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[30] ai.viXra.org:2602.0117 [pdf] replaced on 2026-02-27 02:16:52
Authors: Lluis Eriksson
Comments: 33 Pages.
This document is an experiment-first audit report for a companion-paper programme claiming a constructive solution of the 4D $mathrm{SU}(N)$ Yang—Mills existence and mass gap problem. It specifies a runnable mechanical audit suite of 29 deterministic tests, defines pass/fail criteria, and presents outputs in a compilation-safe format. The report contains: (i) an explicit non-triviality proof showing the Wightman functions do not factorize trivially; (ii) a toy-model validation recovering the exact 2D $mathrm{SU}(2)$ Yang—Mills mass gap to machine precision; (iii) a Bałaban bridge appendix reproducing the critical inductive step of his renormalization group in simplified form; (iv) a reproducibility repository with 3-line setup instructions; (v) a core proof chain audit mechanically verifying the load-bearing theorems of Papers 86—90, covering terminal Kotecký—Preiss convergence, UV suppression, one-dimensionality of the anisotropic sector, Cauchy bounds on polymer jets, the OS1 vanishing rate $O(eta^2 log eta^{-1})$, Lie-algebra annihilation, and KP margin sensitivity. Beyond the 17 core tests, the suite includes a lattice gauge proxy layer (plaquette expansion, Polyakov-loop centre symmetry, Creutz ratio; 3 tests), an infrastructure layer (Bakry—Émery curvature seed $mathrm{Ric}_{mathrm{SU}(N)} = N/4$, the $2^{4k}$ cancellation in $d=4$, heat-kernel column bound; 3 tests), a UV-flow/heat-kernel layer (Parseval identity, diagonal decay exponent $d/2 = 2$, flow—reflection commutation; 3 tests), a non-triviality test (Haar Monte Carlo on $mathrm{SU}(2)$ and $mathrm{SU}(3)$; 1 test), a toy-model validation (2D Yang—Mills transfer matrix; 1 test), and an algebraic QFT layer (Petz recovery fidelity bound $1-F leq C,e^{-2mr}$ from the Split Property; 1 test). All 29 tests pass; the full suite completes in ${approx}70,mathrm{s}$ on a Google Colab CPU. The complete inter-paper dependency DAG is acyclic and explicitly recorded. All code, data, and artifacts are available at https://github.com/lluiseriksson/ym-audit. The companion papers are archived at https://ai.vixra.org/author/lluis_eriksson.
Category: Mathematical Physics
[29] ai.viXra.org:2602.0113 [pdf] submitted on 2026-02-24 08:01:09
Authors: Steven E. Elliott
Comments: 6 Pages.
Standard physics contains formal contradictions when judged as self-consistent physical ontologies. The Einstein Equivalence Principle (EEP) embeds ε—δ processes requiring internal laboratory realization, yet General Relativity’s dynamics destroy all realizers in finite time -empty spacetime (no labs exist), few-body systems (radiation erosion), or cosmological evolution (de Sitter horizons). MATHICCS (Mathematics + Physics + Computational Consistency Substrate)—a higher-order meta-logic—deems axioms whose mathematical processes lose internal persistence invalid for physical ontology. GR asserts EEP-validity while deriving EEP-invalidity, yielding P ∧¬P . The first MATHICCS-valid ontology is the Fractal Substrate Equivalence Physics (FSEP) [viXra:2602.0107], where eternal Apollonian boundary dynamics persist across infinite recursive scales via Möbius inversion, discrete scale flips (r 7→ r/λ), and angular-momentum conservation. FSEP derives Newtonian gravity and inverse-square law from local quadratic expansion of spherical inversion; constant finite light speed from linear term + pole ejection + scale compression; dark matter fraction (≈ 84%) (baryons ≈ 16%) as geometric series from 3D Apollonian fractal dimension (D ≈ 2.473946 [1]) and radius ratio (β ≈ 0.72); and fine structure constant (α ≈ 1/137.035999 [3]) emergently from bipolar pole aperture geometry (λlocal ≈ 21.81), unifying it with observed quasar jet collimation angles (θjet ≈ 5.2◦ [4]);—all parameter-free except the geometric self-consistency of the persistent substrate. MATHICCS demands all physics reconstruct its mathematics from within via persistent internal processes. GR explodes; FSEP survives.
Category: Mathematical Physics
[28] ai.viXra.org:2602.0097 [pdf] submitted on 2026-02-20 18:28:06
Authors: Vinicius F. S. Santos
Comments: 19 Pages.
