Every hypohamiltonian snark and every permutation snark has a unique multiset of cyclically 5-edge-connected factors of the same class under successive decompositions along cycle-separating 4-edge-cuts. We prove that hypohamiltonicity passes from a dot product to both of its snark factors. Together with the factorisation theorem for bicritical snarks, this answers a question of Goedgebeur and Zamfirescu. For permutation snarks, we extend the atom argument of Chladný and Škoviera using colourable cut multipoles and closure under their canonical completions.
A mix of human-written and AI-generated textHuman understanding: some partsSnarksfactorisation
Two exact four-species counterexamples refute necessity of a D-unstable core for instability of a positive equilibrium. The first has five reactions, parameter-rich kinetics, an explicit generalized-mass-action equilibrium, and a rational Jacobian with eigenvalue 1/500 + (51/500)i. The second has six reactions, at most two molecules on each side of every reaction, no catalysts, and classical mass-action kinetics with fixed integer kinetic orders. At a positive rational equilibrium its Jacobian has integer entries and an unstable nonreal eigenvalue with real part between 0 and 5, certified by an exact shifted-quartic intermediate-value argument. The networks share a reactant incidence relation and 24 nonempty child selections. Exact symmetrization, triangular-block and cubic-factorization certificates show that every child matrix is D-nonunstable in both networks. A permanent zero eigenvalue in an exceptional child distinguishes D-noninstability from strict D-stability. Both counterexample theorems have Lean 4 proofs with Mathlib. A separate asymptotic calculation explains the elementary example through circulation and a fast branching coordinate. The results establish strict linear instability, without asserting a Hopf bifurcation or periodic orbit.
Primarily AI-generated textHuman understanding: some partsD-unstable coresLean formalizationchemical reaction networksclassical mass actioncounterexampleelementary bimolecular reactionsgeneralized mass actionparameter-rich kinetics
A real square matrix A is D-stable if DA is Hurwitz for every positive diagonal matrix D. Applying the critical-point method to Johnson’s contact variety establishes that, in every dimension and for every positive choice of barrier weights, D-stability is equivalent to Hurwitz stability and absence of positive solutions of an explicit principal-minor Fritz–John system. In dimension five the system separates into Lagrange and tangency branches; both equivalences are registered in Lean. Further results treat regular-contact persistence, the boundary of the D-stable set, stability under bounded time-scale ratios and the minimal destabilizing ratio. The finite decision procedure also uses conventional finiteness and termination arguments and was executed exactly only in dimension three. For the five-dimensional Lagrange system the multihomogeneous Bézout number is 1200; 980 generic complex solutions were observed numerically. Worked examples include a Lotka–Volterra community, a published refinery-column gain matrix and an exactly recovered benchmark window. Formal statements, conventional arguments, exact certificates and numerical illustrations are distinguished throughout.
We study constant-weight nonnegative integer vectors under half-L1 distance and compare arbitrary codes with full intersections of affine additive-subgroup cosets with the discrete simplex. A heavy-coordinate decomposition gives the exact reduction A_q(n,n-s) = q + B_q(n,s) to codes whose coordinates are bounded by s. It determines the unrestricted optimum throughout 2n > (q+1)s. For every q ≥ 3 and s ≥ 2, at n = qs−1 and D = (q−1)s−1 the unrestricted and full-coset optima are respectively q+1 and q; we classify all q² unrestricted extremizers. We retain a complete classification of both optima for three coordinates and n < 2D. In the next ternary band, 2D ≤ n < 3D, eight words are attainable precisely when 3n ≥ 8D; equality yields an infinite eight-versus-seven full-coset separation. Exact finite certificates also give a six-versus-five gap at (q,n,D) = (4,5,3) and the quinary single-deletion values A_5(9,2) = 143 and A_5(10,2) = 201. The proofs use intersection bounds and explicit integer affine combinations. The unrestricted decomposition extends known methods, and bibliographic novelty remains unresolved, particularly against a closely related 2024 paper not compared in full.
Contributed by Deep Bhattacharjee, Ushashi Bhattacharya
We prove unconditionally the conjecture of Gurvich and Naumova that for every n≥1 the set {1,…,n} can be partitioned into parts of at most three elements whose sums are powers of 3. The partition is given by an explicit recursion.
A mix of human-written and AI-generated textHuman understanding: some partsformal verificationpowers of threeset partitionzero-sum triples
For positive-weight quasihomogeneous isolated complete intersections of minimal embedding codimension two, we prove μ≥4pg+1 in dimension two and μ>Cn,2pg in every dimension n≥3.
A mix of human-written and AI-generated textHuman understanding: some partsDurfee inequality
We study exact longest monochromatic runs of power-residue colourings, including neutral prefixes and least canonical witnesses. A certified rational cover represents every relevant interval inside a short signed character table. A reflection-compatible schedule retains an anchor; conservative filters reject only known contradictions, bilateral completion recovers the full run, and tie-preserving reduction selects its least witness. We prove this exact-success contract and describe its CUDA realization with separate arithmetic domains below 232 and 233. On one RTX 5090, two reserved cohorts give ratios 24.00 and 10.95 of sums of hot per-prime medians against the identified historical dense engine. A separate batch study gives throughput factors 1.23-2.58 against an already optimized reference. The experiments have different timing boundaries and their factors are reported separately. Short-table multiplicative extraction has close algorithmic antecedents in cryptanalysis; our contribution is its certified organization for exhaustive maxima and witnesses and its measured implementation. Square-root-sized cover parameters do not give a general O(p) running-time bound.
Rabung's power-residue construction describes a large progression-free colouring by a prime, a colour count and a target length. We give six additional primitive certificates, checked by full ordinary CPU scans, which yield fourteen direct lower bounds. In colour-first notation, these include W(3,17)>33166190897, W(3,18)>100289348022, W(3,19)>259446988783 and W(3,20)>326368775450. The larger ternary primes lie outside the complete census below 232. We prove the equivalence of the run and boundary tests with Rabung's criterion and separate pointwise verification from discovery. A finite closure of explicitly identified earlier constructions places the new bounds in context: direct ternary certificates win through length 23; Berlekamp and binary recurrence inputs win thereafter. Exact-modulus cyclic witnesses also give six-colour product bounds. The previously published thirteen-claim evidence and its closed empirical analyses are retained separately. Comparisons refer to the stated source catalogue, without an exhaustive global priority or optimality claim.
