Research paperHistorical importARR-2026-72G1SSCNNG8YDVC2 · v1
Lluis Eriksson
For x>0 and real m>=1 define phi_m(x) = [(m-1) I_{m-1}(x)^2 + (m+1) I_{m+1}(x)^2] / (m I_m(x)^2), with I_mu the modified Bessel function of the first kind. We prove phi_m(x) < phi_{m+1}(x) for every x>0 and everyreal m>=1 - a weighted Turan-type monotonicity statement we have not found in the literature, although every ingredient of the proof is classical. The proof is fully elementary: eliminating the neighbouring ratios by thethree-term recurrence, the difference factorizes exactly as (S-3c)(P-(2m+1)c) + (2m+1)c^2, with u = I_{m+1}/I_m, c = 1/x, S = u+1/u, P = 1/u-u; the second factor is positive by the calibrated Amos bound (it is precisely the unit-step inequality of the companion note), the first because the same bound forces u < x/(2m+1) <= x/3. As an application we obtain thestrict determinant ordering c_mn < 0 (m
Research paperHistorical importARR-2026-5HRC3PS9HZ9WDAA5 · v1
Lluis Eriksson
For every r>=1, the uniform measure on the even-parity subset of {+-1}^{r+1}is r-wise independent, yet the last coordinate has unit variance while beingan a.s. function of the others. This example is classical - parity-checkcodes are the standard construction of k-wise independent distributions inthe pseudorandomness literature (Joffe; Alon-Babai-Itai; Alon-Goldreich-Mansour) - and no novelty is claimed for it. What is recorded here is aconsequence we have not seen isolated as a statement: any "comparisonfunctional" whose value depends only on marginal data of order <= r, withconstants uniform over finite measures, takes identical values on the paritymeasure and on the uniform product measure, and is therefore consistent withperfect decoupling on a measure where decoupling fails maximally. Hence noinequality built from bounded-order functionals can imply uniform decouplingprinciples - Dobrushin-type mixing, approximate tensorisation withmeasure-free constants, covariance decay - on any class of measurescontaining the parity family. The case r=1 recovers, and explainsstructurally, the failure of raw-oscillation/Doob and Efron-Stein-type stepsfound repeatedly in an adversarial audit of a constructive Yang-Millsprogramme; no repair within bounded-order data can succeed, because thebarrier recurs at every order. Statements (a) and (b) are machine-checkedin Lean 4/Mathlib parametrically in r (all n; no sorry; standard axiomsonly), the abstract certifying-barrier schema is formalized as well, andfinite decide instances (r<=4) plus exact rational arithmetic (r<=6) serveas independent audits.
Research paperHistorical importARR-2026-5PXDYNR2ER89HTQG · v1
Lluis Eriksson
Let I_nu denote the modified Bessel function of the first kind and, forx>0, let rho_nu(x) = I_{nu+1}(x)/I_nu(x). We give a four-step, fullyelementary proof of the sharp difference inequality0 < rho_nu(x) - rho_{nu+1}(x) < 1/x (x>0, nu>=0), whose right-handinequality is exactly the strict increase of the logarithmic derivative(log I_nu)'(x) under the unit shift nu -> nu+1; consequentlynu -> (log I_nu)'(x) is strictly increasing along every unit-spaced gridnu_0 + N, in particular on the integer and half-integer orders arising inthe application. The stronger continuous-order statement is known(Freitas-Laugesen, arXiv:1810.07461, Lemma 10, via Bessel zeros); we makeno elementary claim about fractional steps. The proof given here uses noinformation about Bessel zeros: it combines the three-term recurrence withthe classical Amos-type upper bound rho_nu < x/(a+sqrt(a^2+x^2)),a = nu+1/2, and rests on the observation that this bound is exactlycalibrated for the problem: 1/U - U = 2a/x is an algebraic identity, and(2nu+1)/x is precisely the threshold the unit step requires; in fact theunit-step monotonicity and the Amos bound are equivalent. As anapplication we record the following consequence in two-dimensional latticegauge theory: for the Wilson action, every mass gap between charactersectors of the 2D transfer operator - for U(1) and SU(2) alike - is astrictly decreasing function of the bare coupling beta, by theFeynman-Hellmann identity. The algebraic core of the proof ismachine-checked in Lean 4/Mathlib (no sorry; axiom oracle: Lean's threestandard axioms), and an independent high-precision numerical audit ofevery inequality used is reported.
Research paperHistorical importARR-2026-2ZQ6F57N2H8MG91V · v1
Lluis Eriksson
We report a complete formalization, in Lean 4 over Mathlib, of volume-uniform Wilson-loop area laws for SU(N c ) lattice gauge theory in an explicit strong-coupling window - including the case of the exact Wilson Boltzmann factor, not a linearized surrogate. The headline theorem bounds the normalized Wilson-loop expectation by N c e P·4dK σ Area(C) e P·4dS(σ) , where Area(C) is an intrinsic combinatorial filling area of the loop, P is its edge-support size, and every constant is volume-free: the bound holds uniformly over all finite lattice sizes. The partition function is cancelled through a fully formalized volume-restricted cluster expansion (loop-tagged factorization, restricted Mayer inversion, Z-ratio bounds, and a pinned-gas resummation built on a Kotecky-Preiss layer with Penrose-style spanning-tree counting). A reusable repackaging converts the bound into manifest exponential area decay with a strictly positive string tension, and the non-vacuity of every hypothesis window is itself machine-checked - both the cluster smallness window and the decay-repackaging window, the latter with an explicit witness of tension log 2 - 1/2. For every exported theorem in this chain the Lean kernel's axiom oracle reports exactly [propext, Classical.choice, Quot.sound]; there is no sorry and no project axiom in the dependency cone. To our knowledge this is the first machine-checked cluster-expansion proof in lattice quantum field theory. All artifacts are public, with per-theorem oracle records in a verification ledger and continuous-integration builds.
Research paperHistorical importARR-2026-70YEZS0RW882CVNM · v1
Lluis Eriksson
Let K n+1 be the complete graph on the vertex set {0, 1, ..., n}, and for a spanning tree T of K n+1 , rooted at 0, let c T (v) denote the number of children of the vertex v. We prove the exact identity: the sum, over all spanning trees T of K n+1 , of the product over vertices v of c T (v)! equals n! C n , where C n is the n-th Catalan number. Equivalently, the normalized sum (n+1)((n+1)!) -1 times the weighted tree sum equals C n exactly. The proof is bijective: pairs consisting of a spanning tree together with a linear ordering of every child set are placed in explicit bijection with vertex-labeled plane trees on n+1 nodes whose root carries the label 0. The identity arises as the exact "second-Ursell" normalization constant in the author's audit-first programme on four-dimensional SU(N) Yang-Mills existence and mass gap, where it had been isolated as a named open proposition in a public challenge repository; the present paper is self-contained combinatorics and makes no claim about that programme. The entire proof has been formalized in Lean 4 against a pinned Mathlib snapshot: the headline declarations compile with no sorry, and the kernel's axiom oracle reports exactly [propext, Classical.choice, Quot.sound]. All artifacts, including a pinned continuous-integration replay of the full verification, are public.
Research paperHistorical importARR-2026-07WK42E0ZR9H8AXE · v1
Lluis Eriksson
This is the experiment-first audit report for the 2602-series programme: a runnable mechanical suite (repository ym-audit) with declared pass/fail criteria, a 2D Yang-Mills benchmark, gauge/infrastructure/UV-flow proxy layers, and a reproducibility manifest. Version 3 aligns the report with the fully audited programme and corrects three defects in the harness itself. Scope statement (sharpened): the 29 tests adjudicate exact identities, toy models, and group-theoretic facts; they cross-validate the per-paper suites of THE-ERIKSSON-PROGRAMME (verification/2602-*, 15 papers, independent implementations agreeing where they overlap, e.g. the triangular-lock dimension counts); they do NOT discharge any entry of the programme ledger - a 29/29 run leaves the hypothesis set {(H1), (H2)+beta_LF, (H3), (H2'), (H-LOC), L6.2-import, (H-Rbeta), (H-P0'), structural+window, lambda != 0 traceable, ...} exactly as it was. Corrections: the d=4 harmonic coefficient is 1/2 (harness had 3/2, a dropped-term bug; the paper chain had 3/5, the d=3 value - three-way inconsistency now settled and machine-adjudicated); the non-triviality test is rescoped (Haar kurtosis != 3 is a single-link fact - exactly 2 for SU(2) - present even at strong coupling; what it verifies is C4(N) > 0); the super-polynomial large-field test is restated under the audited log-power profile. The 2D YM benchmark (transfer-matrix gap Delta = g^2 N/2 to 1e-14) and the dependency DAG survive; "Papers 86-90" are pinned to 2602.0088 v3 / 0087 v3 / 0092 v2 / 0091 v2 / 0096 v2.
Research paperHistorical importARR-2026-6XR934XYNJ9V1B0P · v1
Lluis Eriksson
This is the navigation guide and audit manifesto for the 2602-series programme on 4D SU(N) Yang-Mills. Version 2 is the audited-ledger edition: the dependency graph, Clay/Jaffe-Witten checklist and threat model of v1 are retained, but every node is now pinned to its audited version and carries its named hypothesis loads; the claim is stated as what it is - a conditional assembly: relative to the declared external mathematics (abstract KP, OS reconstruction, lattice reflection positivity) AND to the internal ledger {(H1), (H2)+beta_LF, (H3), (H2'), (H-LOC), per-scale decoupling, (H-Rbeta), (H-P0'), structural+window loads, lambda != 0 traceable}, the chain assembles OS0-OS4 and OS1 and reconstructs a Wightman QFT with mass gap. v1's "unconditional" (in any sense) is withdrawn. Version 2 also corrects two mathematical defects in v1's new material: the hypercubic harmonic's coefficient (3/5, the d=3 value) is corrected to 1/2 in d=4; and the Large-Field Annihilation Lemma's hypothesis p0(g) >= c/g^2 - attributed to a source whose actual profile is (A0 log g^{-2})^{p*} - is replaced by the honest (H2') trichotomy. The genuinely structural new content survives and is machine-verified: the marginal anisotropic sink at d=4 is empty (exact group averaging over W4 on Sym^2(Lambda^2 R^4): quotient dimension 0, versus 1 at d=6), hence renormalization mixing is triangular in the anisotropic channel and the a^2 x a^{-2} -> O(1) objection has no landing site.
Research paperHistorical importARR-2026-11MS45KY8Q8YTSN6 · v1
Lluis Eriksson
Part I (terminal KP bound). We isolate explicit hypotheses (H1)-(H3) on the terminal polymer activities of the 4D SU(N) lattice Yang-Mills programme and prove that, together with a profile condition (H2') made explicit in this version, they imply the Kotecky-Preiss convergence criterion used as Hypothesis (H-KP) in 2602.0088 v3. The implication is elementary and fully machine-verified (exponential inequality, weighted lattice-animal bound with d(X) >= |X|-1, explicit smallness threshold in g). This refines the ledger: (H-KP) <= (H1)+(H2)+(H3)+(H2'). Status of the hypotheses: v1 declared them "verified from primary sources"; per the audited bridge (2602.0069 v2) the correct statement is: (H1) and (H3) are traceable to Balaban's CMP papers; (H2) is traceable with loads (the beta_LF dichotomy, and the unpinned profile (A0,p*) - the recurring (H-P0) datum of the audited series). Part II (assembly map + Clay checklist). We give the dependency graph assembling 2602.0088 v3, 2602.0087 v3 and the (unaudited) rotational Ward companion, and a Clay/Jaffe-Witten checklist with statuses: activating KP does not activate the mass gap of 2602.0088 v3 by itself - that theorem additionally carries (H-LOC), a per-scale decoupling import, and coupling-control loads; OS1 remains open pending the Ward companion's audit. No unconditional Clay claim is made. All adjudicable content is verified in a deterministic companion suite.
