Math and Theory

Graphs with five harmonic boundary vertices, subcubic graphs with six, and minimum-boundary two-trees

2026-11-02T00:00:00Z

Every two-connected graph with at most five Steinerberger boundary vertices has diameter at most two, and a connected graph has exactly five boundary vertices if and only if it is a bridge subdivision of one of 227 reduced graphs of order at most fourteen; this answers Problem 6.1 of Chiem, Dudarov, Lee, Lee and Liu. The proof reduces boundary metrics of every diameter, through the three cones of five-point metrics and a cap preserving local configurations, to an exact certificate that leaves fifteen metrics, excluded by hand. Connected graphs of maximum degree three with exactly six boundary vertices are the bridge subdivisions of 201 reduced graphs. For n ≥ 6 every two-tree on n vertices has at least ⌈n/2⌉ + 2 boundary vertices, with three extremal two-trees at each even n ≥ 8 and, at odd n ≥ 9, a period-eight quartic quasi-polynomial number of them, asymptotic to n⁴/192.

The small-coefficient upper half of the Keller–Klein conjecture

2026-11-01T00:00:00Z

Let X = Σ vᵢεᵢ with independent uniform signs, real coefficients, Σ vᵢ² = 1 and |vᵢ| ≤ a, where 0 < a ≤ 7/100. For every x ≥ 0, P(|X| ≤ x) ≤ 2Φ(x) + 2Φ(a) − 2, equivalently P(X ≤ x) − Φ(x) ≤ Φ(a) − 1/2. Keller and Klein conjectured the two-sided bound |P(X ≤ x) − Φ(x)| ≤ Φ(a₁) − 1/2 with a₁ the largest coefficient; Ivanisvili's computer-assisted manuscript of 21 October 2026 proves the lower half and the full conjecture for a₁ ≥ 0.07, and leaves this range open. The proof follows the analytic route his manuscript proposes, using Prawitz smoothing, coefficient profiles and a compensated alias estimate. Together with his results it implies the full conjecture, conditionally on those unrefereed external results.

The exact last-gradient bound for constant-step gradient descent in the strongly convex case

2026-11-01T00:00:00Z

The exact final-gradient guarantee holds on every real Hilbert space for L > 0, 0 ≤ μ ≤ L, normalized step 0 ≤ h ≤ 2 and every natural horizon. Actual one-dimensional Huber and quadratic functions attain its two branches.

A narrow interval for the universal multiplier step of two-block ADMM

2026-11-01T00:00:00Z

Two-block ADMM with relaxation only in the multiplier update converges for every constant step 0 < τ ≤ 1.886795 on finite-dimensional convex problems whose proximal subproblems attain their minima and which have a saddle point. Both primal blocks, the multiplier and the objective values converge, with no smoothness, strong convexity or bounded-iterate assumption. Fixed compact polytopes in R¹²⁸ have a nonconstant periodic ADMM orbit for every τ ≥ 1.8868840…, so the largest step with universal convergence lies in an interval of width below 0.00009, about a tenth of the version 1.0 interval. For scalar problems the universal range is exactly 0 < τ < 2. Affine-constraint versions are also proved. The earlier assertion of convergence up to τ = 2 by Gao et al. (2022) is shown to be false.

No universal finite collapse of the upper PPT entanglement-cost hierarchy

2026-11-01T00:00:00Z

For every integer q ≥ 1 there is a bipartite density matrix ρ_q with κ_{q+1}(ρ_q) < κ_q(ρ_q) in the upper hierarchy of semidefinite programs of Lami, Mele and Regula, whose logarithms converge to the exact entanglement cost under PPT operations. The witnesses are full-rank product-Bell-diagonal states in local dimension 2^((2q+4)4^q), and the auxiliary matrices are unrestricted. Hence no fixed finite upper level is exact for all states. The proof builds arbitrarily long strictly positive reverse orbits of the absolute partial transpose by a weighted-sign argument and converts full support into a quantitative gap for every shorter chain. Brinster et al. had proved the corresponding all-depth statement for the lower χ hierarchy and left the upper question open. The theorem does not give one fixed state or dimension with infinitely many strict levels.