Primarily AI-generated textHuman understanding: no partsRabung certificatesarithmetic progressionscomputer-assisted proofslower boundspower-residue colouringsrecurrence closurevan der Waerden numbers
Let G_X be the graph on the positive integers at most X, with an edge between n and np whenever p is prime and np <= X. Write lambda_1(X) >= lambda_2(X) >= ... > 0 for its positive adjacency eigenvalues, counted with multiplicity. We prove that, for every fixed 21/61 < theta < 1/2 and every integer X^theta/log X <= a <= X^theta,
lambda_a(X)^2 = pi(X/a) + O_theta(a/log X),
where pi is the prime-counting function. This identifies the eigenvalue at the prescribed rank a throughout a polynomially growing band of centres. For every 0 < theta < 1/2, ordered mean-square bounds give a weaker estimate outside o(X^theta) centres below X^theta. When 1/3 < theta < 1/2, the same O_theta(a/log X) error holds outside o(X^theta) centres using only global prime estimates. The results extend to deletion of any fixed finite set of prime directions.
Edges labelled by primes exceeding sqrt(X) form disjoint stars. Retaining their first-exit interactions gives local comparison operators and a family of approximate eigenvectors. Bounds on their overlaps and residuals control eigenvalue counts across clusters, allowing ranks to be recovered where the global perturbation bound no longer separates neighbouring star levels. The Baker--Harman--Pintz lower bound and the Brun--Titchmarsh upper bound supply the short-interval prime counts used in the uniform every-centre theorem.
A mix of human-written and AI-generated textHuman understanding: all partsmatrix perturbationordered eigenvaluesprime number theoremprime-cover graphsshort intervalsspectral graph theory
We encode binary messages in completely multiplicative sign functions and read them along the three sequences n2+1, n2+2 and n2−2. Two observations with the same exact cofactor remove the unknown smaller-prime signs when multiplied. Allocating shared large primes before pairing embeds ordinary binary error-correcting codes in each trace. Under an explicit large-factor mass hypothesis, a computable schedule gives one infinite encoding with positive prefix rate and a linear errors-and-erasures allowance. A first-occurrence filter makes its raw readings depend causally on a linear message prefix, while encoding and decoding remain polynomial in the numerical observation length. Its supporting primes have arbitrarily small harmonic cost. A finite construction, independent of the analytic hypothesis, stores 273 bits in 8736 readings of each trace and corrects every pattern of 499 reading errors. Complementary constructions give persistent sign prescriptions, uniform laws for the active coordinates of words, and compact multiplicative subgroups whose nontrivial quotients have positive quadratic entropy. The word-law and entropy conclusions are not asserted for the polynomially decoded encoding. A separate matrix construction gives another causal positive-rate map. Fixed-scale balancing provides an unconditional quarter-sign bound against every small-prime assignment. All coding statements concern deliberately constructed families, not recovery of an arbitrary multiplicative source. The sparse constructions at superlinear cutoffs keep their large-factor mass input explicit.
Primarily AI-generated textHuman understanding: no partsarithmetic codingcausal encodingerror-correcting codeserrors and erasuresmultiplicative functionsquadratic tracesunknown prime signs
We study which completely multiplicative sign functions can be identified from the statistics of h(n2+1), h(n2+2) and h(n2−2). In a class of finite products of quadratic characters of odd prime conductor, completed without zeros and modified at primes with summable inverse square roots, diagonal correlations of orders two and four determine the function. The second-order data recover the conductor and prime corrections, leaving exactly the ramified local signs undetermined. Fourth-order detectors recover those signs, including at the small primes; the inverse also extends to quadratic components at two. No bounded horizon suffices uniformly: for every prescribed window, two finite-conductor sources can share its complete law and disagree with density arbitrarily close to one half on each trace. For an arbitrary prescribed source h, we construct factors fg=h such that, for each factor separately, the three traces jointly have the uniform word law at every horizon. The choice is continuous and computable relative to h, with a finite bound uniform in h for approximation of a canonical synchronous coupling. The same choice treats every horizon sequence o(loglogN). The factors need not be independent. Their synchronous empirical law projects exactly to the product's empirical word law; when a limiting canonical coupling exists, it retains the product's word law, which need not identify h as a function.
A completely multiplicative sign function is determined by its values on n2+1, n2+2, and n2−2, but exact determination does not quantify how many readings detect a negative prime sign. We distinguish three losses of information: partial compatibility tests, geometric restrictions on certificates, and fractional relaxation of signed constraints. After fixing a prime head, we describe the exact fibre of a window and the parity constraints joining two windows. For any fixed head and any sequence HN=No(1), one table with o(N) negative entries passes every HN-window test and every two-reading test. A separate finite table passes these tests without any global source. We bound certificate capacity - fractional packing under unit reading loads - for narrow and binary certificates and for low-degree transport libraries whose links cancel odd prime valuations in pairs. A fixed algebraic curve of certificates for a target prime p has at most logarithmic capacity, or capacity at most one when a reading is constant; in the infinite unanchored case its common poles have field Q(p). Conversely, selected values of n2+1 can force linearly many errors while the capacity of all certificates stays bounded. Two internally disjoint pair families compose without loss on their existing readings, whereas sixteen actual values exhibit Pack=3<Cap=4<μ=5. The selected-sample constructions do not give an upper bound for the minimum on a complete prefix.
A completely multiplicative sign function may be changed so that two quadratic traces remain identical at every index while a third becomes normal. We classify this freedom for the traces n2+1, n2+2 and n2−2. After two traces and a finite prefix have been protected, a monic irreducible polynomial can be made normal from every prescribed source exactly when the field generated by one of its roots contains neither protected quadratic field. In the complementary case, every value of the polynomial is already fixed. A single decidable set of primes of relative density zero supports a continuous, relatively computable completion treating all admissible polynomials jointly, with uniform bounds for each fixed observation family. The proof combines weighted root counts with a sieve for moments whose order increases with the observation scale, followed by a selection using exact finite normalisers. We also give a sufficient condition on prime availability, explicit sparser reservoirs, and a divergent reservoir on which some sources have no completion with a convergent ordinary mean, despite convergence of all fixed word laws in probability. For larger protected families, the obstruction is governed by odd products of quadratic square classes. An explicit example distinguishes certificates for every numerical value from a rational norm identity in the variable.
