We machine-check the exact measure and differential bridge between thefour-edge SU(2) crossing chart and the two effective group coordinates used bya finite-dimensional Ward identity. The quotientr(a)=(a2 a4^-1,a1 a3^-1) pushes normalized four-fold Haar measure exactly tonormalized two-fold Haar measure. Two gauge-compensated flows on the originalfour edges intertwine with independent right multiplication of the quotientcoordinates, so first and mixed derivatives of the crossing Wilson worddescend without choosing a gauge. The three Pauli directions are verifiedindividually, their mixed generators close into direct and reverseresolutions, and the two-coordinate Ward theorem lifts to a literal four-edgeintegral with coefficients -1/4 and -1/2. This compensated operator is notidentified with the ordinary-edge mixed operator of Driver-Hall-Kemp: thepaper writes both operators and their distinct reverse resolutions and markstheir comparison as open. The Lean producer has 26 public declarations and 16audited theorems. No weak four-face heat-kernel identity, area derivative, orfull Makeenko--Migdal equation is claimed.
We formalize extended gauge invariance at a simple four-edge SU(2) crossing and connect the geometric edge chart to the two-coordinate chart used by a machine-checked crossing Ward identity. On SU(2)^4 we define the two opposite-edge right actions from the abstract Makeenko--Migdal theorem, prove that they are commuting product-Haar-preserving actions, and show that their common parameter composes to ordinary vertex gauge invariance. The four-edge Wilson word tr2(a3^-1 beta a2 a4^-1 alpha a1) is proved invariant under both half-actions. We construct the explicit quotient r(a)=(a2 a4^-1,a1 a3^-1), a canonical section, and prove existence and uniqueness of the universal factorization for every extended-gauge-invariant complex function. The complete map from the cyclic four-edge chart to physical and gauge coordinates is proved to preserve literal four-fold Haar measure in one public endpoint. Finally, the four-edge Wilson word is identified exactly with the prior two-coordinate crossing word evaluated on r(a). The Lean producer has 56 public declarations in 488 physical lines; all 36 theorems depend only on propext, Classical.choice, and Quot.sound, with no local proof escape. No heat-kernel area derivative or full Makeenko--Migdal equation is claimed.
We formalize a finite-dimensional crossing Ward identity for fundamental SU(2) Wilson words. On every finite edge space SU(2)^E, two distinct coordinates are inserted into a concrete normalized trace word. Lean proves entrywise that right multiplication by each of three explicit SU(2) curves produces the normalized Pauli generators i sigma_a/2, differentiates the trace word once at each selected coordinate, and identifies the Pauli-summed mixed derivative with the rank-two Fierz contraction. Two applications of Haar integration by parts transfer the corresponding mixed generators from a density to the Wilson word. The resulting integral closes exactly on the direct and reverse single-trace resolutions, with coefficients -1/4 and -1/2 in the chosen normalization. The producer contains 19 public declarations in 487 physical lines; all 13 new theorems depend only on propext, Classical.choice, and Quot.sound, with no local sorry, admit, or axiom. This is not a full Makeenko--Migdal area equation: the remaining physical input is a weak four-face identity against crossing-certified extended-gauge-invariant observables. In the program's heat time, generated by the Pauli Laplacian, its coefficient is kappa = 2.
A positive site weight acts on a transfer kernel by congruence, K -> DKD with Dpositive diagonal, not by similarity. Similarity preserves the entire spectrum;congruence preserves strictly less, and strictly more than nothing. We determineboth halves for the kernel of L decoupled Ising bonds. Rigid half: the sign ofevery quadratic form value survives, so definiteness in either sign is acongruence invariant -- the definite case of Sylvester's law of inertia, in aform a machine can check without diagonalisation. Fragile half: the subdominantratio r (second eigenvalue modulus over the Perron root) does not survive atall, and we locate exactly how far it moves. Restricting the L-site kernel tothe two antipodal configurations leaves one Ising bond of coupling bL, of ratiotanh(bL) for b>0: a weight concentrating there fuses the L sites into a singleeffective site carrying L times the coupling. Since tanh(bL) -> 1, no boundr <= rho < 1 holds simultaneously in L and over the whole positive-diagonalcongruence orbit; the obstruction to volume-uniformity is a property of thecongruence, not of any spectral estimate, so an argument bounding r throughcongruence invariants alone cannot produce an L-uniform bound. For arbitraryweights we prove sup_{D>0} r(DMD) = (1-m)/(1+m), where m is the leastoff-diagonal entry: the least correlated pair determines the supremum. (Weevaluate the supremum; we do not classify its maximisers.) The proof isgeometric rather than spectral -- Hilbert's projective diameter is itself acongruence invariant, and Birkhoff's contraction theorem converts it into thebound -- and uses no definiteness, only positivity of the entries. The lowerbound uses no limiting argument about spectra: a two-supported fluctuationvector gives the estimate at each strictly positive epsilon, so no continuity ofeigenvalues is imported anywhere.
We formalize the finite-dimensional integration-by-parts mechanism used inlocal proofs of the Makeenko—Migdal equation. For a measure-preserving realflow on a finite measure space, Lean verifies differentiation under theintegral from an explicit locally uniform integrable majorant and proves thatthe integral of the generator vanishes. A bounded dominated product interfacethen yields integration by parts, and two successive applications transfer amixed pair of generators from a density to an observable with positive sign.These results are instantiated on normalized Haar probability measure ofconcrete SU(2) and on every finite product SU(2)^E, for left and rightmultiplication of one selected edge. To fix the representation normalization,we construct three explicit trigonometric curves in SU(2) and prove entrywisethat their tangents at the identity are i sigma_1/2, i sigma_2/2, and isigma_3/2. The producer contains 30 public definitions, structures, andtheorems in 519 physical lines; its audit contains no local sorry, admit, oraxiom, and the headline results depend only on propext, Classical.choice, andQuot.sound. This closes the Haar integration-by-parts layer. It does not claimthe full four-area Makeenko—Migdal identity: the remaining formal inputs arethe heat-density directional identity, extended gauge invariance at acrossing, and their geometric assembly.
We give a kernel-checked positive-area calculus for the concrete SU(2) classheat kernel used in two-dimensional Yang—Mills theory. With irreducible labeln, dimension n+1, and Casimir c_n=n(n+2)/4, Lean verifies at every positivetime and every derivative order that the infinite spectral jet converges anddifferentiates term by term to the next jet. The hierarchy is packaged as aC-infinity map on the positive-time half-line, using explicit uniform summablemajorants on positive-time neighbourhoods. We then differentiate the literalnormalized-Haar two-face Migdal integral, prove that its left and right areaderivatives equal the first spectral jet at the merged area, and showinfinitesimal invariance under (s,t) -> (s+u,t-u). Finally, every normalizedWilson character satisfies its exact Casimir area ODE as an actual Haarintegral against the infinite heat kernel. The artifact contains 25 publicdefinitions and theorems, no local placeholders, and audited dependencies onlyon propext, Classical.choice, and Quot.sound. We do not claim the four-faceMakeenko—Migdal crossing equation; the remaining inputs are local Lie-groupintegration by parts and certified crossing geometry.