Depth-one Artin–Schreier defect towers are composita

2026-11-01T00:00:00Z

Let k be algebraically closed of characteristic p > 0 and K a perfect henselian subfield of the Hahn field k((y^ℚ)) that contains k(y) and whose elements have no finite accumulation points in their supports. If a nontrivial finite Artin–Schreier tower M/K has depth one, then M is a compositum of Artin–Schreier extensions of K. Nart and Novacoski proved the reverse implication and conjectured this one. For every θ in M outside K, the truncation of θ at the first finite accumulation point of its support generates a nontrivial elementary abelian subextension of K(θ)/K; the extraction adapts an Ore-relation argument of Kedlaya.

Matrix-pencil rank distance as an explicit completion problem

2026-11-01T00:00:00Z

Let A and B be m × n matrix pencils over an infinite field and δ(A, B) the least normal rank of A′ − B′ over pencils A′, B′ strictly equivalent to A, B. For distinct c, d and every x ≥ 0, δ(A, B) ≤ x exactly when an explicit (2m+n) × (2n+m) dilation K_{c,d}(A) is obtained from K_{c,d}(B) ⊕ (t − d)I_x by one constant full-row-rank and one constant full-column-rank matrix. No regularity, rank or spectral condition is imposed, and the anchors may be eigenvalues. Over an algebraically closed field and at anchors outside the spectra, a known block minimal bases theorem turns this into Conjecture 1 of Dodig and Stošić: the bounded rank perturbation problem for matrix pencils is a completion problem.

Counterexamples to the Kronecker-square lower bound for nonnegative rank

2026-11-01T00:00:00Z

Vandaele, Gillis, Glineur and Tuyttens conjectured in 2016 that every nonnegative matrix M satisfies rank₊(M ⊗ M) ≥ rank₊(M) rank(M). For every k ≥ 0 an explicit rational nonnegative matrix C of order k + 11, rank k + 10 and nonnegative rank k + 11 has rank₊(C ⊗ C) ≤ (k + 10)² + 9, so the conjecture fails by at least k + 1. The matrix is the direct sum of a classical 4 × 4 matrix of rank three and nonnegative rank four with a small diagonal perturbation of an all-ones matrix; in the mixed Kronecker blocks a positive rank-one term absorbs a small real perturbation of rank q with only q + 1 nonnegative terms. At k = 0 an exact rational factorization of width 109 beats the conjectured bound 110.

Finite real counterexamples and spectral obstructions for q-permanent monotonicity

2026-11-01T00:00:00Z

An explicit integer Gram matrix B of order 896 and rank three, with row coordinates at most 32 in absolute value, has a negative derivative of its inversion-weighted q-permanent at q = 1, by the exact integer comparison 1000U > 1001·C(896,2)·per(B). The integer matrix A = MB + I is positive definite, its q-permanent decreases between explicit rationals 0 < q₁ < q₂ < 1, and identity padding gives such counterexamples in every order N ≥ 896. This answers the real-symmetric case of the Bapat–Lal monotonicity conjecture on [−1,1] negatively and refutes da Fonseca's conjectures inside the original interval. For bounded-row fixed-rank perturbations I + (t/N)VVᵀ the endpoint derivative is eventually positive for every ordering when tλ_max(Σ) < 1 or tλ₃(Σ) ≤ 1 < tλ₁(Σ), and correlation counterexamples exist with condition number tending to 227.

The sharp level-one Fourier weight at density one eighth

2026-11-01T00:00:00Z

For every Boolean indicator of mean 1/8 on a finite uniform cube, the level-one Fourier weight is at most 3/64. A codimension-three subcube attains the bound in every dimension at least three.

Escaping Nesterov flows and iterates below critical damping

2026-11-01T00:00:00Z

For zero-velocity solutions of X″ + (r/t)X′ + ∇F(X) = 0 and for the Nesterov iteration with momentum k/(k+r), with F convex, its gradient 1-Lipschitz and its minimum attained, boundedness holds for r ≥ 3. On the Hilbert space ℓ²(ℕ;ℝ²) this threshold is sharp for both: for every 0 < r < 3 an explicit block sum of smoothed valleys has a trajectory, and another has iterates, with norms tending to infinity. In the plane, the potential h(√(x²+y²)) − x with the Huber function h has an escaping trajectory from (2,1) for every 0 < r ≤ 2 and bounded trajectories for r > 2. The finite-dimensional range 2 < r < 3 is open.