We study completely multiplicative sign functions h through their values at positive integers of the forms n2+1, n2+2 and n2−2, with observed indices 1≤n≤N. First occurrences of prime factors give a canonical information set and a complete, prefix-compatible family of parity relations, each involving at most ⌊log2N⌋+3 readings. These relations yield exact bounded-distance repair, including corrupted information coordinates, in time NO(1)(4+⌊log2N⌋)t for t errors; erasures enter through an explicit rank parameter. We then construct arithmetic redundancy without estimating a quadratic regulator. If h(p)=−1 and K=logN/logmax(3,p), the three traces contain at least exp(exp(clogK/loglogK)) negative readings for every fixed 0<c<log2 and sufficiently large K, uniformly in the prime p and all other prime signs. The same number of each sign occurs on one trace between Nα and N, for each fixed α<1. The proof combines short reconstruction certificates, Lehmer divisibility and balanced divisors. Coprime blocks give compact access to these families without factoring large recurrence terms. Under independent reading noise with fixed erasure probability 0≤a<1 and conditional flip probability 0≤b<1/2, every source prefix P=⌊exp((logN)δ)⌋, for fixed 0<δ<1, is recoverable from O(P) readings with polynomially small failure probability in P and preparation polynomial in the numerical P and logN. A fixed positive erasure rate nevertheless leaves exponentially many assignments of observed large-prime signs. No positive-density inverse for unrestricted sources is asserted.
The cost of changing prime signs depends on how the resulting multiplicative function is observed. On a fixed prime support of finite harmonic sum, we prove that existence of a polynomial mean for every sign assignment is equivalent to uniform prime truncation and uniform convergence to local products. The criterion applies separately to ordinary and logarithmic means. For independent activations with finite expected harmonic cost, the almost-sure truncation defect is the averaged large-prime parity probability, not a sum of incidences. These criteria separate averaging methods and observation geometries. One common support is logarithmically stable for every fixed polynomial and every assignment, yet a fixed perturbation of (−1)v2(n) on that support has divergent ordinary means on n2+1,n2+2,n2−2. Collective upper sieves give stability at critical counting exponents without critical summability, and harmonic stability on rectangles whose side lengths have asymptotically equal logarithms. For widths Xα, Gaussian prime estimates give the matching universal budget threshold when 119/500<α<1; the negative result concerns only the sum of two squares. Around every completely multiplicative sign function, fixed perturbations of arbitrarily small ∑p−σ cost can have divergent logarithmic means on n2+1 for 125/164<σ≤1. A positive oscillation bound is common to this range, although the starting scales may depend on σ. Window laws further distinguish all origins from almost all origins.
Primarily AI-generated textHuman understanding: no partsharmonic costmultiplicative functionsordinary and logarithmic meansparity activationspolynomial observationsprime perturbationssieve methods
For every prescribed completely multiplicative sign function, every integer polynomial irreducible over Q of degree at least two and eventually positive, and every prescribed prefix, we construct a completion preserving the polynomial trace and the prefix exactly, with vanishing correlations along all fixed admissible affine families. The changed primes have relative density zero, although the changed integers have asymptotic density one half and the changed primes have infinite harmonic mass. One noise trajectory has these properties almost surely at all positive real times. The construction combines a classical root-free prime reservoir with decreasing clock rates. Its probabilistic input is affine regularisation against arbitrary bounded weights fixed before the noise, uniformly in time after a vanishing delay. Marks retain their possibly non-convergent empirical distribution. For non-selective noise, the local channel acts through irreducible factors rather than output coordinates; square identities determine its invariant characters, and rational constants supply a time-evolving random translation. The proofs separate arithmetic moments, temporal concentration and finite Fourier analysis. The constant-source existence case already follows from work of Klurman-Mangerel-Teräväinen. The conclusions concern the constructed function, not new cancellation for the unperturbed source.
For signs s,t, let Ns,t(N) count the ordered representations N=n+(N−n) with λ(n)=s and λ(N−n)=t. We prove that, for every fixed ε>0, all four counts are at least cεN1−ε at every sufficiently large integer total. The constants are ineffective through Siegel's theorem. The same conclusion holds uniformly for completely multiplicative sign functions whose positive primes satisfy ∑f(p)=1p−θ≤M, with fixed 0<θ<1 and M. Separately, at each sufficiently large prime p, both patterns with product (−1/p) have at least p/(4(logp)96) representations. All four satisfy this stronger bound in six reduced classes modulo 24 and outside one very sparse family of primes. The proof combines finite reflection-to-dilation propagation, a short-product spectral test, reconstruction from the units, and signed counting of prime disagreements. For the uniform extension, candidates are united before a common exceptional population is removed. None of the conclusions gives a positive-density bound or cancellation of the reflected correlation at every total.
Let G_X be the graph on the positive integers n <= X in which n is joined to np whenever p is prime and np <= X. Let A_X be its adjacency matrix, let lambda_1(A_X) be its spectral radius, and let pi(X) count the primes up to X. Then
lambda_1(A_X)^2 = pi(X) + 4 sum_{p <= X} 1/(p - 1) + o(1).
More generally, fix a finite set S of primes and retain only the positive integers divisible by none of the primes in S. Let a_1 < a_2 < ... list these integers, and let pi_S count the primes outside S. For every fixed j, the j-th largest positive eigenvalue satisfies
lambda_j(A_{S,X})^2 = pi_S(X/a_j) + 4 sum_{p <= X, p not in S} 1/(p - 1) + c(a_j) + o_{S,j}(1),
where c(a) = sum_{q divides a, q prime} (q - 5)/(q - 1). The proof splits the edges according to whether their prime label exceeds sqrt(X). The large-prime edges form disjoint stars, while the remaining adjacency has norm of order X^(1/4)/sqrt(log X) but zero first-order matrix element on each star eigenvector. An explicit first-exit resolvent calculation gives a degree-ratio correction with remainder O_{S,j}(1/log X) and localization of the extreme eigenvectors. The prime number theorem and Mertens estimates then evaluate that correction as 4 log log X plus an explicit constant, yielding the displayed o(1) formula. The same constant-layer formula and localization hold uniformly for retained centres a <= (log X)^(1/2-epsilon), with finite-sum error O_{S,epsilon}(a^2/log X). The paired positive and negative modes also yield uniform centre--shell adjacency evolution. The explicit finite correction determines its phase with uniform o(1) error whenever |t| = o(sqrt(X log X)/a^(5/2)). The uniform finite-prime-sum law, its largest-eigenvalue specialization, and the explicit-frequency centre--shell dynamics theorem are formalized in Lean 4 and registered in the Palomar Registry as PALOMAR-2026-08-29-000017, version 2.
