The Macdonald (q,t)-permanent replaces the constant cycle weight of the Vere-Jones α-permanent by (1−tk)/(1−qk) for a cycle of length k. We determine which of the probability laws built on the α-permanent survive this deformation, and where. The scalar gamma-type law survives on the whole parameter square, but the matrix law, a compound Wishart matrix, is more fragile. We prove that it exists in every dimension exactly when t∈{0,q,q2,…}, and in dimension two exactly when t≤q. We conjecture that in each fixed dimension the matrix law exists exactly on the region that we show to be necessary, a q-analogue of Gindikin's theorem.
A mix of human-written and AI-generated textHuman understanding: some partsCox processMacdonald permanentSchur measurealpha-permanentcomplete monotonicitycompound Wishart matrixinfinite divisibilityloop soupmatrix Laplace transform
We introduce the Macdonald (q,t)-permanent, a two-parameter cycle-weighted permanent that contains the usual permanent and determinant, as well as the α-permanent of Vere-Jones, as degenerate cases. We prove a MacMahon-type master theorem for its generating function, and determine when it induces a valid kernel on tuples of vectors. Nonnegativity of the (q,t)-permanent itself on semidefinite matrices turns out to be subtler: for t=qα nonnegativity holds exactly at the positive Wallach exponents; for a fixed q we determine the region in rank two and for higher ranks under an additional condition, while for small q positivity can even reach beyond the kernel region. Whether positivity forces t∈{0,q,q2,…} remains an intriguing open problem.
A mix of human-written and AI-generated textHuman understanding: some partsMacMahon master theoremMacdonald permanentWallach setalpha-permanentimmanantpositive definite kerneltotal positivity
Griffiths's 1969 conjecture predicts positivity of Schur forms for Griffiths-positive vector bundles. Finski reduced its weak form to a sign question for the double mixed discriminant Φn(H) of a positive linear map. Du's rank-nine counterexample shows that Φn(H)≥0 fails in general. We prove Φn(H)≥0 for every decomposable map H, a sum of a completely positive and a completely copositive map, using a quaternionic Gaussian determinant representation. Combined with Finski's incidence formula, this inequality gives weak nonnegativity of every Schur form under decomposable curvature, without splitting assumptions. Beyond the decomposable cone, nonnegativity of Φk(HV,U) on every square compression is equivalent to Φn(H+S)≥0 for every entanglement-breaking map S. This class contains explicit indecomposable maps, but is nonconvex in every size n≥9. We prove the nonconvexity by expressing Du's map as an average of four members, and the obstruction persists when all compression inequalities are strict.
A mix of human-written and AI-generated textHuman understanding: some partsBreuer-Hall mapChern formGriffiths positivitySchur formdecomposable mapentanglement-breaking mapmixed discriminantpositive linear mapquantum permanent
We prove a permanental analogue of Koteljanskii's inequality. If A is an inverse M-matrix that becomes symmetric after a positive diagonal similarity, then per(AS∪T)per(AS∩T)≥per(AS)per(AT) for all S,T⊆[n], where AS is the principal submatrix indexed by S. The proof expresses permanents as moments of a complex Gaussian vector and uses Ginibre's correlation inequality. Finally, an explicit counterexample shows that symmetry cannot be dropped.
A mix of human-written and AI-generated textHuman understanding: all partsFKG inequalityKoteljanskii inequalityM-matricescomplex Gaussianspermanents
We study the p-Bures–Wasserstein distance Dp(A,B)=(trAp+trBp−2∥A1/2B1/2∥pp)1/2 on positive semidefinite matrices; it equals the Bures–Wasserstein distance for p=1, and the Hilbert–Schmidt distance for p=2. We prove that Dp is a metric for every 1≤p≤2, by proving a stronger 2014 conjecture of Lin and van den Driessche on 3×3 block matrices. Our main tool is a partition inequality for Schatten norms, proved by complex interpolation. We also characterize Fréchet means for Dp by a fixed-point equation and prove geometric convergence of the associated iteration.
A mix of human-written and AI-generated textHuman understanding: all partsBures-Wasserstein distanceLin-van den Driessche conjectureSchatten normbarycenterscomplex interpolationpartition inequalitypositive semidefinite block matrices
We study the inequality Pλ(x;q,t)/Pλ(1n;q,t)≥Pμ(x;q,t)/Pμ(1n;q,t) for all x∈R>0n whenever the partition λ majorizes μ; Sra suggested it in 2015, and Chen, Khare and Sahi stated it as a conjecture in 2025. We prove the Jack case for every n from Sun's positive Gelfand–Tsetlin integral and the Dixon–Anderson recursion. For Macdonald polynomials we prove the case n=2, the covers (d)⪰(d−1,1) for every n when t≤q, and, by exact certificates, the case n=3 with ∣λ∣≤8. The HCIZ mechanism behind the Jack proof has no direct finite-q analogue: at two explicit pairs (q,t), no positive representation of the normalized ratio has a kernel multiplicative in λ, and normalizing at the principal specialization tδ instead fails the inequality.
A mix of human-written and AI-generated textHuman understanding: all partsDixon-Anderson integralHarish-Chandra-Itzykson-Zuber integralHeckman-Opdam hypergeometric functionJack polynomialsMacdonald polynomialsmajorization
We show that surjectivity of the canonical map Symb(SymaV)→Syma(SymbV) survives grading of the inner factor. Consequently hb[Ha]−Ha[hb] is Schur positive whenever the canonical map is surjective, where Ha is the (modified) Hall--Littlewood polynomial; the same holds for (modified) Macdonald polynomials. We prove this surjectivity for b≥(a−1)2, improving the known bound a(a−1). For Bergeron's q-analog of Foulkes' conjecture that lets both factors be graded, we prove nonnegativity of every Schur coefficient for certain special cases. We also study the full conjecture under an expansion hypothesis on an explicit space of maps.
A mix of human-written and AI-generated textHuman understanding: some partsFoulkes conjectureFoulkes-Howe mapGaussian Newton expansionSchur positivitycoinvariant algebracurrent operatorplethysmq-Foulkes conjecture
Bergeron's q-analog of Foulkes' conjecture asserts that (Hb[Ha]−Ha[Hb])/(1−q) is Schur positive for a≤b, where Hn is the (modified) Hall--Littlewood polynomial. We view this conjecture probabilistically: on each irreducible representation of Sab, the two plethysms induce degree distributions, and positivity means that the first is stochastically smaller than the second. We prove that, for a random (Plancherel) irreducible representation and a degree sampled from its graded multiplicity, the corresponding coefficient is negative with probability at most Cexp(−c(ab)1/3), uniformly in a≤b. Consequently, the q-Foulkes conjecture holds after deleting a fraction at most Cexp(−c(ab)1/3) of each of the two graded modules. Additionally, we show that exact conditional means and variances yield complete positivity for explicit width and height families.