Universal entanglement embezzlement with inverse-quadratic logarithmic infidelity

2026-11-01T00:00:00Z

A target-independent catalyst with local dimension 2^(4L) gives root infidelity at most 125m/L² for every Schmidt probability vector on m entries.

Sharp stability and exact cap optima for Rademacher sums

2026-11-01T00:00:00Z

For normalized Rademacher sums X = Σ bᵢεᵢ with |bᵢ| ≤ a, if every signed sum has magnitude at least x > a/2, the core {i : |bᵢ| ≥ x} is odd and the coefficients lie within squared distance 2 − 2√(1 − (a − x)²) of the flat vector on it, with equality exactly for a saturated core and one residual. At every cap, near-maximizers of the exact support-gap optimum lie within √(2+√2) times the deficit of an admissible vector of one of the two optimal shapes, with an optimal constant, and the exact nonlinear modulus on the saturated branch is determined. Conditionally on the completed Keller–Klein comparison, the maximal Kolmogorov distance K(a) is determined with all optimizers for 1/√5 ≤ a ≤ 1/√3, extending an earlier formula for a ≥ 1/√3.

The exact asymmetric Merino–Welsh region for finite matroids

2026-11-01T00:00:00Z

The theorem characterizes all nonnegative pairs (x,y) for which T_M(x,0) T_M(0,y) ≥ T_M(1,1)² holds for every finite matroid without loops or coloops.

Critical hard-core influence bounds via nonbacktracking collisions

2026-11-01T00:00:00Z

For each fixed Δ ≥ 3, every row of the hard-core influence matrix has absolute sum O_Δ(√n log n) on every n-vertex graph of maximum degree at most Δ, at every activity up to the uniqueness threshold λ_c(Δ) and under every feasible pinning, with an explicit constant. The proof combines exact vertex-deletion identities, a known local squared-message contraction, a self-avoiding-walk transfer in the style of Sinclair, Srivastava, Štefankovič and Yin, and a finite form of the nonbacktracking collision argument of Bencs et al. It recovers, up to a logarithm, an O(√n) bound that was stated in 2025 and withdrawn in 2026. The constant is large, and the mixing-time consequence for Glauber dynamics is conditional on a published proposition.

Gyires's permanent inequality for structured matrices

2026-11-01T00:00:00Z

The preprint proves Gyires's permanent inequality for several structured matrix classes. These include nonnegative circulants of orders five through seven, all-order permutation polynomials A=aI+bQ+cQ², and all-order four-block doubly stochastic matrices. Here Q is any permutation matrix and a,b,c are nonnegative. The four-block result permits unequal block sizes and zero entries. Further results cover one exceptional column or row and every rank-at-most-two doubly stochastic matrix through order fifteen. The supporting sections prove structural, scalar, deletion, balancing, local polynomial, and counting results. Exact counterexamples identify the limits of auxiliary assertions. The principal order-five claims include equality and a sharp sorted-gap bound with coefficient 484. The principal order-five and order-six claims, the all-order permutation-polynomial and four-block theorems, and the exceptional-vector theorem have complete Lean proofs. The rank-two result has a complete Lean proof for orders 2 through 8. The general order-seven circulant result and rank-two orders 9 through 15 remain incomplete in Lean. The unrestricted conjecture remains outside these results.

Periodic obstructions to small-extrapolation Chambolle–Pock

2026-11-01T00:00:00Z

For the unmodified primal-first Chambolle–Pock iteration with extrapolation θ and step product κ = τσ‖L‖², every κ > 2 admits fixed compact polytopes in R⁴ whose iteration cycles without converging for all sufficiently small θ ≥ 0. Any universal sufficient condition κ < c(θ) therefore satisfies limsup c(θ) ≤ 2 as θ ↓ 0, while bilinear and quadratic problems allow κ < 4/(1+2θ). A nonconstant three-cycle of actual proximal updates exists exactly when κ²(1+θ)² − 2κ(1+3θ) − 3 ≥ 0, attained by compact triangles, and a rational seven-cycle has an open set of nonconvergent starting points. The obstructing θ-intervals are very small and the bound two is not shown to be sharp.