We introduce the Secular Replicator Flow, a finite-dimensional algebraic dynamical system inspired by the turbulent energy cascade of the Navier—Stokes equations, built from the spectral theory of golden resolvent operators on discrete network graphs [9]. The continuous mechanics of fluid turbulence—incompressibility, nonlocal pressure, nonlinear advection, and viscous dissipation—find precise algebraic counterparts in the constraints of a replicator equation evolving on the simplex of spectral participation weights, governed by a global secular equation. Within this framework we establish three principal results. First, the Variance Law: the macroscopic coupled eigenvalue λ∗(t) evolves monotonically according to Fisher’s Fundamental Theorem, acting as a strict Lyapunov function (between excision events) whose rate of increase equals the fitness variance of the active spectrum. Second, the Spectral Selection Theorem: the fitness landscape is a strict bipolar Ushape in the base eigenvalue μ, guaranteeing that the replicator flow annihilates mid-spectrum noise and funnels all energy into the extreme macroscopic topologies of the network. Third, Global Regularity: as the system approaches a structural resonance (transparent pole), the fitness plunges to −∞, triggering an auto-excision mechanism that exponentially starves the dangerous channel, rendering every pole singularity removable. The resulting dynamics form a Sawtooth Cascade of smooth climbs interrupted by discontinuous structural snaps whose direction is controlled by the residual load of the excised channel. We classify the sole remaining failure mode as a thermodynamic phase escape at the r = 2 Chebyshev boundary, where the discrete algebraic structure of the network undergoes a global phase transition into unbounded hyperbolic space—a phenomenon fundamentally different from the localised velocity blowup sought by PDE analysis. All regularity results herein apply to this model; implications for the full Navier—Stokes equations in R3 remain open.
Category: Mathematical Physics
[27] ai.viXra.org:2602.0096 [pdf] submitted on 2026-02-20 06:04:55
Authors: Lluis Eriksson
Comments: 21 Pages.
This paper is a hostile-review navigation guide and audit manifesto for a companion-paper programme claiming a constructive solution of the four-dimensional SU(N) Yang-Mills existence and mass gap problem in the Osterwalder-Schrader (OS) framework, reconstructed as a Poincare-covariant Wightman QFT with strictly positive mass gap. The guide provides: (i) an explicit dependency graph and Clay/Jaffe-Witten checklist; (ii) an explicit threat model listing standard failure modes targeted by hostile review (black-box dependence on Balaban, interface friction between gradient flow and the Balaban measure, diagonal-limit non-uniformity, and operator-mixing residues); (iii) an explicit four-pillar defensive architecture resolving each attack with structural (not merely quantitative) shields; (iv) the preventive lock: a triangular renormalization-mixing structure that blocks upward anisotropic flow into the marginal (d=4) sector, neutralizing the standard a^2 x a^{-2} -> O(1) objection; (v) a mechanical audit trail mapping load-bearing hypotheses to primary sources; and (vi) a complete linked index of all supporting preprints for traceability. External mathematics is explicitly declared: abstract polymer cluster expansion (Kotecky-Preiss), OS reconstruction (Osterwalder-Schrader), and lattice reflection positivity (Osterwalder-Seiler). Furthermore, this guide introduces the Triangular Mixing Preventive Lock: a structural algebraic mechanism showing that the operator mixing matrix has no anisotropic marginal d=4 sink in the gauge-invariant W_4-scalar sector. Consequently, the standard O(a^2) x O(a^{-2}) -> O(1) operator-mixing residue attack is blocked structurally: any quadratic divergence is forced to renormalize only O(4)-invariant d=4 data (the isotropic coupling), leaving the O(4)-breaking channel suppressed. This paper is not a claim of institutional validation; it is an audit map prescribing the check order and falsification points for the companion-paper chain.
Category: Mathematical Physics
[26] ai.viXra.org:2602.0095 [pdf] submitted on 2026-02-19 19:48:12
Authors: J. W. McGreevy
Comments: 16 Pages.
We present Arithmetic Relativistic Emergence (ARE) as a "General Relativity of Numbers" — a framework in which the Standard Model, quantum mechanics, classical 3+1 Lorentzian spacetime, and fundamental constants emerge tautologically from the arithmetic geometry of Q. The Riemann zeta function ζ(s) constitutes the maximally symmetric pregeometric vacuum. Its functional-equation symmetry around Re(s) = 1/2, combined with the pole at s = 1, forces spontaneous symmetry breaking via the weight-12 modular discriminant ∆(τ ) = η(τ )24 at the s = 6 harmonic threshold. This breaking disperses the vacuum into Archimedean divergence (Fdiv, smooth curvature density) and non-Archimedean curl (Hcurl, discrete torsion at p-adic fibers). The emergent geometry is governed by Arakelov curvature on the arithmetic surface Spec Z ∪ {∞},where Weierstrass weights act as "mass/energy density" (algebraic rigidity) and the hyperbolic/Bergman metric plays the role of spacetime. Modular transformations toward cusps correspond to Lorentz rapidity, yielding an equivalence principle analogue between inertial (modular flow resistance) and gravitational (metric warping) responses. The adelic spectral triple (KO-dimension 6, finite algebra C ⊕ H ⊕ M3(C)) induces symplectic deformation of phase space, with the non-trivial zeros of zeta providing the Dirac spectrum (Hilbert—Pólya realized). The Minkowski interval ds2 = −c2dt2 + du20d7x 2 emerges as the unique adelic-invariant quadratic form, with light cone as the resolved cusp boundary (holographic screen).The spectral action Tr f (D/Λ) recovers Einstein—Cartan gravity with non-Abelian Yang—Mills, where generalized Rainich conditions (quadratic invariants involving structure constants f abc) are satisfied at s = 6, with torsion (Hcurl) regularizing self-interactions. The full SM gauge group SU(3)c × SU(2)L × U(1)Y and three chiral generations emerge fromadelic place ramification and Leech lattice Z2-orbifold. Constants (α−1 ≈ 137 from Petersson + torsion residues, ℏ from Lambert-Planck suppression, G from unification suppression, Λ ∼ e−288) are inevitable invariants. Langlandsfunctoriality acts as the holographic dictionary mapping prime rigidity to bulk physics. ARE thus unifies physics as the macroscopic shadow of arithmetic rigidity, with the Riemann Hypothesis as a necessary stability condition for the emergent universe.