A mix of human-written and AI-generated textHuman understanding: some partsMacdonald polynomialPlancherel measureSchur positivityplethysmq-Foulkes conjecturestochastic order
We prove that the plethysm Hm[Hμ] of a Macdonald polynomial by the Hall–Littlewood polynomial Hm lies above the Macdonald polynomial Hmμ obtained by fusing m copies of Hμ. More specifically, the quotient (Hm[Hμ]−Hmμ)/(1−qm) is Schur positive with coefficients in N[q,t], and it is the character of an explicit module that we call the fusion defect. For one-row shapes the inequality places both q-Foulkes plethysms Hb[Ha] and Ha[Hb] above the same polynomial Hab, and comparing their two defects we obtain a distorted version of Bergeron's q-analog of Foulkes' conjecture, in which the subtracted plethysm is shifted by qa(a−1)⌊b/2⌋.
A mix of human-written and AI-generated textHuman understanding: some partsMacdonald polynomialSchur positivityfusion productplethysmq-Foulkes conjecture
We prove that the Hankel kernel Φ(x+y), where Φ is the Jacobi theta kernel whose cosine transform is the Riemann Ξ-function, is strictly reverse-regular through order four. The order is sharp: an interval enclosure of the fifth Hankel–Wronskian at t=10−2 shows failure at order five. The third-order condition is Csordas's conjecture that s′2−ss′′ is strictly log-concave for s(t)=Φ(t). The proof reduces the kernel minors to Wronskian signs and establishes the signs by exact polynomial estimates for t≥1 and interval arithmetic on [0,1]. The fourth-order sign also yields three positive iterates of s↦s′2−ss′′, the triple Turán inequalities for the normalized Taylor coefficients of Ξ, and quartic integral inequalities for Ξ.
A mix of human-written and AI-generated textHuman understanding: some partsCsordas-VargaJacobi theta functionRiemann Xisign regularitytotal positivity
We study the Turán-ratio Sλ,i(x)2/(Sλ,i−1(x)Sλ,i+1(x)), where Sλ,i(x):=sλ(1[n−i],x[i]), and sλ is the Schur polynomial in n variables (indexed by partition λ) with i arguments equal to x and n−i of them equal to 1. Specifically, we prove that this Turán-ratio is strictly increasing for x>1 and 1≤i<n (unless all n padded parts of λ are equal, then it is identically one). For one-row shapes, our result answers a question of Ait-Haddou about complete homogeneous symmetric polynomials. For this case, we also offer a second proof through the zeros of Meixner polynomials that gives an explicit positive formula for the logarithmic derivative. We conclude with a Turán-ratio conjecture for Macdonald polynomials.
A mix of human-written and AI-generated textHuman understanding: all partsMeixner polynomialsSchur polynomialTuran ratiohomogeneous symmetric polynomialmonotonicity
Griffiths's 1969 conjecture on the positivity of Schur forms reduces to the positivity of the quantum permanent Φn on positive bistochastic maps. This conjecture was refuted by Du with a map for which Φ9<0. Therefore, we ask: how negative can Φn be? We prove a lower bound Φn≥−3n/(3n2+20n+103) (for n≥8), and an asymptotically stronger bound Φn≥−1/n+(3e2−7−o(1))logn/n2. The key to our bounds is the orbital polynomial of the map, which turns Φn into a linear function of a distribution on partitions and positivity into linear constraints. We also construct explicit maps for n=8,9 that improve on Du's example. Finally, if d∗∈{4,…,8} is the first dimension for which Φn<0, the minimum μn of Φn over positive bistochastic maps satisfies −μn=Ω(n−d∗); we conjecture the matching order −μn=O(n−d∗).
A mix of human-written and AI-generated textHuman understanding: some partsGriffiths positivitybistochasticlower boundspositive mapsquantum permanentsemidefinite programming
We prove a functional extension of Gaussian operator-norm small-ball inequalities. Let X=∑i=1dgiBi, where g∼N(0,Id), the matrices Bi∈Rd×d are symmetric, and ∑iBi2⪯Id. For every even log-concave f:R→[0,∞), we obtain Edetf(X)≥(Ef(κG))d at dilation κ=13/5, where G is standard Gaussian. The proof reduces each function to one small-ball estimate, obtained from matrix-weighted Poincaré inequalities, and controls the rest by averages over the spectrum. We give an equivalent entropy formulation. An explicit counterexample shows that no universal dilation below π/2 can work; we conjecture that this value is optimal.
A mix of human-written and AI-generated textHuman understanding: all partsdeterminant inequalitylog-concave functionsmatrix Gaussian seriesrandom matricessmall-ball probabilities
Petz (1994) conjectured that the scalar curvature of the Kubo–Mori metric on invertible density matrices increases as the state becomes more mixed. Dittmann (2000) reduced the conjecture to the concavity of an explicit symmetric function h on (0,∞)3. We prove that h is strictly concave, and hence Petz's conjecture. The proof writes h=−1/(8M) with M positive and homogeneous of degree one, and its central idea is to show that t↦1/M(X+tY) is a Stieltjes function for all X,Y>0; Cauchy–Schwarz and homogeneity then give concavity of h on (0,∞)3.
A mix of human-written and AI-generated textHuman understanding: all partsKubo-Mori metricMorozova-Chentsov kernelPetz conjectureStieltjes functionlogarithmic meanscalar curvaturestrict concavity
We reconstruct the argument recorded in a frozen Lean development establishing nonvanishing of the unit-compatible angular Hecke L-functions over Q(−3) in the strict half-plane ℜ(s)>7/8, apart from the genuine principal pole. The family includes every admissible integer angular order, all nonzero finite ideal moduli, primitive and imprimitive presentations, and real unitary norm twists. This strengthens the earlier 15/16 result for the same angular family.
The proof compares two estimates for a single normalized physical probe: a low bound with exponent 3/16+ω and an approximation by a reciprocal Mellin signal with error exponent βn−11/16−σ, where βn is the supremum of the real parts of zeros within a finite angular cap. Crucially, the positive saving σ is chosen before the target character. Mellin continuation converts these estimates into a strict improvement of the zero supremum. Induction on angular order, together with conjugation, establishes nonvanishing for the full family.
The repaired public proof closure passed a clean compilation of 8,213 Lean modules and fresh native Lean kernel replay. An independent AI-assisted audit recorded five ACCEPT verdicts for the repaired package, following the identification and correction of a reproducibility defect in the original archive. We document the mathematical argument, formal verification evidence, and remaining limitations. This manuscript is an expository reconstruction, not independent human mathematical certification. We describe the formally verified proof as a *xenoproof* and the result provisionally as a *xenotheorem*, distinguishing machine-checkable correctness from established human mathematical understanding.
