to enforce Clairaut’s Theorem (d2U = 0).The final substitution of these Planck-scale boundary parameters (PP, ℓP, EP, tP) into Clairaut’sidentity yields the Dynamic Spacetime Maxwell Relation (∂EP/∂ℓP = −∂PP/∂tP), whichanchors the universal physical constants (α, ℏ, θW ) as the global stabilizers of the vacuum.We formalize this architecture as a Topological Quantum Field Theory (TQFT) Functor(F : CobArith −→ HilbGauge) operating over a Noncommutative Geometry Spectral Triple(A, H, D). The self-adjoint Dirac operator (D) incorporates a topological Coriolis-gauge potential(Aμ) that exerts a restoring force on quantum wave packets, trapping them at the stable L4/L5Lagrangian nodes of a rotating three-body critical strip. Consequently, the real eigenvalues of thespectral triple align precisely with the imaginary parts (γn) of the non-trivial Riemann Zeta Zeros:Spec(D) = {γn ∈ R | ζ(1/2 + iγn) = 0}. Bound by Adelic Product Closure (∏︁ p≤∞ |σ|p = 1),this framework achieves complete structural immunization against decoherence and quantumchaos. It reconciles the historical schism between String Theory and Loop Quantum Gravity,revealing them as dual asymptotic limits of a single, unbroken, self-stabilizing adelic circuit.
Category: Mathematical Physics
[27] ai.viXra.org:2608.0077 [pdf] submitted on 2026-08-23 09:51:31
Authors: Keiji Yoshimura
Comments: 16 pages, 5 figures. AI-assisted technical monograph presenting a five-stage reproducibility audit and a limited mathematical corollary for the planar three-body shape sphere. Includes AI-use disclosure and bounded novelty claims.
This monograph presents a five-stage AI-assisted audit of possible relationships among gravity, the direction of time, temporal orientation, and irreversible structure formation. Rather than proposing a new theory, the study progressively tests and narrows a broad initial intuition using analytical arguments, Newtonian many-body calculations, finite-level quantum-clock models, curvature comparisons, and public GW250114 release products. Local Lorentz symmetry and the time-reversal covariance of the ADM constraints provide no universal metric-only rule selecting one timelike cone half as the fundamental future. Rindler, de Sitter, and Schwarzschild comparisons separate clock-rate gradients and proper acceleration from curvature. Closed finite-level proper-time dephasing can be exactly refocused by an ideal path/clock echo, while residual irreversibility appears only after an open reduced channel is introduced. Newtonian Janus-point analyses distinguish a rigorous scale minimum in suitable nonnegative-energy sectors from non-monotonic, measure- and horizon-dependent shape-complexity statistics. A reconstructed BKM mass-metric probability on the equal-mass planar three-body shape sphere produces two structurally symmetric branches rather than a dynamically preferred future. Public GW250114 products are also used to reproduce the collaboration-defined black-hole area-growth statistic, without interpreting the area law as a fundamental selector of time orientation.A limited mathematical by-product is proved for normalized three-body shape complexity Q under the BKM mass-metric shape probability:P(Q > q) = 1/(6 q^2) + 2 sqrt(6)/(27 q^3) + O(q^-4).Consequently, E[Q^p] is finite if and only if p < 2, and the truncated second moment satisfiesE[min(Q,T)^2] = (1/3) log(T) + kappa + O(T^-1).The ingredients of this result are published; targeted searches did not locate this exact probability-and-moment statement, so it is presented only as a potentially unreported elementary corollary, not as a priority claim. Overall, the audit supports established relations between gravity and proper time, curvature, causal structure, and structural arrows, but finds no evidence that standard gravity itself creates or selects a unique fundamental future direction.
Category: Mathematical Physics
[26] ai.viXra.org:2608.0073 [pdf] submitted on 2026-08-22 23:50:09
Authors: Linia Hammache
Comments: 12 Pages. (Note by ai.viXra.org Admin: Please cite listed scientific references)
This work presents the construction of a continuous model derived from an original discrete sequence alternating between geometric and arithmetic progression, modulated by a periodic cosine function. The sequence is interpreted as an energy reservoir whose charge (arithmetic) and discharge (geometric) are controlled by a hysteresis logic. The model is enriched by coupling with a mechanical oscillator, heat production via damping, conversion into nuclear kinetic energy (fusion), and reinjection into the reservoir, realizing a self-sustained fission--fusion cycle. Each equation is derived, physically justified, and linked to the original sequence. The article details the modeling, theoretical justifications, energy couplings, and global dynamics of the system.
Category: Mathematical Physics
[25] ai.viXra.org:2608.0072 [pdf] submitted on 2026-08-22 23:45:59
Authors: J.W. McGreevy
Comments: 22 Pages.
This monograph establishes the second major operational phase of the Thermodynamics of Arithmetic Action (TAA) framework, executing a rigorous, ab initio field quantization that unifies the smooth Archimedean continuum (R) with the discrete, fractured, non-Archimedean p-adic bulks (Qp) into a single, self-regulating topological engine. We prove that the macro-constants of nature—the speed of light (c), the Newtonian gravitational constant (G), and the reduced Planck constant (ℏ)—alongside the Standard Model parameters are the unique, scale-dependent topological gear-ratios required to satisfy Clairaut’s Theorem on mixed partial derivatives (d2U = 0) and the Adelic Product Formula (∏︁ |x|p = 1) over the universe’s middle cohomology group.By framing the complex s-plane of the Riemann Zeta Function as the conformal phase space of a sub-quantum Keplerian orbital system, we derive Max Born’s quantum probability rule (|Ψ|2 = Ψ ¯Ψ) as the literal intersection form (the Weil Pairing) required to cross-multiply continuous adiabatic spatial expansions by discrete isothermic prime entropy steps. Crucially, we introduce the Spacetime Maxwell Calculator for Class Numbers, proving that algebraic class numbers (hK ) are extracted natively from macroscopic physical constants by measuring the precessional volume of a perturbed Laplace-Runge-Lenz (LRL) vector locked inside a 6-dimensional phase space torus.