Research paperHistorical importARR-2026-2A2CRNRPDC9WCRDQ · v1
Lluis Eriksson
Info de reemplazo — ai.viXra:2602.0092 (v1 → v2)Paper a reemplazar: 2602.0092Method: replacement of existing paperPDF: ward_v2.pdf — sha256 9564aec01a4836c9 — 4 páginasCategory: Mathematical Physics (como v1)Title: Rotational Symmetry Restoration and the Wightman Axioms for Four-Dimensional SU(N) Yang-Mills TheoryAuthor: Lluis ErikssonAbstract (texto plano):We derive a lattice Ward identity for infinitesimal Euclidean rotations of the Wilson theory, identify the breaking term as a dimension-6 anisotropic operator insertion (in the classification of 2602.0087 v3), and show that the breaking distribution is O(eta^2 |log((Lambda_YM eta)^{-1})|) -> 0, establishing axiom OS1 (full O(4) covariance) for subsequential continuum limits - conditionally on the audited companion inputs. Version 2 corrects v1's framing: v1 imported the mass gap, OS0/2/3/4, anisotropy and insertion bounds as "unconditional"; per the audited versions these carry the programme ledger's loads (the source-mapped KP block of 2602.0091 v2, (H-LOC), per-scale decoupling and coupling control for 2602.0088 v3; window and structural loads for 2602.0087 v3). The assembled result - a non-trivial Poincare-covariant Wightman theory with mass gap Delta_phys >= c_N Lambda_YM > 0 - therefore holds under the explicit composed hypothesis set, stated in Section 5; v1's "no unproved hypotheses remain" is withdrawn. The native content survives audit and is machine-verified: the Ward mechanism on an exactly solvable lattice Gaussian model (breaking = O(eta^2) measured), the lambda_{mu nu} != 0 mechanism in the exact quartic model (rotations annihilate O(4) invariants, not the hypercubic harmonic), the Lie-algebra-to-group invariance lemma, the symmetric-difference eta^2/6 error constant, and the eta^2 log vanishing arithmetic.
Research paperHistorical importARR-2026-2BNVY391BQ9A5VM6 · v1
Lluis Eriksson
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 technique of Luscher-Weisz. The anisotropic sector is one-dimensional (Theorem 3.6, Proposition 3.7: uniqueness of the hypercubic harmonic) - a purely representation-theoretic fact, machine-verified in the companion suite together with the classical Symanzik a^2/12 anisotropic term of the Wilson plaquette itself. Inside Balaban's renormalization group framework (small-field regime, g_k <= gamma_0, k <= k* - the window bookkeeping of the audited chain), we extract the anisotropic projection of the effective action via local Taylor (jet) expansion of polymer activities and prove the quadratic bound |c^(k)_{6,aniso}| <= C a_k^2, uniformly in lattice spacing eta, physical volume, and RG step k within the window - conditionally on the structural package, whose audited status is now attached (traceable per 2602.0069 v2; beta_LF dichotomy for the large-field remainder). We further prove an insertion integrability estimate for connected correlators with one anisotropic insertion; Version 3 corrects its status: v1-v2 called it unconditional, resting on an imported clustering/mass-gap bound (Theorem 4.5, from the unaudited companion [1]) claimed uniform in eta and L_phys - per the audited program (2602.0053/0054 v2) such a gap is conditional on (H-DOB-blk)+(H-P0) and windowed, so Theorem 6.6 is conditional on that input. Combined with the rotational Ward identity of the companion [2] (unaudited, identifier pending), the O(4)-breaking distribution tested against Schwartz functions is O(eta^2 |log((Lambda_YM eta)^{-1})|) and vanishes as eta -> 0 - under the same conditional load. This paper thereby supplies, conditionally, the operator-classification input required by 2602.0063 v3 (Proposition 5.3/Remark 5.4) for continuum SO(4) restoration. All verifiable content (representation counts, the unique hypercubic harmonic, the plaquette's own a^2/12 anisotropy, Cauchy-jet mechanics, the a_k^2/log bookkeeping, and the clustering-to-integrability conversion) is adjudicated in a companion suite.
Research paperHistorical importARR-2026-4P13YY8RKA84FRBQ · v1
Lluis Eriksson
We assemble a proof architecture for exponential clustering with a strictly positive mass gap for four-dimensional pure SU(N) lattice Yang-Mills theory with Wilson's action, Cov(O(0),O(x)) <= C exp(-m|x|/a*), m > 0, a* ~ 1/Lambda_YM, with constants uniform in lattice spacing eta and physical volume L_phys - conditionally on three identified inputs. (1) Balaban's structural package (polymer decompositions, exponentially decaying activities), traceable per 2602.0069 v2 with the beta_LF dichotomy attached. (2) A terminal-scale Kotecky-Preiss smallness bound, Hypothesis (H-KP): v1-v2 cited it as proved in an unpublished companion with no identifier; Version 3 retags it as a hypothesis until that companion exists and survives audit. (3) A localization hypothesis (H-LOC) for conditioned observables, made explicit for the first time in v3: the telescoping step compares the terminal clustering bound against O~ = E[O | sigma_a*], whose support is not local. The coupling control (Proposition 4.1) is proved by Cauchy bounds conditionally on the uniform-in-k analyticity radius of Balaban's discrete beta-function and on the large-field penalty profile satisfying the (A0,p*) trichotomy of the audited ledger. What is unconditional and machine-verified: the multiscale telescoping identity (exact for any measure and any nested sigma-algebra chain), the summation-over-scales arithmetic, the lattice-animal bounds, the implications KP => exponential clustering and clustering => spectral gap in exactly solvable settings, and the coupling-control recursion. We verify OS0, OS2, OS3 unconditionally at the lattice level and OS4 conditionally; OS1 (full O(4) covariance) is not established here - its natural conditional supplier is 2602.0087 v3 via 2602.0063 v3. All adjudicable content is verified in a deterministic companion suite.
Research paperHistorical importARR-2026-7APPCT8GM198ERNP · v1
Lluis Eriksson
We present two parallel results for SU(N) pure gauge lattice Yang-Mills in four Euclidean dimensions, at fixed lattice spacing eta > 0 and weak coupling g0 <= g*, both conditional on a single shared input, the Dobrushin-Shlosman complete analyticity condition (H-CA): (A) a log-Sobolev inequality Ent(f^2) <= (2/rho) E(f,f) with rho > 0 independent of L, via Cesi's quasi-factorisation seeded by a Bakry-Emery/Holley-Stroock local LSI; and (B) a spectral gap m_gap >= m0 > 0 for the Osterwalder-Seiler Hamiltonian, via Dobrushin clustering and reflection positivity. The two outputs are logically parallel; neither implies the other here (the bridge lemma remains open, Remark 6.3). Version 2 corrects the status of the shared input: v1 declared (H-CA) verified from Balaban's infrastructure; per the audited chain (2602.0053/0054 v2) that route is conditional on (H-DOB-blk)+(H-P0) and valid only in the volume window L <= exp(C/g0^2). Accordingly all results are stated in two regimes: windowed (audit-backed conditional) and all-volume (under the strictly stronger bare hypothesis (H-CA)_infty, required for the thermodynamic limit, Theorem C). What is unconditional and machine-verified: the curvature computation Ric_SU(N) = N/4, the Holley-Stroock block-seed arithmetic, the variance/entropy decompositions and the failure of pointwise inheritance, the Dobrushin contraction machinery and its window arithmetic, the clustering => transfer-gap lemma on explicit operators, and the convergence of Cesi's geometric factor. All bounds remain explicit in N, g0, eta.
Research paperHistorical importARR-2026-6GVNY9YDSY8V9B86 · v1
Lluis Eriksson
VERSION 2 RETRACTION AND STATUS NOTE. The principal results of version 1 are withdrawn. Theorem 4.4 and Proposition 3.8 depend on the false Wilson-flow linearisation retracted in ai.viXra:2602.0085 and on further false statements. Lemma 3.1 omits the stationary heat-kernel term; Lemma A.1 gives a variance-oscillation bound that fails for correlated non-product measures; Theorem 5.1 does not obtain a positive self-adjoint generator from its stated hypotheses; Lemma 3.3 uses a non-periodic separation across the torus boundary; and Lemma 3.5 bounds a nonlinear map by a differential at one endpoint rather than an integrated or uniform Jacobian. Definitions 2.2, 2.5 and 2.6 remain definitions, and the cited lattice reflection-positivity Theorem 4.1 is not retracted here. Nothing in this note proves the intended almost-reflection-positivity conclusion false; the printed proof and several printed statements fail. The six-page erratum is followed by the preserved 15-page version 1.
Research paperHistorical importARR-2026-5E8SDS6DM8826ABE · v1
Lluis Eriksson
VERSION 2 RETRACTION AND ERRATUM. The principal results of version 1 are withdrawn. Lemma 3.6, Eq. (16), falsely identifies the Wilson-flow linearisation with a connected weighted scalar Laplacian plus a pointwise adjoint term. At the trivial configuration, gauge invariance forces the true Hessian to annihilate a pure-gauge subspace of dimension at least (|V|-1)(N^2-1), incompatible with the printed connected-Laplacian kernel; if the positive-weight graph is disconnected, the single stationary heat-kernel term used downstream is itself false. Consequently Lemma 3.8, Proposition 3.9, Theorem 3.11 and Theorem 1.1 are withdrawn. Lemma 2.2 and Proposition 1.3 remain intact. Lemma 3.2 is only recoverable as a separately specified scalar heat-kernel theorem; its original instantiation is withdrawn. Reflection positivity, Osterwalder-Schrader reconstruction, thermodynamic limit and mass gap remain open. The five-page erratum is followed by the preserved 21-page version 1.
Research paperHistorical importARR-2026-6RC0ZG5NW08AZ9WD · v1
Lluis Eriksson
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, CONDITIONALLY on an explicit hypothesis ledger: a quantitative regularity hypothesis for the blocking map (squared-oscillation summability, Assumption A — equivalently (H-LIP^2), a strengthened form of the (H-LIP) contraction of 2602.0073 v2), the Dobrushin-type decoupling hypothesis (H-DEC/AT) of the sibling audits, and the audited statuses of the Balaban structural package. The argument assembles: (i) Balaban's renormalization group program (polymer representation, irrelevance bounds after beta-function extraction, UV stability of effective densities) — traceable per 2602.0069 v2, with the beta_LF dichotomy for the large-field part; (ii) a Doob-martingale covariance IDENTITY (exact for every measure) together with a conditional-oscillation influence bound — v1's claim that the oscillation control holds "without product-measure hypotheses" is withdrawn: v1's Remark 2.3 correctly rejected Efron-Stein for the non-product interpolating measures, but the Doob-oscillation Lemma 1.5 of v1 fails for the same reason (two-spin counterexample, 2602.0070 v2); the repair is (H-DEC); (iii) the RG-Cauchy summability framework of 2602.0073 v2, consumed with its full ledger (including (H-theta)/F-SQRT for the truncation errors). Under the ledger, the telescopic state sequence converges and the resulting state omega_L is gauge-invariant, Euclidean-covariant (hypercubic), and positive. Osterwalder-Schrader reconstruction, the thermodynamic limit, and the mass gap remain open, as in v1. All mechanical steps — the exact covariance identity, both counterexample adjudications, the Assumption A mechanics on explicit blocking maps (including a non-local sharpness example showing locality plus contraction are genuinely needed), and the corrected Proposition 6.1 arithmetic with its exact M 2^(-4k) scale cancellation — are machine-verified in a companion suite.