Structural selfadhesivity does not imply a numerically selfadhesive realization

2026-11-01T00:00:00Z

An explicit conditional-independence model on five variables, with 26 of the 80 elementary statements, admits for every overlap an adhesive polymatroid gluing whose two restrictions both induce it, yet no real polymatroid inducing it is selfadhesive in Matúš's numerical sense. This answers negatively Question 4.18 of Boege, who asked whether every structurally selfadhesive structural semigraphoid is induced by a selfadhesive polymatroid; his computation gave a positive answer for four variables. The positive part is certified by 32 integer polymatroids, the negative part by an exact identity among 51 nonnegative conditional mutual informations valid for every real inducing profile.

Exact three-uniform Berge-path saturation numbers and a counterexample to a conjectured formula

2026-11-01T00:00:00Z

English, Graber, Kirkpatrick, Methuku and Sullivan conjectured that the three-uniform Berge-P_m saturation number is ⌈(n − ⌊n/b⌋)/2⌉ for all m ≥ 10 and n ≥ b = 2a_m + 1. Two disjoint loose 8-cycles with two isolated vertices form a Berge-P₁₀-saturated hypergraph on 34 vertices with 16 hyperedges, one below the conjectured 17. The preprint determines the saturation number on the whole conjectured range: it is one less than the conjectured value exactly when b < n < 2b and n ≡ 2 (mod 2(m − 2)), and the extremal hypergraph at those orders is unique.

Exact Berge saturation numbers of the subdivided star in every uniformity and its minimality among trees

2026-11-01T00:00:00Z

Let B_a be the star K_{1,a} with one edge subdivided. For all a, r ≥ 3 and n ≥ (r − 1)a + 1 the r-uniform Berge saturation number, in which any absent r-set may be added, is sat_r(n, Berge-B_a) = ⌊a(n + 1)/((r − 1)a + 1)⌋, attained by disjoint loose stars; it is a for max(a + 2, r + 1) ≤ n < (r − 1)a + 1. Saturated hypergraphs within k hyperedges of the minimum consist of loose stars with at least a spokes and at most (r − 1)((r − 1)a + 1)(k + 1) − 1 further vertices once n ≥ ((r − 1)a + 1)((r − 1)(k + 1) + 1), sharply for k = 0. For a ≥ 5 every tree T ≇ B_a on a + 2 vertices has sat₃(n, Berge-T) ≥ ⌈(a + 1)(n − 2)/(2a + 3)⌉, so B_a is the unique tree of its order with the least asymptotic lower three-uniform saturation coefficient, a/(2a + 1), and the unique minimizer for n ≥ 3(a + 1)(2a + 1); this is the Berge analogue of the theorem of Faudree, Faudree, Gould and Jacobson for graphs.

Strong minimum-rank multiplicativity from unextendible product constraints

2026-11-01T00:00:00Z

A complex matrix subspace annihilated by an unextendible family of product functionals doubles minimum rank under tensor products: if every nonzero member of T has rank at least r, every nonzero member of S ⊗ T has rank at least 2r, and the minimum rank is exactly doubled when S contains a rank-two matrix. For the Tiles bound entangled state ρ, this gives Schmidt number SN(ρ^⊗N) = 2^N for every N ≥ 1, so N copies cost exactly N ebits to prepare by LOCC with zero error. The two-copy case and nonsharp exponential growth were known.

A quartic entropy deficit for random rankings

2026-11-01T00:00:00Z

A distribution on rankings of n labels has entropy at most (1 − ε⁴/144) log(n!) if every pairwise comparison has probability at distance at least ε > 0 from 1/2. The proof includes all finite sizes and actual position-valued permutations. It assumes neither independent comparisons nor transitive preferred directions.

Order-polynomial positivity for width-two posets with a chain of at most five elements

2026-11-01T00:00:00Z

For every finite poset P that is the union of two chains A and B with |A| ≤ 5, the chain B being arbitrarily long and the comparabilities between the chains arbitrary, every coefficient of |P|! Ω_P(t) is a nonnegative integer, where Ω_P is the order polynomial; this is a case of Conjecture 7.1 of Ferroni, Morales and Panova. The linear coefficient is at least (a − 1)!(b − 1)! m/(a + b)!, with m the number of elements of B comparable with some element of A, so it is positive exactly when the comparability graph is connected. The proof reduces the linear coefficient on each of 64 chambers to a polynomial in the gaps between boundaries, certified nonnegative by exact square certificates and 194,112 point identities, and lifts it to all coefficients by the ideal–filter reduction of Ferroni, Morales and Panova.