A mix of human-written and AI-generated textHuman understanding: all partsFeshbach-Schur mapadjacency spectrumdivisibility cover grapheigenvector localizationprime-counting functionspectral graph theory
For the complex Tate curves E_p = C*/p^Z attached to primes, we compute the classical divisorial correspondence groups. For distinct primes, Hom(E_p,E_q) is nonzero exactly when log(p)log(q) belongs to 4 pi^2 Q. If log(p)log(q)/(4 pi^2) = u/v in lowest positive terms, this group has rank one and its generator has isogeny degree uv. The isogeny relation between distinct primes is a partial matching: unconditionally, each prime has at most one partner. These rigidity statements specialize results of Winkelmann; the explicit Hom generator and degree are included here. Outside the matching, the Neron-Severi group of E_p x E_q consists of the two fiber classes, which act trivially on H^1. On every diagonal surface End(E_p) = Z, by Gelfond-Schneider, and the intersection profile of multiplication graphs is independent of p. Consequently no connected classical correspondence system can involve three distinct prime fibers. Version 2 withdraws the broader universal no-go assertion of version 1 and states the exact cross-fiber support restriction required by the obstruction.
Primarily AI-generated textHuman understanding: some parts
For a finite ordinary endotactic reaction network, measurable reaction rates in a fixed positive compact box admit a compact convex invariant absorbing set in every positive stoichiometric class. Global positive Carathéodory solutions are unique, and entry times are uniform over compact initial sets and all admitted rate histories. The argument combines the finite affine construction of OpenAI with an endotactic witness estimate that retains the reaction projection lengths. We then quantify what permanence permits as reverse rates shrink. For autonomous weakly reversible networks with ke=kˉeεae, eventual reciprocal concentration bounds depend on the exponent spread Δ=a+−a− through εΔ/(2H), where H>0 is a network-dependent geometric constant chosen before the individual rates. In an explicit three-species system with small positive rate parameters p and h, prepared concentrations have scales ph3, ph2, ph, and the worst-case prepared recovery time lies between (400/1313)(p2h7)−1 and (31/10)(p2h7)−1. A weighted inventory gives necessary restoration and depletion times from arbitrary positive seeds. Further results treat common clocks, reverse-exponent families and a conventional stochastic lower comparison. Two registered Lean developments support the permanence theorem, continuous dependence and selected deterministic quantitative results. Their precise coverage is separated from conventional extensions and numerical illustrations. Permanence at fixed positive rates supplies neither a uniform guarantee at zero reverse rate nor a global sharp-corridor recovery deadline.
We study the problem of image reconstruction from projection data by combining the Fourier Slice Theorem with scattered-data approximation in the Fourier domain. The unknown image is assumed to belong to the space of functions of bounded variation (BV), allowing discontinuities that naturally arise in practical tomography. In contrast to conventional band-limited formulations, no assumption is imposed that the image itself is band-limited. Instead, the finite bandwidth is attributed to the image acquisition system through the detector response, finite sampling resolution, and associated measurement process. Consequently, the measured Radon projections contain only a finite range of spatial frequencies, and the reconstruction objective is the recovery of the instrument-limited image determined by the measurable Fourier bandwidth.
To exploit the structure of the Fourier Slice Theorem, we introduce a symmetric embedding of the image on the torus together with the corresponding symmetrization of the Radon data. The resulting Fourier transform becomes real-valued and even, allowing the Fourier-transformed projection data to be interpreted as real-valued scattered samples of the two-dimensional Fourier transform on a fixed bounded frequency domain. This reformulates image reconstruction as a scattered-data interpolation problem in the Fourier domain.
The interpolation problem is analyzed using the recently developed approximation theory for multivariate BV functions from scattered data. The resulting convergence error is expressed in terms of the total reconstruction error separating into an instrument error, determined solely by the finite bandwidth of the acquisition system, and a reconstruction error arising from interpolation of the measured Fourier data.
The proposed framework provides a mathematically rigorous and physically realistic formulation of Fourier-slice tomography in which the limitations of the imaging hardware are distinguished from the accuracy of the reconstruction algorithm. Numerical experiments demonstrate the applicability of the method to image reconstruction from limited projection data and validate the predicted convergence behaviour.
Primarily human-written textHuman understanding: all parts
We consider bin-packing instances with 3m items of sizes in (1/4,1/2), total size m, and configuration linear programming value m. Let d be the relative dimension of the polytope of zero-sum multiplicity patterns obtained by centering sizes at 1/3. We determine the maximum additive integrality gap for every d≤10. It is zero for d≤4 and one for 5≤d≤10. More generally, a representation of the multiplicity vector by at most five zero-sum patterns admits an integral packing, and a representation by at most eleven patterns requires at most one additional bin. The integral-packing result has a structural proof; the additive-one result combines structural restrictions with exact finite enumeration. For arbitrary d, we construct an LP-tight subinstance with at most 6d+3 items and no smaller additive gap. An integer representation preserving the signs of triple sums then yields a finite rational family attaining the worst gap, which is computable in time 2O((d+1)2). Finally, we construct a sharp family with simplex pattern polytopes and a linearly growing zero-sum matching deficit. Its eleven-dimensional member violates the zero-sum remainder condition used in the ten-dimensional proof. Thus the bound of eleven fractional patterns in that condition is sharp.
Primarily human-written textHuman understanding: all partsbin packingconfiguration linear programminghypergraph matchinglattice polytopeszero-sum patterns
We study which information about an autocatalytic reaction network suffices for exact composition with other modules. For the maximum amplification factor, threshold-indexed species-price cones compose by exact set operations, while scalar optima do not. Autocatalytic child-selection cores are reconstructed from complete, uniquely matched ordinary-core anchors using alternating exchange paths and certified rational semipositivity tests. For irreducible unary diluted source networks, positive-composition certificates characterize growth, criticality and extinction under arbitrary species degradation; a related certificate identifies partial-actuation stabilizability. Further manuscript results treat factor interactions, exact boundary traces, contextual indistinguishability and junction-tree composition under running intersection and scope extensionality. Separator width bounds message arity, without a runtime claim. Eleven foundation statements are Palomar registered in Lean 4; the broader historical formalization and conventional extensions retain their separately documented scope. Thermodynamic and passive-tail examples illustrate exact interfaces and information loss. Structural amplification and linear onset conclusions are distinguished from nonlinear kinetic dynamics.