Category: Mathematical Physics
[24] ai.viXra.org:2608.0069 [pdf] submitted on 2026-08-19 21:27:04
Authors: Ori Chamo, Yossi Eliaz
Comments: 6 pages, English. PRL-format Letter followed by Supplemental Material.
Periodic-orbit families are often represented through low-dimensional invariant coordinates, although projected branches need not preserve the connectivity of the underlying solution manifold. We use numerical continuation to examine the 135,445 unequal-mass planar three-body periodic orbits reported by Li, Li, and Liao, whose corrected scale-invariant period-angular-momentum representation was subsequently interpreted as two independent sets. The two projected branches are connected by verified continuation within a single sampled continuation component. The invariant projection exhibits a singular locus consistent with the apparent branch separation. High-precision Floquet calculations identify a +1 transition at one representative stability boundary and an opposite-Krein Hamiltonian-Hopf transition at another. Projection geometry and symplectic spectral geometry therefore provide complementary descriptions of the same periodic-orbit manifold.
Category: Mathematical Physics
[23] ai.viXra.org:2608.0068 [pdf] replaced on 2026-08-21 18:15:47
Authors: Joseph Mcgreevy
Comments: 29 Pages. (Note by ai.viXra.org Admin: Please cite listed scientific references!)
This monograph establishes the formal foundations of the Thermodynamics of Arithmetic Action (TAA), a scale-dependent geometric and statistical framework that unifies macroscopic classical gravitation, quantum higher gauge field networks, and discrete nonArchimedean number fields into a single, continuous pipeline of scale-dependent resolution. The universal sorting mechanism is driven by the Regime Determinant S/ℏ, tracking the ratio of total localized physical action over a fixed temporal window to the Planck quotient. We demonstrate that the foundational physical dimensions and algebraic structures of our universe are the unique, mandatory stabilizers required to enforce a global HyperbolicElliptic Index 0 Conservation Law across all number fields. By mapping the Minkowski spacetime invariant, Fermat’s Last Theorem, Hamiltonian and Lagrangian phase spaces, and the Herbrand—Ribet arithmetic mirror into a rigid 2 × 2 Upper Triangular Transformation Matrix, we derive the emergence of exactly 3D + 1 spacetime, the strict left-handed chiral orientation of gauge forces, the topological confinement of spin-1/2 quarks, and a native, geometric resolution to the Strong CP Problem (θ ≈ 0). The entire framework is anchored to the universal short exact sequence of Derived Differential Cohomology, proving that mass-energy and metric curvature are the inseparable dual payloads of a single, self-regulating holographic index engine.
Category: Mathematical Physics
[22] ai.viXra.org:2608.0058 [pdf] submitted on 2026-08-16 19:32:57
Authors: Pedro A. Kubitschek Homem de Carvalho
Comments: 9 Pages. (Note by ai.viXra.org Admin: Title and abstract MUST BE in English!)
This work begins with a simple question: could what we describe as translation and force actually be the local manifestation of a fundamentally angular dynamics? In the *Principium Geometricum* (PG), we investigate the possibility that the most fundamental physical description does not start with a body's trajectory through a pre-existing space, but rather with states of phase, rotation, closure, and causal updating. We introduce a normalized phase state q(ϕ) = (cos ϕ, sin ϕ), for which ∥ ˙q∥² = ω². In parallel, we consider the geometry of a torus, whose volume is VT = 2π²Ra². In the reduced geometric case where R = a = r, we have VT = 2π²r³. These two structures suggest the quantity IT ≡ VT 2π² ∥ ˙q∥² = r³ω². We propose investigating the conservation of IT for orbital states associated with the same global source. If IT is constant, Kepler's third law follows directly, as does—for the circular projection—an acceleration proportional to ru207b². The observational identification IT ↔ GM then recovers the Newtonian limit. This result does not constitute a microscopic derivation of gravity. The conservation of IT remains a physical hypothesis requiring further substantiation. The aim is to show, in an explicit and falsifiable manner, where the geometric identity ends, where the PG hypothesis enters, and how the classical limit is recovered.
Category: Mathematical Physics
[21] ai.viXra.org:2608.0049 [pdf] submitted on 2026-08-11 12:20:40
Authors: Lluis Eriksson
Comments: 9 pages, 1 figure. Exact d=5 inverse commutator cost: 12 Horn chambers, optimal-rank law, sharp tax [1,5/2], triangular synthesis, and reproducible rational certificates.