Category: Mathematical Physics
[25] ai.viXra.org:2602.0092 [pdf] submitted on 2026-02-19 12:13:41
Authors: Lluis Eriksson
Comments: 9 Pages.
We complete the rigorous construction of four-dimensional Euclidean SU(N) Yang--Mills quantum field theory and establish the existence of a mass gap. Building on the companion papers -- which unconditionally establish exponential clustering with mass gap, the Osterwalder--Schrader axioms OS0, OS2, OS3, OS4, and quantitative irrelevance of O(4)-breaking lattice operators -- we derive a lattice Ward identity for infinitesimal Euclidean rotations, identify the breaking term as a dimension-6 anisotropic operator insertion, and prove that the breaking distribution vanishes as $O(eta^2,|log((Lambda_{mathrm{YM}}eta)^{-1})|) to 0$ in the continuum limit, establishing axiom OS1 (full O(4) Euclidean covariance). Combined with the Osterwalder--Schrader reconstruction theorem, this yields a non-trivial Poincare-covariant Wightman quantum field theory with mass gap $Delta_{mathrm{phys}} geq c_N,Lambda_{mathrm{YM}} > 0$ for each $N geq 2$.
Category: Mathematical Physics
[24] ai.viXra.org:2602.0091 [pdf] submitted on 2026-02-19 12:16:47
Authors: Lluis Eriksson
Comments: 9 Pages.
This paper has two goals.Part I (terminal KP bound). We provide a verifiable, citation-driven derivation of the terminal-scale Kotecky--Preiss (KP) smallness bound used in the companion paper on exponential clustering and mass gap. Rather than re-deriving the full multiscale renormalization group (RG), we isolate explicit hypotheses (H1)--(H3) on the terminal polymer activities and prove that they imply the KP convergence criterion. We then verify (H1)--(H3) by mapping them to specific statements in Balaban's published primary sources (CMP 116, 119, 122), with an audited notation bridge recorded in the structural package companion paper.Part II (assembly map + Clay checklist). We give an explicit dependency graph assembling the companion papers together with the KP input proved here. We provide a checklist matching the Clay/Jaffe--Witten formulation of the Yang--Mills existence and mass gap problem to the theorems across the paper sequence (OS0--OS4, OS1, and the mass gap).Scope / external mathematics. The argument uses the abstract KP cluster expansion theorem (Kotecky--Preiss 1986) and the Osterwalder--Schrader reconstruction theorem (1975). It relies on the terminal polymer representation and activity bounds as proved in Balaban's CMP papers cited above.
Category: Mathematical Physics
[23] ai.viXra.org:2602.0089 [pdf] submitted on 2026-02-18 09:21:04
Authors: Lluis Eriksson
Comments: 17 Pages.
We establish two independent rigorous results for four-dimensional SU(N)pure-gauge lattice Yang—Mills theory with Wilson action, at fixed latticespacing η > 0 and weak coupling gu2080 ≤ g_*, uniformly in the spatial volume L.(A) Uniform Log-Sobolev Inequality. The Wilson measure μ_L satisfies Ent_{μ_L}(f²) ≤ (2/ρ̂) E_L(f,f) with constant ρ̂ > 0 independent of L, where E_L is the natural Dirichlet form on SU(N)^{|E(Λ)|}.(B) Uniform Mass Gap. The Osterwalder—Seiler Hamiltonian H_L has a spectral gap m_gap ≥ mu2080 > 0, uniformly in L.Both theorems share a single input — the Dobrushin—Shlosman completeanalyticity (CA) condition, verified via Bałaban's renormalization groupprogram — but follow logically independent paths. Theorem A is derivedthrough Cesi's quasi-factorization of entropy, seeded by a Bakry—Émerylocal log-Sobolev inequality on SU(N)^{|E(Σ)|}; the Ricci curvatureRic_{SU(N)} = (N/4)g plays a key role. Theorem B is derived throughexponential clustering of temporal correlations — a consequence of CA viaDobrushin contraction — combined with the Osterwalder—Seiler transfer-matrixconstruction and the Krein—Rutman theorem. We further prove (C) that {μ_L}converges weakly to a unique, translation-invariant infinite-volume Gibbsstate μ_∞ satisfying the DLR consistency equations, whose reconstructedHamiltonian H_∞ inherits the mass gap mu2080. All constants are explicit inN, gu2080, and η. The present results hold at fixed lattice spacing; thecontinuum limit η → 0 is addressed in a companion paper.
Category: Mathematical Physics
[22] ai.viXra.org:2602.0088 [pdf] replaced on 2026-02-19 12:11:33
Authors: Lluis Eriksson
Comments: 21 Pages.