Finite-rank Makeenko-Migdal equations generate products of Wilson traces at self-intersections. For SU(2), this apparent multitrace obstruction closes exactly on single traces, but the statement is normalization-sensitive: the traceless Lie algebra contributes a finite-rank correction that disappears for U(2) and must not be dropped. We give a Lean 4/Mathlib formalization of the complete group-algebraic closure mechanism on Mathlib's concrete special unitary matrix group. With normalized trace tau(A)=Tr(A)/2 and normalized anti-Hermitian Pauli directions X_j=i sigma_j/2, the kernel checks the Casimir identity, the rank-two Fierz identity, the induced crossing contraction, and the SU(2) trace-skein identity tau(g)tau(h)=(tau(gh)+tau(gh^{-1}))/2. Consequently, the finite-SU(2) crossing term tau(g)tau(h)-tau(gh)/4 equals tau(gh)/4+tau(gh^{-1})/2. We then formalize a universal local interface with four cyclically ordered branch holonomies, an independent orientation on each branch, the two opposite-strand words, and precisely the two direct/reversed reconnections. Its corrected crossing term closes on those reconnections for every branch assignment and orientation choice. A recursive theorem also extends the reduction to products of arbitrarily many fundamental traces. The identities are classical; the contribution is a concrete, kernel-checked normalization bridge from Pauli contraction to the single-trace closure used in finite-rank loop equations. We do not claim a formal derivation of the Yang-Mills area derivative, planar loop geometry, or the full Makeenko-Migdal equation.
BOND REFLECTION, AND THIS ONE IS ABOUT THE MEASURE. A whole path X of 2m+2slices has a past half past(X) and a future half; write rev(X) for the futurehalf READ BACKWARDS FROM THE FAR END, which is the reflected copy. Let F be acomplex observable of an entire half. Then for every L, every m, everystrictly positive source weight w and every beta >= 0, sum over whole paths X of conj(F(past X)) * F(rev X) * W(X) >= 0the sum running over all whole paths and W being the ordinary Gibbs weight.This is the finite-volume Osterwalder-Schrader reflection-positivityinequality for this model at odd separation. The sum is UNNORMALISED, which isthe form the axiom is about; dividing by the partition function is division bya positive number and preserves the sign. The Gram matrix of a finite familyof such observables, with complex coefficients, satisfies it as well. It isobtained by proving that assembling a past half, a crossing bond and areversed future half is a BIJECTION onto paths, and that the weights multiplywith exactly one crossing factor.SITE REFLECTION, AND THIS ONE IS NOT, YET. Through a site the two halves SHAREthe middle slice, so the assembly is not a product of two independent halvesand its bijection is a different statement. What is proved there is thehalf-chain form, its collapse identity and its Gram positivity -- at EVERYbeta, negative coupling included -- but NOT its identification with the pathmeasure. For that geometry the object remains a candidate, and the statustable says so row by row.ON THE BETA HYPOTHESIS. beta >= 0 is proved SUFFICIENT and is not provednecessary here; at L = 0 it could not be, since the bare kernel is then thescalar 1. The witness that would make the boundary exact from one site upwardsis authorised by a gate and is not written.THE MECHANISM, IN ONE SENTENCE. Summing out the interior of a half sends F toa vector indexed by its boundary slice alone, after which the bond casereduces to positive semidefiniteness of the kernel -- machine-checked in thecompanion paper -- and the site case to a weighted squared norm, which uses noproperty of the kernel at all.WHERE THE TWO REFLECTIONS DIFFER, AND IT IS NOT COSMETIC. Through a bond thetwo halves are disjoint and meet through one kernel factor, so positivityneeds the kernel itself positive semidefinite -- which, for L >= 1, holdsexactly when beta >= 0. Through a site the halves SHARE the middle slice; theform is then a sum of squared moduli divided by the weight of that slice, soit is non-negative for EVERY beta, negative coupling included, and no propertyof the kernel is used at all.WHAT THE SOURCE WEIGHT DOES. In the companion paper the weight was handled bycongruence: conjugation by sqrt(w) cannot change the sign of a quadratic form.Here it is not conjugated away, it is SUMMED away -- and what is left overdiffers between the two geometries. In the bond case every source weight isabsorbed into the collapse and the bare kernel remains; in the site case theshared boundary slice survives as a factor 1/w(sigma), because each halfcarries that slice's weight and the product would count it twice. Differentreasons, same conclusion: nothing in the hypotheses depends on w beyondpositivity.WHAT THIS IS NOT. No reconstruction: the physical Hilbert space as thequotient of the past algebra by the null space of this form is not built.Nothing here concerns uniformity in the extent, SU(N), the continuum limit, orthe Yang-Mills mass gap.ON THE PRE-REGISTRATION. Four gates were committed before a line of the modulewas written; the status table reports what happened to each. Two of them reada minimum eigenvalue rather than sampling observables, which is the instrumentthe previous campaign's autopsy said was needed.