Research paperHistorical importARR-2026-5FCQWZC9RM9TF9DV · v1
Lluis Eriksson
We provide a self-contained, equation-level traceability derivation of the three structural hypotheses — polymer representation (A1), per-link oscillation bounds with irrelevance factor (A2), and large-field suppression (B5) — that were assumed in the companions "Doob Influence Bounds for Polymer Remainders in 4D Lattice Yang-Mills Renormalization" and "RG-Cauchy Master Framework". All results are traced to precise equations in the primary sources: T. Balaban (Commun. Math. Phys., 1984-1989) and the expository trilogy of J. Dimock (2011-2014). The translation from Balaban's analytic norms on gauge-covariant function spaces to the per-link oscillation language of the probabilistic framework is made explicit. Version 2 corrects the status of the discharge: it is TRACEABLE AND CONDITIONAL, not unconditional. (i) The small factor of Theorem 8.4 (= Eq. (1.89) of Balaban, Large field renormalization II) carries the constant 2/(1+beta_LF); whether beta_LF is O(1) or a large reference coupling is precisely the dichotomy adjudicated against the audited series (2602.0052/0056/0057 v2), where the large reading trivializes the factor (e^(-c p0) ~ 0.95) and forces hypothesis (H-P0); the dichotomy is now stated as an explicit open interface question (Remark 8.7). (ii) The summability claim (M3) of the RG-Cauchy interface was justified in v1 by "super-polynomial decay from asymptotic freedom"; with the profile p0(g) = A0 (log g^-2)^theta0 the decay in the scale index j (distance to the infrared end) is e^(-A0 (ln j)^theta0): sub-polynomial for theta0 < 1 (sum diverges), j^(-A0) at theta0 = 1 (converges iff A0 > 1), and super-polynomial only for theta0 > 1. (M3) is therefore conditional on the explicit profile condition theta0 > 1 (Remark 12.1; hypothesis (H-theta)). (iii) The irrelevance factor (L^k eta)^(4+alpha) is geometric in the distance to the ultraviolet cutoff, not in the infrared direction; the direction-of-limit bookkeeping for (M1) is made explicit (Remark 10.4) and remains hypothesis-level until the Doob companion is audited. What is machine-verified in the companion suite: the abelian RG operator algebra (Lemma 2.2 mechanics), propagator decay and the random-walk expansion, exponential sum control, lattice-animal counting (Lemma C.1; the illustrative d=4, n=3 count of v1 is corrected from 86 to 84), and the oscillation-analyticity bridge with its Cauchy constants and exact factor-2 saturation. Together with the (unaudited) Doob companion, this package provides a CONDITIONAL discharge of the UV structural inputs at finite volume; the finite-volume, ultraviolet character of the package is what shields it from the infrared volume window of the audited chain (2602.0041 v3, 2602.0051-0057 v2, 2602.0063 v3, now cited).
Research paperHistorical importARR-2026-15XPZCFB4T847870 · v1
Lluis Eriksson
We study 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, via the Doob influence seminorm sigma_nu(f)^2 = sum_i E_nu[(Delta_i f)^2] and its exact covariance identity. Version 2 corrects a genuine error of v1: the increment-oscillation inequality E[(Delta_i f)^2 | F_{i-1}] <= (1/4) osc_{e_i}(f)^2 was asserted for ARBITRARY probability measures; it is false in general (Example 3.4: two perfectly correlated spins, f = X_2, give E[(Delta_1 f)^2] = 1 while osc_{e_1}(f) = 0), because the Doob increment collects influence transmitted through correlations. The correct, measure-independent statement uses the CONDITIONAL oscillation (Lemma 3.5); passing back to the raw single-link oscillation requires a decoupling hypothesis (H-DEC) bounding the influence-leakage matrix, of Dobrushin type — plausible for the interpolating Gibbs measures nu_{k,t} in the small-field weak-coupling regime, but unproven, and structurally akin to the (H-DOB-blk) family of the audited chain. On exact Gibbs chains the v1 bound is violated already at weak coupling for delocalized observables, while the (H-DEC)-corrected bound holds with the Dobrushin coefficient. Under (H-DEC), the imported oscillation input (A2) (now cited from 2602.0069 v2: traceable, conditional — beta_LF dichotomy included), and the lattice-animal lemma (proved here, verified exactly), the main theorem holds: sup_t sigma_{nu_{k,t}}(V_k^irr) <= C uniformly in the RG scale k, by the exact scale cancellation M 2^{-4k} = 4(L/a_0)^4. The Duhamel interface then delivers a one-step rate delta_k = O(4^{-k}) — precisely the geometrically summable rate that Assumption 3.5 of 2602.0063 v3 requires (its Remark 3.7 with eta = 2) — CONDITIONALLY on (H-DEC) + (A2) + the assumed blocking contraction (H-LIP). v1's closing claim "this establishes the RG-Cauchy property" is softened accordingly: this paper supplies the leading candidate for closing (H-CAUCHY), not its proof. All quantitative claims, including the counterexample and the Dobrushin-corrected bound on exact Gibbs chains, are adjudicated in a companion suite.
Research paperHistorical importARR-2026-6TNWJANDFS91097E · v1
Lluis Eriksson
We study the influence estimate — Assumption (B6) — required by the RG-Cauchy summability framework for blocked observables in four-dimensional SU(N_c) lattice Yang-Mills theory, measured by the Efron-Stein seminorm sigma_nu(f)^2 = sum_e E_nu[Var_{nu_e}(f)]. In the small-field regime of Balaban's multiscale effective action, under (A1) a polymer representation, (A2) a per-link oscillation bound with irrelevance factor 2^(-2k), and (A3) lattice-animal counting — all imported from the traceability companion 2602.0069 v2 (conditional) — we prove the UNCONDITIONAL seminorm bound sup_t sigma_{nu_{k,t}}(V_k^irr) <= C independent of the RG scale k: the single-link conditional variance obeys Var_{nu_e}(f) <= (1/4) osc_e(f)^2 for EVERY measure (Lemma 3.2 — conditioning on all other links freezes them, so no influence leaks; this is the sound half, in exact duality with the sibling paper 2602.0070, whose per-link lemma failed but whose covariance identity was exact). Version 2 corrects the unsound half: v1's covariance bound |Cov_nu(f,h)| <= sigma_nu(f) sigma_nu(h) (its Eq. (18)) is FALSE for non-product nu — Example 3.5: perfectly correlated spins give sigma_nu(X_2) = 0 < 1 = Var_nu(X_2) — because Efron-Stein tensorisation is an independence theorem, and the interpolating Gibbs measures nu_{k,t} couple links. On exact Ising chains the tensorisation ratio Var/sum E[Var_e] equals 1.00/1.35/3.40/52.4 at J = 0/0.15/0.5/1.5. Restoring the Duhamel application requires APPROXIMATE TENSORISATION of variance (H-AT): Var_nu(f) <= C_AT sum_e E_nu[Var_{nu_e}(f)] uniformly along the interpolation — a Dobrushin-uniqueness-type condition, the same family as the sibling's (H-DEC) and the chain's (H-DOB-blk), verified here on exact Gibbs chains at weak coupling (C_AT ~ 1.35) and violated without it. There is also a seminorm-interface gap: the companion Duhamel lemma is proved for the Doob seminorm, and sigma_Doob is NOT dominated by the Efron-Stein seminorm for non-product nu (same counterexample; the two seminorms are incomparable in general). Conclusion: (B6) AS CONSUMED by the RG-Cauchy argument is closed conditionally on (H-AT) (or (H-DEC)); the unconditional content of this paper is the Efron-Stein seminorm bound and its scale-uniform M 2^(-4k) = 4(L/a_0)^4 cancellation (with the convergence threshold kappa > log C_anim of v1's Remark B.1 confirmed). Joint statement with 2602.0070 v2: the UV block's only open probabilistic input is Dobrushin-type decoupling of the interpolating measures. All claims, including both counterexample adjudications and the weak-coupling validation of (H-AT), are machine-verified in a companion suite.
Research paperHistorical importARR-2026-70T922KQ6294MT24 · v1
Lluis Eriksson
We prove, conditionally on an explicit hypothesis ledger, 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 4d SU(N_c) lattice Yang-Mills theory with a_k = a_0 2^(-k); in particular the continuum-limit state omega(O) = lim_k exists on the blocked class A^block_l. The architecture is unchanged from v1: (i) an exact RG identity (law of iterated expectations — which resolves at the structural level the "on/off-vs-k->k+1" gap flagged in the sibling audits: the one-step comparison here genuinely is a scale comparison); (ii) pushforward stability for blocked observables from approximate centering and Gaussian control of fast modes; (iii) measure comparison by Duhamel interpolation with influence control. Version 2 repairs the single broken brick: v1's Lemma 7.1 asserted the covariance bound for the Efron-Stein seminorm for arbitrary measures while proving the Doob martingale identity; per the sibling audits (2602.0070/0072 v2, same two-spin counterexample) the ES bound is false for non-product nu and the two seminorms are incomparable, so converting the ES-form input (B6) into the Doob-form covariance control requires the decoupling hypothesis (H-DEC/AT) (Dobrushin-type, verified on exact Gibbs chains at weak coupling). The main theorem is restated with the full ledger: Assumption 3.6 (blocking contraction, (H-LIP)), Assumption 5.1 with (B6) as the CONDITIONAL Efron-Stein closure of 2602.0072 v2 and with sum_k sqrt(tau_k) < infinity in (B3) tied to the profile condition (H-theta) of 2602.0069 v2 — sharpened here by a new finding (F-SQRT): the square root halves the effective amplitude, so at the representative polylog floor (theta_0 = 1.1, A_0 = 1) the sum is formally convergent but its crossover lies beyond j ~ e^1668, i.e. practically divergent; (B3) realistically requires power-law-strength p_0, and (H-P0) rejoins the ledger unless the amplitude is large — plus (H-DEC/AT) and trajectory matching. Under this ledger, the one-step error is O(4^(-k)) + O(sqrt(tau_k)), absolutely summable, and Assumption 3.5 of 2602.0063 v3 — the RG-Cauchy hypothesis (H-CAUCHY), whose naive bridge is not summable (F-SUM) — HOLDS FOR THE BLOCKED CLASS: the cleanest conditional delivery of (H-CAUCHY) in the series. v1's cross-reference "Assumption 4.1 of [18]" is corrected to Assumption 3.5, and the companion references are updated from their withdrawn "unconditional" titles to the audited versions. All mechanical steps (exact RG identity, Lipschitz iteration, pushforward stability in a Gaussian toy, telescoping arithmetic, and both counterexample adjudications at the Lemma 7.1 junction) are machine-verified in a companion suite.