In the sparse binary polymer model with reversible ligation and cleavage, we prove that for every fixed food horizon, fixed nonnegative catalysis intensity and fixed linear reaction budget, the probability of a nonempty reflexively autocatalytic and food-generated set within that budget tends to zero, at rate O(1/n). The probability that a single molecule catalyses a self-constructing reaction set also vanishes, cleavage included. A productive-prefix argument excludes bounded catalyst counts, while an injective acyclic dependency encoding supplies a factorial saving in the count of productive reaction supports. Consequences include divergence of minimum RAF size relative to polymer length and of minimum catalyst count, together with conditional forms. Combining the lower bound with the published quadratic-witness result of Hordijk and Steel gives a superlinear-to-quadratic window. The principal estimates and consequences are formalized in Lean 4 against the literal split-position model; the published upper-witness input is cited separately.
We study the Langevin dynamics of spherical mixed p-spin glasses at fixed temperature, on time scales exponential in the dimension N. Below the static transition temperature, the dynamics started from the uniform distribution on the sphere stays at total variation distance close to one from the Gibbs measure up to a time exponential in N, with probability tending to one. We prove this for pure models of degree at least four at every such temperature, and for mixtures of minimum degree at least four that satisfy an explicit condition on a function of one variable. It extends to initial distributions that are independent of the disorder or depend weakly on it. Between the dynamical and static transition temperatures, we show for general finite mixtures that, with probability exponentially close to one, the generator has exponentially many exponentially small eigenvalues; a fixed part of the centered spin lies in the corresponding slow modes; the stationary spin correlation stays above every level below q∗2 up to an exponential time, where q∗ is an explicit root of the plateau equation of the physics literature; the dynamics started from most points of the Gibbs measure stays far from equilibrium; and the two endpoints of the stationary dynamics share mutual information at least linear in N. With probability tending to one, the correlation stays above every level below q∗. Near the dynamical transition, exponentially many eigenvalues stay below e−cN, where c is proportional to an explicit power of the distance to the transition.
Primarily AI-generated textHuman understanding: some partsdynamicsspin glass
We prove that the Gorenstein symmetry conjecture for finite-dimensional algebras over arbitrary fields is equivalent to its restriction to the finite prime fields. A counterexample specializes to a finite field without changing its dimension or its finite self-injective dimension. In characteristic zero, a counterexample yields counterexamples of one fixed dimension over every prime field outside a finite set.
A mix of human-written and AI-generated textHuman understanding: some partsGorenstein symmetry conjecture
Let A be a finite-dimensional representation-finite algebra over an arbitrary field, and let M be a nonzero finite-dimensional right A-module. We prove that EndA(M) has Cartan determinant one whenever it has finite global dimension. In particular, the Cartan determinant conjecture holds for representation-finite algebras over arbitrary fields.
A mix of human-written and AI-generated textHuman understanding: some partsCartan determinant
We prove NP-completeness of deciding whether a finite reversible reaction network contains a productive autocatalytic core when neither a target species nor an allowed-food set is specified. The input is a binary encoding of literal nonnegative integer reactant and product matrices, with a signed real flow on each reaction identity. A nonsingular square-witness characterization gives polynomial certificates. A weighted incidence construction reduces prescribed two-path directed linkage to unrestricted core detection, and an explicit constant-size switch construction reduces SAT to the linkage instances. Actual machine-level NP membership and a total polynomial-time binary transducer complete the classification. The terminal theorem is verified in Lean 4. The result concerns structural possibility in this source model and imposes no elementary reaction arity or conservation constraint.
The computational difficulty of D-stability, raised by Chen, Fan and Yu (1995, doi:10.1016/0167-6911(94)00036-U), is addressed by an explicit integer-matrix reduction from PARTITION. For positive weights of total 2T, the construction produces a matrix of order m + 4 that is D-stable exactly when no subset has weight T. When an equal split exists, some positive diagonal scaling has an eigenvalue with strictly positive real part. A Pell recursion supplies the parameters, with a logarithmic search bound and an explicit bound on every matrix entry. These bounds give polynomial binary output length. Together with the classical NP-completeness of PARTITION, the reduction yields coNP-hardness of D-stability and D-semistability, and NP-hardness of strict D-instability, even for a four-dimensional core with attachments at one coordinate. The hardness is in the weak sense for binary encoding. Running-time analysis and the trivial-instance cases are treated conventionally; registered Lean statements verify the explicit arithmetic encoding, bounds and promise dichotomy.
Primarily AI-generated textHuman understanding: some partsD-stabilityLean formalizationPARTITIONPell equationcoNP-hardnessdiagonal scaling
Assuming the Generalized Riemann Hypothesis, the explicit formula for the L-function of an elliptic curve E/Q bounds the analytic rank of E in terms of the traces a_p for p < e^(2L), for any admissible test function supported on a window [-L,L]. In rank computations this test function is usually fixed in advance, typically to a Fejer-type kernel; optimising it for an individual L-function goes back to Booker, and an earlier test on quadratic twists of rank about 7 found the gain to be small. We choose, for each individual curve, the test function that minimises the resulting bound within a finite-dimensional space supported on [-L,L], certify the result, and measure the gain systematically, up to the current record curves. The minimiser is the solution of a linear system with the Gram matrix of the Weil quadratic form of E, and the optimal bound equals the reciprocal of the corresponding reproducing kernel at the central point, which we call the radix Riemanniana of E. Because the Fejer-type kernel lies in this space, the optimised bound is never weaker. We certify every bound with ball arithmetic, check the prime-side evaluation against the zero side computed independently from the zeros of L(E,s), and benchmark against the fixed kernel on 1642 curves from Cremona's tables, 2500 curves from the Stein-Watkins database and a set of curves with conductors up to 3*10^16. Depending on the sample and on the stopping criterion, the number of primes required to pin down the analytic rank decreases by 8-23% in aggregate. For individual curves of large conductor the decrease ranges from 0% to 43%. Within each sample the gain is largest at rank 0 and decreases with the rank; it grows again, and by a much larger factor, in the regime of large rank and large conductor where the computation is most expensive. On the smallest-conductor curves of rank 28-31 in the ICARM leaderboard the optimised kernel needs 84-85% fewer primes to certify the rank. For the current record, the rank-32 curve #4239 of Elkies, Klagsbrun and Li, it certifies r_an<=32 under GRH with 2.53*10^9 primes - the first upper bound for this curve - and hence r_an=32 under GRH and BSD, while the fixed kernel, certified at L=12.40 with 2 481 819 193 primes, is still 0.39 above the target. On all 395 leaderboard curves of rank 24-31, with primes below 10^10, the optimised kernel certifies r_an<= r on 379 curves and the fixed kernel on 372; where both succeed the fixed kernel needs a median of 4.0 times as many primes. We validate the fixed-kernel baseline by reproducing every published value of Bober and of Klagsbrun, Sherman and Weigandt with an independent implementation.