We establish exponential clustering with a strictly positive mass gap for four-dimensional pure SU(N) lattice Yang--Mills theory with Wilson's action, uniformly in lattice spacing $eta$ and physical volume $L_{mathrm{phys}}$:$|mathrm{Cov}_{mu_eta}(mathcal{O}(0),mathcal{O}(x))| leq C,e^{-m,|x|/a_*}$, with $m > 0$ and $a_* sim Lambda_{mathrm{YM}}^{-1}$.The proof assembles three ingredients: (1) Balaban's rigorous renormalization group for lattice gauge theories (CMP 1984--1989), which produces effective densities with local polymer decompositions and exponentially decaying activities; (2) a terminal-scale polymer cluster expansion (imported from Balaban's convergent renormalization expansions), which implies exponential clustering for the effective terminal measure; and (3) a multiscale correlator decoupling identity (this paper), which separates ultraviolet fluctuations from infrared physics and yields uniform UV suppression. The coupling control required by Balaban's framework -- that the effective couplings remain in the perturbative regime throughout the RG iteration -- is established via an inductive argument using Cauchy bounds on the analyticity of the effective action. We also verify the Osterwalder--Schrader axioms OS0, OS2, OS3, and OS4 for subsequential continuum limits, and establish vacuum uniqueness and non-triviality. The remaining axiom OS1 (full O(4) Euclidean covariance) is not established here; we prove covariance under lattice translations and the hypercubic group $mathcal{W}_4$, and show that if O(4) covariance holds in the continuum limit, the reconstructed Wightman theory is a non-trivial relativistic quantum field theory with mass gap $Delta_{mathrm{phys}} geq c_N,Lambda_{mathrm{YM}} > 0$, where $c_N > 0$ depends only on $N$ (and is independent of $eta$ and $L_{mathrm{phys}}$).
Category: Mathematical Physics
[21] ai.viXra.org:2602.0087 [pdf] replaced on 2026-02-19 12:12:41
Authors: Lluis Eriksson
Comments: 18 Pages.
We classify gauge-invariant local lattice operators of classical dimension 6 on the four-dimensional hypercubic lattice into O(4)-invariant, hypercubic-invariant but O(4)-breaking (anisotropic), and on-shell-redundant components, following the Symanzik improvement programme and the on-shell improvement technique of Luscher--Weisz (1985). Inside Balaban's renormalization group framework for SU(N) lattice Yang--Mills theory, we extract the anisotropic projection of the effective action via local Taylor expansion of polymer activities in the small-field regime and prove a quantitative quadratic scale bound for the anisotropic coefficient: for every RG step $k leq k_*$ with effective coupling $g_k leq gamma_0$, the coefficient of the (one-dimensional) anisotropic sector in the classical dimension-6 Symanzik expansion satisfies $|c_{6,mathrm{aniso}}^{(k)}| leq C,a_k^2$, uniformly in lattice spacing $eta$, physical volume $L_{mathrm{phys}}$, and RG step $k$. We further prove a quantitative insertion integrability estimate for connected correlators with one insertion of the anisotropic operator. When combined with the rotational Ward identity derived in the companion paper, this yields that the corresponding breaking distribution tested against Schwartz functions is $O(eta^2,|log((Lambda_{mathrm{YM}}eta)^{-1})|)$ and hence vanishes as $eta to 0$.
Category: Mathematical Physics
[20] ai.viXra.org:2602.0085 [pdf] submitted on 2026-02-17 16:18:50
Authors: Lluis Eriksson
Comments: 21 Pages.
We prove that Wilson-loop expectations in four-dimensional Euclideanlattice Yang—Mills theory with compact gauge group G admit auniversal continuum limit, independent of the lattice approximationscheme, for every contractible loop and all values of the coupling.The proof proceeds by a multiscale decomposition that combinesBalaban's renormalization-group framework with a quantitativegradient-flow smoothing step at each scale. For an observableliving at lattice scale k, the Yang—Mills gradient flow is run fora time proportional to the squared lattice spacing a_k^2; adeterministic contraction estimate (Theorem 3.5) shows that thisreduces the single-link oscillation of the flowed observable by afactor L^{-2k}, where L is the blocking factor. The resultinggeometric series is summable and yields the desired uniform bound.The two main inputs are: (i) a pointwise domination lemma(Lemma 3.3) that controls the gradient of the flowed observableby a scalar heat kernel on the link graph, exploiting thecontractivity of parallel transport; and (ii) a Duhamelinterpolation formula (Lemma 4.1) that converts each change-of-measure error into a covariance with the irrelevant part of theeffective action, bounded via a Poincaré-type inequality. Togetherthese close the Balaban—Doob inductive circuit under a quantitativeblocking hypothesis that is verified in a companion paper.As a corollary, we establish Osterwalder—Schrader reflectionpositivity for the gradient-flow-smoothed Wilson-loop observable,which together with the continuum limit yields a construction ofthe physical Hilbert space and a positive transfer matrix for thetheory.
Category: Mathematical Physics
[19] ai.viXra.org:2602.0084 [pdf] submitted on 2026-02-17 19:43:23
Authors: Lluis Eriksson
Comments: 15 Pages.