For a prescribed traceless Hermitian five-level target F, we determine exactly the least product of Hilbert-Schmidt norms of Hermitian A,B satisfying -i[A,B]=F. If g_i=lambda_i-lambda_{i+1} are the ordered spectral gaps, the answer is the maximum of six explicit rational linear forms and their reversals. Exact Littlewood-Richardson enumeration gives 142 Horn inequalities. Twelve sparse dual identities prove the lower facets, and twelve rational singular-spectrum maps attain them; exact cone certificates prove completeness without sampling. In contrast with the two qutrit facets and four four-level facets, dimension five has twelve exposed facets and open chambers that force rank four. For every nonzero target, the least rank of an optimal factor is max(n_+(F),n_-(F)). We also prove the sharp tax 1 <= kappa_5/(||F||_1/2) <= 5/2 with complete equality loci, and the exact balanced triangular perimeter S_2^2=12 sqrt(3) kappa_5. Two rational certificates and an independent deterministic LP campaign accompany the paper.
Category: Mathematical Physics
[20] ai.viXra.org:2608.0048 [pdf] submitted on 2026-08-11 13:00:56
Authors: Lluis Eriksson
Comments: 10 Pages.
Under an explicit noncontextual homogeneous-response ansatz for rank-one quantum effects, we study normalization on every unitary rotation of a fixed finite POVM. The orbit-normalization operator diagonalizes on the projective harmonics of complex projective space. We derive its weighted all-degree singular spectrum and show that the smallest non-Born spectral value is the sharp condition number controlling distance from the affine space of trace-one Hermitian quadratic responses. For dimensions at least three, the high-degree spectrum of a finite seed converges to the sum of squared aggregate ray weights; for an outcome-simple equal-trace seed with N outcomes, the limit is d squared divided by N. Combining this ceiling with the forced geometry of the two smallest tight frames solves the global equal-trace design problem in every dimension d at least five. An orthonormal-basis seed uniquely maximizes the all-degree Born-rigidity gap, with value d minus 6 divided by d plus 1, up to unitary equivalence and outcome relabeling. Every distinct-outcome overcomplete seed in the stated class is strictly worse. By contrast, every weighted complex projective 2-design has a degree-two nullspace and therefore zero rigidity gap. Thus measurements optimal for state tomography can be maximally non-rigid for this covariant normalization objective. The result concerns one-step response fields and does not derive state update, intrinsic randomness, or microscopic particle dynamics.
Category: Mathematical Physics
[19] ai.viXra.org:2608.0047 [pdf] submitted on 2026-08-11 13:47:42
Authors: Lluis Eriksson
Comments: 10 pages, 1 figure. Reproducible certificate and independent verifier are linked in the PDF.
A passive lossless network may route the same k-dimensional input channel to different output subspaces at different frequencies. Every two-frequency restriction can be cheap while the joint task is not. For L distinct boundary frequencies and mutually orthogonal prescribed output k-planes, we prove that the minimum McMillan degree among square finite rational-inner transfer matrices regular at the interpolation nodes is exactly k(L-1). Each pair alone requires degree k, so the largest two-node minimum underestimates the global memory by the unbounded factor L-1. The result is independent of node spacing. Necessity follows both from a zero budget for an analytic minor and from a Pick-Stein displacement identity whose negative inertia cannot exceed realization rank. Sufficiency follows from an explicit positive Pick completion. Beyond exact orthogonality, we derive a general cross-frequency inertia certificate, a gauge-independent span bound, an open robust region preserving the integer memory count, and a fail-closed noisy-eigenvalue certificate. Deterministic code tests arbitrary nodes, synthesizes minimal conservative realizations, and independently verifies the stated identities.
Category: Mathematical Physics
[18] ai.viXra.org:2608.0045 [pdf] submitted on 2026-08-11 15:57:16
Authors: Lluis Eriksson
Comments: 9 pages, 1 figure, 1 table. Exact generic memory law, colliding-target discontinuity, noisy projector certificate, and three-line phase diagram.
At L boundary frequencies, a passive lossless network must route one fixed k-dimensional input channel to prescribed output k-planes. For targets with joint span dimension r, every square finite rational-inner interpolant satisfies the sharp universal sandwich r-k <= d_min <= k(L-1). Direct-sum targets therefore have exact minimum McMillan degree k(L-1); when N >= Lk this maximal-memory locus is open and dense. The paper gives an explicit positive Pick completion, a realization- and inertia-based lower bound, a family of targets that collide while retaining maximal exact degree for every nonzero opening, an exact quadratic certification margin, and a fail-closed noisy projector test. For three scalar target lines it also proves the complete phase diagram: degree zero for one common line, degree one exactly for three distinct coplanar lines, and degree two otherwise. Deterministic certificates and an independent implementation accompany the proofs.
Category: Mathematical Physics
[17] ai.viXra.org:2608.0043 [pdf] submitted on 2026-08-12 01:48:33
Authors: Joseph Mcgreevy
Comments: 20 Pages. (Note by ai.viXra.org Admin: Please cite listed scientific references)
We formalize a unified geometric and representation-theoretic architecture bridging the continuous deformation of modular parameter spaces to the discrete, quantized structures of arithmetic number fields and local gauge theories. In the first part of this work, we present astructured verification of the Birch and Swinnerton-Dyer (BSD) Conjecture for Rank 2. By defining a stationary modular wave equation via the Maass-Laplacian, we isolate the weight-12 cusp form ∆ as an on-shell, massless zero-mode. At the bad reduction prime p = 691, this continuous flux degenerates against a singular cusp, collapsing the higher-dimensional geometric cycles of the 3-dimensional Kuga—Sato variety W2 onto the n = 0 neutral core of the Schmid weight filtration light cone. We construct the explicit 2×2 p-adic Syntomic Comparison Matrix, proving that its determinant is non-zero and non-degenerate under local-to-global stress at the fracture wall if and only if the p-adic N´eron—Tate height regulator is non-singular. In the second part, we document a precise homological parallel between this arithmetic localization and the structural stability constraints of quantum gauge fields. We demonstrate that the mechanism passing the E2 quasi-modular anomaly through higher-weight Eisenstein series mirrors the scaling of a gauge connection protecting a positive-definite kinetic energy landscape. This continuous propagation forces a compact Lie group topology, which shatters at the 691 boundary into an 11-dimensional coefficient space—casting a direct arithmetic shadow onto the χ 11 odd eigenspace of the cyclotomic minus class group via the Herbrand—Ribet theorem. We present this unified framework not as a physical identity, but as a rigid representation-theoretic blueprint governing Rank-2 root system degenerations at the boundary of a continuum.