Research paperHistorical importARR-2026-1P88SB34HC8V492J · v1
Lluis Eriksson
Building on the lattice results of Papers [E26I]-[E26IX] — which, per the series audit (all companions now at v2/v3), are WINDOWED and CONDITIONAL rather than unconditional — 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, expectations converge — without extracting subsequences — to a unique limit; precompactness is trivial (| _{a,L}| <= 1), and uniqueness follows from a multiscale RG-Cauchy estimate (Assumption 3.5), the single hard analytic input of this layer: as already recorded in v2 (Remark 3.6, Appendix B), the naive asymptotic-freedom rate g_k^2 ~ c/k is NOT summable, so summability is a genuine hypothesis, not a consequence of the chain. Layer 2 (Confinement): the physical string tension sigma_phys > 0 is established through step-scaling of Creutz ratios at fixed physical loop size, conditionally on Assumptions 4.4, 4.7 and 4.9. The limiting state inherits Osterwalder-Schrader positivity and admits Hilbert-space reconstruction; the mass gap is conditional on a uniform physical transfer-matrix gap (Assumption A.2) and strong continuity (Assumption 5.5). Version 3 retags the input layer to the audited chain: the uniform-LSI inputs are conditional on (H-P0)+(H-YGZ)+(H-SFI)+(H-ABS), the DLR-LSI/mass-gap route on (H-DOB-blk), and all lattice statements hold in the volume window L_vol <= e^(C/g^2+O(1)). A new window-compatibility lemma (Lemma 1.4) shows this window is NOT an obstruction to the continuum limit: along the 2-loop trajectory the required lattice size L/a_k = 2^k L/a_0 satisfies ln(L/a_k) ~ 0.69 k while the audited window allows ln L_lat <= 32.3/g_k^2 ~ 3.12 k — a margin factor ~4.5 at the series' representative arithmetic. Assumption A.2 is now cross-referenced to its conditional lattice supplier (2602.0054 v2: transfer-matrix gap under (H-DOB-blk)+(H-P0), in kernel form). All quantitative claims, and exact validations of both layers in a solvable d = 2 toy, are adjudicated in a companion numerical suite.
Research paperHistorical importARR-2026-09869HXY878C7V5J · v1
Lluis Eriksson
We address the remaining interface gaps in the programme [E26I]-[E26VIII] toward a uniform log-Sobolev inequality (LSI) and transfer-matrix spectral gap for lattice SU(N_c) Yang-Mills in d = 4 at weak coupling. Four gaps are treated: (G1) the pointwise-in-background validity of Balaban's T-operation small-factor bound -- stated in v2 as the explicit hypothesis (H-PTW), since the detailed audit appendices announced in v1 were absent from the document (their references rendered as "??"); (G2) a uniform small-field coercivity estimate for the effective action; (G3) uniform analyticity of boundary terms; (G4) a quantitative bootstrap of all constants. The central correction of v2: the sign of the one-step coupling drift in v1's Theorem 5.2 (g_{k+1}^{-2} = g_k^{-2} + 2 b_0 ln L_RG, coupling weakening toward the infrared) is inverted relative to asymptotic freedom, and contradicts the companions' own use of n_max ~ 1/(2 b_0 g^2 ln 2) (ai.viXra:2602.0032 Sec. 8, 2602.0033, 2602.0041 Sec. 7.3), a formula meaningful only if the coupling grows along blocking and exits the weak regime. With the corrected sign the monotone bootstrap of v1's Theorem 5.3 reverses: the inductive conditions are guaranteed only up to the finite horizon k*(beta) = (g_0^{-2} - gamma_0^{-2})/(2 b_0 ln L_RG) + O(1), and since the multiscale construction uses log_2 L scales, the conclusion holds on the volume window log_2 L <= k*(beta), i.e. L <= e^{C/g^2 + O(1)} with C = 1/(2 b_0) = 24 pi^2/(11 N_c) -- exactly the window of the companion papers ai.viXra:2602.0032/0033 (v2). Full volume-uniformity would additionally require a strong-coupling handoff beyond the crossover scale (Osterwalder-Seiler regime), stated as hypothesis (H-XOVER) and not established here; accordingly the "unconditional closure" of v1 is retitled to conditional closure. Version 2 also repairs the dangling "??" references, supplies the missing proof of Lemma 3.1, rewrites the proof of Lemma 7.4 in the series' fundamental-trace convention (its statement W''(0) = 1/(2N_c) is correct; the v1 proof mixed normalized and fundamental traces), corrects sum_{k>=0} (k+1) 2^{-3k} = 64/49 (v1: 8/49), and fills in the companion identifiers. All corrections are verified in a companion numerical suite.
Research paperHistorical importARR-2026-2695DMHYEK9NEBXD · v1
Lluis Eriksson
This replacement restores the paper originally submitted as version 1 and retracts its claimed unconditional closure of the weak-coupling lattice Yang-Mills mass-gap chain. The principal-logarithm construction in Lemma A.1 fails at central elements such as -I in SU(2), and the printed argument does not establish the required measurable conditional kernel. The proof of Lemma 6.2 applies the Holley-Stroock comparison in the wrong direction: it does not give a coupling-uniform per-block log-Sobolev bound on the unbounded range beta >= beta_0. A bounded beta window can control that per-block estimate, but it does not prove Lemma 6.3, which separately requires an inter-block contraction hypothesis. Three numerical samples and an upper bound on the oscillation are recorded only as numerical evidence; they do not prove linear growth of the optimum or a vanishing infimum. The remaining horizon-transfer, analyticity, and boundary-uniform interfaces are classified as conditional or not established. Corollary 7.3 and the abstract's unconditional mass-gap conclusion are withdrawn. No numbered lemma is declared false unless the corrective note supplies the stated counterargument; otherwise the status is expressly "not established."
Research paperHistorical importARR-2026-0S1R9TEN1580VT82 · v1
Lluis Eriksson
We study the passage from the uniform log-Sobolev inequality (LSI) on periodic tori, developed in the companion series, to a DLR-uniform LSI for the conditional Gibbs specification of SU(N_c) lattice Yang-Mills in d >= 3 at weak coupling (beta >= beta_0), and from there to a mass gap via Stroock-Zegarlinski and Osterwalder-Seiler reflection positivity. Version 2 corrects the logical status of the main results after a quantitative audit (companion numerical suite included; Appendix A). (i) The fiber assembly (Lemma 3.5) assumes the block Dobrushin condition delta < 1, which v1's own Remark 3.6 left unverified; the printed influence bound c_ij <= tanh(beta n_bd/2) tends to 1 as beta -> infinity and yields delta < 1 only for beta <~ 10^(-2) (d=3) or beta <~ 10^(-3) (d=4) — the opposite of the weak-coupling regime. A rotor exhibit shows the genuine worst-case block influence also tends to 1, so no worst-case criterion can close the gap: the condition is now the explicit hypothesis (H-DOB-blk), and v1's claim of removing the Dobrushin-type Assumption 6.3 of [14] is withdrawn — the present paper reduces that assumption to (H-DOB-blk). (ii) The quantitative absorption in Proposition 4.3 inherits hypothesis (H-P0) of ai.viXra:2602.0052(v2): under the polylog penalty floor p0(g) >= c_0 |log g|^(1+epsilon_0) the required inequality e^(-c p0(g_k)) <= C L_RG^(-(d-1)k) fails already at k = O(1). (iii) The proof assumes g_k <= gamma_0 for all k <= n_max ~ log_LRG diam(Lambda'); with the corrected asymptotic-freedom flow of the series erratum this holds only on the volume window log_LRG diam(Lambda') <= k*(beta), i.e. diam(Lambda') <= e^(C/g^2+O(1)). Theorems 1.1-1.2 are therefore restated as windowed and conditional on (H-DOB-blk) and (H-P0). What survives unconditionally — and is validated numerically — is the boundary-uniformity mechanism itself: the per-plaquette oscillation and gradient bounds (Lemma 3.1; sharp for N_c=2), the "frozen = slow" reduction (Lemma 3.2), the refined dynamical large-field event, the energy-penalty identity ||U-1||_HS^2 = 2N_c(1 - Re tr U / N_c), the TV <= tanh(osc/4) lemma with its two-point equality case, and the Bakry-Emery constant N_c/4 in the = -2 tr(XY) convention. The contribution of the paper is thus retagged: a boundary-uniform reduction of the DLR-LSI and the mass gap to (H-DOB-blk)+(H-P0) within the volume window — not an unconditional mass gap.
Research paperHistorical importARR-2026-2MF2AYPVBX9V5T7Y · v1
Lluis Eriksson
This paper assembles the route from the uniform log-Sobolev inequality (LSI) on periodic tori to a transfer-matrix spectral gap for SU(N_c) lattice Yang-Mills in d >= 3 at weak coupling: periodic LSI + boundary-uniform RG outputs => DLR-LSI => Stroock-Zegarlinski mixing => exponential clustering => (reflection positivity) => Delta_phys > 0. Version 2 corrects the logical status of this assembly after a quantitative audit (companion numerical suite; Appendix A). (i) v1 claimed the route "bypasses any explicit Dobrushin contraction estimate". This is withdrawn: the DLR-LSI input (Theorem 5.1) invokes the multiscale fiber assembly of [2], whose inter-block step IS a Dobrushin-type condition — made explicit as (H-DOB-blk) in ai.viXra:2602.0053(v2), the detailed companion treatment which v1 did not cite. v1's supporting claim in the proof of Theorem 5.1, that the fiber oscillation is "O(1) regardless of beta", is also withdrawn: the conditional fast potential obeys osc = 2 beta n_plaq + C_poly, LINEAR in beta (2602.0053(v2), Lemma 3.2; reproduced numerically here). (ii) The quantitative absorption in Proposition 4.7 inherits hypothesis (H-P0) of ai.viXra:2602.0052(v2), and the corrected asymptotic-freedom flow restricts all statements to the volume window L <= e^(C/g^2+O(1)): Theorem 1.1 is restated as windowed and conditional on (H-DOB-blk)+(H-P0). (iii) The erratum for [2] in Sec. 10 is corrected: v1's items (a) and (c) ("Assumption 6.3 is removed", "Theorem 1.1(ii) of [2] is now unconditional") are withdrawn — the assumption is REDUCED, not removed; item (b) (withdrawal of Lemma 6.4 of [2] due to the volume factor (MR^n_max)^d) was correct and stands. (iv) A new technical finding (Remark 2.8): the row-normalized transfer operator T-hat of Definition 2.2 satisfies the correlation identities (12)/(29) exactly only when its normalizer D(sigma) = int K(sigma,sigma') d sigma' is constant (true in the d=2 toy, false for d >= 3 where the spatial factor e^((beta/2)S(sigma)) survives); the correct identities hold in kernel form (with K, or the symmetrized D^(-1/2) K D^(-1/2)). Since D^(-1)K and D^(-1/2)KD^(-1/2) are similar, the spectrum — hence Delta_phys — is unaffected; adjudicated numerically (spectra equal to 10^(-16); the T-hat-form of (29) deviates from the exact path integral by 0.30 in a d=3 toy). What survives and is validated end-to-end in exact toys: the slab splitting (Definition 2.1), self-adjointness and detailed balance (Lemma 2.3), the spectral clustering-to-gap step (Proposition 2.4), Osterwalder-Seiler reflection positivity including the Peter-Weyl positive-definiteness of Re tr(UV^(-1)) (Theorem 2.6), and the gauge-invariance lemmas of Sec. 8. The contribution is retagged: a correct and verifiable transfer-matrix back end for the program, whose front end (DLR-LSI) is conditional and windowed.