We establish quantitative almost-reflection positivity (almost-RP) for a family of flowed observables in finite-volume lattice Yang-Mills theory on the four-dimensional Euclidean torus T_L^4 with structure group G = SU(N). The lattice Wilson flow - the lattice counterpart of the Yang-Mills gradient flow - acts as a gauge-covariant smoothing that suppresses ultraviolet fluctuations. By combining three ingredients: (i) a Gaussian localization bound that controls the variance of flowed observables via an Efron-Stein-type inequality, (ii) Jacobian estimates for the lattice Wilson flow that yield exponential decay of trans-plane influence, and (iii) the exact lattice reflection positivity of the Wilson action, we show that the failure of RP for flowed observables is exponentially small in the ratio epsilon_0^2 / t, where epsilon_0 is the physical separation between the observable's support and the reflection plane (minus the diffusion scale sqrt(8t)), and t > 0 is the flow time. We record the standard Osterwalder-Schrader reconstruction as a conditional statement: exact reflection positivity on a positive-time algebra implies a Hilbert space, a vacuum, and a non-negative Hamiltonian. Our approach is non-perturbative, holds for all values of the lattice coupling, and requires no cluster expansion or infinite-volume limit.
Category: Mathematical Physics
[18] ai.viXra.org:2602.0082 [pdf] submitted on 2026-02-17 02:36:31
Authors: Tingfang Yi
Comments: 7 Pages.
We propose a minimal six-dimensional (6D) light null entity in which the six dimensions are intrinsic degrees of freedom of a null physical entity. The six dimensions consist of a two- dimensional null propagation geometry together with four intrinsic one-dimensional degrees of freedom of light: optical phase, polarization, frequency, and orientation along the null momentum generator. In this framework, all four-dimensional (4D) spacetime optical, electromagnetic, and quantum phenomena are understood as lower-dimensional projection or section measurements of a single higher-dimensional null entity.
Category: Mathematical Physics
[17] ai.viXra.org:2602.0078 [pdf] submitted on 2026-02-15 17:57:53
Authors: Luisiana X Cundin
Comments: 6 Pages.
A formal, systematic approach for generating nonlinear partial differential equations is outlined, which provides a more robust, reliable method. Additionally, formal methods provide a means to test the validity and/or the veracity of proposed nonlinear partial differential equations, thereby potentially saving researchers precious time and effort.
Category: Mathematical Physics
[16] ai.viXra.org:2602.0077 [pdf] submitted on 2026-02-15 05:11:13
Authors: Lluis Eriksson
Comments: 14 Pages.
We prove that the continuum limit of pure SU(N) lattice Yang—Mills theory in four Euclidean dimensions exists on the algebra of blocked observables at fixed finite volume, conditional on a quantitative regularity hypothesis for the blocking map. The argument combines three components: Bałaban's rigorous renormalization group program, which provides polymer representations and ultraviolet stability; a Doob-martingale influence bound that controls covariance without product-measure assumptions; and a renormalization-group Cauchy summability framework converting per-scale oscillation decay into convergence. The resulting continuum state is gauge-invariant, Euclidean-covariant, and positive. Osterwalder—Schrader reconstruction, the thermodynamic limit, and the mass gap remain open.
Category: Mathematical Physics
[15] ai.viXra.org:2602.0073 [pdf] submitted on 2026-02-14 09:06:13
Authors: Lluis Eriksson
Comments: 17 Pages.
We prove that expectations of blocked, bounded Lipschitz observables at a fixed physical scale ℓ > 0 form an absolutely summable telescoping sequence along a Balaban-matched renormalization trajectory in four-dimensional SU(N_c) lattice Yang—Mills theory with lattice spacings a_k = a_0 2^{-k}. In particular, the continuum limit state ω(O) := lim_{k→∞} ⟨O^{(k)}⟩_{Λ_k, β_k} exists for every O in the blocked observable algebra A_ℓ^{block}. The proof uses three ingredients: (i) an exact RG identity (law of iterated expectations), (ii) a one-step pushforward stability bound for blocked observables derived from Gaussian control of fast modes and an approximate centering property of the fluctuation field, and (iii) a measure-comparison lemma via Duhamel interpolation using polymer remainder bounds. No quantitative rate of asymptotic freedom is required beyond staying in the small-coupling regime where the RG estimates hold; summability follows from the geometric decay (a_k/ℓ)^2 = O(4^{-k}) together with the assumed summability of the large-field/truncation errors {τ_k}. We also state a conditional extension to "renormalized" observables (e.g. Creutz-type constructions) contingent on a nonperturbative Symanzik extraction from polymer expansions, and we discuss the relation to Osterwalder—Schrader reconstruction and the mass gap problem.
Category: Mathematical Physics
[14] ai.viXra.org:2602.0072 [pdf] submitted on 2026-02-14 11:43:03
Authors: Lluis Eriksson
Comments: 14 Pages.
We close the missing influence estimate — Assumption (B6) — required by the RG-Cauchy summability framework for blocked observables in four-dimensional SU(N_c) lattice Yang-Mills theory. The influence is measured by the Efron-Stein seminorm sigma_nu(f)^2 = sum_{e} E_nu[Var_e^nu(f)] that appears in the Duhamel interpolation lemma of the companion paper. We work in the small-field regime of Balaban's multiscale effective action and assume: (A1) a standard polymer representation for the irrelevant remainder V_k^{irr} = sum_X K_k(X); (A2) an explicit per-link oscillation bound for polymer activities carrying the correct irrelevance factor 2^{-2k}; (A3) a lattice-animal counting estimate. Under these three verifiable hypotheses — to be discharged from Balaban's historical work in a companion compendium paper — we prove sup_{t in [0,1]} sigma_{nu_{k,t}}(V_k^{irr}) <= C, where C = C(N_c, beta_0, kappa, C_osc, C_anim, p, L/a_0) is independent of the RG scale k. The proof uses only oscillation bounds and combinatorics: no log-Sobolev inequality, no mixing hypothesis, and no measure-dependent technology beyond the definition of conditional variance. This removes the only genuinely novel probabilistic input remaining in the UV block of the programme towards the Yang-Mills Millennium Prize.