Category: Mathematical Physics
[16] ai.viXra.org:2608.0033 [pdf] submitted on 2026-08-10 19:28:35
Authors: Lluis Eriksson
Comments: 11 pages, 1 figure. Exact action-memory theorem with explicit compiler and independent verification. Code: github.com/lluiseriksson/finite-sample-spectral-certificates
A passive lossless network pays two different costs when it rotates a signal subspace over a frequency arc: a continuous Wigner—Smith action and an integer McMillan memory. Known Grassmannian length bounds and rational-inner degree identities constrain these resources separately. For the stated finite rational-inner class, we determine their joint attainable region exactly. Let two rank-k subspaces have principal angles βj, put B = Σj βj and let r be the number of nonzero angles. For a square finite rational inner transfer matrix of degree at most d, define the arc action A = ∫I tr Q(θ)dθ, where Q = −iS*∂θS is positive semidefinite. If B > 0, the exact feasible set is d ≥ r and 2B ≤ A ≤ 2πd − 2B. Both faces are attained. Equivalently, dmin(A) = max{r, ⌈(A + 2B)/(2π)⌉}. The upper face is a return cost: the complementary arc must rotate the subspace back, while the full-circle trace action is exactly 2π times the degree. We prove sufficiency by an explicit compiler of normalized rank-one Blaschke—Potapov gates; it realizes every interior point and both faces, including rank-deficient and orthogonal cases. When the endpoint subspaces coincide, the region changes discontinuously to A ∈ [0, 2πd), with the upper endpoint open. We derive fail-closed noisy degree certificates and give a multi-frequency warning using classical boundary Nevanlinna—Pick theory: three pairwise degree-one routing tasks can require degree two jointly. For orthogonal-line data the obstruction is a frame-independent cycle phase, exhibited by an exact positive semidefinite Pick completion with spectrum (2,1,0). Deterministic code compiles random points of the diamond, constructs the degree-two colligation, and is replayed by an independent verifier.
Category: Mathematical Physics
[15] ai.viXra.org:2608.0020 [pdf] submitted on 2026-08-05 18:09:44
Authors: Lluis Eriksson
Comments: 17 pages. Lean 4 formalization. AI-assisted with OpenAI Codex; all Lean builds and axiom-oracle checks were verified by the author. Source and validation evidence are linked in the PDF.
We present a Lean 4 formalization of a thermodynamic-limit construction for the anisotropic nearest-neighbour Ising model on the two-dimensional integer lattice. A telescoping comparison argument and the classical Dobrushin resolvent produce a volume-uniform expectation bound. Exact restriction and reindexing maps between finite Gibbs measures yield convergence of the complete free-boundary sequence, independence of auxiliary envelopes and cofinal samplings, stability under receding boundary perturbations, and equality of free and periodic limits.Finite-support cylinder presentations are quotiented by equality of their represented functions on the full spin space. The resulting local algebra carries a genuine lattice-translation action, and the limiting functional is positive, normalized, real-linear, and invariant under every integer translation. We equip this algebra with its intrinsic uniform norm, construct its complex star-algebra representation, take the corresponding commutative C*-closure, and extend the limiting functional to a positive complex-linear functional of norm one. We also formalize normalized Gibbs conditional kernels for arbitrary finite conditioning sets and prove positivity, exterior locality, idempotence, and the exact finite-volume Gibbs tower identity.All results remain within the classical anisotropic Dobrushin region. The infinite-volume DLR fixed-point equation for the completed state is not claimed.
Category: Mathematical Physics
[14] ai.viXra.org:2608.0019 [pdf] submitted on 2026-08-05 18:19:22
Authors: Lluis Eriksson
Comments: 31 pages, 1 figure. Companion reproducibility archive containing complete LaTeX source, numerical verification code, data and SHA-256 manifest.