WHAT IS PROVED, STATED BEFORE ANYTHING ELSE. The reflected two-point form ofthe Gibbs measure, at the two ENDS of a path, for REAL observables of a SINGLEslice, is non-negative under the parity and coupling hypotheses stated below-- at even separation for every beta, and at every separation exactly for beta>= 0 -- and the Gram matrix of a finite family of such observables is positivesemidefinite under the same hypotheses. THIS IS NOT YET THEOSTERWALDER-SCHRADER AXIOM, which quantifies over observables of the wholepast half-chain and over complex ones. The half-chain algebra, the reflectionmap and the sesquilinear form are not built here; the scope section says whatis missing and why it is a construction rather than a further inequality.Eleven papers in this lane end with REFLECTION POSITIVITY IS UNTOUCHED. Thistouches the endpoint form of it, and the interesting part is what the spatialsource weight does -- namely nothing. Earlier papers do prove statements aboutthe coupled kernel; specGap < lambda holds there too. What is new is that thisresult is UNCHANGED by the weight: same hypothesis, same conclusion, and noconstant that depends on w. Every earlier coupled-kernel statement contains anumber that moves when w does. Positivity is the SIGN of a quadratic form, andconjugation by sqrt(w) is a congruence, so there is nothing for the weight tomove.TWO REFLECTIONS, TWO HYPOTHESES. The reflected two-point sum is with v the dressed observable, and the two cases have genuinely differentcontent. Through a SITE (N even) the sum is a square, hence non-negative forEVERY beta -- negative coupling included -- with nothing used but symmetry ofthe kernel. Through a BOND (N odd) it is , which needs Kitself positive semidefinite; that holds exactly for beta >= 0, by aninduction on the extent.AND THE SECOND HYPOTHESIS IS ACTIVE. At beta < 0 and odd separation thesingle-site sign observable gives the reflected sum in closed form, 2(e^beta -e^-beta)^N, which is negative. Congruence is invertible, so the same witnessdivided by sqrt(w) shows the COUPLED kernel is indefinite below zero for EVERYstrictly positive weight, at every extent with at least one site. At L = 0there is nothing to witness -- one configuration, and there the decoupledkernel is 1 while the coupled one is the positive scalar w(empty) -- and thatexception is recorded rather than left to be found. The boundary beta = 0 issharp and exhibited, not inferred.WHAT THIS IS NOT. Neither the full axiom (above) nor any reconstruction: thephysical Hilbert space as the quotient of the past algebra by the null spaceof this form is not built. And the degeneration of specRatio(L) under a ringweight is MEASURED, not proved; no theorem here or elsewhere in the lane saysthe weight destroys uniformity, only that no uniform bound is proved for it.Nothing here concerns uniformity in the extent, SU(N), the continuum limit, orthe Yang-Mills mass gap.ON THE PRE-REGISTRATION. Three ACTIVE gates were committed before any Lean waswritten, and the gates section reports what happened to each. A fourth ispreserved there and not counted: the original Gate B, which failed because ofa design error of ours, and which two of the three active gates replaced.
We formalize in Lean 4 the thermodynamic limit of bounded local Gibbs expectations for a periodic lattice gauge model in a uniform Kotecky-Preiss regime. The proof treats the complete finite-volume sequence: an exact one-volume marked expansion cancels the extensive far gas algebraically, common-window terms are transported exactly, and the remaining boundary contribution is bounded by an existing volume-uniform pinned cluster tail. The resulting explicit Cauchy modulus tends to zero, so completeness constructs an infinite-volume positive normalized real local state. On the intrinsic integer-coordinate local-observable algebra, the state carries a genuine additive action of Z^d and is invariant under every integer translation, including inverses. For SU(2), Haar probability measure, and the physical Wilson plaquette energy Re tr(U), the hypotheses are discharged throughout the explicit punctured intervals 0 < |beta| <= 10^-5 in d=2 and 0 < |beta| <= 10^-6 in d=4. We construct a genuine centered free-boundary exhaustion and prove that its complete cofinal sequence converges to the same state as periodic boundary conditions. The normalized finite-volume two-plaquette truncated-correlation bound also passes to the state under explicit eventual realization and separation hypotheses. We do not claim arbitrary boundary conditions, a C*-algebraic state, a continuum limit, Osterwalder-Schrader reconstruction, or progress on the continuum Yang-Mills mass-gap problem.
Every rate in this lane so far has been a fixed-extent rate. The gap paper provedstrict spectral separation at each extent and said plainly that it was notuniform; the modulus paper gave that separation a number, specRatio(L), andreported measurements saying the number tends to 1 outside the disordered region.A geometric bound whose rate tends to 1 is empty in the volume limit, so nothingin the lane survived L -> infinity, and the word CLUSTERING was never used.PROVED. For the DECOUPLED kernel - the transfer kernel at constant source weight- the modulus is specRatio = tanh(beta) at EVERY extent, with L nowhere in it.Both directions: an operator bound by induction on the extent, and attainment bythe single-site observable the extent paper already built. Composing with themodulus paper's endpoint, the normalised Gibbs two-point function obeys|E[A(X_0)A(X_N)]| <= C_A tanh(beta)^N past one threshold serving everyobservable - a bound whose RATE contains no L, and therefore the first statementin this lane that survives the volume limit.WHY THE PROOF IS NOT THE SPECTRAL DECOMPOSITION. The decoupled kernel is aproduct over sites, so its spectrum is a product; that route needs the spectrumof a Kronecker power, which the library does not carry. It is not needed. Themodulus paper proved that specGap is the GREATEST norm ratio on the fluctuationsector, so bounding it above is an operator inequality and nothing else, and thatfalls to induction: the even part of an observable keeps its mean zero andinherits the rate, the odd part keeps nothing and gets only Schur's test, and thetwo recombine EXACTLY, because tanh(beta) Z = D with Z the row sum and D the oddeigenvalue of a single bond.NOT PROVED, AND A JUDGE THAT FAILED. The COUPLED kernel is untouched. Before anyof this was written we pre-registered two falsifiable predictions. The first -that the decoupled rate is exactly tanh(beta) at every extent - passed to 1e-16,and authorised the work above. The second - that the coupled uniformity boundaryis the Onsager curve - failed on one of eight pre-registered cells, and it staysfailed: that claim is reported as NOT ESTABLISHED, not softened. At constantsource weight the spatial slices are independent, so what is proved here is astatement about a product measure; that is exactly why it is reachable, and it issaid in the paper rather than left to be noticed. Reflection positivity isuntouched, and nothing in this paper is a claim about SU(N), the continuum limit,or the Yang-Mills mass gap.