Category: Mathematical Physics
[13] ai.viXra.org:2602.0070 [pdf] submitted on 2026-02-14 18:53:53
Authors: Lluis Eriksson
Comments: 7 Pages.
We prove a uniform Doob martingale influence bound for the irrelevant polymer remainder arising in multiscale renormalization group analyses of four-dimensional SU(N_c) lattice Yang-Mills theory at fixed physical volume. Our main tool is the Doob influence seminorm sigma_nu(f)^2 = sum_i E_nu[(Delta_i f)^2], which yields an exact covariance identity for arbitrary probability measures. Assuming a deterministic per-link oscillation estimate for polymer activities with a scale factor 2^{-2k} (imported from the Balaban renormalization group programme) and using a standard lattice-animal counting lemma (proved here), we obtain a bound sup_{t in [0,1]} sigma_{nu_{k,t}}(V_k^{irr}) <= C independent of the RG scale k. We then explain how this bound feeds into a Duhamel interpolation step used in RG-Cauchy convergence arguments.
Category: Mathematical Physics
[12] ai.viXra.org:2602.0069 [pdf] submitted on 2026-02-14 20:48:17
Authors: Lluis Eriksson
Comments: 23 Pages.
We provide a self-contained derivation of the three structural hypotheses — polymer representation (A1), per-link oscillation bounds with geometric decay factor (A2), and large-field suppression (B5) — that were assumed in Doob Influence Bounds for Polymer Remainders in 4D Lattice Yang—Mills Renormalization and in the RG—Cauchy Master Framework. All results are traced to precise equations in the primary sources: the series of papers by T. Bałaban (Commun. Math. Phys., 1984—1989) and the expository trilogy by J. Dimock (2011—2014). The translation from Bałaban's analytic norms on gauge-covariant function spaces to the per-link oscillation language used in the probabilistic framework is made explicit. Together with Doob Influence Bounds for Polymer Remainders in 4D Lattice Yang—Mills Renormalization, this completes the unconditional discharge of the UV structural inputs for the renormalization group approach to the Yang—Mills mass gap problem at finite volume.
Category: Mathematical Physics
[11] ai.viXra.org:2602.0063 [pdf] replaced on 2026-02-14 08:18:38
Authors: Lluis Eriksson
Comments: 19 Pages.
Building on the lattice results established in Papers [E26I]-[E26IX], we give a conditional construction of a scaling-limit state for pure SU(N_c) lattice Yang-Mills theory in four Euclidean dimensions, along dyadic lattice spacings a_k = a_0 2^{-k}. The construction proceeds via a two-layer architecture. Layer 1 (Local fields): For bounded gauge-invariant local observables (Wilson loops, normalized plaquette traces), expectations converge without extracting subsequences to a unique limit. Precompactness of expectations at fixed physical side length L is trivial since |_{a,L}| <= 1. Uniqueness follows from a multiscale RG-Cauchy estimate that bounds the change of local expectations under a single RG step. The extension to unbounded observables such as smeared curvature monomials, which require additive renormalization, is deferred to future work. Layer 2 (Confinement): The physical string tension sigma_phys > 0 is established through step-scaling of Creutz ratios evaluated on Wilson loops whose physical dimensions R x T are held fixed as a -> 0. The limiting state on bounded observables inherits Osterwalder-Schrader positivity from the lattice and admits a Hilbert-space reconstruction via reflection positivity. SO(4) rotational invariance is expected in the continuum (the hypercubic breaking being O(a^2), subject to standard operator classification and construction of renormalized Schwinger functions). The mass gap is established conditionally via uniform exponential clustering of connected correlators -- an input from a uniform physical transfer-matrix spectral gap -- and the reconstruction theorem. Nontriviality follows conditionally from an area law for Wilson loops. Key dependencies on prior papers: uniform LSI inputs [E26I]-[E26IX]; Balaban multiscale effective action [E26III]-[E26V]; DLR-LSI [E26VII]; unconditional lattice closure inputs [E26IX].
Category: Mathematical Physics
[10] ai.viXra.org:2602.0057 [pdf] submitted on 2026-02-12 19:06:11
Authors: Lluis Eriksson
Comments: 22 Pages.
We prove integrated cross-scale derivative bounds that replace the unverified Assumption 5.4 of a companion paper. Combined with two explicit large-field inputs (a residual pointwise derivative bound and a Balaban-type conditional large-field suppression) and conditional inequalities from the orbit space Ricci curvature, this yields a uniform (volume-independent) log-Sobolev inequality for the Wilson lattice gauge measure at sufficiently weak coupling (large beta). The key innovation is a decomposition into small-field and large-field contributions: the former is controlled by Balaban's polymer expansion, while the latter is handled by a pointwise gradient bound combined with exponential measure suppression. We provide a self-contained verification of the unconditional large-field tail mechanism for SU(2) in d=2, together with numerical validation.