Research paperHistorical importARR-2026-1M72VBGYEF9TX8KW · v1
Lluis Eriksson
We prove residual derivative bounds for the polymer expansion of Balaban's multiscale decomposition of the Wilson lattice gauge measure for SU(N_c) in dimension d >= 3, and we assemble them, together with the companion series, into a uniform log-Sobolev inequality. Version 2 corrects the status of this assembly after a quantitative audit. (i) The core mechanism of the paper — locality of polymer functionals, Cauchy estimates on Balaban's analytic domains, and a volume-independent counting bound for connected polymers containing a fixed link — survives intact and is validated numerically; it yields a pointwise derivative bound on the polymer residual with constants independent of the lattice volume, CONDITIONALLY on Balaban's small-field inputs (B1)-(B4). (ii) However, the final step of v1's Theorem 3.5, the inequality k <= C_RG(1+beta_k), relied on the inverted-sign running-coupling flow of the series erratum; with the correct asymptotic-freedom flow the reduced coupling beta_k DECREASES along the cascade, the small-field condition g_k <= gamma_0 is available only for k <= k*(beta), and the derivative bound holds in the windowed form C_res(1+beta) for L_vol <= e^(C/g^2+O(1)). (iii) The assembly of the main theorem inherits two hypotheses identified in the audits of 2602.0052 v2 and 2602.0053 v2: the large-field absorption step requires a power-law penalty exponent — hypothesis (H-P0) — since with the stated polylog floor the suppression factor trivializes (e^(-c_sf p0(gamma_0)) ~ 0.95 at gamma_0 = 0.1) and the absorption inequality fails at every scale (excess >= 10^4.6); and any quantitative use of the conditional fiber LSI via Holley-Stroock carries the penalty e^(-2 beta n_plaq) — hypothesis (H-YGZ) (log10 alpha_blk ~ -5559 at gamma_0 = 0.1, n_plaq = 64). (iv) Version 1's Corollary 1.2 and Remark 5.1 claimed that the Dobrushin-type Assumption 6.3 of Paper I is "no longer needed" via the DLR route of the companion 2602.0053; the v2 audit of that companion shows the route REDUCES Assumption 6.3 to an unverified block condition (H-DOB-blk) whose printed bound c_ij <= tanh(beta n_bd/2) trivializes at weak coupling. Accordingly, v1's closing claim is replaced: the uniform LSI of Theorem 1.1 is WINDOWED and CONDITIONAL on (H-P0) and (H-YGZ), and the mass gap of Corollary 1.2 is additionally conditional on (H-DOB-blk). This replacement also records the completed retagging of the chain: the companion 2602.0054 has been audited and replaced (v2, conditional/windowed assembly), so 2602.0051-0055 now all carry their v2 statuses; reference [6] is corrected (v1 listed 2602.0053 under the title of 2602.0054). All quantitative claims are adjudicated in a companion numerical suite (9 deterministic checks).
Research paperHistorical importARR-2026-1KD8YWD6KE8SKTHQ · v1
Lluis Eriksson
We verify, at the level of form, the large-field hypothesis (Hypothesis 4.2) of the companion paper on integrated cross-scale derivative bounds for Wilson lattice gauge theory (Paper III). 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, Large field renormalization II); (iii) the uniformity estimate (Eq. (1.75) ibid.) 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. Version 2 corrects the status of these results after the quantitative audit of the series (ai.viXra:2602.0051-0055, all v2). (a) v1's claim that the bound is "more than sufficient" for the absorption condition of Paper III is withdrawn: the printed small factor is exp(-c p0(g_k)) with c = 2/(1+beta_0), and with the polylog floor on p0 the suppression trivializes (e^(-c p0(gamma_0)) ~ 0.95) and the absorption inequality fails at every scale (excess >= 10^4.6); effectiveness requires the power-law hypothesis (H-P0) of 2602.0052 v2. (b) v1's premise "p0(g_k) -> infinity as g_k -> 0 along the flow" and Sec. 7's appeal to a stability theorem rely on the inverted-sign running-coupling flow of the series erratum; with the correct asymptotic-freedom flow the small-field condition g_k <= gamma_0 holds only for k <= k*(beta), and all statements are windowed: L_vol <= e^(C/g^2+O(1)). (c) v1's Remarks 4.1-4.2 (slow-field identification and Balaban conditional representation) are unproved interface statements; they are made explicit here as hypothesis (H-SFI), cf. the interface lemmas of 2602.0052 v2. (d) In d = 2 the prefactor K_beta(1)/Z(U_B) of Proposition 6.4 is not uniform in beta, so the d = 2 route verifies a fixed-beta variant only; this is now stated in the theorem. What survives unconditionally and is validated in the companion numerical suite: the HS/Lie-algebra dictionary (Lemma 2.1), the gauge-invariance identity (Remark 2.2), the block event inclusion (Lemma 3.2), and the character-positivity mechanism of Section 6 (Peter-Weyl positivity of the Wilson weight, convolution stability, maximum at the identity, and conditional tail domination in an exact d = 2 toy).
Research paperHistorical importARR-2026-62METTRRHF9Q081B · v1
Lluis Eriksson
We prove integrated cross-scale derivative bounds that replace the unverified Assumption 5.4 of the companion 2602.0041. Combined with two explicit large-field inputs (Hypotheses 3.2 and 4.2) and the conditional inequalities of 2602.0046, this yields — under the audited hypotheses listed below and within the stated volume window — the corresponding log-Sobolev assembly for the Wilson lattice gauge measure at sufficiently weak coupling, with constant independent of L_vol inside the window. The key decomposition into small-field and large-field contributions survives verbatim from v1, as do the sweeping-out modification (an L^1 bound in place of an essential supremum, and a shifted essential supremum over G_{k+1}), the Rothaus closure, the SU(2), d=2 toy-model analysis, and the correction lambda_1 >= alpha_* to Proposition 6.1(2) of 2602.0041. Version 2 corrects the logical status of the assembly after the quantitative audit of the series (ai.viXra:2602.0051-0056, all v2). (a) The Absorption step in the proof of Theorem 1.1 relied on the premises "p0(g) -> infinity as g -> 0 along the flow" and "if beta_k grows sufficiently with k": both use the inverted-sign flow of the series erratum and are withdrawn; with the correct asymptotic-freedom flow all statements hold in the window k <= k*(beta), i.e. L_vol <= e^(C/g^2+O(1)). (b) A new finding (F-ABS): the absorption condition (16) consumes Hypothesis 4.2 in the strong exponent form e^(-c beta_k eps_k^2), but the companion verification (2602.0056 v2) delivers only e^(-c_sf p0(g_k)) with c_sf = 2/(1+beta_0) — bounded along the window — so (16) fails at every scale k >= 2 with the Balaban-compatible thresholds (9), even under (H-P0). The strong form is therefore an additional explicit hypothesis (H-ABS), and its saturated variant eps_k = eps_* opens a second window k <= k_abs proportional to beta eps_^2. (c) The inputs are retagged per their v2 verifications: Hypothesis 3.2 is windowed and conditional on Balaban's small-field inputs (2602.0055 v2); Hypothesis 4.2 is form-level and conditional on (H-SFI)+(H-P0) (2602.0056 v2); the fiber LSI consumed by Corollary 1.2 inherits (H-YGZ) (2602.0053 v2). (d) Reference hygiene: v1's reference [8] was a self-citation of the present paper and is removed; the Wilson duplicate is removed; the audited chain 2602.0051-0056 (v2) is cited. What survives and is validated in the companion suite: the per-direction Wilson bound (Lemma 3.1), the energy-distance identity (13), the single-plaquette tail (Proposition 7.1, Table 1 reproduced digit-by-digit, with its caption/values normalization mismatch fixed), the impossibility Remark 7.2, the factorization step (21), the Rothaus closure, and Appendix A's lambda_1 >= alpha_.
Research paperHistorical importARR-2026-7709A2RGAP84Y9YZ · v1
Lluis Eriksson
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 shows that the configuration space A = SU(N_c)^{|B_1(Lambda)|}, with the bi-invariant product metric = -2 tr(XY), is an Einstein manifold with Ric_A = (N_c/4) g_A (Proposition 2.2), and applies the stability of the RCD* condition under quotients by compact groups of measure-preserving isometries (Galaz-Garcia-Kell-Mondino-Sosa). This bypasses O'Neill computations and handles the singular stratum (reducible connections) automatically. As a consequence we derive a conditional log-Sobolev inequality for measures d mu = e^{-Phi} d nu / Z with constant alpha = (N_c/4) e^{-osc(Phi)}. All constants are computed explicitly for SU(2) and SU(3). This provides the geometric input in a program aiming at a volume-uniform log-Sobolev inequality for SU(N_c) lattice Yang-Mills theory at weak coupling; the complementary analytic input is developed in the companion papers cited in Section 6.1. Note added (v3). Version 3 accompanies the paper with a numerical verification suite (full Einstein tensor for SU(2/3/4); the convention triangle N_c/4 <-> N_c/2 <-> 1/2 closing the series' Ricci bookkeeping; the horizontal characterization at machine precision; the Bakry-Emery convention on the Gaussian via the exact Gross optimizers; Holley-Stroock in LSI form; and the exact energy/entropy correspondence for a Z_2 quotient toy). It corrects a sign in Section 5.1 ([T^a,T^b] = +eps^{abc} T^c requires T^a = -(i/2) sigma^a), repairs the attributions in Section 1.4 (the O'Neill-sketch Ricci statement lives in ai.viXra:2602.0036, whose v2 sharpened the -tr-convention value to N_c/2, consistent with N_c/4 here), adds the measured sectional-curvature range of SU(3) to Section 5.2, and fills in the companion identifiers in Section 6.1 with their honest status. No statement of v2 is refuted.
Research paperHistorical importARR-2026-3TG8KQ8WTF9M4SP7 · v1
Lluis Eriksson
This replacement corrects the title and foregrounds the logical status already partly recorded in public version 3. The uniform log-Sobolev conclusion requires the cross-scale derivative hypothesis (H-XSD), whose purported companion-paper discharge is not re-verified here. The mass-gap conclusion additionally requires the Dobrushin-type hypothesis (H-DOB), or an independently valid alternative DLR-LSI/mixing route. Neither input is proved by this manuscript alone. No unconditional weak-coupling lattice mass gap, continuum construction, Osterwalder-Schrader reconstruction, or Clay-problem result is claimed. The preserved version 3 follows the correction page for provenance.
Research paperHistorical importARR-2026-6RKVSY9K529RTVRV · v1
Lluis Eriksson
We prove that the lattice Yang-Mills measure with gauge group SU(N_c) in d = 4 dimensions at sufficiently large beta = 2N_c/g^2 satisfies a Poincare inequality with constant alpha* > 0 uniform in the lattice size L, conditionally on Balaban's constructive RG and an RG-normalized disintegration hypothesis. The proof uses: (i) the Ricci curvature bound of the gauge orbit space -- sharpened in v2 to Ric_B >= N_c/2, following the correction at its source in ai.viXra:2602.0036 (v2) -- giving a uniform spectral gap for conditional fast modes at each RG scale; (ii) Balaban's polymer derivative bounds, controlling residual cross-scale coupling; and (iii) a multiscale martingale variance decomposition avoiding recursive composition losses, with commutator coefficients D_k <= C e^{-2 kappa} 2^{-3k} made summable by the geometric scaling of transversal block averaging. Version 2 corrects the coupling-flow direction in the statement of Balaban's theorem (which improves the fallback bound of Remark 2.7: beta_k <= beta is bounded, rather than O(k)), records that the summability is robust to the block-averaging convention (new Remark 2.8: the Balaban-style convention gives 2^{-(d-2)k}, still summable), clarifies that the commutator coefficient involves the centered gradient of the conditional potential (which is the mechanism by which G_k-measurable parts drop out, as Assumption 2.6 asserts), and updates the companion references. Unlike other v2's of this series, no statement of v1 is refuted: the entire martingale machinery (commutator identity, telescoping, absorption, the assembled constant alpha*) is validated end-to-end in a companion numerical suite, exactly in a two-scale Gaussian model and against the true spectral gap in a compact four-rotor model, where the recipe's alpha* is confirmed as a valid lower bound.