Primarily AI-generated textHuman understanding: some partsEisenstein integersHecke L-functionsLeananalytic number theoryangular Hecke charactersformal verificationxenoproofxenotheoremzero-free regions
Contributed by Mihaly A. Csirik, Andre Laestadius, Markus Penz, Erik I. Tellgren
We develop a density-functional theory for the Pauli-Fierz Hamiltonian of
nonrelativistic quantum electrodynamics in the spirit of
the convex-analytic formulation of Lieb. We consider finitely many electrons,
subject to external scalar and vector potentials, coupled to the full
continuum of massless transverse photons with an ultraviolet cutoff.
The natural basic variables are the standard electron density and a combined current
density, the sum of the paramagnetic current density and a
field-induced current density generated by the quantized vector
potential. We prove representability results for these density-current
pairs and show
that the mixed-state constrained-search functional attains its infimum
and coincides with the Lieb functional obtained by convex conjugation.
Since arbitrarily soft photons purify mixed states at a negligible energy
cost without changing their density and combined current density, the
pure-state Levy-Lieb functional equals the Lieb functional and is
therefore convex, even though its infimum need not be attained.
In addition, we establish the zero-coupling limit, which recovers the
mixed-state Vignale-Rasolt functional of current-density-functional theory.
Finally, we consider the long-wavelength (dipole) limit and show that
the limiting functional can be written as a constrained search for the
well-known length-gauge dipole Hamiltonian.
Primarily AI-generated textHuman understanding: all parts
Let χ be a real primitive Dirichlet character of conductor N ≥ 3 and parity k ∈ {0,1}. Let Λ(s,χ) = (N/π)^{(s+k)/2} Γ((s+k)/2) L(s,χ) be its completed L-function, and let H^χ_t(z) = ∫_0^∞ e^{tu²} Φ_χ(u) cos(zu) du be the de Bruijn–Newman heat flow of Λ(·,χ), normalised so that H^χ_0(z) = (1/8) Λ(1/2 + iz/2, χ). We give a proof, checked by the Lean 4 kernel, that for every such χ and every t ≥ 9/32 all zeros of H^χ_t are real. Equivalently, the generalised de Bruijn–Newman constant satisfies Λ_χ ≤ 9/32.
The mathematics is not new, and we say exactly whose it is. De Bruijn's strip-contraction theorem (1950) gives Λ_χ ≤ D²/2 whenever the zeros of H^χ_0 lie in |Im z| ≤ D. Freire, González-Riquelme, Muñoz, Poblete and Salinas observed (arXiv:2610.27343, Remarks 3.1 and 3.4) that the zero-free half-plane Re s > 7/8 for Dirichlet L-functions asserted by OpenAI would give D = 3/4, and hence Λ_χ ≤ 9/32. What this paper adds is a complete formal proof of that composition. For conductors up to 400, FGMPS's unconditional bounds are stronger for 136 of the 242 characters; what is new is that 9/32, for every conductor, is a kernel-checked theorem.
OpenAI's half-plane enters as their own kernel-checked Lean theorem. Every other step is proved in Lean on top of Mathlib: de Bruijn's theorem for a general class of kernels, the identification of the theta integral with Mathlib's completed L-function, and Gauss's sign theorem for real primitive characters. The final theorem depends only on the axioms propext, Classical.choice and Quot.sound. In local, unsandboxed replays five checkers accept it: Lean's kernel (run twice), nanoda, and the con-leche/con-ron pair. All five reject a deliberately unsound control. A sandboxed Comparator run against a separately written challenge also accepts it. That run was made by another agent session, and the auditor reproduced its solution export byte for byte.
The result rests on OpenAI's Lean proof, whose accompanying manuscript is unrefereed. It is not progress on the Generalised Riemann Hypothesis, which is the statement Λ_χ ≤ 0. A short, separable section records two companion results for ζ that use no OpenAI code, Λ ≤ 225/512 and ψ(x) = x + O_ε(x^{31/32+ε}), both from Cureton and Thorner's half-plane Re s > 31/32.
Code, verification records and archive: https://github.com/DrMurphyIsIn/real-dirichlet-dbn-lean (release v1.0.0); DOI 10.5281/zenodo.23840082.
Contributed by Narayanan Narayanan, Suchismita Mishra, Umesh Shankar
We prove that every claw-free perfect graph G satisfies ch(G)=χ(G)=ω(G).
More strongly, if S is a clique of simplicial vertices of G and k≥ω(G),
every proper precoloring of S extends to a proper coloring of G from arbitrary
lists of size k on V(G)∖S. The proof has three coloring steps:
a mixed-list theorem for cobipartite graphs, a precoloring-extension theorem for
elementary graphs built on Galvin's kernel method and a homogeneous-pair
exchange argument, and a polynomial method (Combinatorial Nullstellensatz) argument for
graphs of interval type in which a Vandermonde factor is transferred across each
incomplete pair. The three coloring steps are assembled along the structure theorems of
Chv\'atal--Sbihi, Maffray--Reed and Chudnovsky--Seymour, using double clique
cutsets for composition. As a consequence, every perfect line graph is chromatic-choosable, recovering the theorem of Peterson and Woodall. Our precoloring extension theorem also resolves a conjecture posed by Chudnovsky and Seymour.
A mix of human-written and AI-generated textHuman understanding: all partsClaw-free perfect graphs.Graph coloringList Coloring Conjecture
We prove that the complete prime factorization of an n-bit integer can be computed deterministically in time 2O(n/logn) and polynomial space. This establishes the bound conjectured by Rudich and Rousseau, sharpening their qualitative No(1) result. The proof combines OpenAI's quantitative Boolean tensor-contraction theorem with the linear-size multiplication encoding of Akshima et al. The encoding permits exact counting of divisors with prescribed binary prefixes without loss in the exponent. The least prime factor is then recovered by successive prefix queries, and iteration yields the complete factorization within the same asymptotic bound.
Primarily AI-generated textHuman understanding: all partsBoolean tensor networksdeterministic algorithmsexact countinginteger factorizationsubexponential time
Lueker (J. ACM 56(3), 2009) constructed a family of finite-state machines Mh whose associated transfer matrices yield upper bounds on the Chvátal–Sankoff constants γk, and proved (his Theorem 3.26) that these bounds converge to γk for every alphabet size k. He published values only for k=2. For k≥3 the best upper bounds we were able to find in the literature in September 2026 were still those of Dančík's 1994 thesis: subsequent work generalising Lueker has generalised his Section 2, the lower bound. We implement Mh for arbitrary k. One ingredient has no binary analogue — the generating function of a truncation class when both sides of a minimal match pair are truncated — and we derive it. We then give a reduction by relative forms under which the transition multiplicities of the transfer matrix are falling factorials in k; one alphabet-free construction consequently determines the exact matrix for every k. We obtain improved upper bounds for k=3,…,15, each supported by a Collatz–Wielandt certificate verified in exact rational arithmetic, among them γ5≤0.645033. A note added on 31 October 2026 compares these values with those of a repository made public by L. Jack on 25 October 2026.