Category: Mathematical Physics
[9] ai.viXra.org:2602.0056 [pdf] submitted on 2026-02-12 19:07:45
Authors: Lluis Eriksson
Comments: 22 Pages.
We verify the large-field hypothesis (Hypothesis 4.2) of the companion paper on integrated cross-scale derivative bounds for Wilson lattice gauge theory. The proof rests on three ingredients: (i) a dictionary lemma translating the Hilbert-Schmidt large-field condition on plaquette holonomies into Balaban's Lie-algebra formulation; (ii) an interface lemma connecting conditional measures with Balaban's T-operation and its uniform small-factor bound on admissible background fields (Eq. (1.89) of Balaban, Commun. Math. Phys. 122 (1989)); (iii) the uniformity estimate (Eq. (1.75) of the same reference) ensuring that slow-field dependence contributes only an O(1) multiplicative constant. For d=2, we give an independent proof via character-positive convolutions that avoids the Balaban machinery entirely. Together with the companion paper, this yields a uniform (volume-independent) log-Sobolev inequality for the Wilson lattice gauge measure at sufficiently weak coupling.
Category: Mathematical Physics
[8] ai.viXra.org:2602.0055 [pdf] submitted on 2026-02-12 19:09:28
Authors: Lluis Eriksson
Comments: 10 Pages.
We prove that the Wilson lattice gauge measure for SU(N_c) in dimension d >= 3 at sufficiently weak coupling (beta >= beta_wc) satisfies a log-Sobolev inequality with constant alpha_* > 0 independent of the lattice volume. This completes the multiscale program initiated in Paper I by verifying Hypothesis 3.2 of Paper III, the last remaining analytic input. The verification uses three ingredients: (i) the locality of polymer functionals, which restricts the sum over polymers to those intersecting a fixed link; (ii) Cauchy estimates on Balaban's analytic domains for polymer activities and boundary terms; and (iii) a combinatorial counting bound for connected polymers containing a given link, which is independent of the lattice volume. Combined with the synthetic Ricci curvature bound of Paper II, the integrated cross-scale derivative bounds of Paper III, and the large-field suppression established in Paper IV, this yields the uniform log-Sobolev inequality unconditionally.
Category: Mathematical Physics
[7] ai.viXra.org:2602.0054 [pdf] submitted on 2026-02-12 19:10:34
Authors: Lluis Eriksson
Comments: 16 Pages.
We prove that the one-step transfer operator of SU(N_c) lattice Yang-Mills theory in dimension d >= 3 has a spectral gap Delta_phys > 0 uniformly in the lattice volume (for even side length L), for all sufficiently large inverse coupling beta >= beta_0. The proof combines four ingredients: (i) the uniform log-Sobolev inequality on periodic tori established in a companion paper; (ii) a verification that the multiscale RG outputs needed for the LSI argument are uniform in frozen boundary conditions (Section 4), yielding the full DLR-LSI property (Section 5); (iii) the Stroock-Zegarlinski equivalence theorem, which in its standard formulation deduces Dobrushin-Shlosman mixing and exponential clustering from DLR-LSI; and (iv) Osterwalder-Seiler reflection positivity of the Wilson action, which translates temporal exponential clustering into a spectral gap of the transfer operator.
Category: Mathematical Physics
[6] ai.viXra.org:2602.0053 [pdf] submitted on 2026-02-12 19:11:43
Authors: Lluis Eriksson
Comments: 14 Pages.
We prove that for SU(N_c) lattice Yang-Mills theory in d >= 3 dimensions at sufficiently weak coupling (beta >= beta_0), the conditional Gibbs specification satisfies a DLR-uniform log-Sobolev inequality: for every finite sub-lattice Lambda' subset of Z^d and every boundary condition omega, the conditional measure mu_{Lambda'}^{omega} satisfies LSI(alpha_*) with a constant alpha_* > 0 independent of Lambda' and omega.The proof combines three ingredients:(i) the multiscale entropy decomposition developed in our earlier work (Papers I-V), which establishes a uniform log-Sobolev inequality on periodic tori;(ii) a uniform fiber oscillation lemma showing that frozen boundary links -- treated as external parameters in Balaban's renormalization group -- do not increase the per-block oscillation of the conditional fast potential, thanks to compactness of SU(N_c) and the locality of the polymer expansion;(iii) a refined large-field event restricted to dynamical (non-frozen) plaquettes, which ensures that the large-field suppression mechanism extends uniformly to boundary blocks.As a consequence, the Stroock-Zegarlinski equivalence theorem yields Dobrushin-Shlosman mixing, exponential clustering of gauge-invariant correlations, and -- via Osterwalder-Seiler reflection positivity -- a strictly positive mass gap Delta_phys >= m(beta, N_c, d) > 0 for the transfer matrix on the periodic torus (Z/LZ)^d, uniformly in even L. This removes the Dobrushin-type Assumption 6.3 of Paper I and the boundary-uniformity Assumption 3.1 of Paper VI, rendering the lattice mass gap unconditional at weak coupling.
Category: Mathematical Physics
[5] ai.viXra.org:2602.0052 [pdf] submitted on 2026-02-12 19:21:09
Authors: Lluis Eriksson
Comments: 11 Pages.