Two companion papers were left carrying the same debt from opposite sides. Thegap paper proved that every eigenvalue of the coupled slice other than thePerron eigenvalue is STRICTLY smaller in modulus, and said plainly that thisprovides NO MODULUS of separation. The bridge paper proved that the Gibbscorrelations of the spatial system are matrix elements of a self-adjointtransfer operator, and obtained geometric decay only UNDER A CONTRACTIONHYPOTHESIS IT DID NOT DISCHARGE. The missing step is identical in both: finitelymany strict inequalities are not an operator-norm bound.PROVED. We construct specGap, the largest |mu| over the eigenvalues differentfrom the Perron eigenvalue lambda, and prove specGap < lambda. We then prove theoperator bound: for every observable u orthogonal to the Perron vector,||Ku|| <= specGap*||u||. That is exactly the hypothesis the bridge papercarried, so its bound becomes unconditional. The bound is SHARP: whenever thestate space has at least two points, some nonzero fluctuation observable attainsit. The argument splits at specGap = 0, where the maximising index need notsupply a NON-PERRON eigenvector (it does supply the Perron one), hence none inthe fluctuation sector. Stated about an object too: the set of Rayleighnorm ratios on the fluctuation sector has a greatest element, equal to specGap(same two-point hypothesis: with fewer, that set is empty).A WARNING WE STATE BEFORE ANYONE ELSE HAS TO. specGap < lambda is NOTspecGap < 1: the kernel is unnormalised, so both are typically far above one andspecGap^N GROWS. On its own the unnormalised bound controls growth, it does notexhibit decay. The rate that is below one is the RELATIVE one,specRatio = specGap/lambda < 1, and the decay statement is that the fluctuationcontribution is suppressed by specRatio^N RELATIVE to the Perron scalelambda^N. Both forms are proved; only the second is called decay.The step that does not follow from the inequalities is the one abouteigenvectors AT lambda: geometric simplicity, proved in the Perron paper for anarbitrary eigenvector rather than a positive one, makes them INVISIBLE to afluctuation observable, so the top term of the spectral sum vanishes instead ofmerely being bounded.NOT PROVED. specGap DEPENDS ON THE EXTENT, and nothing here bounds it away fromlambda uniformly. Direct diagonalisation gives specGap/lambda = 0.9205, 0.9829,0.9964, 0.9992 at L = 2,3,4,5 for one parameter pair: a geometric bound whoserate tends to 1 is empty in the limit, and that is reported, not hidden. Thenormalised Gibbs EXPECTATION is now bounded too: splitting the dressed constantobservable along the Perron direction bounds the partition function below withno eigenbasis index identified, so at a fixed extent the two-point function isbounded by C*specRatio^N past an explicit threshold. That rate depends on theextent, so this is not clustering. The bound and its attainmentare both proved, which is what it means for specGap to be the operator norm onthe fluctuation sector; what is NOT done is introducing that norm as a definedobject and proving an equation about it. And the threshold N_0 does not dependon the observable: it is built from the dressed CONSTANT observable, so ONE N_0serves every fluctuation observable at once and only C sees A. Reflectionpositivity is untouched, and nothing in this paper is a claim about SU(N), thecontinuum limit, or the Yang-Mills mass gap.
Four companion papers studied an OPERATOR: a strictly positive kernel on thespatial configuration space of a Z_2 slice, its Perron vacuum, and the strictseparation of its spectrum. None of them exhibited a MEASURE. That omission isthe kind this programme is built to notice: a transfer operator is a matrixuntil something says which Boltzmann weights it transfers, and until then avacuum is an eigenvector and a gap is a statement about eigenvalues. Neither isyet a statement about a statistical-mechanical system.PROVED. We define the two-dimensional Gibbs weight of the spatial system fromBoltzmann factors alone - a spatial factor at every time slice, a time-bondfactor between consecutive slices - and prove the DRESSING IDENTITY: that weightequals the path weight of the SYMMETRISED kernel multiplied by a factorsupported entirely on the two boundary slices. Hence every UNNORMALISED Gibbstwo-point sum of the spatial system is a matrix element of an iteratedSELF-ADJOINT transfer operator between boundary-dressed observables, and theNORMALISED expectation is the RATIO of two such matrix elements - the numeratoralone is not the correlation. The generic half holds for an arbitrary symmetrickernel on an arbitrary finite type; no positivity and no structure of theconfiguration space enter it. We further show that the operator the bridge landson is the one the companion papers analysed, that the fluctuation sector isinvariant, and that under an explicit contraction hypothesis the connectedtwo-point function decays geometrically in the time separation.NOT PROVED, AND THIS IS THE POINT OF THE LAST SECTION. The contractionhypothesis is carried as a theorem hypothesis and is NOT discharged. Thecompanion papers prove a STRICT gap with no modulus, and a strict inequalityamong finitely many eigenvalues does not by itself produce the operator-normbound a decay rate requires. Converting one into the other needs the spectralmaximum over the fluctuation sector, which is not constructed here. Inparticular NOTHING UNIFORM IN THE SPATIAL EXTENT is obtained, and none issuggested: with r = r(L) approaching 1, the bound is empty in the limit.Reflection positivity is not addressed. Nothing in this paper is a claim aboutSU(N), the continuum limit, or the Yang-Mills mass gap.
A companion paper supplied the vacuum of the coupled Z_2 slice at every spatialextent: a strictly positive eigenvector, unique up to scale, carrying thespectral radius. It listed PERIPHERAL SEPARATION as out of scope, and thatomission is not cosmetic - without it |mu| <= lambda leaves mu = -lambda open,and no gap follows at all.This paper closes it, and then draws the consequence that matters, which isnegative.PROVED. For a strictly positive kernel on a finite nonempty type, -lambda is notan eigenvalue, hence every real eigenvalue other than the Perron eigenvalue isSTRICTLY smaller in absolute value. Specialised to the coupled slice: at everyextent L, every beta, and every strictly positive source weight, the transferoperator has a strict spectral gap. The proof of peripheral separation avoidsthe equality case of the triangle inequality, which is where the classicalargument spends its effort: writing u = |w|, p = u - w and q = u + w, one getsA p = lambda q and A q = lambda p, so a nonzero p would make A p strictlypositive, hence q strictly positive, hence w nonnegative, hence p = 0.The separation is then extended from the real eigenvalues to ALL of them. Thecoupled kernel is conjugate by a positive diagonal to its symmetrised form,which is symmetric; and a real symmetric kernel has real eigenvalues, by acomputation that pairs the eigenvector against its image twice and is tworearrangements of a double sum. So there are no complex peripheral eigenvaluesleft to exclude, and the strict gap is a statement about the whole spectrum.That composition is itself a single machine-checked theorem(coupled_gap_all_eigenvalues), not a step left to the reader. We also deliverthe vacuum in Euclidean normalisation, norm(Omega) = 1 with T Omega = Omega.NOT PROVED, AND THIS IS THE TITLE. The gap is STRICT, not QUANTITATIVE: thetheorem provides no modulus of separation, and in particular nothing uniform inL. Direct numerical diagonalisation shows the subdominant ratio running0.9205, 0.9829, 0.9964, 0.9992 at L = 2,3,4,5 for beta = 0.8, gamma = 1.2 -collapsing towards 1. That computation is reported as measured and unproved, andno theorem here depends on it; its role is to say that a paper reporting onlythe positive half would be reporting the half that does not matter.Nothing in this paper is a claim about SU(N), the continuum limit, or theYang-Mills mass gap.