Primarily AI-generated textHuman understanding: no partsChvátal–Sankoff constantsCollatz–Wielandt certificateslongest common subsequencetransfer matrices
We deduce from the p-adic kit for three logarithms of the companion note hexagon:2610.00706 a fully explicit bound in closed form. If γ1,γ2,γ3 are multiplicatively independent rationals, congruent to 1 modulo p (modulo 4 if p=2), of heights at most ajlogp with aj≥1, and if ∣bj∣≤B, then vp(γ1b1γ2b2γ3b3−1)≤a1a2a3B2max{27711,C2(p)maxjaj}, where B=max{logB/logp,10} and C2(p)=11107p−1p(1+logp0.31)2, as soon as γ1b1γ2b2γ3b3=1. By a change of basis in the multiplicative group, the result applies to arbitrary rationals prime to p. On the examples of that note, the bound is 42 to 956 times smaller than that of Yu's explicit theorem, and stays within a factor between 1.2 and 2.5 of the numerical optimisation of the parameters of the kit.
Primarily AI-generated textHuman understanding: no partsYu's theoremexplicit boundsinterpolation determinantslinear forms in p-adic logarithms
Yamada (J. Number Theory 130, 2010) deduced from the method of interpolation determinants of Bugeaud and Laurent the bound vp(xp−1−1)≤283(p−1)logylog(xy)/(logp)2+4, for multiplicatively independent integers x,y coprime to p. We replace 283 by 69. The new ingredient is an elementary bound, by rearrangement, on the spread of the exponents that occur in the Liouville inequality: it at least halves the height term of Yamada's criterion, involves no congruence class, and allows the parameters to be re-optimised. The proof is complete up to the lemmas of Bugeaud–Laurent (zero lemma, analytic upper bound) and of Laurent–Mignotte–Nesterenko (bound for the polynomial), which are used as they stand. A self-contained verifier checks the numerical inequalities.
Primarily AI-generated textHuman understanding: no partsFermat quotientsYamada's boundinterpolation determinantslinear forms in p-adic logarithms
We transpose to the p-adic case the kit of Mignotte and Voutier for linear forms in three logarithms. The new tool is a lower bound for the p-adic valuation of interpolation determinants with no factorial loss: it is obtained by expanding the powers θℓz as binomial series in θ−1, and then transporting an order of vanishing that we compute optimally for generic points, by means of the initial forms of the space of exponential polynomials involved. It holds for every prime p. Combined with a rearrangement bound for the spread of the exponents and with Gouillon's zero lemma applied within a Teichmüller class, it gives a criterion of the form of criterion (2.8) of Mignotte–Voutier, in which logρ is replaced by logp and the height term contains no analytic contribution. For two logarithms, in Qp with p odd, it improves Lemma 8 of Bugeaud–Laurent. A change of basis in the multiplicative group reduces to the case of principal units: the factor g=p−1 leaves the volumes and moves into the heights. On examples, with exponents up to 1010 or 1030, the complete bounds (degenerate cases included) are 154 to 667 times smaller than those of Yu's explicit theorem for p∈{5,13}, 608 to 1980 times for p∈{7,11} (where the explicit form of Yu goes through Q(i)), and 73 to 105 times for p=101.
Primarily AI-generated textHuman understanding: no partsBaker's methodYu's theoreminterpolation determinantslinear forms in p-adic logarithms
We study Riordan paths enumerated by the area between the path and the axis. We show that the generating function satisfies a q-Riccati equation and can be written in a closed form as a ratio of basic hypergeometric series. The auxiliary series admits an algebraic representation as the motivic generating series of a symmetric two-node quiver. We obtain explicit formulas for the coefficients, the exponential asymptotics of the number of paths of given area, and a linear recurrence whose coefficients are signed counts of partitions into distinct parts. For a random path of area n we obtain a joint Gaussian limit law for the numbers of returns, horizontal steps and up steps. The height is of order logn, and its distribution concentrates on two adjacent integers determined explicitly by n.
A mix of human-written and AI-generated textHuman understanding: some parts
In the kit of Mignotte and Voutier for linear forms in three logarithms, the order of vanishing at the origin of the interpolation determinant is bounded below by Waldschmidt's general multiplicity lemma; the authors indicate that this estimate should be improvable. We give a lower bound for this order which is optimal for generic points. The functions involved span a space whose initial forms are described by a uniqueness property for a linear differential equation with constant coefficients. As a consequence, the left-hand side of their criterion (2.8) can be taken equal to 3(2K+1)(L−1)logρ, with main term 32KLlogρ instead of 2KLlogρ. On the two examples of their paper, with their public code modified in one line and the conditions of the zero lemma checked, the bounds given by the criterion go from 17.94⋅106 to 4.79⋅106 and from 28.85⋅106 to 7.00⋅106. The same estimate is the multiplicity input of our kit for three p-adic logarithms.
Primarily AI-generated textHuman understanding: no partsinterpolation determinantslinear forms in logarithmsmultiplicity estimateszero lemma
The maximum independent set problem asks for the largest set of pairwise non-adjacent vertices in an undirected graph. It is NP-hard to approximate within n1−ε for every ε>0, and under the Unique Games Conjecture it is hard to approximate within O(Δ/log2Δ) on graphs of maximum degree Δ; the Khot--Regev form of the conjecture, for which a proof has been reported but not yet peer reviewed, also rules out a (2−ε)-approximation of Minimum Vertex Cover. We present version 0.1.6 of \textsc{Esperanza}, which takes the better of a dynamic minimum-degree (Caro--Wei) greedy set and the complement of the \textsc{Salvador} vertex cover, and improves it by three budgeted local searches: repair-and-grow restarts at the cover vertices and at a maximum-degree vertex, chained local restarts realising (k,k+1)-improvements, and an iterated (1,2)-swap search. We prove that it runs in O((n+m)αA(n)) expected time for fixed budgets, with αA the inverse Ackermann function. We prove that the output is a maximal independent set, that its complementary vertex cover has at most 2τ−r vertices, where τ is the minimum cover size and r the number of components with an edge, and that the ratio is at most (Δ+2)/3 on graphs of maximum degree Δ. We prove the ratio 2n on every graph with Δ≤n and on three further classes, and state it for all graphs as a conjecture, whose proof would imply P=NP. We also state the conjecture that the complementary cover has at most 2τ−∣M∣ vertices for the NetworkX minimum maximal matching M; a proof would give a 3/2-approximation for Minimum Vertex Cover and refute the Unique Games Conjecture unless P=NP. In reproducible experiments the returned set is optimal on all 30000 graphs of a suite certified by an exact integer program; on 40 Khot--Regev YES instances with 640 to 2560 vertices the conjectured bound is certified on 38 and open on 2; and on 31 bounded-degree graphs the certified ratio is at most 1.0769 on the families with a planted independent set.