Three-uniform Berge-C₄ saturation: linear bounds with coefficients 5/8 and 3/4, the cone spectrum and exact small values

2026-11-01T00:00:00Z

The three-uniform Berge-C₄ saturation number s(n), in which any absent triple may be added, satisfies max(⌈(n − 1)/2⌉, ⌈(5n − 3623)/8⌉, ⌈(5n + 89)/8 − √(270n + 1891)/4⌉) ≤ s(n) ≤ ⌈(3n − 9)/4⌉ for every n ≥ 9, so 5/8 ≤ liminf s(n)/n ≤ limsup s(n)/n ≤ 3/4; the previous explicit upper bound was n − 2. The preprint classifies the saturated cones for n ≥ 5, whose sizes are exactly the integers from ⌈(3n − 5)/4⌉ to n − 2 together with n − 1 when n ≡ 1 (mod 3), proves that a vertex reached from every other vertex by paths of lengths one and two forces 3(n − 1)/4 hyperedges, and gives s(n) for n ≤ 13 from exhaustive certificates.

Ultra-log-concavity of domination sequences of powers of paths and cycles

2026-11-01T00:00:00Z

The domination counts of every natural power of a path or cycle form an ultra-log-concave sequence. The proof includes all natural graph orders and powers, the binomial normalization, and the boundary graph cases.

The multi-marginal converse of Minty's theorem fails in the plane

2026-11-01T00:00:00Z

Bartz, Bauschke, Phan and Wang asked whether the converse half of Minty's theorem extends to several marginals: does maximal c-monotonicity for the cost Σ⟨xᵢ, xⱼ⟩ force the sum image to be the whole space? For three marginals on ℝ², three integer points form a c-monotone set to which no tuple with sum zero can be added, because four compatibility inequalities have a positive weighted sum equal to a sum of squares plus 23/26. Every maximal extension, which exists by Zorn's lemma, therefore has a sum image that omits the origin and is not convex.

The upper saturation spectrum of Berge triangles in triple systems

2026-11-01T00:00:00Z

For every n ≥ 8, a Berge-triangle-free triple system on n vertices with more than ⌊(n−1)²/8⌋ edges lies inside a paired product F(n,p): p disjoint pairs, the other n − 2p vertices, and all triples formed by one pair and one other vertex. Saturated systems above this threshold are therefore paired products, and the saturation spectrum there consists exactly of the sizes p(n − 2p) up to Győri's maximum ⌊n²/8⌋. The threshold is sharp, since a saturated system with exactly ⌊(n−1)²/8⌋ edges that is not a paired product exists for every n ≥ 8. About n/4 sizes lie in the window and n/4 − O(√n) of them are missing, so the missing sizes near the top are not finitely many uniformly in n.

Blocking sets of PG(2,13) invariant under a fixed-point-free Singer subgroup meet some line in five points

2026-11-01T00:00:00Z

Let H be the subgroup of order three of a Singer cycle group of PG(2,13); it fixes no point. Every H-invariant blocking set of PG(2,13) meets some line in at least five points, and the 21-point blocking set of Blokhuis, Brouwer and Wilbrink, invariant under a conjugate of H, meets every line in at most five points. So five is the least possible largest line intersection of an H-invariant blocking set, for every fixed-point-free collineation of order three. This settles a symmetric case of the plane of order 13 in Erdős Problem 1159; blocking sets without this symmetry are not covered.

A rank bound for represented totally positive unit squareclasses

2026-11-01T00:00:00Z

Let K be a totally real algebraic field, possibly of infinite degree, and L a totally positive definite quadratic lattice over its ring of integers with Q(L) ⊆ O_K, not necessarily classical. Then every family of pairwise distinct classes of totally positive units modulo squares of units represented by L has at most rank L members, and the bound is attained when the field has enough classes. This answers negatively Question 3.4 of Daans, Kala, Man, Widmer and Yatsyna, who proved the bound 2n − 2 and asked whether n + 1 classes can occur. The bound is a short consequence of standard facts on finite reflection groups: unit-valued vectors give integral reflections, and conjugate reflections carry the same unit class. As a corollary, a universal lattice has rank at least the number of totally positive unit classes.