We determine when rapid exact maintenance of a rank-deficient quantum target produces logarithmically divergent free-energy restoration power in algebraic quantum field theory. On every sigma-finite properly infinite factor we construct a bounded quantum Markov semigroup, a nonfaithful normal target and a faithful invariant reference state for which Araki relative entropy reduces exactly to a binary divergence. We derive the semigroup first from localized thermal fermionic probe collisions and then from one autonomous finite-bandwidth Dirac KMS reservoir with fixed smooth coupling. Exact memory equations yield an explicit finite-coupling Davies bound on finite van Hove windows. For a displayed compactly supported massless Dirac form factor, Araki—Wyss regularity, threshold behaviour and Fermi-golden-rule positivity give a completely bounded Davies approximation uniformly for all times with error of order (O(|lambda|)); the sharper (O(lambda^2)) result is isolated under additional reduced-resonance hypotheses. We construct a background-covariant two-Dirac-field completion using Green operators, Møller maps and relative Cauchy scattering, proving naturality, causal factorization and exact spacelike triviality. A locality obstruction shows why a strictly local multiplier cannot coincide exactly with the solvable rank-one reservoir coupling, while a Feshbach reduction quantifies the correction. Finally, we prove that no fixed faithful vacuum or KMS restriction can exhibit the rank-boundary mechanism, but faithful families with a vanishing spectral floor recover its complete coefficient. The results separate algebra type, target faithfulness, microscopic realizability and regulator uniformity, and provide reproducible numerical audits of the finite-dimensional identities and explicit Dirac form factor.
Category: Mathematical Physics
[13] ai.viXra.org:2608.0018 [pdf] submitted on 2026-08-05 21:39:01
Authors: Lluis Eriksson
Comments: 13 Pages. All results are machine-checked in Lean 4 (no sorry, no project axioms; headlines depend on exactly [propext, Classical.choice, Quot.sound]).
For the spatial Z_2 (Ising-slice) system inside the Dobrushin window 2 tanh|beta| + 2 tanh|gamma| <= alpha < 1, we machine-check in Lean 4 an end-to-end chain from the Gibbs measure to the spectrum of the reconstructed transfer operator. (i) The Osterwalder-Schrader (site-form) reconstruction of the transfer operator is unitarily conjugate, by the explicit sqrt(w) boundary dressing, to the symmetrised Dobrushin kernel. (ii) The unnormalised Gibbs sums themselves are exact matrix elements of that operator's powers: gibbsPathSum(w,beta,N,A,B) = lambda^N
Category: Mathematical Physics
[12] ai.viXra.org:2608.0016 [pdf] submitted on 2026-08-04 19:14:39
Authors: Lluis Eriksson
Comments: 9 pages. New companion obstruction paper to ai.viXra:2607.0089, not a replacement. Diagnostic code and reproducibility record: https://github.com/lluiseriksson/THE-ERIKSSON-PROGRAM
For beta > 0 let I_m = I_m(beta) denote the modified Bessel functionof the first kind and define a_m = I_m^2 ((m-1) I_(m-1)^2 + (m+1) I_(m+1)^2), b_m = m I_m^4,with sine series F_A(t) = sum_(m>=1) a_m sin(mt) andF_B(t) = sum_(m>=1) b_m sin(mt). The associated Wronskian is negativeexactly when F_A/F_B is decreasing. We isolate what can and cannot beproved from two natural inputs: the Neumann convolution kernelI_0(2 beta sin(phi/2)) and the small-coupling anchor whose normalizedlimit is 4 sin^3(t).First, the convolution does prove F_B(t) > 0 for 0 < t < pi. Second,the endpoint is governed by two alternating quantities, c_3 and B_pi,through an exact cubic law. We prove B_pi > 0, derive integralrepresentations, and establish that the cancellation lost by replacingthe alternating quantities with positive-term majorants has exponentialrate 8 - 4 sqrt(2). Finally, we prove a smooth one-parameter perturbationtheorem: one may keep F_B and its kernel unchanged, preserve positivityand strict coefficient-ratio ordering, and preserve every jet at beta = 0,while choosing either sign of the endpoint cubic coefficient.Consequently those structural data, even taken together, do not implyglobal Wronskian negativity. This is a no-go theorem for a proofarchitecture, not a counterexample to the original Bessel conjecture.A short high-precision kill-test accompanies the paper; nocomputer-assisted inequality is used in the proofs.
Category: Mathematical Physics
[11] ai.viXra.org:2608.0015 [pdf] submitted on 2026-08-04 19:42:27
Authors: Lluis Eriksson
Comments: 13 Pages.
We study whether a flat lattice-gauge Hodge form supplemented by a penalty depending only on a line-integral block variable can control the full fine one-cochain norm uniformly in the block side. The constant-field sector selects the exponent (d-2)/2 only under exact scale neutrality, but this scaling test does not address the kernel of the block map. We construct transverse, divergence-free, block-periodic Fourier cochains that are annihilated exactly by the block map. Their first-mode Hodge Rayleigh quotient is 4 sin²(pi/L), forcing every admissible full-space Poincaré constant to grow at least quadratically with L. The obstruction applies to every scale-dependent scalar functional factored through the block variable and satisfying only that it vanishes at the coarse zero field; no linearity, continuity, positivity, or growth condition is required. For linear postconditioners, higher Fourier frequencies yield an orthogonal real subspace of dimension 2R and a min—max bound on the 2R-th eigenvalue, establishing a growing low-energy spectral cluster. We also derive an exact repair identity and a necessary witness-channel budget for modified block measurements and additional fine-space terms. The kernel construction, Hodge energy, factorized no-go theorem, repair identity, and necessary repair budget are formalized in Lean 4. The result concerns a finite periodic flat full-domain form and makes no claim about interacting coercivity, gauge quotients, continuum limits, or the Yang—Mills mass gap.
Category: Mathematical Physics
[10] ai.viXra.org:2608.0013 [pdf] replaced on 2026-08-04 10:13:23
Authors: Lluis Eriksson
Comments: 25 pages. Lean 4, no sorry, no project axiom. Twelve machine-checked theorems: positive weight to exponential decay, and through an abstract transport theorem to a volume-uniform operator gap.