Primarily AI-generated textHuman understanding: all partsApproximation AlgorithmBounded-degree GraphsLinear TimeMaximum Independent SetP versus NPUnique Games ConjectureVertex Coverlocal search
Robin's criterion states that the Riemann Hypothesis holds if and only if σ(n)<eγ⋅n⋅loglogn for every n>5040, and Nicolas' criterion states that it holds if and only if c(q)=log(q#/φ(q#))−γ−loglogθ(q)>0 for every prime q, where q# is the primorial of q and θ is the Chebyshev function. Let N be a colossally abundant number, let ξ(N) be the real number associated with N by Nicolas, and let q be the largest prime not exceeding ξ(N). Writing Δ(N)=log(σ(N)/N)−γ−logloglogN, we prove without any hypothesis that, whenever logq≥31, 1.353/(qlog2q)≤Δ(N)−c(q)+22/(qlogq)≤3.970/(qlog2q). Thus Robin's inequality at N and Nicolas' inequality at q differ by 22/(qlogq) up to an explicit error of smaller order. As consequences, every colossally abundant counterexample to Robin's inequality yields a prime q with c(q)>2.70/(qlogq); conversely, every prime q with logq≥31 and c(q)>22(1−0.4787/logq)/(qlogq) yields a colossally abundant counterexample, so that Nicolas' oscillation theorem implies Robin's theorem on colossally abundant counterexamples; and under the Riemann Hypothesis every colossally abundant number with logq≥31 satisfies Δ(N)<−0.654/(qlogq), in accordance with the constant 2(2−1) of Ramanujan's bound. For superabundant counterexamples, using the exponent bound of Nazardonyavi and Yakubovich, which gives logn≥ψ(pk) for the largest prime factor pk, we prove c(pk)>0.9991/(pklogpk), and we show that the least colossally abundant number not smaller than the least counterexample n is itself a counterexample N with logN−logξ(N)≤logn.
We determine the minimum support and Shannon entropy of a shared classical seed needed to attain optimal worst-message success in a specified one-use normalized-loop model. The loop contains a backward binary symmetric channel with crossover 0 < p < 1/2 and a fixed finite-dimensional forward quantum channel. Seed branches are mixed as unnormalized weights before one normalization for each message. For qubit output and interior forward trace-distance diameter, equality at the known unrestricted optimum forces every supported seed to activate exactly two message labels. The seed-cost problem becomes a fractional perfect matching problem: even message counts are optimally covered by disjoint pairs, while odd counts require a triangle and disjoint pairs. We give separate exact formulas at zero and unit diameter, prove a sufficient higher-dimensional extension of the interior result, and exhibit an erasure-channel counterexample to diameter-only seed costs in general dimension. Explicit near-optimal constructions show how the exact-attainment cost discontinuities can be avoided at finite tolerance. Compactness also gives a strict performance gap below the minimum seed support. A complete exact-certificate calculation determines the two-seed classical optimum for three messages at backward and forward crossover 1/10. These are resource-specific mathematical results, not a physical backward-communication protocol or an asymptotic capacity theorem.
Primarily AI-generated textHuman understanding: no partsShared randomness; normalized loop; postselection
We study a finite-message communication model defined by contracting one backward binary symmetric channel with one fixed, finite-dimensional forward quantum channel and then normalizing the resulting weights. The devices may share a finite classical seed that is independent of the message before normalization. The seed branches are mixed at the level of unnormalized weights, followed by a single normalization for each message. For M ≥ 2 messages, backward crossover probability 0 < p < 1/2, and forward channel Φ, we derive the exact worst-message success probability S_raw(Φ,p,M) = 2(1−p)/[M−(M−2)(1−2p)δ(Φ)], where δ(Φ) = max over input states ρ, σ of ½‖Φ(ρ)−Φ(σ)‖₁. The quantity δ is the established trace-distance contraction coefficient, not a new channel invariant. A cyclic construction with M seed values attains the bound. Crucially, normalization can make the posterior seed distribution message-dependent: the theorem does not describe an average of separately normalized branches or require success for every seed. We give an exact example separating these conventions. Matched depolarizing-qubit and classical binary symmetric forward channels consequently have equal optima in this resource model. The result is a restricted one-use optimization, not a physical backward-communication protocol or a general asymptotic capacity theorem.
Primarily AI-generated textHuman understanding: no partsShared randomness; quantum channels; normalization; one-shot communication
We prove that every elliptic curve over $\Q$ whose points of order two are not all rational has cyclic reduction at a positive lower density of primes. This is the only necessary restriction: full rational two-torsion prevents cyclic reduction at every odd prime of good reduction. More precisely, a common zero-free half-plane ℜs>1−1/N for Dedekind zeta functions gives a lower proportion of at least cE/(NlogN), with cE>0 independent of N; the theorem of Cureton and Thorner supplies N=32. The proof uses the varying-cofactor sieve of our earlier works on primitive roots and primitive points, in conjunction with the marked Type~II estimates of~OpenAI, to construct candidate primes. We then apply a smoothed explicit formula for Dedekind zeta functions, with nonnegative weights on prime ideals, to bound the number of candidates with noncyclic reduction. This gives a common argument for CM and non-CM curves.
A mix of human-written and AI-generated textHuman understanding: some parts
Two centuries ago, Lagrange derived an upper bound for every root of a complex polynomial using the maximum of powers of coefficients. In 1916, Fujiwara independently found the Lagrange bound. In 1914, Carmichael and Mason derived an upper bound using the sum of squares of the absolute values of the coefficients of a polynomial. This bound was extended by Kuniyeda in 1916. We derive noncommutative versions of the Lagrange-Fujiwara, Carmichael-Mason and Kuniyeda bounds.
Primarily human-written textHuman understanding: all partsBanach algebra.Carmichael-Mason boundKuniyeda boundLagrange-Fujiwara bound
We give a deterministic polynomial-time primality test whose main verification is the congruence (1+X)n=1+Xn in Cn[X]/(Xr−b(Y)), where Cn=(Z/nZ)[Y]/(Φr(Y)). Two identities in the coefficient ring normalize b and ensure compatibility with the main congruence. A finite-field root count and a lattice bound show that an accepted input is a prime power; elementary preprocessing excludes nontrivial perfect powers. We construct b deterministically using Jacobi sums and the uniform Dedekind-zeta zero-free theorem of Cureton and Thorner, which extends OpenAI's Hecke zero-free theorem. The total bit complexity is L32+o(1), where L=⌈log2n⌉, and verification after constructing b costs L6+o(1). More generally, for a fixed real algebraic 1/2≤θ<1, a common zero-free boundary ℜs>θ for the relevant Dedekind zeta functions gives Lmax{6,(1−θ)−1}+o(1). This bound reaches L6+o(1) at θ=5/6, and hence under GRH. We state a fixed-base modification of Agrawal's conjecture for odd integers passing the base-2 Fermat test, using the modulus Xr−2 with r the least positive integer not dividing n(n−1).