Logarithmic influence bounds and variance counterexamples for Boolean decision-tree covariance

2026-11-01T00:00:00Z

For a finite Boolean decision tree under uniform independent bits, the sum of absolute child covariances, weighted by node reach probabilities, is at most the sum of each coordinate influence times the base-two logarithm of its global query-node count. Counts zero and one receive weight zero. No coordinate repeats on a root-to-leaf path. A read-k tree therefore has absolute covariance at most I(f) log₂ k. The known dummy-prefix construction attains equality for powers of two. A four-variable witness and a nine-variable witness refute the coefficient-one logarithmic variance proposal. Every query in the latter witness is relevant. The formalization contains no Fourier-entropy theorem.

Diagonal concavity does not imply concavity for sums of elementary symmetric polynomials

2026-11-01T00:00:00Z

For positive M, let f=1+Mσ₁+σ₂+Mσ₃+σ₄ in four variables. Its fourth root is concave on the positive diagonal exactly when M≥1. It is concave on the full positive orthant exactly when M=1. Every M>1 therefore disproves Lachaume’s diagonal criterion with strictly positive coefficients.

Counterexamples to a best-of-n win-rate bound at every budget

2026-11-01T00:00:00Z

For every n≥2 and δ>0, there is a finite positive reference distribution whose largest atom is at most δ and whose best-of-n policy violates the proposed divergence–win-rate curve. A four-response best-of-two example gives exact probabilities and a strict logarithmic certificate. The proof uses finite interval bounds and gives a quantitative positive gap. It does not determine the sharp replacement curve.

Real positive-definite counterexamples to extended q-permanent monotonicity

2026-11-01T00:00:00Z

For every real parameter q>1, a real symmetric positive-definite matrix has a negative q-permanent derivative at q. An explicit matrix of order five decreases from q=14 to q=15. Correlation matrices with no zero entries and integer-entry examples are also obtained. The dimension can depend on q. The real-symmetric question on [−1,1] remains outside the result.

Counterexamples to a late-iterate ordering conjecture for generalized Hofstadter functions

2026-11-01T00:00:00Z

For every k≥4, some input n satisfies Fₖ^(2k+1)(n)>Fₖ₊₁^(2k+2)(n). The proof constructs inverse iterates whose exact gap is k−3. The finite witness F₄⁹(132)=8>7=F₅¹⁰(132) gives the first instance. The result concerns infinitely many parameters, not infinitely many violating inputs for each fixed parameter.

A rational five-cycle for two-block ADMM with multiplier step below two

2026-11-01T00:00:00Z

An exact period-five orbit for multiplier-only two-block ADMM at step 199/100.

Sharp root log-concavity of alternating Motzkin sums

2026-11-01T00:00:00Z

Let Tₙ be the alternating sum of Motzkin numbers through index n. The sequence Tₙ^(1/n) is strictly log-concave on its tail starting at five. The inequality reverses at center five, so no earlier tail is log-concave. This strengthens Zhao’s proposed tail starting at nine. The source recurrence and finite sum are linked in Lean.

Sharp log-concavity for alternating Motzkin sums

2026-11-01T00:00:00Z

A robust bounded-forcing recurrence theorem and corrected scaled corollary, with applications to alternating Motzkin-type sums.

Sharp universal Rényi threshold for iid sums on finite Sidon sets

2026-11-01T00:00:00Z

A sharp universal Renyi-order threshold for iid sums supported on finite strong-B2 sets.

Unbounded coalesced-mass certificates for resolution trees

2026-11-01T00:00:00Z

A coalesced dyadic clause-mass functional has exact supremum n+1 on the stated unrestricted resolution-tree class and is unbounded across dimensions.

Ordered-instance allocation transfer with exact extra-copy preservation

2026-11-01T00:00:00Z

A set-valued ordered-instance transfer preserving utility improvement, the exact extra-copy count, per-good caps and coverage.

Leaf deletion can move the unique mode of a tree domination polynomial by two

2026-11-01T00:00:00Z

Deleting a leaf can move the unique mode of an explicit tree domination polynomial by two.

A sharp universal skewness bound for unicyclic currents

2026-11-01T00:00:00Z

The net marked-edge current on every finite bidirectional Markov ring with at least three states, positive rates and unequal forward and reverse rate products satisfies −1/16 ≤ C₃/C₁ ≤ 1. The two-channel two-state case has the stronger interval [1/4,1]. This ratio is not standardized skewness.