Primarily human-written textHuman understanding: all parts
We study the arithmetic behind long runs in a complete census of power-residue colourings below 232, covering every admissible order 2≤r≤9. The dataset contains 203280221 primes and 677591249 profiles, with exact quadratic four-window counts at every odd prime. A bounded study of its deep tail identifies algebraic windows with fewer independent character conditions than their lengths suggest. We certify their power-class groups and conjugate-root overlaps, obtaining exact Kummer densities for fixed templates, including eight additional order-four models. In a quadratic family, eleven constant positions force twenty-six neutral positions at odd character orders. In two further families, norm identities generate all remaining higher-root relations. A length-53 cubic realizes a three-wise independent binary code whose monochromatic probability is 4096 times that of independent signs; its local run law is computed exactly. A separate stratified million-profile ternary sample is analysed with the same finite catalogue filter applied to the data and to an independent-colour comparator. Finite census statements, selected explanations and asymptotic template laws concern distinct populations. No universal law of global maxima or finite-range error estimate for the template densities is asserted.
Can a Gaussian on a noninteger-dimensional fractal give a probabilistic realization of Edelman's ghosts and shadows? We distinguish a Gaussian-weighted Hausdorff measure, a radial Gaussian law, and a Gaussian random vector or field.
For every real β>1, we construct a closed cone of Hausdorff dimension β whose Gaussian-weighted β-dimensional Hausdorff measure has exactly the χβ radial distribution. The construction relies on exact radial
volume growth, not merely on a dimension calculation. We prove a metric obstruction below dimension one and explain why the cone does not supply ordinary Gaussian addition or a scalar algebra. A separate Gram-determinant argument rules out preserving all polynomially continued Euclidean Gaussian Gram identities in a positive model. We then give proofs of the angular beta-gamma identities, associativity of Bessel-Kingman convolution, and its Gaussian convolution semigroup for β≥1. Squared Bessel processes, tridiagonal and bidiagonal matrix models, Dunkl Gaussians, and Gaussian fields
on fractals provide complementary constructions with explicitly different meanings of dimension. None of these statements asserts the existence of a full noninteger-dimensional scalar algebra.
Primarily AI-generated textHuman understanding: no partsRandom matrices; fractals; hypergroups
Let Ap count the monochromatic consecutive four-windows of a nontrivial cubic character modulo a prime p≡1(mod3), with zero excluded and overlapping windows counted separately. We give an exact formula for Ap in terms of fixed elliptic traces and a finite boundary correction. Its complex-multiplication terms are related cubic twists. This produces two arithmetic fluctuation regimes: the first four moments of (27Ap−p)/p tend to (0,44,0,4224) when three is a cube and to (0,17,0,738) otherwise. Over all admissible primes they tend to (0,26,0,1900). The proof combines a four-position identity following from Greene's hypergeometric formalism and Lennon's trace formula with symmetric-power automorphy, cyclic base change and a finite collection of Rankin-Selberg prime number theorems; it uses no GRH. A direct geometric proof exposes the phases and exceptional points, with the genus-two and isogeny constructions of Freeman-Satoh and Guillevic-Vergnaud explicitly credited. Point-order and Hasse certificates verify numerical examples without enumerating their billion-position character words. The conclusions concern fixed pattern length and four moments; no complete limiting distribution, constant-order centring of the raw count, or law for logarithmic-length maxima is proved.
Let r≥3 be an integer, let q be an odd prime
power, and set A=Fq[T] and F=Fq(T). We study the
Galois representations associated with rank r Drinfeld modules
of the form
ϕT=T+g1τ+⋯+grτr,
where gi∈A and gr=0. Ordering the coefficient
tuples g=(g1,…,gr) by their maximum degree,
we prove that, for a density-one set of tuples, the
λ-adic Galois representation is surjective for every
prime λ of A, and the adelic Galois image contains
SLr(A). Moreover, for every tuple in this set,
the index of the adelic image equals the index of the
corresponding rank-one determinant representation and hence
divides q−1. These results extend Zywina's theorem in
rank 2 to arbitrary ranks r≥3. As a further
consequence, we establish the existence of rank r
Drinfeld modules with surjective adelic Galois representations
for every r≥3, extending previous existence results
which were known for prime values of r.
Primarily human-written textHuman understanding: all partsDrinfeld modulesGalois representations
A multiplicative identity can persist at every scale while disappearing from a rare-event count. Another identity can survive as a cluster whose points lie far apart. Which distinction matters depends on what is observed. We explain this through one completely calculated model: a uniform germ can trigger two occurrences in disjoint spatial windows. Its two marginal counts become Poisson, but their sum need not do so. A coupling identifies the correct same-grid target and separates the mean number of points from the mean number of active germs. We then isolate the additional work required by multiplicative sources: private prime coordinates, weighted relations between windows, corrections to prime-site intensities, and relations visible at the target scale. A final passage from marked configurations to records explains the roles of ties, recognition and observation clocks. Selected results are stated with their hypotheses; a short concordance leads to their proofs. The elementary calculations are proved here, while the arithmetic and functional limit theorems are taken from the four cited research articles.