Two companion papers established, for a Z_2 lattice gauge slice with a spatialcoupling, that the elementary route to the vacuum stops, and that the naturalreplacement - the Hilbert projective metric - is blind to the coupling anddegenerates in the volume. Both had to work around the same absence: the pinnedmathlib carries no Perron-Frobenius theorem. The first paper could therefore onlysay the vacuum had become unavailable; the second had to build its dominationbound from scratch, and could exhibit the vacuum in closed form only at two sites.This paper discharges that dependency. For a strictly positive kernel on a finitenonempty type we prove, in Lean 4 with mathlib: a strictly positive eigenvectorEXISTS; its eigenvalue is strictly positive; any two strictly positiveeigenvectors are proportional and share their eigenvalue; and every realeigenvector for that eigenvalue is a scalar multiple of it. Together with thedomination theorem of the companion paper this gives the Perron statement thelane needs: the eigenvalue is the spectral radius.The existence proof does not use a fixed-point theorem, because the pinnedmathlib revision contains none. It maximises r over the compact set of pairs(r,x) with x in the simplex and r x <= A x; maximality forces equality, since astrict inequality anywhere would let one further application of A produce anadmissible pair with a larger r. The bound that keeps the set compact is obtainedby summing the constraint: r = r * sum x <= sum (A x).The application is the point. At EVERY spatial extent, and for EVERY strictlypositive weight on the source configuration - the class that contains the coupledkernel of the first paper - the vacuum exists, is unique up to scale, and carriesthe spectral radius. The obstruction of that paper was never an absence; it wasan unavailability, and it was an unavailability of one route rather than of theobject.NO SPECTRAL GAP IS PROVED HERE, uniform in the volume or otherwise, and none isclaimed. Nothing in this paper is a claim about SU(N), the continuum limit, orthe Yang-Mills mass gap.
A companion paper proved that when a spatial coupling is switched on in a Z_2lattice gauge slice, the transfer kernel loses constant row sums, so the uniformvector is no longer fixed and the elementary route to the vacuum stops. Thestandard replacement, when row sums fail, is the Hilbert projective metric: astrictly positive kernel contracts it, and the contraction factor bounds thesubdominant spectral ratio. This paper asks what that replacement gives here, andanswers in Lean 4 with mathlib.It gives no coupling-sensitive and no volume-uniform information, for twoindependent reasons, and both are proved. First, BLINDNESS: the projectivecross-ratio is invariant under multiplication by any nowhere-zero function of thesource configuration alone. The coupled kernel is exactly such a product, so themetric assigns the interacting and the non-interacting kernels the same diameterat every spatial extent - the route cannot see the coupling at all. Second,VOLUME DEGENERATION: two constant configurations realise the cross-ratioe^(4 beta L), so every admissible projective diameter is at least 4 beta L andevery contraction factor obtainable this way is at least tanh(beta L), which lieswithin 2 e^(-2 beta L) of the trivial bound 1. At the one place where the truthis known - the decoupled kernel, whose subdominant ratio the companion papercomputes to be exactly tanh beta at every L - this route already returnstanh(beta L) instead. The degeneration is the method's, not the model's.We then hand over the object the elementary route stopped producing, at thesmallest interacting size. In the character basis the coupled two-site kernelsplits into two 2x2 blocks, and we exhibit a strictly positive eigenvector inclosed form, together with a second exact eigenpair. The identity A - B = 4between the two decoupled even-sector eigenvalues drives every estimate. Theblindness is proved two-sided, so it covers the symmetrised conventionw^(1/2) K w^(1/2) as well, and the positive eigenvector is proved to dominateevery eigenvalue, real or complex - so its eigenvalue is the spectral radius,which is the Perron statement this development needs and proves without aPerron-Frobenius theorem in the library.NO VOLUME-UNIFORM STATEMENT ABOUT AN INTERACTING SYSTEM IS PROVED HERE, AND NONEIS CLAIMED; the general-L behaviour is recorded separately as measured andunproved. Nothing in this paper is a claim about SU(N), the continuum limit, orthe Yang-Mills mass gap.
For beta > 0 let I_m = I_m(beta) denote modified Bessel functions ofthe first kind, and set a_m = I_m^2 [(m-1) I_(m-1)^2 + (m+1) I_(m+1)^2], b_m = m I_m^4, F_A(t) = sum_(m>=1) a_m sin(mt), F_B(t) = sum_(m>=1) b_m sin(mt), E(t) = F_A(t)/(2 F_B(t)).The global ratio-monotonicity problem for the surface expansion of atwo-dimensional SU(2) lattice gauge observable is: (i) F_B > 0 on(0,pi), and (ii) E' < 0 on (0,pi), for every beta > 0. Bothstatements are proved. Positivity of F_B has two exact proofs. Ratiomonotonicity is reduced to exact algebraic identities andoutward-rounded interval certificates: small and compact beta arehandled by pair identities and interval Taylor models;20 <= beta <= 1000/9 by a direct Wronskian cover; andbeta >= 1000/9 has three certified moving-edge lambda lanes. Theremaining lambda >= 3 lane is closed by the exact identityE'/(-sin(t/2)/2) = Q + X_full, where Q > 19/20, an exactmain--mirror--rest decomposition, and a division-free covariancecertificate proving X_main > -1/20 on two adjacent rectangles thatcover the full angular interval. The exact near and far relay marginsare positive. All load-bearing production and independent replaytranscripts are checked for exact rational coverage, dependencyhashes, strict outward-rounded decision endpoints, and byte equality.The structural core is exact: E is, as an algebraic identity, the meanof cos(psi) under the midpoint law of a four-step killed von Misesbridge; the generating kernels reduce, via the Neumann additiontheorem, to two-dimensional integrals of a single Bessel functionwhose saddle deficit is an exact sum of two squares; and exact saddlecancellations yield the coefficients of the verified closedsecond-order law E = cos(t/2)(1 - c(t)/beta) + O(beta^-2), c(t) = (4 cos^2(t/4) - 1)/(2 cos(t/4) cos(t/2)).Three certified negative results (interval arithmetic, twoimplementations, nested enclosures) kill every monotone full-pathcoupling, with an exact mechanism at threshold beta |cos t| = 3/2. Atthe pi endpoint we also give exact identities for the cubiccoefficient c_3 (telescoped alternating form, integral form, parity)together with its verified prefactor law. Every claim is labelledexact / certified / verified; the machine-checked lemmas are Lean4/Mathlib, machine-checked modulo classical Bessel inputs carried asnamed hypotheses.