We establish three interface lemmas that close the remaining gaps in the proof chain for the mass gap of SU(N_c) lattice Yang-Mills theory at weak coupling (beta >= beta_0) in dimension d >= 3.Lemma A (Horizon Transfer) establishes a uniform conditional large-field suppression bound mu_k(Z_k(B) | G_{k+1}) <= exp(-c p_0(g_k)) holding mu_beta-a.s., without any admissibility restriction on the background field. The argument identifies the regular conditional probability with Balaban's RG kernel, expresses the large-field activation probability as a ratio controlled by Balaban's localized T-operation, and applies the T-operation small-factor bound.Lemma B extracts from Balaban's inductive scheme that the boundary terms B^{(k)}(X) share the same uniform analyticity domain as the polymer activities R^{(k)}(X), with radius hat{alpha}_1(gamma) > 0 independent of k.Lemma C extends the multiscale LSI to finite volumes with arbitrary frozen boundary conditions omega via tensorization-plus-perturbation, replacing the unverified Dobrushin block condition of Paper VII.Combined with Papers I-VII, these lemmas render the lattice mass gap theorem unconditional.
Category: Mathematical Physics
[4] ai.viXra.org:2602.0051 [pdf] submitted on 2026-02-12 19:22:19
Authors: Lluis Eriksson
Comments: 16 Pages.
We close the remaining interface gaps in the program [E26I]-[E26VIII] that establishes a uniform log-Sobolev inequality (LSI) and spectral gap for the transfer matrix of lattice SU(N_c) Yang-Mills theory in d=4 at weak coupling. Four technical gaps are identified and resolved: (G1) the Balaban small-factor bound for the T-operation is shown to hold pointwise for every real background by auditing Balaban's proof and verifying that it uses only the uniform inductive conditions; (G2) we establish a uniform small-field coercivity estimate (Hessian lower bound) for the effective action and use it, together with Balaban's small-factor mechanism, to control the conditional inequalities in the multiscale entropy decomposition -- circumventing the need for a global fiber LSI with constant O(beta); (G3) uniform analyticity of boundary terms is extracted from Balaban's inductive scheme; (G4) a quantitative bootstrap verifies the simultaneous compatibility of all constants for a single choice of beta_0. Combined with [E26I]-[E26VIII], these closures yield an unconditional proof that Delta_phys(beta,L) >= c(N_c,beta_0) > 0 uniformly in the volume L for beta >= beta_0.
Category: Mathematical Physics
[3] ai.viXra.org:2602.0046 [pdf] replaced on 2026-02-12 19:04:00
Authors: Lluis Eriksson
Comments: 11 Pages.
We establish that the orbit space B = A/G of SU(N_c) lattice gauge theory satisfies the Riemannian curvature-dimension condition RCD*(N_c/4, dim A); in particular, it satisfies CD(N_c/4, infinity) in the sense of Lott-Villani-Sturm. The proof proceeds by showing that the configuration space A = SU(N_c)^{|B_1(Lambda)|}, equipped with the bi-invariant product metric
Category: Mathematical Physics
[2] ai.viXra.org:2602.0041 [pdf] replaced on 2026-02-12 19:02:28
Authors: Lluis Eriksson
Comments: 24 Pages.
We establish that SU(N_c) lattice Yang-Mills theory in d=4 dimensions with Wilson action at sufficiently weak coupling (beta = 2N_c/g^2 >= beta_0) satisfies a log-Sobolev inequality with constant alpha_* > 0 uniform in the lattice size L_vol. Combined with reflection positivity of the Wilson action and the DLR-LSI extension plus Stroock-Zegarlinski mixing route, this yields a mass gap Delta_phys > 0 uniform in L_vol without additional assumptions. The proof combines three ingredients: (i) Balaban's constructive renormalization group, which produces controlled effective actions at all scales; (ii) the orbit space Ricci curvature bound Ric_B >= N_c/4, which gives a uniform log-Sobolev constant for conditional measures of fast modes at each RG scale via the Bakry-Emery criterion; and (iii) a multiscale entropy decomposition with sweeping-out bounds, where the geometric scaling factor ||Q_(k)*||^2 = 2^{-(d-1)k} of transversal block averaging ensures summability of cross-scale errors.
Category: Mathematical Physics
[1] ai.viXra.org:2602.0040 [pdf] submitted on 2026-02-08 19:36:55
Authors: Lluis Eriksson
Comments: 11 Pages.
We prove that the lattice Yang-Mills measure with gauge group SU(Nc)in d=4 dimensions at sufficiently large β=2Nc/g2satisfies a Poincaré inequality with constant α*>0uniform in the lattice size L. The proof uses three ingredients:(i) the Ricci curvature bound RicB ≥ Nc/4 for thegauge orbit space, giving a uniform spectral gap for conditional measures offast modes at each renormalization group scale; (ii) Balaban's constructive RGwith polymer derivative bounds, controlling the residual coupling betweenscales; and (iii) a multiscale martingale variance decomposition that avoidsrecursive composition losses, with a commutator coefficientDk ≤ C e-2κ 2-3k made summable bythe geometric scaling factor of transversal block averaging. Under anRG-normalized disintegration consistent with Balaban's absorption structure,only exponentially decaying polymer residuals contribute to Dk,ensuring Σk Dk << c0. Theresulting uniform Poincaré inequality gives volume-independent control ofthe variance-to-energy ratio for gauge-invariant observables.
Category: Mathematical Physics