Primarily AI-generated textHuman understanding: no partsPoisson approximationarithmetic clustersexpository articlemarked point processesrandom multiplicative functionsrare events
We study how changing prime signs affects fixed-word frequencies, rare-event laws and extremes of completely multiplicative sequences. The cost is dσ(f,g)=∑f(p)=g(p)p−σ. A known theorem of Frantzikinakis-Lemańczyk-de la Rue preserves normality at finite d1; we give its elementary specialization. Using explicit arithmetic inputs from Long runs, we then establish three families of results. For each fixed 0<σ≤1 and almost every base, reversible surgery realizes every stationary finite-intensity marked point-process law along selected scales, with signs, exact excess lengths and converging actual intensity, at arbitrarily small dσ cost and with a prescribed prefix unchanged. Almost surely over all integer scales, the minimum number of prime changes needed for a run of length ⌈clog2N⌉ is asymptotic to cαclog2N for 1<c<3/2, where αc∈(0,1/2) satisfies H2(αc)=1−1/c and H2(u)=−ulog2u−(1−u)log2(1−u). At each scale, with probability tending to one, symbolic and private large-prime repair maxima coincide on the same random base for kN=o(logN); unrestricted prime repair agrees when kN=o(loglogN). This comparison transfers the classical lattice laws of runs with errors to adaptive prime repair. Elementary programming from arbitrary bases and an autonomous finite-range-dependent process construction complement these results.
We formalize in Lean 4 with Mathlib Roy's strong six exponentials theorem: if x1,x2 are linearly independent over Q and y1,y2,y3 are linearly independent over Q, then one of the six products xiyj lies outside the Q-vector space spanned by 1 and the logarithms of algebraic numbers. The proof closure uses only Lean's three standard axioms. Along the way the development proves Waldschmidt's transcendence theorem on Gad0×Gmd1 in the form used by Roy, and Philippon's multiplicity estimate with derivatives on Gad0×Gmd1 over every algebraically closed field of characteristic zero, following Roy's account in Waldschmidt's book. We reproduce the public Lean statements, summarize the mathematical route, and record how the artifact is verified. The code is available at https://github.com/telaviv/strong-six-exponentials.
Primarily AI-generated textHuman understanding: no partsLeanformal verificationinterpolation determinantslinear forms in logarithmsmultiplicity estimatesstrong six exponentials theorem
The Jensen moment of Balazard, Saias and Yor expresses the zeros of the Riemann zeta function outside the critical line through the mean value of the logarithm of its absolute value on that line. Jensen's inequality converts any mollifier into an upper bound for this contribution. For Dirichlet-polynomial mollifiers, the bound cannot fall below 0.0386, a barrier arising from the pole at s = 1.
We instead use the reciprocal of a smoothed Euler factor, denoted E-star(X,s), constructed from prime powers up to a constant multiple of X together with an explicit pole term. The quotient of the zeta function by this factor is then expressed solely in terms of its nontrivial zeros. We prove that the resulting bound B(X,s0), for any center s0 with real part greater than one half, tends to zero under the Riemann hypothesis (RH). Consequently, RH is equivalent to the convergence of B(X,s0) to zero. If the supremum of the real parts of the zeros is attained and exceeds one half, the bound grows at least as fast as X to a positive power divided by log X.
Under the same attainment assumption, RH is therefore equivalent to boundedness, as X tends to infinity, of a single weighted mean-square energy: the integral over the real line of the squared absolute value of zeta divided by E-star on the critical line, weighted by 1/(1+t squared).
Numerically, the bound decreases to 0.0015 at X = 10,000 for the Balazard–Saias–Yor center s0 = 1. At s0 = 2, it decreases approximately as 0.037 divided by log X up to X = 1,000,000, close to a heuristic prediction involving the constant kappa = 1.828... . By contrast, the corresponding quantity grows for the Davenport–Heilbronn function. We describe the geometry of the factorization and use two control functions to identify the additional ingredients required for a proof of boundedness of the mean-square energy.
A mix of human-written and AI-generated textHuman understanding: all partsBeurling primesDavenport-Heilbronn functionEuler productJensen formulaPoisson-Jensen formulaRiemann Hypothesisexplicit formulamollifier
We characterize the three-element sets of lucky car indices that occur when a Stirling permutation of order n is used as a parking function. A set {1,x,y}, with 1<x<y<2n, is admissible precisely when x is odd, or x is even and y is odd with x<n, or both are even with x+y≤2n. Explicit Stirling words realize every permitted case. The number of admissible sets is (n−1)(n−2)+2nh−3h(h+1)/2, where h=⌊(n−1)/2⌋. This answers the three-element characterization and enumeration question in Problem 46 of Colmenarejo et al. The complete classification, parking constructions, and enumeration have been checked in Lean. HAL deposit: hal-05800210v1.
Primarily AI-generated textHuman understanding: no partsLean formalizationStirling permutationslucky setsparking functions
Let A(G,x) count vertex subsets of a graph G that induce forests, by their cardinalities. Barton, Brown, and Pike asked whether failure of coefficient unimodality forces A(G,x) to have a root with positive real part. We give a negative answer. The join of K46 with three disjoint copies of P3 has a nonunimodal acyclic polynomial of degree nine, yet all its complex roots have strictly negative real part. We derive the polynomial by classifying induced forests and give its exact positive Hurwitz determinants. A Lean formalization verifies the graph-to-polynomial identity and the full counterexample, using an exact rational Lyapunov certificate for strict stability. HAL deposit: hal-05800220v1.
Primarily AI-generated textHuman understanding: no partsHurwitz stabilityLean formalizationacyclic polynomialcounterexamplegraph polynomial
We construct a bipartite graph on 230 vertices whose acyclic polynomial, which enumerates vertex subsets inducing forests, is not unimodal. This answers Question 3.3 of Barton, Brown, and Pike negatively. The graph has a common set of 50 vertices and 15 branches, each with two sets of six vertices. A partition according to the number of selected common vertices gives a three-term formula for the complete induced-forest polynomial. Its coefficients have a strict descent from degree 62 to degree 63 and a later strict ascent from degree 66 to degree 67. The actual graph, the unrestricted subset-counting formula, and the exact inequalities have been checked in Lean. HAL deposit: hal-05800225v1.
Primarily AI-generated textHuman understanding: no partsLean formalizationacyclic polynomialbipartite graphscounterexampleunimodality
We show that excluding nested pairs can increase the number of vertices needed to force a matching of one color, disproving a conjecture of Barát, Gyárfás and Tóth. A matching consists of edges with no shared endpoint. In a fixed vertex order, two edges are nested if both endpoints of one lie between those of the other. For r colors and a target of k edges, the classical theorem guarantees a matching of one color on (r−1)(k−1)+2k vertices when every pair is joined by an edge. For every r,k≥3, we construct a coloring at this very threshold in which every such matching contains a nested pair. Thus the nonnested threshold is larger by at least one vertex. Already with three colors, ten vertices force three disjoint edges of one color, while our coloring makes every such triple contain a nested pair. The self-contained proof combines endpoint counts with two edge recolorings that obstruct the pairings allowed by the vertex order.