Exact conditional contraction for Pauli channels and depolarization in every dimension

2026-11-01T00:00:00Z

The conditional hockey-stick contraction coefficient is determined exactly for every stochastic Pauli qubit channel and for depolarization throughout its physical parameter interval in every dimension. The formulas include all boundary cases and every hockey-stick parameter γ ≥ 1, with arbitrary finite quantum references.

Exact three-setting correlation bounds at fixed steering centre

2026-11-01T00:00:00Z

The exact least upper bound of the optimized three-setting squared correlation value is determined for every two-qubit steering-centre length. The threshold for a value greater than one is c < 2/3. Attainment, nonattainment and the singular marginal boundary are included.

A positive constant-recursive sequence with nonquasipolynomial power ranks

2026-11-01T00:00:00Z

Exact minimal recurrences for every power of a rank-thirteen sequence, nonquasipolynomial ranks, and infinitely many power-rank profiles at each rank at least three.

A sum-of-squares proof of the four-qubit Higuchi–Sudbery entropy bound

2026-11-01T00:00:00Z

The average von Neumann entropy of the three two-qubit marginals of a pure four-qubit state is at most log(2) + log(3)/2. The Higuchi–Sudbery state attains this bound. The proof uses all sixteen complex amplitudes, actual reduced-matrix eigenvalues, an exact sum-of-squares identity, and an explicit maximizing state.

The three-state root-fidelity entropy bound

2026-11-01T00:00:00Z

The entropy of the weighted three-by-three root-fidelity matrix bounds the Holevo quantity of every three-state ensemble in arbitrary finite dimension. The result includes mixed and singular states and zero probabilities.

A 64-point spectral tile in F₂⁹ with no periodic complement

2026-11-01T00:00:00Z

An explicit 64-point set E ⊂ F₂⁹ is spectral and tiles by translation, but has no periodic tiling complement: its complements are exactly the 512 translates of one aperiodic eight-point set. In particular no linear subspace complements E, and F₂⁹ fails the periodic tiling property of Fan and Zhang. The construction places eight three-dimensional subspaces in parallel six-dimensional fibres; the shifts disjoint from E are the 28 pair sums of an affine eight-circuit, which form the line graph of K₈ and contain no three-dimensional subspace. This answers negatively the stronger subspace-complement part of Question 8.1 of Ferguson and Iosevich and its equivalent Question 8.2. Whether every spectral 64-point subset of F₂⁹ tiles remains open.

Sharp escape of complex domination roots in square flowers

2026-11-01T00:00:00Z

Sharp complex-root escape for square flowers, with complete flower and book real-root classifications and multiplicities.

A three-state thermal counterexample to a heat-current kurtosis conjecture

2026-11-01T00:00:00Z

A three-state model with two positive-temperature baths satisfies local detailed balance and has positive total off-diagonal transition rates. Its dominant tilted-generator eigenvalue has fourth-to-second derivative ratio −3470947982622836/176627429595846691. This refutes the thermal nonnegativity proposal in its finite spectral formulation. The proof constructs a smooth local eigenvalue branch, proves strict dominance over distinct complex eigenvalues, and calculates the exact derivatives.

An entropy increase for a structurally disassortative DomiRank graph

2026-11-01T00:00:00Z

A connected graph with 14 vertices and 32 edges has negative degree–log-degree edge covariance. All resolvent scores are positive on [251/1000,63/250] inside the stable domain. Its entropy nevertheless increases by more than 1/10000 in natural-logarithm units. This refutes the source’s assertion for every positive-score interval.

An unbounded obstruction to projection AM–GM in Loewner order

2026-11-01T00:00:00Z

Three rank-one orthogonal projections in dimension three violate the symmetrized-product Loewner AM–GM bound. No finite real multiplier repairs it, even with a positive-definite mean. The examples do not satisfy the additional nonsingular block structure of the motivating coordinate-descent application.

A pure-state matched-basis information inequality violation in dimension five

2026-11-01T00:00:00Z

A dimension-five pure-state matched-basis information inequality violation, with a conventional all-prime companion.

A polygon enclosure fails for sums of normal matrices

2026-11-01T00:00:00Z

A polygon enclosure fails for a specified pair of normal matrices and its complete transposition graph.