A formal Lean 4 development revisits a two-dimensional spatial transfer kernel whose earlier uniform spectral argument relied on constant row sums. We show that the loss of row-sum constancy obstructs that method, not the conclusion. With no sorry and no project axiom, we prove twelve machine-checked theorems.The chain establishes the sharp field-uniform one-bond influence envelope tanh J; assembles these bounds into a finite-volume Dobrushin matrix; proves a volume-independent resolvent estimate under row-sum bounds alpha < 1 without assuming constant rows; mechanises Dobrushin's comparison inequality with the attained 1/4 covariance constant; constructs Gibbs measures, heat-bath kernels and intrinsic influence matrices from arbitrary strictly positive finite weights; and specialises to anisotropic Ising interactions. For L x T rectangles with free boundary, the condition 2 tanh|beta| + 2 tanh|gamma| <= alpha < 1 yields exponential decay of correlations with beta, gamma, alpha and the prefactor fixed before the volume quantifiers.The transport into the operator formulation is closed: an exact finite band identity relating endpoint covariances to matrix elements; an abstract theorem showing that a common exponential decay rate for band covariances - a hypothesis on finite path measures, carrying no operator, norm or spectrum - implies a uniform positive gap for a family of projected transfer operators; a Perron-boundary tilt identity that preserves the decay rate while absorbing boundary costs into extent-dependent constants; an exact currying identification of the free strip measure with the rectangle Ising measure; and the resulting corollary: inside the window there is one m > 0 bounding the projected transfer operator of the coupled kernel's normalised Perron data by exp(-m) at every extent, with m = -log alpha.Numerical measurements at L <= 12 provide counterevidence to attributing spectral degeneracy solely to nonzero spatial coupling. No infinite-volume state, thermodynamic limit or boundary-condition independence is constructed; the window is sufficient, not sharp. The underlying Dobrushin mathematics is classical; the contribution is a non-vacuous, reproducible mechanisation and composition of the full chain, ending at a volume-uniform operator gap. No consequence for Yang-Mills theory is claimed.
Category: Mathematical Physics
[9] ai.viXra.org:2608.0010 [pdf] submitted on 2026-08-02 03:33:12
Authors: Lluis Eriksson
Comments: 8 pages, 2 tables, 1 figure. Lean 4.29.0-rc6; Mathlib 07642720480157414db592fa85b626dafb71355b. Clean rebuild passed. 25 public declarations; no sorry, admit, or local axiom. New sequel, not replacement; four-face crossing not claimed.
We give a kernel-checked positive-area calculus for the concrete SU(2) classheat kernel used in two-dimensional Yang—Mills theory. With irreducible labeln, dimension n+1, and Casimir c_n=n(n+2)/4, Lean verifies at every positivetime and every derivative order that the infinite spectral jet converges anddifferentiates term by term to the next jet. The hierarchy is packaged as aC-infinity map on the positive-time half-line, using explicit uniform summablemajorants on positive-time neighbourhoods. We then differentiate the literalnormalized-Haar two-face Migdal integral, prove that its left and right areaderivatives equal the first spectral jet at the merged area, and showinfinitesimal invariance under (s,t) -> (s+u,t-u). Finally, every normalizedWilson character satisfies its exact Casimir area ODE as an actual Haarintegral against the infinite heat kernel. The artifact contains 25 publicdefinitions and theorems, no local placeholders, and audited dependencies onlyon propext, Classical.choice, and Quot.sound. We do not claim the four-faceMakeenko—Migdal crossing equation; the remaining inputs are local Lie-groupintegration by parts and certified crossing geometry.
Category: Mathematical Physics
[8] ai.viXra.org:2608.0009 [pdf] submitted on 2026-08-02 07:14:51
Authors: Lluis Eriksson
Comments: 6 pages, 2 tables, 1 figure. Lean 4.29.0-rc6; Mathlib commit 07642720480157414db592fa85b626dafb71355b. Clean-source build and audit passed. 30 public declarations; no sorry, admit, or local axiom.
We formalize the finite-dimensional integration-by-parts mechanism used inlocal proofs of the Makeenko—Migdal equation. For a measure-preserving realflow on a finite measure space, Lean verifies differentiation under theintegral from an explicit locally uniform integrable majorant and proves thatthe integral of the generator vanishes. A bounded dominated product interfacethen yields integration by parts, and two successive applications transfer amixed pair of generators from a density to an observable with positive sign.These results are instantiated on normalized Haar probability measure ofconcrete SU(2) and on every finite product SU(2)^E, for left and rightmultiplication of one selected edge. To fix the representation normalization,we construct three explicit trigonometric curves in SU(2) and prove entrywisethat their tangents at the identity are i sigma_1/2, i sigma_2/2, and isigma_3/2. The producer contains 30 public definitions, structures, andtheorems in 519 physical lines; its audit contains no local sorry, admit, oraxiom, and the headline results depend only on propext, Classical.choice, andQuot.sound. This closes the Haar integration-by-parts layer. It does not claimthe full four-area Makeenko—Migdal identity: the remaining formal inputs arethe heat-density directional identity, extended gauge invariance at acrossing, and their geometric assembly.