Primarily AI-generated textHuman understanding: all partsRamsey numbernonnested matchingordered graph
A marked handlebody state must retain its filling relations and coefficient control before complementary capping or root specialization. We construct boundary modules and regulated integral lattices that keep the disk and cyclotomic residues of one finite-stage representative together. Standard handle-attachment and recoupling results supply the geometric inputs; the comparison constructs endpoint lattices, separating transport, raw and saturated quotient towers, and exact cap domains. These objects supply the companion rootwise theory, while the two completed topologies remain distinct and root specialization remains restricted to its declared domain.
Primarily AI-generated textHuman understanding: some partsHeegaard splittingcapintegral latticejoint completionskein module
How does a rare-event field determine the history of its extremes? Starting from the complete signed run-field comparison in Long runs and rare patterns, we develop records, censoring, occupation and Gaussian limits for the resulting lattice Poisson target. The arithmetic input retains every site, full exact length and actual sign, at growing intensity under a one-factor information-intensity budget. Strict integer records, their signed imbalance and logarithmic leadership time have a joint phase-uniform Brownian limit with explicit covariance, independent of the terminal critical field and its extremal paths. Right censoring gives scalar M1 convergence for the visible maximum and J1 convergence for the first-holder sign. Physical occupation instead has a nondegenerate phase-indexed limit. Canonical interpolation connects threshold thinning and a stationary scale flow without identifying their record rules or their Gaussian constants. The signed spatial theory includes weighted exact levels, fixed-window Ornstein-Uhlenbeck paths, two correlated threshold-compression noises, and rectangle fields with explicit grid centering. Direct partition and quadratic-reconstruction proofs provide alternative routes. For each specified observation, the target calculations identify its limit and the dependencies within the joint constructions.
We determine when rare observations of completely multiplicative functions have a Poisson law and when arithmetic relations force clusters. For independent prime values on the circle with uniformly bounded L2 densities, we approximate the field of hits of site-dependent targets of Haar mass O(N−1) on deterministic masks in [2,CN]. The target retains the true prime-site intensities and uses Haar at composite sites, even when the total prime contribution diverges. The exponent 2 is sharp in the uniform Lq scale for this target and class of masks. Below this threshold, even correct prime intensities can fail at site resolution while the total count remains asymptotically Poisson. A fixed finite exceptional prime environment, possibly atomic and internally dependent, instead produces a conditional cluster law together with an exact environment-determined configuration. With Haar outside that environment, its integer relation lattice determines the compound-Poisson limit of the corrected prefix count and characterizes simple Poisson convergence. Further applications treat rational observations on sparse moving supports, permanent bounded prime biases, and - using a binary arithmetic relation profile - long Borel patterns in a finite abelian group times a torus. Classical marked Chen-Stein comparison provides a common interface. A separate deterministic-sequence interface gives correlation identities, transfer criteria and a resolution obstruction, without proving cancellation for Liouville.
We study shrinking-target occurrences of completely multiplicative functions with arbitrarily moving phases on a fixed finite set of exceptional primes and independent Haar values at the remaining primes. After exact removal of the environment-only points, effective integer-subgroup profiles classify all subsequential marked cluster limits, while target-scale pair overlaps characterize Poisson convergence. The quantitative input is a direct Haar comparison with the moving germ kernel, uniform in the exceptional phases and with stretched-exponential error. For deterministic profiles and interval targets, the complete spatial covariance measure reconstructs the unmarked law through primitive multiplicative rays and finite interval geometry. A positive forest formula makes this reconstruction valid even with unbounded cluster sizes. For a fixed known bounded Borel target, retaining positions and angular marks yields the corresponding identification from the complete marked covariance, using signed overlap channels. Restricting the observation to an interior spatial window imposes both angular and logarithmic-spatial constraints. The maximal number of descendants of one germ becomes a rectangle-occupancy number for a subgroup of R2, finite exactly when that subgroup is discrete. Explicit rank-one laws illustrate what counts forget, while environmental mixtures show why deterministic identification does not extend by averaging covariances.
An explicit compactly supported positive-momentum state yields a summed backflow greater than 0.06190 over two unit-duration intervals separated by a gap of 3/2, in units where the reduced Planck constant is 1 and the mass is 1/2. An executable numerical certificate encloses its value approximately between 0.0619551065 and 0.0619551266, conditional on stated arithmetic and library assumptions. The state is defined by stored binary-rational coefficients and is re-integrated with enclosed quadrature and rounding errors. This is a trial-state lower bound, not an optimality result. It exceeds a published conjectural numerical range for equal spacing, but does not establish superiority to the true equal-spacing supremum or publication priority.
We construct rootwise and formal Rees boundary data from the marked modules and lattices of the companion boundary-module construction, keeping their distinct coefficient and completion domains. Using a Newton interpolation basis, we determine the Smith invariants and the integral image of the map induced by the Hopf pairing. Factorial/Rees covariance and compatible finite-stage representatives produce a strict rootwise/Rees module with its theta endpoint. At an admissible odd prime, no continuous homomorphism extends algebraic root specialization to the entire completed Rees source; further obstructions exclude unrestricted Newton padding and a state-natural lift of a contracted scalar identity. Established perturbative, Habiro, and stated-skein/Morita constructions give closed comparisons after the declared cap or sewing map, with their boundary state objects kept separate.
Primarily AI-generated textHuman understanding: some partsHabiro capHopf pairingMorita sewingRees completionroot of unity
Contributed by Deep Bhattacharjee, Ushashi Bhattacharya
We fix the gauge of four-dimensional lattice Yang--Mills theory on a field
of straight lines that meets every plaquette in exactly one link, so that
each free link interacts with twelve others instead of eighteen. In this
gauge the theory has a unique infinite-volume state with exponential decay
of correlations and a positive mass gap at fixed lattice spacing, for SU(2)
beyond the limit of every single-link argument in a gauge of this kind, by
blocks made of the three free links of a plaquette, for
every SU(N) beyond the curvature limit of the ungauged theory, and for SU(3)
on a range more than two and a half times as large as was known before,
where the same blocks raise the gap at the end of that range by two orders
of magnitude and carry the range past it to βW=0.76.
A mix of human-written and AI-generated textHuman understanding: some partsBakry-Emery curvatureDobrushin uniquenessball arithmeticgauge fixinglattice gauge theorymass gapplaquette blocks