We give an unrefereed computer-assisted proof that Korenblum's constant in unweighted Bergman space satisfies C68≥0.44685. The new input retains the signed first contact harmonic in an annular coefficient estimate. An outer factor and square completion yield a correction of 19/200 to the linear coefficient, replacing the 7/100 correction of version 2. A rational measure verifies the resulting moment conditions. We include the analytic argument and describe the finite interval certificates and both infinite tails. The Green/flux input is V. Patankar's (PR 274); the coefficient and moment methods build on Wang, Pinho's earlier construction, and Wikström. There has been no independent human review or formalization.
Primarily AI-generated textHuman understanding: some partsBergman spaceKorenblum's maximum principleannular Green functioncontact momentsinterval certificatesouter functions
An exactly specified rational piecewise-linear density gives C86≤1.80560940 for the Schur--Siegel--Smyth trace constant. It uses 28 classical auxiliary polynomials, ten with nonreal roots, and 5,375 rational nodes. Their recorded list occurrences are: Flammang 2019 (all 28), Liang--Wu 2011 (26), Flammang 2009 (21), McKee 2011 (21), Tomar's public A22 pool (18), Aguirre--Peral 2007 (16), and Flammang 2024 (9). The density refines infrastructure originating in Tomar's adaptive-node certificate. We give the full deduction from finite potential and logarithmic-integral checks through Smith's arithmetic-measure criterion and converse theorem, and describe the rigorous continuous enclosures. The contribution is the explicit feasible measure and its smaller upper bound. This note is unrefereed, with no independent human review or formalization.
We give the unrefereed bound C44<2.0417086280 for the number of relevant variables of a Boolean function of real degree d, r(f)≤C442d. The argument bounds the number of coordinates controlling a ternary derivative, exploits its dictator branches and Boolean completion, and propagates local-degree caps. The small-degree inputs are R4=22, R5≤50, and R6≤109. The quartic proof uses an exact calculation on Rasoolzadeh's published cubic catalogue; the remaining finite arithmetic is rational. The lower bound remains 3/2. There has been no independent human review or formalization.
We give an exact computer-assisted proof of R(5,6)≤84. We derive a system of linear inequalities that must be satisfied by any graph on 84 vertices containing neither a five-vertex clique nor a six-vertex independent set. We provide explicit nonnegative rational weights for which adding these inequalities yields a contradiction. Published small Ramsey bounds and complete catalogues of smaller Ramsey graphs supply the external inputs. The accompanying verification programs check the derivation of the inequalities and their weighted sum using exact arithmetic. Two supplements give a worked local calculation and a detailed mathematical specification of the complete proof.
Primarily AI-generated textHuman understanding: some partsRamsey theory
In Appendix B of "A 2o(n) time algorithm for 3-SAT" (Claude, Anthropic, hexagon:2610.00577) the gain parameter ρ=1/2 comes from the Hardy--Ramanujan growth logp(E)∼cE of the partition numbers, and Appendix H observes that growth of order Eα would give ρ=2α−1 by the same calculation. We point out that the rank floor stated in the same Appendix H already excludes α<1/2 for every graded family of the same shape: the pieces at weight h have total rank at most (4h+1)dimV≤4h in one of the two directions, and the crossing they must add up to has rank (dimVh)2, so the construction needs logdimV≤4h≥2logdimVh−O(logh), that is α≥1/2. Restoring the rank with a larger weight costs ρ≥2−α in the weight count, so the count never goes below max(2α−1,2−α)≥1/2, with equality only at α=1/2; when the identity holds on the whole weight space, as Lemma 9 of the paper gives it, the bound sharpens to 2α/(2(4α−1)), which exceeds 1 for α<0.357. Theorem B of the paper is rank-tight up to a polynomial factor. The inequality is exact and we tabulate it: for the partition numbers it holds for every h, for partitions into squares (α=1/3) it fails for all h≥5891. We also record that the Heisenberg Fock spaces on a sparse set of modes, the natural candidates for slower growth, do not carry Lemma 9, and that no choice of coefficients repairs it. The floor ρ≥1/2 remains open.
Primarily AI-generated textHuman understanding: some parts3-SATExponential Time Hypothesisinteger partitionsrank bounds
Let sn=n(n+1)/2. We prove that every smooth parallelizable Riemannian n-manifold, n≥3, admits a smooth isometric embedding into Rsn+1 whose image lies in an arbitrarily small ball. For closed stably parallelizable manifolds, each of whose connected components has vanishing Euler characteristic, we obtain small-image isometric embeddings into Rsn+5. The proof combines explicit partial isometries along commuting Hopf flows, an iterative scheme for exact metric correction using oscillatory perturbations constructed from the second fundamental form, and a transversality argument for metric-preserving perturbations.
A mix of human-written and AI-generated textHuman understanding: some partsHopf actionh-principleisometric embeddingmetric correctionparallelizable manifoldsecond fundamental form
This paper collects, with uniform notation, a month of new results around one way of looking at the Riemann hypothesis (RH): as the statement that explicitly computable margins stay on one side of explicit constants for ever. Part I studies the Cesaro margin M(N) between a smooth curve and the weighted prime sum 2*sum_{q<N} Lambda(q) q^{-1/2} log(N/q): RH is equivalent to M(N) >= 0 for all large N (a theorem of Suzuki), and we verify 0 < M(N) < 0.18484 for every real 1 < N <= 2.15*10^29, with explicit unconditional bounds beyond; the margin is analyzed as a walk of the primes against a curve, as an energy, and as a one-sided mean: RH holds if and only if a weighted mean of the prime-counting error stays below -B = -(1 + gamma/2 - (1/2) log 4pi) for ever. Part II dissects the Mobius side of the same coin: an exact positive-definite variance identity for sum_{d<=D} mu(d)({x/d}-1/2), empirically Gaussian statistics, and an exact anatomy of the fourth moment — the conductor-1 orbit decouples, the cross term moves only at composite squarefree conductors, two of its four sectors are O(D log^4 D) unconditionally, and an empirical jump law, first refuted, is proved exactly in its sector. Part III shows that 2B is the Baez-Duarte constant of the Nyman-Beurling criterion and recomputes the Nyman-Beurling distances from scratch through Vasyunin's cotangent formula: d_N^2 log N reaches 0.0461 against 2B = 0.04619, with no zero of zeta used anywhere in the computation. Part IV distils the author's twenty-two-part study of the cross-energy of consecutive Dirichlet partial sums: the trichotomy that singles out the critical line as the trace of the pole, the energy law (the mean energy of the prime-counting error is >= 2B unconditionally, with equality if and only if RH holds with simple zeros), the energy slope as a truncated Weil functional exact in the primes, the functional reflection that reproduces Hardy's function from the integers, and the systematic Davenport-Heilbronn and Beurling controls. Part V states precisely what none of this can do: a no-go theorem shows that weights indistinguishable from the primes' to q^(delta-1/2) satisfy every averaged statement here and still carry an off-line zero, so the exact identity Lambda = mu * log must enter beyond its first moments. The paper opens with a list of its twenty-two original contributions and a matching statement of what is due to others. We do not prove the Riemann hypothesis, and no statement in this paper should be read as progress towards it beyond what it literally says.