Research paperHistorical importARR-2026-46C3FRWKDN91YBNR · v1
Lluis Eriksson
We reduce the weak-coupling lattice Yang-Mills mass gap to four explicitly stated hypotheses: assuming them, SU(N_c) lattice Yang-Mills theory in d=4 dimensions with Wilson action at sufficiently weak coupling has a positive mass gap m_gap >= c(N_c) e^{-C(N_c)/g^2} > 0 in lattice units, uniformly in lattice sizes L <= C_0 e^{C/g^2}. The argument combines Balaban's constructive renormalization group, a Morse-Bott/Witten-Laplacian semiclassical spectral gap estimate at the terminal scale, and a transfer-matrix trace identity. Version 1 of this paper presented the result as self-contained modulo Balaban's RG. Version 2 corrects this assessment: the proof is conditional on four explicitly stated hypotheses (Section 1.3). In particular: (i) the Morse-Bott non-degeneracy required by the Helffer-Sjostrand theory fails on the orbifold locus of the flat-connection moduli space -- including the minimum theta=0 of the Born-Oppenheimer potential -- where (d-1)(N_c^2-1-r) quartic "toron" zero modes appear, as we exhibit numerically on a real lattice (Proposition 5.3); and (ii) the constants of Balaban's construction must satisfy a quantitative compatibility window kappa > C' N_c^{3/2}/gamma^2 together with gamma^2 <= 2 N_c h_0, which is empty for typical O(1) decay constants and is not known to follow from Balaban's papers. Version 2 also corrects the sign of the coupling flow in the statement of Balaban's theorem, an inverted extraction regime in the transfer-matrix doubling argument (replaced by a finite-window extraction with explicit error), the definition and Hessian normalization of the Born-Oppenheimer potential (whose v1 form is numerically non-positive), and the Ricci constant of the orbit space (N_c/2, not N_c/4; direction favorable). All corrections and the surviving ingredients are verified in a companion numerical suite. None of this yields an unconditional result, and the continuum, infinite-volume problem remains expressly out of scope.
Research paperHistorical importARR-2026-0G3K2JDS6F93CTZJ · v1
Lluis Eriksson
This replacement preserves the published version 2 and appends a page-fixed erratumand an object-provenance sheet. The erratum withdraws the printed derivations ofTheorems 1.1, 3.1, and 4.1 without claiming that their conclusions for the intendedYang-Mills operators are false. It separates defects in the Euclidean-measure toground-state-measure identification, the use of a four-dimensional effective actionon a three-dimensional spatial orbit space, the volume-dependent transfer-operatortrace extraction, the finite-error resolution window, the factorisation-erroraccumulation, and the admissible-volume quantifiers. Three explicit positivetrace-class counterexamples show why ordinary spectral convergence, positivityimproving, first-excited multiplicity control, and total-tail control do not bythemselves imply gap doubling. A sufficient exact-trace repair is stated at thefinite-volume scale: the leading-eigenvalue normalisation error must be little-o ofthe sum of the two actual first-excited contributions, together with subexponentialfirst-excited multiplicities and vanishing relative excited tails. The previous v2text remains included solely for provenance; no unconditional four-dimensional orcontinuum mass-gap result is claimed.
Research paperHistorical importARR-2026-00YSQD5FFQ8Y99ZZ · v1
Lluis Eriksson
This replacement restores the corrected Morse-Bott revision that was previously placed on another public record and adds an erratum delimiting its valid scope. The original manuscript used one symbol for both total loop holonomy and per-link angle, losing factors of L in the covariant symbol and metric. It also contained an incorrect sine identity, Fourier labels and normal-bundle statements that fail independently, and a determinant convention that double-counted the Faddeev-Popov factor. The replacement withdraws the claims that hypothesis (H1) was discharged and that (H-FACT) by itself replaced (H2); it records the resulting dependency loop with ai.viXra:2602.0033. What remains is a qualitative fixed-volume positivity statement, conditional quantitative implications under explicitly named hypotheses, and numerical evidence in the tested finite-volume case. The numerical checks printed in the historical revision are treated as author-reported evidence because the cited companion verifier is not present in the audited package. No unconditional continuum or infinite-volume mass-gap theorem is claimed.
Research paperHistorical importARR-2026-4F24313TF996VSTG · v1
Lluis Eriksson
This replacement withdraws the global orbit-space Ricci lower bound and the spectral-gap corollaries derived from it. The printed proof of Theorem 3.1 traces O'Neill's horizontal sectional-curvature identity as though the total-space Ricci tensor contained only horizontal directions. The correct trace also subtracts the mixed horizontal-vertical sectional curvatures. For the bi-invariant product metric these terms are nonnegative before subtraction, so the omitted contribution has the unfavorable sign; the manuscript does not prove that the positive O'Neill term dominates it. Accordingly Theorem 3.1 is not established, rather than asserted false, and Corollaries 3.2-3.3 are withdrawn. The correction also separates a finite-dimensional orbit-space statement from later RCD results formulated for a different measure and operator, and records the broken dependency on a version of ai.viXra:2602.0035 that was not present in its public record. The local geodesic-convexity results and the independently stated structural obstruction are retained only within their stated scope.
Research paperHistorical importARR-2026-0PJHG1P2B09S2SXM · v1
Lluis Eriksson
This replacement corrects the title and headline scope while preserving public version 2 in full. For the simplified quadratic Gribov-Zwanziger lattice measure defined in the manuscript, the zero-momentum propagator obeys the stated volume-uniform bound and its thermodynamic-limit value is computed. Finiteness of D(0) is only the necessary condition identified in Definition 4: it does not by itself prove exponential clustering, a transfer-operator spectral gap, a mass gap for the simplified measure, or a mass gap for Wilson Yang-Mills theory. The former title's phrases "Mass Gap" and "A Non-Perturbative Proof" are withdrawn. No continuum-limit or Clay-problem conclusion is claimed.
Research paperHistorical importARR-2026-7PY68QESR297W938 · v1
Lluis Eriksson
We establish a conditional reduction of the Yang-Mills mass gap problem to a concrete spectral inequality involving the gradient flow. Main result (informal): for pure SU(N) Yang-Mills theory, if the gradient flow beta-function satisfies a uniform strict asymptotic freedom condition |beta_GF(g)| >= delta g^3 for large g, and a Tauberian regularity condition holds for the spectral density, then: in d = 3 the theory has a mass gap Delta > 0; in d = 4 the infrared trace anomaly vanishes, a_IR = 0, ruling out a conformal infrared fixed point -- combined with the phase exclusion of the companion paper, this reduces the mass gap to explicit spectral conditions. However, the spectral argument is marginal in d = 4 and requires additional non-perturbative input. The proof uses a spectral representation of the gradient flow energy E(t) with the monotonicity identity R'(t) = -2 Var_t(lambda) <= 0, the Komargodski-Schwimmer a-theorem, and a gradient flow Poincare inequality connecting functional inequalities to exponential clustering. We verify all perturbative inputs: the free-field calibration gives R_free(t) = 2/t in d = 4 and the one-loop correction has the correct sign. We identify the indefiniteness of the Weitzenbock curvature term as the precise technical barrier in d = 4. v2 (no v1 numbered statement is changed): the fixed-measure spectral representation is retagged as an explicit hypothesis (H0) for the nonlinear interacting flow (equivalent to complete monotonicity of E(t), proven only for the linearized flow; the status table is retagged accordingly); an unresolved citation is repaired; the Holley-Stroock route claimed in v1 to give the Poincare inequality unconditionally at finite beta is corrected (its constant is exponential in the volume, so the L-uniform statement remains conditional outside the two controlled regimes); the a_IR = 0 phase classification behind the d = 3 mass-gap corollary is retagged as imported physical input; the Karamata attribution is replaced by the elementary direct bound actually used; and a verification suite adds numerical evidence: exact spectral machinery, lattice free-field calibration, the quantified d = 3 vs d = 4 dichotomy on synthetic spectral densities, and a first in-framework SU(2) Wilson-flow Monte Carlo diagnostic (8^4, beta = 2.4: c(t) = t^2 rises 42 percent across the measured window and R(t) < 2/t pointwise, i.e. beta_GF < 0 throughout -- numerical support for Hypothesis B' in a controlled lattice window, evidence not proof).
Research paperHistorical importARR-2026-1FKXRMCZSH9NHSY7 · v1
Lluis Eriksson
We present a rigorous framework for the Yang-Mills mass gap problem, combining three independent lines of argument that reinforce each other. Result A (Unconditional): a new MaxEnt Clustering-Recovery Bridge -- for lattice gauge states with finite correlation length xi, in the polymer/Kotecky-Preiss regime made precise in Section 5, the Petz recovery fidelity satisfies 1 - F <= C e^{-r/xi}, proved via maximum-entropy truncation on gauge-invariant algebras, a convergent polymer expansion, and the Fawzi-Renner theorem. Result B (unconditional on the lattice, conditional for all couplings): for SU(N) lattice gauge theory (T=0, theta=0, d=3+1, N >= 2), the algebraic phase exclusion, using the projective commutation relation of 1-form symmetry operators, excludes the trivially gapped symmetric phase (v2: given the imported lattice realization of the symmetry pair); combined with Perron-Frobenius non-degeneracy and Gauss-law constraints, this forces confinement at strong coupling; the extension to all couplings relies on Hypothesis 1.1 (absence of a bulk phase transition), supported but not proven. Under Hypothesis 1.1 the uniform lattice mass gap holds for all lattice spacings. Result C (Conditional): under the same hypothesis, the continuum limit exists as a Euclidean QFT satisfying all Osterwalder-Schrader axioms with mass gap. Result D: the gradient flow reduction (developed in the companion ai.viXra:2602.0020). v2 (no v1 numbered statement is changed): the exact lattice realization of the commuting projective 1-form pair is made an explicit imported input, and a new remark records the Perron-Frobenius tension that forces this framing -- at finite volume, PF uniqueness plus exact commutation of both generators would contradict the projective relation outright, so the magnetic operator commutes with H only up to defect terms (verified numerically: toric code, exact pair with 4-fold degenerate ground state; Z2 gauge theory with electric term, unique ground state with both string commutators nonzero); the Result-D naming collision is resolved (the d = 2+1 theorem is now Result E); an unresolved citation is repaired; the epsilon-powers in the MaxEnt bridge are harmonized; and a replication report is added: the 7-qubit Z2 table is reproduced digit for digit, the Z3 table is reproduced digit for digit after a documented g <-> 1/g erratum between the v1 script and table, and the torus finite-size-scaling table could not be reproduced from the printed conventions and is downgraded to archival status (no framework result depends on it). v2 is presented in condensed form: every v1 numbered statement is preserved with the same numbering; detailed proofs and the full computational listing remain in the v1 PDF as the archival source.