Two companion developments verified an Osterwalder-Seiler reconstruction end toend for a lattice gauge chain whose spatial slice is a single point. Every stepof that chain begins by knowing the vacuum, and in the one-dimensional case thevacuum is free: the normalised transfer kernel has constant row sums, so theuniform vector is fixed, and T*Omega = Omega follows from normalisation alone.This paper asks what survives when the slice acquires spatial extent, andanswers in Lean 4 with mathlib.The algebraic half survives untouched. With time bonds only, the row sums of thetransfer kernel are constant for every spatial extent L, so the uniform vacuumpersists on a space of dimension 2^L; and the single-site sign observable is aneigenvector whose normalised eigenvalue is exactly tanh beta, with L free in thestatement.The uniform vacuum does not survive. Switching on a coupling between sites inside aslice makes the spatial weight depend only on the source configuration, so itfactors out of the sum over the target and the row sums becomeconfiguration-dependent. We exhibit two explicit configurations of a two-siteslice with different row sums, and conclude that no constant row sum exists: theuniform vector is not fixed, so T*Omega = Omega is FALSE for it. The vacuumbecomes a Perron vector that row-sum normalisation no longer supplies in closedform, and every later step of the reconstruction loses its starting point.We state plainly what the positive half is and is not. The decoupled system is Lnon-interacting copies of a two-state system, and the rate it yields - theeigenvalue tanh beta of the single-site sign mode - is independent of L fortrivial reasons, so it is physically empty and is recorded only because itisolates which half of the construction survives. NO GAP FOR THE COUPLED SYSTEM IS PROVED HERE, AND NONE IS CLAIMED.Nothing in this paper is a claim about SU(N), the continuum limit, or theYang-Mills mass gap.
The Osterwalder-Seiler reconstruction passes from a reflection-positive measureto a Hilbert space by quotienting out the null space of the reflected pairing.In a companion development that step was present but did nothing: the pairingthere was definite, so the quotient was the identity, and that paper says so inits own abstract. This paper supplies the missing case, in Lean 4 with mathlib.For the Z_2 lattice gauge chain we take half-space observables of two timeslices - a four-dimensional space - and form the reflected pairing directly fromthe Boltzmann weights. For beta > 0 the reconstructed physical space istwo-dimensional. Integrating out the future collapses four observables onto twostates, and that collapse is the null space. We prove: the pairing factors through anindependently defined reconstruction map Phi; its self-pairing rearranges into amanifest sum of two non-negative terms, from which positivity and the null spacefollow together; the null space is EXACTLY ker Phi, not merely non-empty; anexplicit non-zero observable lies in it; and the quotient is isomorphic to thephysical space BY THE MAP Phi ITSELF, not by a dimension count.The degeneracy is the mechanism the reconstruction exists to handle, and the oneexpected to reappear in systems with larger half-space algebras. What is notclaimed: this is twotime slices and not m; still Z_2, one variable per slice, fixed finite size, andnot volume-uniform; Z_N for N > 2 is untouched; and the completion step of thereconstruction is trivial here because every space in sight isfinite-dimensional, which we state rather than present as work done. Nothing inthis paper is a claim about SU(N), the continuum limit, or the Yang-Mills massgap.
We give a finite-dimensional Osterwalder-Schrader reconstruction for an anisotropic Ising model on open rectangular strips and prove a spectral gap whose rate is uniform in the finite spatial extent. For temporal coupling beta, spatial coupling gamma, and parameters satisfying 0 < alpha < 1 and 2 tanh|beta| + 2 tanh|gamma| <= alpha, one number m > 0 works for every spatial length. Reflection positivity is proved directly for the Gibbs measure; the null space of the reflected form is identified with the kernel of an explicit boundary-collapse map; the OS quotient is linearly equivalent to the finite boundary-vector space; and the transfer operator forced by the site and bond forms is intertwined with a symmetrised transfer matrix.Dobrushin comparison supplies positive Perron data and a projected operator norm bounded by exp(-m) for every spatial length. Consequently, connected reconstructed correlations decay at the same common rate. Lean 4 checks thecomposition without assuming reflection positivity, Perron data, a vacuum, a gap, or clustering at the public endpoint. This replacement extends the point-slice, fixed-size v1 to finite strips with one rate uniform in spatial size. It does not construct an infinite-volume operator or Hamiltonian, provea unique particle excitation or relativistic mass shell, or claim a Yang-Mills mass gap.
We machine-check, in Lean 4 with no sorry and no project axioms, theOsterwalder-Seiler reflection positivity of a lattice gauge theory with finiteabelian gauge group. The development is organised so that the three ingredientsare separated and each is proved on its own: an analytic step, a geometric step,and the single place where a property of the Boltzmann factor is actually used.The analytic step is that a crossing kernel of the formK(x,y) = sum_i c_i phi_i(x) conj(phi_i(y)) with c_i >= 0 is positivesemidefinite, that this class is closed under products, and that it is closedunder conjugation by a positive diagonal. Formulating the hypothesis as anon-negative combination of characters rather than as non-negativity of Fouriercoefficients removes any need for Bochner's theorem on a finite abelian groupand for the Schur product theorem: the development uses no spectral and nomatrix-positivity API.The geometric step is a splitting of the configuration space across thereflection plane under which the reflection is the swap and the Gibbs weightfactors as w(x) w(y) K(x,y). We prove that the Osterwalder-Seiler pairing of anobservable of one half against its reflection is then exactly the quadratic formof w(x) K(x,y) w(y), so that reflection positivity follows from the analyticstep.The physical step is the instance. For Z_2 the Wilson factor exp(beta s),s = +-1, expands in the two characters with coefficients(exp(beta) +- exp(-beta))/2, both non-negative exactly when beta >= 0; so theZ_2 Wilson crossing kernel is positive semidefinite at non-negative coupling. Asingle endpoint combines a gauge system with a nontrivial time reflection, aconcrete splitting, that weight at positive coupling, and the conclusion; itsplaquette straddles the reflection plane, so the entire Gibbs weight is thecrossing kernel. It is a two-edge system, and a full temporal box is nottreated. For Z_N with N > 2 the coefficients are discrete Bessel-type sums andtheir non-negativity is not established here.We are explicit about what is absent: no Gelfand-Naimark-Segal quotient, notransfer operator, no identification of a Euclidean correlator with a matrixelement, and therefore no mass gap. Nothing here is a claim about SU(N), thecontinuum limit, or the Clay problem.