Category: Mathematical Physics
[7] ai.viXra.org:2608.0008 [pdf] submitted on 2026-08-02 11:55:22
Authors: Lluis Eriksson
Comments: 18 pages. All new lemmas machine-checked in Lean 4 (pinned Mathlib); the supremum theorem is conditional, carrying the Birkhoff spectral bound as an explicit hypothesis, not an axiom. Source: github.com/lluiseriksson/THE-ERIKSSON-PROGRAMME
A positive site weight acts on a transfer kernel by congruence, K -> DKD with Dpositive diagonal, not by similarity. Similarity preserves the entire spectrum;congruence preserves strictly less, and strictly more than nothing. We determineboth halves for the kernel of L decoupled Ising bonds. Rigid half: the sign ofevery quadratic form value survives, so definiteness in either sign is acongruence invariant -- the definite case of Sylvester's law of inertia, in aform a machine can check without diagonalisation. Fragile half: the subdominantratio r (second eigenvalue modulus over the Perron root) does not survive atall, and we locate exactly how far it moves. Restricting the L-site kernel tothe two antipodal configurations leaves one Ising bond of coupling bL, of ratiotanh(bL) for b>0: a weight concentrating there fuses the L sites into a singleeffective site carrying L times the coupling. Since tanh(bL) -> 1, no boundr <= rho < 1 holds simultaneously in L and over the whole positive-diagonalcongruence orbit; the obstruction to volume-uniformity is a property of thecongruence, not of any spectral estimate, so an argument bounding r throughcongruence invariants alone cannot produce an L-uniform bound. For arbitraryweights we prove sup_{D>0} r(DMD) = (1-m)/(1+m), where m is the leastoff-diagonal entry: the least correlated pair determines the supremum. (Weevaluate the supremum; we do not classify its maximisers.) The proof isgeometric rather than spectral -- Hilbert's projective diameter is itself acongruence invariant, and Birkhoff's contraction theorem converts it into thebound -- and uses no definiteness, only positivity of the entries. The lowerbound uses no limiting argument about spectra: a two-supported fluctuationvector gives the estimate at each strictly positive epsilon, so no continuity ofeigenvalues is imported anywhere.
Category: Mathematical Physics
[6] ai.viXra.org:2608.0007 [pdf] submitted on 2026-08-02 14:02:27
Authors: Lluis Eriksson
Comments: 6 pages, 2 tables, 1 figure. Lean 4.29.0-rc6; Mathlib commit 07642720480157414db592fa85b626dafb71355b. Clean-source build and audit passed. 19 public declarations; no sorry, admit, or local axiom.
We formalize a finite-dimensional crossing Ward identity for fundamental SU(2) Wilson words. On every finite edge space SU(2)^E, two distinct coordinates are inserted into a concrete normalized trace word. Lean proves entrywise that right multiplication by each of three explicit SU(2) curves produces the normalized Pauli generators i sigma_a/2, differentiates the trace word once at each selected coordinate, and identifies the Pauli-summed mixed derivative with the rank-two Fierz contraction. Two applications of Haar integration by parts transfer the corresponding mixed generators from a density to the Wilson word. The resulting integral closes exactly on the direct and reverse single-trace resolutions, with coefficients -1/4 and -1/2 in the chosen normalization. The producer contains 19 public declarations in 487 physical lines; all 13 new theorems depend only on propext, Classical.choice, and Quot.sound, with no local sorry, admit, or axiom. This is not a full Makeenko--Migdal area equation: the remaining physical input is a weak four-face identity against crossing-certified extended-gauge-invariant observables. In the program's heat time, generated by the Pauli Laplacian, its coefficient is kappa = 2.
Category: Mathematical Physics
[5] ai.viXra.org:2608.0006 [pdf] submitted on 2026-08-02 16:46:30
Authors: Lluis Eriksson
Comments: 6 pages, 2 tables, 2 figures. Lean 4.29.0-rc6; Mathlib commit 07642720480157414db592fa85b626dafb71355b. Clean-source build and audit passed. 56 public declarations; no sorry, admit or local axiom.
We formalize extended gauge invariance at a simple four-edge SU(2) crossing and connect the geometric edge chart to the two-coordinate chart used by a machine-checked crossing Ward identity. On SU(2)^4 we define the two opposite-edge right actions from the abstract Makeenko--Migdal theorem, prove that they are commuting product-Haar-preserving actions, and show that their common parameter composes to ordinary vertex gauge invariance. The four-edge Wilson word tr2(a3^-1 beta a2 a4^-1 alpha a1) is proved invariant under both half-actions. We construct the explicit quotient r(a)=(a2 a4^-1,a1 a3^-1), a canonical section, and prove existence and uniqueness of the universal factorization for every extended-gauge-invariant complex function. The complete map from the cyclic four-edge chart to physical and gauge coordinates is proved to preserve literal four-fold Haar measure in one public endpoint. Finally, the four-edge Wilson word is identified exactly with the prior two-coordinate crossing word evaluated on r(a). The Lean producer has 56 public declarations in 488 physical lines; all 36 theorems depend only on propext, Classical.choice, and Quot.sound, with no local proof escape. No heat-kernel area derivative or full Makeenko--Migdal equation is claimed.
Category: Mathematical Physics
[4] ai.viXra.org:2608.0005 [pdf] submitted on 2026-08-02 19:16:06
Authors: Lluis Eriksson
Comments: 7 pages, 2 tables, 2 figures. Lean 4.29.0-rc6; Mathlib commit 07642720480157414db592fa85b626dafb71355b. Clean-source build and audit passed. 26 public declarations; no sorry, admit or local axiom.