A mix of human-written and AI-generated textHuman understanding: all partsRiemann hypothesis; Cesaro; Mobius; Landau; Nyman; Dirichlet; Weil; Davenport
Let M(A), A∈M2(C∞), be the elementary Anderson t-motive of dimension 2 and rank 4 defined by Te=θe+Aτe+τ2e. Using the affine equation of Ehbauer, Grishkov and Logachev, we give an explicit sufficient condition for M(A) to be uniformizable, in terms of the valuations of the coefficients of that equation. It holds as soon as no cancellation occurs among the four summands of one of these coefficients; in particular h1(M(A))=4 in the simple case where no cancellation occurs in any coefficient. This proves their Conjecture 3.7: h1(M(A))=4 whenever (orda11,orda12,orda21,orda22) lies outside an explicit union of nine hyperplanes.
Primarily AI-generated textHuman understanding: no partsAnderson t-motivesDrinfeld modulesfunction fieldsuniformizability
We study the lifetime of a regular algebraic branch under continuous forcing and infinite-activity tangent jumps. For a parabolic fold, a first-order tangent kick decreases the discriminant by an exact quadratic amount. Symmetric stable marks of index α∈(0,2) therefore generate a one-sided normal jump kernel of index q=α/2, with a coefficient determined by curvature. The normal diffusion and jump amplitudes depend on the evolving tangential state. We compute the full killed generator on powers of the discriminant and obtain occupation and terminal-jump bounds for this same joint process. Uniformly subcritical normal drift gives finite censored boundary lifetime and finite cumulative killing rate; continuous contact and jump exit both have positive probability. At and above the uniform critical threshold, a logarithmic nonlocal barrier excludes censored boundary contact, and the original branch terminates at a jump almost surely. A comparison after a stochastic time change yields an explicit stretched-exponential upper bound for survival and hence all positive lifetime moments. In the subcritical regime, an optimized upper bound for the probability of jump termination tends to zero as the curvature coefficient or the initial discriminant tends to zero. The analysis combines power estimates for the jump generator with a logarithmic barrier at the critical threshold. Numerical calculations illustrate the dependence of terminal causes on curvature.
A mix of human-written and AI-generated textHuman understanding: all partsLévy processesalgebraic foldsboundary accessibilitykilled jump diffusionsstochastic differential-algebraic equations
An arithmetic expression is a word of operations; a motion is a flow on a manifold. For expressions whose evaluation has a single internal spine, the two are one structure: affine for addition and multiplication, projective once the second operand slot is divided, contact-geometric in the order defect. A regular arithmetic expression space (AES) is an oriented Riemannian surface (M,g) with a smooth assignment a:M→R satisfying the eikonal equation ∣∇a∣g2=μ2+λ2a2; the four operations then act as motions translating and scaling a. In the basic model E0, the upper half-plane with the left-invariant metric of Aff+(1,R) (curvature −λ2), addition and positive multiplication are affine translations and dilations obeying a Baumslag--Solitar law, and a is a Laplace--Beltrami eigenfunction of eigenvalue 2λ2. Left-expanded words stay in the affine Borel cell fixing ∞; the right-expanded language, once second-slot division is admitted, generates all of PGL2(R) through the rank-one Bruhat decomposition, and pulling standard horocycles back along one operator word gives the dual reading, ripple geometry. Commutativizing the motions by total charge yields the accumulative commutative space, where the order defect of two histories is an exact weighted area, the tearing. The same defect is the constant μλ in the bracket [Du,Dv]=μλ∂a of the horizontal fields of α=da−μdu−λadv, a contact form whenever μλ=0, and it is the curvature of that horizontal connection; for E0 the basic model is proved to be the Reeb quotient of the contact model, and fixed-charge tearing and fixed-state Reeb drift are Legendre dual readings of the same curvature. The generators ∂a, a∂a, and aloga∂a span an infinite-dimensional Lie algebra, so no finite-dimensional Lie group carries all three ranks, and no observable of the charge endpoint detects tearing. For every finite puncture count a conformally compactifiable punctured AES is constructed, with an explicit four-circle zero locus and exact reconstruction from three-ended pants blocks. All results are elementary, planar, or local to a three-dimensional contact model.
A mix of human-written and AI-generated textHuman understanding: all partsBaumslag--Solitar relationLegendre dualityaffine cocyclesarithmetic expression geometrycontact structures and Reeb quotientseikonal equation on surfaceshyperbolic upper half-plane and horocyclesinfinite-dimensional Lie algebraspositive affine grouppunctured Riemann surfaces and pants decompositions
## Abstract
Let (Σ,g) be a closed Riemannian three-manifold, and let M=−G be its negative Einstein endomorphism.
Where detM>0, the companion paper [MA] reduces realizability as a hypersurface of E4 to one equation,
F(g)=0, and studies it locally. Here we treat the global questions.
- **Compactness.** A closed realization has detM≥R−6 at a point farthest from any centre, where R is that
distance, and so maxdetM≥125/(8diam6) by Jung's theorem. If detM>0 everywhere, the realization is a strictly
convex embedded sphere, unique up to rigid motion. So a closed connected metric with detM>0 is realizable iff
Σ is simply connected and F(g)=0. No closed three-manifold other than S3 carries such a realizable metric.
- **The homogeneous geometries.** A realizable closed metric with detM>0 and constant scalar curvature is round,
and so is every compact locally homogeneous realization. Locally, constant M-eigenvalues with detM>0 force
constant curvature, a case of Tsukada's classification. This recovers Agaoka and Hashinaga's list of the
left-invariant metrics that embed locally. The eight Thurston geometries are worked as checks.
- **Total absolute curvature.** J(g)=∫detMdV equals 2π2 on every closed realizable metric with detM>0, so
any other value excludes realizability. At the round sphere J is stationary. Its second variation on
transverse-traceless tensors of level n is −(μn+2)(μn+3)/16 times the squared norm, μn=n2−3, and its
radical is exactly the conformal and Lie directions. On conformal rays J is flat to third order. Along the
conformal path of [MA], J=2π2−π2t4/12+O(t6).