We give a machine-checked reduction of the finite-dimensional fluctuation integral in Balaban's CMP116 large-field analysis. Starting from the physical block constraint Q, the formal development constructs a sparse right inverse E and the constraint-elimination operator C = I - EQ. It proves QE = I, QC = 0, C² = C, the exact sparse norm ||EB|| = M^(d-1)||B||, and the volume-independent bound ||C|| ≤ 1 + M^(d-1) for d ≥ 3. An exact physical/CMP116 isometry transports C to finite Gaussian coordinates without norm loss. The same development constructs the physical localization projector P_Z0, evaluates the complex quadratic Gaussian, localizes its determinant to rank |I(Z0)|, performs the outer Gaussian integration, and absorbs both costs into an explicit exp(c|Z0|) factor.Two corrections exposed by formalization are central. First, the useful domination occurs after Gaussian integration rather than through an unavailable pointwise supremum in the fluctuation field. Second, the localized quadratic matrix is A = -alpha_5 P_Z0. In the exactly identified trivial-background sector, the terminal Lean theorem inserts the concrete C, the flat Hessian, complement localization, and covariance root directly into the printed source Gamma_k = C^T Delta_k (C P_Z0^c)(C^(k))^(1/2), returning an explicit Cauchy bound without an ambient-volume factor. CMP116, however, requires the base Hessian at a generally nontrivial small background Ubar. We do not construct D²S_Wilson(Ubar) or the random-walk estimate (2.16), and therefore do not prove the physical domination, (2.26), hraw, hRpoly, a continuum limit, or a mass gap. The contribution is an auditable reduction that closes the constraint and Gaussian layers and identifies the first genuinely missing interacting construction.
We present an end-to-end Lean 4/Mathlib formalization of the exact evaluation of the two-dimensional SU(2) heat-kernel lattice model on certified finite combinatorial disk cellulations. The development starts from normalized Haar probability on the concrete matrix group SU(2). It identifies its transport to S^3 with the canonical spherical measure, proves an all-order orbital integration formula, derives translated character convolution, and passes from finite character sums to the infinite heat-kernel semigroup by dominated convergence. A genuine shared-edge integral then yields the two-face Migdal move.The geometric layer is independent of any reduction tree. A cellulation stores vertices, paired half-edges, cyclic face words, incidence, Euler characteristic, and positive face areas. Connected dual graphs admit certified elimination schedules, every valid schedule reduces to the heat kernel at total area, and all schedules give the same amplitude. For the original edge model, a rooted spanning tree produces a measurable, product-Haar-preserving gauge equivalence SU(2)^E ≃ SU(2)^(V{r}) × SU(2)^(ET). A compatible tree-cotree construction then retains the exterior holonomy rather than integrating it out. For every certified physical disk cellulation, the boundary-conditioned original-edge amplitude is exactly the SU(2) heat kernel at the total face area. Coefficient extraction gives, for every irreducible label n, the normalized exterior-boundary identity E_P[W_n(H_boundary)] = exp[-n(n+2)(sum_f t_f)/4], where H_boundary is the retained holonomy of the complete exterior boundary word. The universal record is demonstrably inhabited: a concrete three-spoke disk has (V,E,F)=(4,6,3) and derived dual graph K_3. A reproduced audit covers 177 audited declarations, explicitly including both headline theorems, and finds only propext, Classical.choice, and Quot.sound in their dependency cones. The analytic solution is classical. The contribution is a concrete kernel-checked composition from Haar measure and characters to physical edge variables, gauge fixing, tree-cotree elimination, and the exact boundary-observable endpoint.
Inside a Lean 4 formalization programme for four-dimensional SU(Nc) lattice Yang—Mills, we report two machine-checked negative results and the machine-checked infrastructure that makes them meaningful. The positive substrate is: (i) a fixed-volume Combes—Thomas chain for self-adjoint coercive finite-range lattice operators, instantiated on the flat gauge-fixed covariance of the physical shell, with coercivity constant c = min(1,a)/C_P fed by a proved fixed-volume flat Hodge/block-Poincaré inequality; and (ii) the concrete adjoint model of SU(n) — su(n) with the trace inner product, dim_R su(n) = n² − 1, and the isometric transport to Euclidean coordinates — so that the abstract adjoint-model interface has a concrete nontrivial inhabitant and the flat-lane results can be instantiated with the genuine matricial adjoint model. The first wall states that, under the block normalization actually used by the formalized chain, every flat Hodge/block-Poincaré constant obeys L^d/L² ≤ C_P on the fine torus of side LNu2032, hence the volume-uniform Poincaré gate is provably false for d ≥ 3 and Nc ≥ 2, and no positive coercivity constant survives all volumes through this route. The route consumed by the fixed-volume endpoint is therefore closed by theorem. A second wall stands in the fluctuation sector. For d ≥ 3 and a transported half-period square-wave mode on the exact fine side (2M)Nu2032, the formalization proves ||QA||² ≤ (2M)u207b¹||A||², the exact identity = 8((2M)Nu2032)u207b¹||A||², and therefore a Rayleigh numerator at most 9(2M)u207b¹||A||². Every quotient Poincaré constant is thus at least 2M/9, so the volume-uniform fluctuation-sector gate is also provably false for every positive Nu2032, d ≥ 3, Nc ≥ 2, and every adjoint model. Everything stated here is checked by Lean 4 against a pinned Mathlib, with zero sorry, zero project axioms, and a committed axiom-oracle transcript. A dependency record, theorem-artifact map, and reproduction instructions expose the complete proof chain. Both walls concern the current unscaled line-integral block map with the current unweighted coarse norm; neither gate is claimed to be necessary, equivalent, or exhaustive for Yang—Mills theory. No claim toward a continuum construction or a mass-gap theorem is made.
For SU(N_c) Wilson lattice gauge theory on d-dimensional periodic tori (d >= 2), we present a machine-checked (Lean 4, pinned Mathlib) renormalization-group interface for two-plaquette truncated correlators, in which every structural ingredient is a theorem rather than a postulate: the scale transformation is a concrete decimation map, defined once -- measurable, local, and gauge-covariant -- and its induced pushforward preserves probability; the effective measures are its literal iterated pushforwards of the Wilson Gibbs measure; the multiscale decomposition of the correlator is proved by telescoping, never carried as data; the terminal scale of the decomposition is a fixed index kTerm(n) = n with typed range 1 <= kTerm(n) <= n, which excludes, in the type, the circular depth-zero layer in which an infrared clause would hypothesize the bound being sought; the conditional decay conclusion is stated in the physical distance 2^n u with a single constant pair (C, m) quantified before every torus base and depth; and the terminal observable is operationally support-certified: the infrared object consumed by the interface equals a base-measure integral of an explicitly composed pullback observable whose dependence is contained in a transported support set, for which the separation lower bound 2^n(2u) - (2^n + 1), strictly positive on the whole interface window, is proved. The design is deliberately adversarial: four natural naive formulations are presented together with the explicit countermodels that defeat them -- scalar relabeling of known decay, sink flows on measures, clamped scales and per-volume constants, and depth-zero circularity -- and with the typed repairs that exclude each. The central hypothesis, PhysicalTerminalScaleWilsonGate, is an open proposition: no witness is provided, and the infrared/ultraviolet bounds it demands of the actual Wilson measure are exactly the open analytic mathematics (Balaban-type single-scale estimates). The final theorem is conditional: a witness of the gate yields |Cov(2^n u)| <= C e^{-m 2^n u} with one pair (C, m), m > 0, for every base M_0 >= 4 and every depth n >= 1. No mass-gap claim, no claim of gate satisfiability, and no thermodynamic or continuum limit is made or implied.