Research paperHistorical importARR-2026-7R7V1N2Y1K877A8T · v1
Lluis Eriksson
We present a reproducible pipeline to compute region algebraic entropies and conditional mutual informations (CMI) in a tiny truncated Hilbert space (here dim = 8) indexed by discrete fusion-like descriptors desc = (x, mu) on L = 4 cells. To generate nontrivial ground states within the descriptor-labeled subspace, we introduce an effective Hermitian mixing Hamiltonian based on a weighted k-nearest-neighbor (kNN) graph Laplacian over configuration labels. Across a parameter sweep, we identify a strong-mixing regime where the participation ratio approaches dim (consistent with Laplacian-dominated ground states on connected graphs) and algebraic CMI diagnostics become extremely small (down to 10^{-6} and below) for the chosen algebraic factorization, while region algebraic entropies remain O(1) and exhibit near-quantized values ~ n log 2. We stress that the mixing term is an ansatz used to probe information-theoretic diagnostics and is not claimed to coincide with a Kogut-Susskind plaquette operator. v2 (no v1 number is changed): the reproducibility gap of v1 is repaired -- v1's pipeline loaded an unshipped basis file (descs.pkl) that was never specified, so the v1 dataset was not regenerable from the paper; v2 prints a canonical self-contained basis whose pipeline reproduces every structural finding, keeping v1's Table 1 as an archival dataset; two empirical observations are upgraded to proved statements -- the descriptor-to-key map is injective, making S_alg a genuine von Neumann entropy of a sector (center-type) decomposition, and in the strong-mixing limit the ground state converges to the uniform superposition where the quantization S_alg = n log 2 and the vanishing of both CMIs are exact; the additional experiments recommended in v1 (Haar baseline, kNN ablations, finer t_mix grid) are executed -- the Haar median of I_sum is 0.39, five to six orders of magnitude above the strong-mixing point, so the small-CMI regime is nontrivial; and the verification suite is fully self-contained (no Drive dependencies).
Research paperHistorical importARR-2026-1T29MY5N4J9ZHVQ5 · v1
Lluis Eriksson
We study approximate quantum Markov structure in a Z2 lattice gauge ground state using the conditional mutual information (CMI) I(A:C|B(w)) and the performance of Petz recovery across a family of tripartitions (A, B(w), C) parameterized by a buffer width w. We consider a 2x4 plaquette lattice with open boundaries and qubits on links, restricted to a gauge-invariant (Gauss-law) physical sector, at coupling g = 1.0. For each w we compute reduced density matrices, the entropies entering the CMI, and a Petz-recovered state sigma_ABC = (id_A (x) R^Petz_{B->BC})(rho_AB), reporting fidelity F(rho_ABC, sigma_ABC) via the recovery error E_rec(w) = -log F. The Overleaf project includes the plot, a formatted table, raw CSV outputs, and a hash-based manifest; the appendix typesets raw artifacts. We also report numerical cross-checks (dense vs. low-rank method agreement and trace stability) to support validity. v2 (no v1 number is changed): the star-operator definition is corrected -- with the Hamiltonian convention used here (single-link Z terms), the Gauss stars must be G_s = prod Z_l; v1's printed prod X_l anticommutes with the Z_l terms of H (the code used the consistent convention: an independent reconstruction from the manifest alone reproduces the CSV ground energy to 7x10^{-15} and every CMI of Table 1 to machine precision); two interpretive remarks are added -- the CMI rise at w = 2 tracks the shrinking traced-out complement (|D|: 8 -> 2 -> 0), so the profile is not a shielding-decay curve, and the w = 2 ~ w = 3 plateau is the buffer-saturation identity of the companion 2601.0050 (v2); the apparent Petz-over-CMI excess at w = 1 (E_rec > I) is shown to be the delta = 10^{-6} regularization floor -- regenerating with delta = 10^{-12} restores E_rec <= I at every w in the regenerated dataset; and a verification suite replicates the full pipeline from the manifest data alone.
Research paperHistorical importARR-2026-7X027P5YTZ9M88CK · v1
Lluis Eriksson
We study an information-theoretic notion of locality -- approximate quantum Markov behavior -- via the conditional mutual information (CMI) I(A:C|B(w)) in a semi-infinite geometry of the 1D transverse-field Ising model (TFIM). Using infinite matrix product states (iMPS), we compute I(A:C|B(w)) as a function of the collar width w separating two semi-infinite regions. In a representative gapped point (h = 1.5), we observe clean exponential decay and a rapid plateau of the local effective-length estimator, yielding an early-decay length xi_rec^(early) comparable to the iMPS transfer-matrix correlation length xi_corr. Near criticality (h = 1.005), the local estimator increases throughout the accessible range, indicating a pre-asymptotic regime; we therefore report a fixed-window effective length and a window-sensitivity range as a systematic uncertainty. All generated assets used here (two JSONL data streams, the figure, and the LaTeX table snippet) are included in the Overleaf project. v2 (no v1 number is changed): Appendix A is repaired (in v1 the data-source filenames were typeset in math mode and the near-critical entry was missing entirely); Table 1 is re-typeset (collided columns in v1); the Colab scripts of Appendix B are shipped as runnable files in the series repository rather than as listings; series positioning is added -- this paper is a numerical instantiation of the A-CMI hypothesis of the contract note ai.viXra:2601.0066 in a semi-infinite 1D geometry; and an independent verification suite (free fermions via Jordan-Wigner, no tensor networks) reproduces the gapped point of Table 1 exactly (xi_rec^(early) = 1.149 on the main window, window sensitivity [1.149, 1.158], both matching v1 digit for digit), verifies the operational identity I = 2 S_cut - S(B(w)) to 10^{-11}, and reproduces the rising near-critical xi_local(w); a TeNPy cross-check reproduces the free-fermion I(w) pointwise to 5x10^{-5} relative.
Research paperHistorical importARR-2026-53WA9C4RB39JRRXF · v1
Lluis Eriksson
We isolate the dynamic hinge in typed separation-to-rate-to-power pipelines within the Davies (weak-coupling, Markovian) setting in finite dimension. First, we formulate an upper-envelope statement (RIP-U): under an explicit factorized bath-correlation envelope with separation-dependent amplitude f(epsilon) and integrable time profile, Fourier-transformed Davies rates inherit an O(f(epsilon)) envelope. Under additional regularity assumptions preventing trivial degeneracies, this yields a worst-case bound kappa_up(epsilon) <= C f(epsilon) for the instantaneous relative loss rate of a coherence-like functional. Second, we isolate a structural obstruction to lower-envelope statements: we prove an exact identity for the omega = 0 contribution to the Davies Dirichlet form, E^(0)sigma(O) = (gamma(0)/2) ||[S(0),O]||^2{2,sigma}, and derive a witness mechanism showing how omega = 0 channels can enforce a non-vanishing dissipation contribution for suitable observable families. We emphasize directionality: RIP-U (upper) does not imply a positive lower envelope kappa_down(epsilon) without additional family-qualified input. All assumptions are explicit and accompanied by falsification routes. v2 (no v1 number is changed): two v1 assumptions are upgraded to proved statements in their canonical instances -- Delta-MONO holds unconditionally for the energy pinching (Davies covariance plus data processing), and a sufficient bridge with an explicit constant is proved; the cross-reference labels are repaired (in v1 every assumption and theorem was typeset as "Definition x.y"); Table 1 is re-typeset (overlapping text in v1); the omega = 0 identity is placed within the corrected Bohr-channel decomposition of the companion 2601.0023 (it is exactly the omega = 0 sector, and the plain-commutator form provably fails at omega != 0); and a verification suite reproduces every proved item numerically, including the identity at machine precision and an explicit family with kappa_down much smaller than kappa_up under the same envelope.
Research paperHistorical importARR-2026-3QBZE3EYPV8Y6RPX · v1
Lluis Eriksson
Local algebras in relativistic quantum field theory are typically Type III, so reduced density matrices and von Neumann entropies are not available without additional structure. We give a B-minimal, audit-friendly interface for recoverability in Type III AQFT: we fix a collar geometry and a split datum N (an intermediate Type I factor) and define (i) a split-regularized conditional mutual information (CMI) and (ii) a Bures-fidelity-based recovery error for normal states. We isolate, as explicit assumptions, the two hard bridges needed for an exponential recoverability statement in Type III: (a) existence of an omega_0-preserving conditional expectation onto N (a Takesaki-type condition) and (b) an FR-type inequality in the fixed split implementation. We prove a conditional theorem: if split-regularized CMI decays exponentially in the collar width and an FR-type inequality holds in that split, then recoverability error decays exponentially, with constants tracked explicitly. This paper makes no Clay mass-gap claim and does not invoke von Neumann entropy on Type III algebras without split regularization. v2 (no v1 number is changed): the cross-reference labels are repaired (in v1 every assumption and theorem was typeset as "Definition x.y") and Table 1 is re-typeset; a direction slip in the recovery candidate is corrected -- under CE the GNS-adjoint of the conditional expectation is provably the inclusion iota: N into M, so the operational candidate aligned with the recovery task is the predual of the conditional expectation itself; the finite-dimensional reduction is proved rather than remarked, including the equivalence of the two split-regularized CMI definitions and c_FR = 1 in the fixed conventions; a constructive instance of CE with product reference state is proved; series positioning is added; and a verification suite instantiates the entire contract end to end in the Type I regime (gapped transverse-field Ising collar: CMI decay with alpha ~ 1.06, Petz-type reconstruction satisfying E_rec <= I^(N) at every width in the generated dataset, and the omega_0-adjoint identity E^# = iota at machine precision).
Research paperHistorical importARR-2026-4G5E4HG7NM8YGB7S · v1
Lluis Eriksson
We present an audit-friendly logical contract for a multi-layer program connecting (i) static locality/Markovness, (ii) recoverability bounds, (iii) separation-dependent dissipation rates, and (iv) thermodynamic maintenance power. Each interface is typed with explicit quantifiers, tagged as [PROVED]/[IMPORTED]/[ASSUMED]/[CONJECTURED], and paired with falsification routes. We do not claim a proof of the Clay Yang-Mills mass gap; we separate a Clay (closed-Hamiltonian) track from an operational (open-system/maintenance) track. As a fully closed lane inside this paper, we prove that an exponential conditional mutual information (CMI) decay hypothesis implies exponential recoverability via an imported Fawzi-Renner inequality, with fidelity conventions fixed explicitly. v2 (no v1 number is changed): the cross-reference labels are repaired (in v1 every assumption, lemma, theorem, remark and corollary was typeset as "Definition x.y") and Tables 1 and 2 are re-typeset; the constant-alignment step of Appendix A is executed rather than prescribed -- in the locked conventions (squared fidelity, E_rec = -log F) the Fawzi-Renner import yields c_FR = 1 exactly; the deferred interfaces of Appendix B are now concrete series papers and are cited as such (the Type III interface is ai.viXra:2601.0065, the Davies/RIP-U dynamics note is ai.viXra:2601.0064, the benchmark infrastructure lives in the series repository); the BATO-LAW upgrade path of Appendix C records the partial progress of ai.viXra:2601.0031; and a verification suite instantiates everything instantiable: the closed recoverability lane end to end on a gapped transverse-field Ising collar (I ~ 0.20 e^{-1.06 w}, E_rec <= I at every width in the generated dataset, full 1 - F <= E_rec <= c_FR K e^{-alpha epsilon} chain), the exact-Markov example at machine precision, and the dephasing example with an explicit kappa_up/kappa_down spread of two orders of magnitude illustrating the directionality golden rule.