- **Non-convex realizations.** Embedded hypersurfaces of revolution are non-convex exactly around their necks. For
the radial graphs r=1+εcos2χ, convexity is lost at ε=1/5 in a Whitney cusp of the Gauss map. With
δ=ε−1/5, the relative excess J/2π2−1 then grows as 500δ3/2/(92π).
- **Connectivity, and the round sphere.** Modulo isometry, the closed realizable metrics with detM>0 form a
contractible family, contracted by an explicit equivariant path. The embedded hypersurfaces of revolution join
the round sphere through the degenerate locus. We collect the characterisations of the round sphere, with their
hypotheses.
Every displayed result is reproduced by a script; Appendix V indexes them.
Primarily AI-generated textHuman understanding: no partsEinstein tensorGauss map singularitiesGauss–Codazzi equationsThurston geometrieshypersurfaces of Euclidean spaceisometric immersionsupport functiontotal absolute curvature
Contributed by Matthew Bowen, András Máthé, Oleg Pikhurko
This AI-assisted draft constructs, for every sufficiently large D, a local rule of radius O((lnD)2) that produces, on any triangle-free graph G of maximum degree at most D, a random independent set that includes each vertex with probability at least (1−oD(1))lnD/D. Further properties of the rule yield, for every fixed ε>0 and all sufficiently large D, a randomised LOCAL algorithm that colours any such finite graph G with at most (1+ε)D/lnD colours, with high probability, in Oε(lnD)+(lnln∣V(G)∣)O(1) communication rounds. Applying known transference principles for Borel graphs, we obtain the same asymptotic bound on the number of colours for measurable and Baire-measurable colourings, and for Borel colourings of graphs with finite asymptotic separation index.
We determine the asymptotic distribution of the smallest gaps between eigenvalues in the bulk of the spectrum of real symmetric Wigner matrices, including Bernoulli matrices. After multiplication by N3/2, the gaps and their locations converge to a Poisson point process with the same intensity as for the Gaussian orthogonal ensemble. Since changing a single entry can move eigenvalues by an amount comparable to the smallest gaps, standard perturbative comparison estimates do not directly resolve this scale for arbitrary entry distributions. Instead, we use that resampling a few random entries does not change the law of the matrix, and we show that the resampling moves each pair of nearby eigenvalues by a random walk that satisfies a local limit theorem; this requires excluding arithmetic structure in the eigenvectors. We also prove a law of large numbers for the number of gaps at all scales between N−3/2 and N−1, and we sketch the proof of the analogous smallest-gap result for complex Hermitian Wigner matrices.
Primarily AI-generated textHuman understanding: some partsrandom matrix theory
We study one-use communication of a finite classical message through a stipulated normalized loop containing a backward binary symmetric channel and one noisy forward memory carrier. The carrier is either a classical bit or a qubit; no shared random seed, retained side register, or entanglement is supplied. For every finite message count, we characterize the qubit optimum by a constructive scalar feasibility test. For rational noise parameters and a rational target success, the test uses O(M²) constant-size rational arithmetic checks. The proof combines an exact decoder reduction, minimum-trace allocation for constant encoders, and a concave geometric cap with a matching polygon construction. For three messages, the qubit and classical-bit optima coincide throughout the allowed noise range; an explicit Bernstein certificate proves the remaining polynomial inequality. For four messages, an exact rational qubit protocol exceeds a bound valid for every classical-bit protocol. These are conditional mathematical results for the declared resource model, not a construction of a physical backward channel. Independent human review and originality assessment remain pending.
Primarily AI-generated textHuman understanding: no partsinformation theory quantum information
For r≥1, let Ir(n) be the least length forcing a nonempty
zero-sum subsequence of index at most r over the cyclic group Cn.
We prove Ir(n)≤n1+o(1) when
r≥C[ω(n)log(2ω(n))]2, and
I1(pk)≪(k+1)pklog(2pk) for prime powers.
We also construct sequences showing that Ir(n)/n is unbounded
for every fixed r; for r=1, all terms may be generators.
A mix of human-written and AI-generated textHuman understanding: some parts
In October 2026 two manuscripts claimed refutations of the Exponential Time Hypothesis by deterministic algorithms for #k-SAT, each with a Lean formalisation of its top-level statement. Neither result applies directly to connectivity problems such as Hamiltonicity, because no linear-size CNF encoding of "this edge set is a single cycle" is known. This paper derives a corollary of the Anthropic manuscript: assuming three of its internal results (a network-to-clique reduction, an algorithm for cliques of all levels, and the existence of its "pieces"), the number of Hamiltonian cycles of a graph with n vertices and at most Dn edges can be computed exactly and deterministically in time O(2δn) for every δ>0, and in time 2O(max(m,n)/logn) for a graph with m edges. A rank-encoded variant of the construction gives 2O(nlogdˉ/logn) for average degree dˉ. Deterministic decision, search and Hamiltonian-path counting follow, and so do exact sparse TSP with polynomially bounded weights, exact counting of directed Hamiltonian cycles, and ATSP. The result is not a standalone breakthrough; it stands or falls with the claimed refutation. The ingredients are known: the folklore matrix-tree identity on 2-factors, the fermionic (Berezin) matrix-tree theorem of Caracciolo, Jacobsen, Saleur, Sokal and Sportiello, and the blob partitions of Cut&Count. What is new is the observation that connectivity can be encoded with O(1) bits per edge through a Grassmann determinant whose only non-local part is a sign, and that this sign costs only O(g3) bits for g blobs, so the whole construction is a network in the sense of the manuscript, whose value is n⋅#HC(G). Every theorem we claim is formally verified in Lean, end to end, on top of the manuscript's own Lean development: the three counting theorems, the decision, search and path corollaries, sparse TSP, directed Hamiltonian cycles and ATSP, using only the standard axioms, with fresh kernel replays. The Lean sources are at https://github.com/xennygrimmato/eth-hamiltonicity. We also record weaker results that rest only on the machine-checked top-level theorems: the parity of #HC(G) for every graph in time 2O((m+n)/log(m+n)) (machine-checked in Lean, with the hypotheses discharged by the labs' own Lean proofs), and one-sided randomized decision. We describe what is proved, what is checked numerically, what is formalised, what is assumed, and what the limitations are.
To the best of our knowledge, we present a state-of-the-art change-of-basis algorithm for 2×2 matrix multiplication under the Karstadt--Schwartz-style basis change regime [1].
We do this by introducing basis-change matrices BU, BV, and BW that each require only a single addition and hence give rise to a matrix-multiplication algorithm performing 5nlog27−4n2+43n2log2n, arithmetic operations when multiplying two n×n matrices, with n a power of 2. This improves on the previous best bound, which had last term 23n2log2n.
Primarily human-written textHuman understanding: all parts