SUPERSEDED BY AI.VIXRA:2607.0039V1 FOR THE ORIGINAL-EDGE AND TREE-COTREE CLOSURE. This replacement preserves public version 1 and corrects its publication-level scope. Version 1 establishes the compact-group analytic chain, heat-kernel reductions, schedule independence and reduced-model results, but pages 7 and 9 leave open the bridge from the post-gauge-fixed evaluator to the original-edge physical integral for arbitrary cellulations. ai.viXra:2607.0039v1 supplies the missing original-edge gauge fixing, simultaneous face-holonomy transport and compatible tree-cotree closure and is therefore the authoritative source for the full exact simple-loop area-law theorem at the stated finite two-dimensional SU(2) heat-kernel scope. Neither paper constructs four-dimensional continuum Yang-Mills theory or proves a four-dimensional mass gap. The preserved version 1 follows the two-page notice unchanged.
Amos's upper bound for the modified Bessel function ratio, rho_n(x) = I_{n+1}(x)/I_n(x) < x/(n + 1/2 + sqrt((n+1/2)^2 + x^2)) = B_n(x), is a classical theorem, and its derivation through the qualitative theory of the associated Riccati equation is an established technique. This paper contributes, to our knowledge, the first formalization: a complete, machine-checked Lean 4 proof of the bound for every integer order n >= 0 and every x > 0, over the power-series definition of I_n carried in the same pinned development, with axiom oracle [propext, Classical.choice, Quot.sound] and no analytic hypothesis of any kind. The formalized route runs through the Riccati equation rho_n' = 1 - ((2n+1)/x) rho_n - rho_n^2 (itself derived from the formalized series calculus), the observation that B_n is exactly the positive root of the Riccati quadratic, a small-argument zone bound uniform in n obtained from pure geometric tail estimates, and a first-crossing barrier argument in a transformed variable psi_n = x(1/rho_n - rho_n) whose structural feature — every touch of the critical level forces rho_n' = 0, so the barrier never needs to be differentiated — is the simplification this formalization contributes. As corollaries, the unit-step inequality, the strict monotonicity of the logarithmic derivative across orders (in deriv form), and a phi-monotonicity step used by a lattice-gauge surface expansion all become unconditional theorems. The theorem is proved for the in-core power-series definition of the integer-order modified Bessel function; no formal identification with an external special-functions library object, and no extension to noninteger order, is claimed.
The Amos-type upper bound on the modified-Bessel ratio, I{nu+1}(x)/I_nu(x) < x/(nu + 1/2 + sqrt((nu+1/2)^2 + x^2)), has a distinguished algebraic property: its right-hand side U satisfies the exact calibration identity 1/U - U = (2nu+1)/x. From that identity alone — by ordered-field algebra, with no further analytic input — follow a unit-step inequality rho_nu - rho{nu+1} < 1/x for consecutive ratios, the strict increase of the log-derivative (log I_nu)' = rho_nu + nu/x across orders, and the strict monotonicity of a phi-sequence arising in a two-dimensional lattice-gauge surface expansion. We formalize this calculus in Lean 4: a single module defines the bound once (AmosBound) and proves the calibration engine and four consequence theorems through that one definition, together with two rational satisfiability witnesses whose Amos hypothesis holds by exact Pythagorean arithmetic; all eighteen Lean statements of the development pass the axiom oracle with exactly [propext, Classical.choice, Quot.sound] against a pinned Mathlib. A certified companion (256-bit interval arithmetic, self-contained series-plus-tail enclosures, committed transcript) certifies the bound provably strictly at all 1206 points of a pre-registered grid covering the arguments the applications consume. A Bessel interface completes the closure: integer-order I_n is defined by its power series in the same pinned development, with positivity, the three-term recurrence, the termwise-differentiated derivative identity I_n' = I_{n+1} + (n/x) I_n, and the logarithmic-derivative identity (log I_n)' = rho_n + n/x all proved as theorems, so the consequence theorems — including the unit step read as strict log-derivative monotonicity, in deriv form — hold for genuine Bessel ratios with the Amos bound as the single remaining hypothesis. The scope is stated exactly: the Amos bound itself remains a classical cited theorem taken as hypothesis — this paper unifies its three previously scattered uses in our formal development into one named proposition with one oracle and one certified numerical witness, and no downstream result changes its verification class.
We present a machine-checked quantitative toolkit for cluster expansions of polymer systems with excluded regions (holes), in the discrete cube geometry of Balaban-Dimock renormalization-group analyses. Five Lean 4 theorems, checked against a pinned Mathlib revision, provide: (i) the identity sum_T prod_v c_T(v)! = n! C_n for child factorials over spanning trees of the complete graph K_(n+1), with the rooted-tree majorant 4^n as corollary; (ii) a marked-root leaf summation for the tree-graph majorant of an Ursell-type expansion with holes, with the moment constant M paid once at the root and closed leaf ratio 4M^2 per additional vertex, together with its Catalan-sharpened form M^(2n+1) C_n (a gain of order n^(3/2) in the n-th coefficient); and (iii) a target-preserving orderwise bound in which the target union itself survives until the modified-metric exponential is extracted. A certified companion (interval arithmetic, 120-bit precision, committed transcript with a committed reproduction witness) tabulates the smallness gate and encloses every derived constant. Non-vacuity is machine-checked: a concrete hole family satisfying every hypothesis is exhibited in Lean, and the two distinct hypothesis sets among the polymer-facing theorems are both instantiated at it with a strictly positive weight. Each claim is labelled with its verification layer: exact (Lean theorem), certified (interval transcript), or paper-level.
SUPERSEDED BY AI.VIXRA:2607.0089V1 FOR THE TERMINAL GLOBAL-SIGN CLAIM; INDEPENDENT REPLAY PENDING. This replacement preserves public version 1 and records the later terminal-claim paper without overstating its audit status. Version 1 proves positivity of F_B, the ratio sign for 0 < beta <= 3, exact bridge and single-Bessel reductions, certified negative results, asymptotic structure and a two-scale closure map; pages 11-13 leave the global sign as a quantified conjecture. ai.viXra:2607.0089v1 claims the global ratio-monotonicity theorem through exact identities and outward-rounded interval certificates. It supersedes this record for that terminal claim, but independent replay of every exact-to-Arb handoff and interval regime remains pending in this audit. No PASS is inferred from a printed or green transcript alone. The preserved version 1 follows the two-page notice unchanged.
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
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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).
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.
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.
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.
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.