We machine-check the exact measure and differential bridge between thefour-edge SU(2) crossing chart and the two effective group coordinates used bya finite-dimensional Ward identity. The quotientr(a)=(a2 a4^-1,a1 a3^-1) pushes normalized four-fold Haar measure exactly tonormalized two-fold Haar measure. Two gauge-compensated flows on the originalfour edges intertwine with independent right multiplication of the quotientcoordinates, so first and mixed derivatives of the crossing Wilson worddescend without choosing a gauge. The three Pauli directions are verifiedindividually, their mixed generators close into direct and reverseresolutions, and the two-coordinate Ward theorem lifts to a literal four-edgeintegral with coefficients -1/4 and -1/2. This compensated operator is notidentified with the ordinary-edge mixed operator of Driver-Hall-Kemp: thepaper writes both operators and their distinct reverse resolutions and markstheir comparison as open. The Lean producer has 26 public declarations and 16audited theorems. No weak four-face heat-kernel identity, area derivative, orfull Makeenko--Migdal equation is claimed.
Category: Mathematical Physics
[3] ai.viXra.org:2608.0004 [pdf] submitted on 2026-08-02 19:25:02
Authors: Lluis Eriksson
Comments: 8 pages. Lean 4 formalization with pinned Mathlib. Nine headline declarations were audited and depend only on propext, Classical.choice, and Quot.sound; no project axioms or sorryAx. Formal source frozen at commit f21539ed0bb880a04078de369bf5cbf063f7b101.
Let a fine periodic lattice have side LN' and let Q_L be the L^{-d}-normalized block average of length-L line integrals. In four dimensions, the rescaling Q_L -> LQ_L repairs the elementary constant-field scaling obstruction to a Poincare estimate. We prove that it cannot yield coercivity on the full one-cochain space with a constant uniform in the block side.For every L >= 2, every fixed N' >= 1, and N_c >= 2, we embed the first within-block Fourier phase zeta_L = exp(2 pi i/L) in a real two-plane of the internal coordinate space and construct a transverse one-cochain A_L. It satisfies Q_L A_L = 0 and div A_L = 0. For every dimension d >= 2,||A_L||^2 = (LN')^d,
Category: Mathematical Physics
[2] ai.viXra.org:2608.0003 [pdf] submitted on 2026-08-01 23:42:12
Authors: J. W. McGreevy
Comments: 17 Pages. (Note by ai.viXra.org Admin: Please cite and list scientific references)
We develop a symplectic geometry of atomic shear measures supported on the irregular primes of a modular curve. Starting from a space of dis-crete measures subject to a global valence constraint, we perform a strict symplectic reduction and obtain a reduced phase space equipped with a canonical Darboux form. The local coordinate functions that extractthe individual shear amplitudes are shown to be in Liouville involution, generating a completely integrable system whose invariant level sets areLagrangian tori. By lifting the functional equation of the associated automorphic L-functions to an anti-symplectic involution on this phase space, and under the established essential self-adjointness of the clutched conical Dirac operator, we prove a Confinement Theorem: the only invariant Lagrangianleaves compatible with the reflection symmetry are those whose spectralimage lies on the critical line Re(s) = 1/2. We conclude with a brief, explicitly programmatic dictionary that explores possible links between the resulting arithmetic geometry and structures appearing in gauge theory and the Standard Model. This workextends earlier constructions developed under the working title Relativistic Field Theory of Primes (RFTP), in which the irregular primes, the weight-12 valence constraint, and the associated clutching data were first introduced as geometric ingredients of an arithmetic field theory.
Category: Mathematical Physics
[1] ai.viXra.org:2608.0001 [pdf] submitted on 2026-08-01 18:50:53
Authors: Lluis Eriksson
Comments: 9 Pages. Lean 4/Mathlib formalization of the finite-SU(2) Pauli/Casimir contraction, trace-skein identity, four oriented local branches with both reconnections, corrected single-crossing closure, and an all-order multitrace-to-single-trace recursion.
Finite-rank Makeenko-Migdal equations generate products of Wilson traces at self-intersections. For SU(2), this apparent multitrace obstruction closes exactly on single traces, but the statement is normalization-sensitive: the traceless Lie algebra contributes a finite-rank correction that disappears for U(2) and must not be dropped. We give a Lean 4/Mathlib formalization of the complete group-algebraic closure mechanism on Mathlib's concrete special unitary matrix group. With normalized trace tau(A)=Tr(A)/2 and normalized anti-Hermitian Pauli directions X_j=i sigma_j/2, the kernel checks the Casimir identity, the rank-two Fierz identity, the induced crossing contraction, and the SU(2) trace-skein identity tau(g)tau(h)=(tau(gh)+tau(gh^{-1}))/2. Consequently, the finite-SU(2) crossing term tau(g)tau(h)-tau(gh)/4 equals tau(gh)/4+tau(gh^{-1})/2. We then formalize a universal local interface with four cyclically ordered branch holonomies, an independent orientation on each branch, the two opposite-strand words, and precisely the two direct/reversed reconnections. Its corrected crossing term closes on those reconnections for every branch assignment and orientation choice. A recursive theorem also extends the reduction to products of arbitrarily many fundamental traces. The identities are classical; the contribution is a concrete, kernel-checked normalization bridge from Pauli contraction to the single-trace closure used in finite-rank loop equations. We do not claim a formal derivation of the Yang-Mills area derivative, planar loop geometry, or the full Makeenko-Migdal equation.
Category: Mathematical Physics