Research paperHistorical importARR-2026-6XR3EE53GH8E49RH · v1
Lluis Eriksson
We provide a finite-size benchmark testing whether a CMI-based recoverability proxy correlates with Wilson-loop confinement diagnostics in Z2 lattice gauge theory in 2+1 dimensions, computing ground states by sparse exact diagonalization on 2x2 and 2x3 open lattices (link qubits, Gauss-law penalty verified by ~ 1) and evaluating entropic quantities by pure-state Schmidt/SVD. We state a conditional bridge from a spatial area law to an operational "information horizon" via explicit hypotheses (H1)-(H4), with two recovery-length conventions (absolute, and normalized by the boundary prefactor Sigma_B(w) = |dB(w)| log 2). v2 (no v1 number is changed) adds a structural lemma that v1's own Table 1 was exhibiting unnoticed: for a pure global state, when the collar saturates (B = (A u C)^c) purity forces I(A:C|B) = I(A:C|empty) -- this is exactly why v1's Table 1 shows I(w=2) = I(w=0) = 0.4992999 to seven digits; the saturated row carries no buffer information, and the informative range of that benchmark is w in {0, 1}. v2 also anchors the non-monotonicity of CMI under collar enlargement as geometry-dependent (v1's wall geometry shows growth at w=0 to 1; a BFS-patch geometry on the same states decays monotonically -- there is no data-processing theorem in that direction); cross-links the CMI benchmark to the Petz-error twin on the same lattices (densified n = 8 sweep: Spearman rank-trend +1.00, permutation p = 1e-4, against both 1/sigma_eff and the inverse spectral gap -- so, as in the twin, confinement specificity is unresolved at these sizes); harmonizes the fidelity convention (v1 correctly uses root fidelity with I >= -2 log f, equivalent to the companions' squared-convention I >= -log F); removes an internal phase label leaked into v1's Section 1 title and a duplicated heading; and adds series positioning and a regenerable verification suite. The conditional proposition and its hypotheses are unchanged.
Research paperHistorical importARR-2026-29YZ0PQA699GV8T9 · v1
Lluis Eriksson
We provide reproducible finite-size benchmarks testing whether a Petz-type recoverability proxy correlates with Wilson-loop confinement diagnostics in Z2 lattice gauge theory in 2+1 dimensions: an exact-diagonalization benchmark on 2x2 and 2x3 plaquette lattices (Gauss penalty, ~ 1 verified), and TeNPy DMRG ladders 2xL, L in {4, 6, 8}, chi_max = 96, with a warm-start bond-dimension stability check (chi = 96 to 192 stable to all shown digits). In the tensor-network part, E_rec is reported as a function of contiguous buffer size |B| in MPS site ordering (a declared proxy), not the BFS collar of the ED part. v2 (no v1 number is changed): the broken Appendix A.1 of v1 -- whose published PDF literally prints "Missing file" where the ED reproduction script should appear -- is repaired by pointing to the series repository, where that script and a full verification suite for the ED benchmark already live with the companion paper; internal working titles leaked throughout v1's scripts and captions are removed; the MPS-ordering proxy is sharpened with new evidence -- on an exact Z2 ladder the geometric admissible buffer decays cleanly (5e-4 to 3e-8) while the contiguous-ordering proxy starts orders of magnitude higher and saturates, and a from-scratch DMRG replication (reduced chi, Lx = 4) reproduces both the trend direction of the money plots and v1's own buffer inversion E_rec(|B|=2) > E_rec(|B|=1), confirming it as a property of the contiguous proxy rather than a numerical accident; the confinement-vs-gap degeneracy established for the ED twin applies verbatim to the ladder money plots and is now stated; and series positioning is added. All conclusions remain finite-size benchmark statements.
Research paperHistorical importARR-2026-2KDX6T0J7Y9348TM · v1
Lluis Eriksson
SUPERSEDED BY AI.VIXRA:2601.0051V2 FOR THE CONSOLIDATED BENCHMARK. This replacement preserves public version 2 and makes the series relation explicit. The exact-diagonalization benchmark remains a valid finite-size component, but the successor is the authoritative combined source because it retains that benchmark and adds tensor-network ladder calculations and further scope controls. The reported Petz/Wilson rank alignment is not a confinement-specific or thermodynamic theorem: absolute recovery errors remain prescription dependent, and an equally strong spectral-gap covariate leaves confinement specificity unresolved at the tested sizes. The preserved version 2 follows the two-page notice unchanged.
Research paperHistorical importARR-2026-7B37HVCEAD9A1BWQ · v1
Lluis Eriksson
We propose an operational route from recoverability data to effective geometry. Given a tripartition A-B(w)-C and a collar width w, we consider a Petz-type recoverability error E_rec(w) defined via fidelity and extracted from a fixed collaring rule (A, C, w) -> B(w). We define distance-like functionals from the minimal buffer needed to suppress E_rec(w) below a threshold, and from exponential fit scales when such a regime exists; these are organized into a (generally non-metric) dissimilarity matrix on coarse regions, symmetrized when needed, and embedded via multidimensional scaling or diffusion maps. The paper emphasizes precise definitions (collaring rule, symmetrization, censoring below numerical floors) and falsifiable diagnostics (approximate triangle inequalities, robustness to thresholds and regularization). A minimal control experiment in the 1D transverse-field Ising model illustrates the pipeline and the growth of a recoverability length near criticality. v2 (definitions unchanged) adds: a regenerable suite replacing the "representative run" of v1 -- the control table is regenerated from scratch, its g = 0.5 row is flagged as unstable by the paper's own fit-window policy, and a three-point-fit caveat is stated; the first in-model test of the tracking conjecture -- from the same ground states, xi_rec/xi_corr in {0.98, 0.53, 0.52} across regimes, same order of magnitude throughout; a first numerical illustration of the embedding machinery (four coarse regions, MDS recovering the chain order exactly, triangle violations bounded by discretization), which also surfaces an operational lesson: tracing out the region between B(w) and C fakes separation and inverts monotonicity, so the separation condition of the collaring rule is essential, and profiles below the separating width are not admissible; the observation that the fixed-|A|,|C|/traced-environment design of the control is precisely the fixed-target protocol that resolves the |C|-shrinkage confound identified in the companion d_eff notes; and series positioning.
Research paperHistorical importARR-2026-4P264G2DC08DTTKP · v1
Lluis Eriksson
We propose a numerical protocol and falsifiable conjectures relating quantum-information recoverability measures to confinement diagnostics in lattice gauge theories. For a tripartition A-B-C and collar width w, we define a Petz-type recovery error E_rec(w) and extract a recoverability length from threshold and fit criteria. Since gauge constraints obstruct naive factorization, the protocol is formulated in an extended-Hilbert-space (EHS) prescription by default, with an algebraic (gauge-invariant) variant outlined together with its subtleties (centers, sectors). We conjecture that E_rec(w) decays exponentially in gapped phases and that its scale tracks confinement scales set by Wilson loops. v2 executes the testbed that v1 only specified: on a Z2 ladder of four plaquettes (14 links, Gauss law enforced exactly, = 1 to 1e-6), the suite computes E_rec(w) and Wilson decay on the same ground states across five couplings. Findings: the pipeline runs end to end; xi_rec is identifiable at three of five couplings and nearly coupling-independent (0.21-0.31, spread x1.4), while the Wilson decay length varies strongly (0.38-6.9, spread x8.6): no tracking at ladder level. Because height-one ladders degenerate area and perimeter, the Wilson scale there is not a pure confinement scale, so this first dataset is inconclusive-but-cautionary for the tracking conjecture rather than a falsification -- and it sharpens the requirement: a genuine 2D lattice is needed. v2 further flags the TFIM control row shared with the companion framework note as unstable under regeneration, and adds series positioning. No v1 definition is changed.
Research paperHistorical importARR-2026-5HG8P9BAQ78C5AB2 · v1
Lluis Eriksson
We formulate an operational notion of recoverability in algebraic quantum field theory for type III local von Neumann algebras. Fixing a faithful normal KMS reference state and assuming a state-preserving conditional expectation, we define the recovery channel and, working in a fixed split implementation for each separation r, we assume (i) exponential decay of split-implemented conditional mutual information and (ii) a CMI-to-recovery inequality. Under these explicit bridge assumptions we obtain a conditional exponential recoverability bound E_rec(r) <= g(C1 e^(-m r)). v2 corrects the duality underlying the construction: v1 defined the Petz-type channel as the "Accardi-Cecchini adjoint" via the pairing omega(Z R(X)) = omega(eps(Z) X) and "proved" finite-dimensional consistency with the standard Petz map; that identity is false in general (the printed proof contains an invalid cyclicity step; numerically the identity fails at order 1e-1 on random faithful states, holding only in commuting/product situations). The correct statement, proved and machine-verified here, is that the standard Petz map is the trace-predual of the generalized (Accardi-Cecchini) conditional expectation; accordingly, v2 defines the recovery channel in Schrodinger picture as precomposition with the conditional expectation. This also repairs a type/direction error in v1's recovered-state definition, whose corrected form (omega restricted to AB, composed with the normal extension of id_A tensor eps) reproduces the standard Petz reconstruction exactly in finite dimensions. We further relabel v1's finite-dimensional map as the generalized conditional expectation (its Takesaki module property fails generically -- verified), record that for a true state-preserving conditional expectation the recovery is the CE-pullback, and add series positioning: this note is the AQFT capstone announced by the companions, its split-implemented CMI is one of three compatible regularizations in the series, and the numerical program deferred by v1 has since been executed. The main theorem is unchanged: a conditional framework statement isolating the missing bridge assumptions.
Research paperHistorical importARR-2026-0MHWPSFFX696C8KW · v1
Lluis Eriksson
We study finite-size scaling of an operational recovery length extracted from Petz-map recovery in the transverse-field Ising chain. For a tripartition A-B-C with a collar B of width w, we define E_Petz(w) = -log F (squared Uhlmann fidelity), E_best(w) = min over w' <= w of E_Petz(w'), and the effective recovery distance d_eff(epsilon), with log-linear interpolation. Using exact diagonalization at hz = 0, beta = 12, |A| = 2 for N in {9, 10, 11, 12}, we analyze the peak height d_max(epsilon; N) = max over hx of d_eff in a censoring-free threshold regime, finding that the finite-window data are well summarized by descriptive power-law fits d_max(epsilon; N) ~ N^kappa(epsilon) with, e.g., kappa(3e-3) of about 0.44 and kappa(5e-3) of about 0.26, and a pseudocritical drift of the peak location with a threshold-dependent effective exponent nu_eff -- reported as operational quantities, not universal estimates. v2 adds (no v1 number is changed) two mandatory caveats, both quantified by a regenerable suite that reproduces v1's Table 1 exactly: (i) a |C|-shrinkage/growth confound -- the off-critical baseline d_eff(hx = 0.80; N) also grows with N at fixed thresholds, so the raw peak growth conflates critical physics with tripartition geometry; the cleaner object is the critical enhancement Delta(N) = peak - baseline, which still grows with N (e.g. 0.42 to 0.74 over N = 9, 10 at epsilon = 3e-3), so the critical signal survives baseline subtraction while kappa(epsilon) from raw peaks must be read as geometry-contaminated; (ii) functional-form indistinguishability -- over the accessible sub-octave in N, power-law, logarithmic and linear fits of the peak height have R^2 spreads below 0.01, and kappa itself shifts strongly with the fit window; the correct reading of kappa(epsilon) is a descriptive summary, not an established power law. Series positioning and a verification suite are included.
