All works

math.GM — General Mathematics

Uniform irreducibility and Galois groups of generalized Laguerre polynomials

Contributed by Khai-Hoan Nguyen-Dang

We prove that the generalized Laguerre polynomial $H_{n,s}$ is irreducible over $\mathbb Q$ and has Galois group containing $A_n$ for every positive integer $s$, once $n$ exceeds an effective absolute constant. Equivalently, every numerator and denominator in the exponential Pad\'e table has these properties once its own degree is sufficiently large. The exceptional positive integral pairs form an effectively computable finite set. The proof combines integral traces at degenerate fibers, Pad\'e resultants, and an energy inequality for Galois-invariant positive semidefinite forms on the roots.

cs.CC — Computational Complexity

Class Numbers of Imaginary Quadratic Orders in the Polynomial Hierarchy

Contributed by Frank Vega

For $d \geq 3$ with $-d \equiv 0, 1 \pmod 4$, let $h(-d)$ be the number of classes of primitive positive definite binary quadratic forms of discriminant $-d$. We prove that $d \mapsto h(-d)$ is computable in deterministic polynomial time with an oracle for a language in the second level $\Sigma_2^{\text{P}}$ of the polynomial hierarchy. The proof uses only reduction theory of binary quadratic forms, Lagrange's theorem and Hermite normal forms; it does not assume the generalized Riemann hypothesis, does not take a generating set of the class group as input, and uses no randomness. A $\Sigma_2^{\text{P}}$ oracle finds a generating set of $O(\log d)$ forms, and an NP oracle then determines the lattice of relations, whose index is $h(-d)$. Consequently, if $\text{P} = \text{NP}$, class numbers of imaginary quadratic fields are computable in polynomial time. As an application, the number $D_n$ of representations of $n$ by the ternary form $8x^2 + 2y^2 + 16z^2$ in Tunnell's congruent number criterion equals $2h(-4n)$ for square-free $n \equiv 2 \pmod 8$ and vanishes for square-free $n \equiv 6 \pmod 8$; hence $D_n$ lies in $\text{FP}^{\Sigma_2^{\text{P}}}$ on even square-free $n$, and an exact counting reduction from $\#\text{SAT}$ to $D_n$ would collapse the polynomial hierarchy to $\Delta_3^{\text{P}}$.

Author comments: In this new version, we have added a computer-based numerical verification.

math.NT — Number Theory

Selected character moments, zero-free half-planes for Hecke L-functions over number fields, and Quasi-Riemann Hypothesis

Contributed by Baiying Liu

In this paper, we obtain a zero-free half-plane $\Rea s>B_2$ for every finite-order Hecke $L$-function over every number field, where \[ B_2=\frac{777-\sqrt{1081}}{852} =0.873381966613207519929\ldots. \] We adapt selected character-sum estimates and finite moment inductions to ideal-character sums over arbitrary number fields and restore ordinary sixth powers. Two finite moment inductions give an ordinary product estimate with length condition $n_1+n_2+2z\le M$ for sufficiently fine original prime factors. Applying this estimate to zero detection and the averaged numerator in the cubic-theta comparison gives the stated boundary. We determine the exact infimum of the resulting comparison system and verify its scalar inequalities by rational Bernstein bounds. The proof uses the earlier theta, moment and continuation theorems over number fields.

math.RT — Representation Theory

Nonpolynomial Hall numbers for tame and wild algebras

Contributed by Qiyue Tang

We give examples showing that fixed-triple Hall polynomials may fail to exist for tame nonhereditary algebras and for wild algebras, both hereditary and nonhereditary.

math.NT — Number Theory

Landau--Siegel zeros over varying number fields

Contributed by Baiying Liu

Let $K$ be a number field of degree $n$ and absolute discriminant $D_K$. We give a direct arithmetic proof that every real zero $\beta\in(0,1)$ of $\zeta_K$ satisfies \[ (1-\beta)\{\log D_K+n\log(2n)+6n+1\}\geq\frac1{1370}. \] The bound is explicit for arbitrary varying fields, including nonnormal fields, and gives $(1-\beta)\log D_K\gg1/\log(2n)$ for $K\ne\Q$. Exceptional-zero exclusion over each fixed base field was obtained by Banks and, independently, Frei--Kai. The proof here combines OpenAI's determinant method, Masser's zero estimate as used by Kerr, and Banks' selected-embedding calculation. The resulting bound is expressed directly in the degree and discriminant of $K$, without introducing those of its normal closure. Stronger nonvanishing conclusions follow from recent half-plane results and positivity. The contribution here is an arithmetic proof with fully displayed dependence on the varying field, using neither those half-plane results nor earlier zero-exclusion results for real Dirichlet $L$-functions or Hecke characters.

math.NT — Number Theory

GRAM OBSERVATIONS AND THETA-SERIES RECONSTRUCTION OF EUCLIDEAN LATTICES

Contributed by Changxia Lin

A finite collection of lattice-vector lengths probes a positive-definite Gram matrix through rank-one linear functionals. We use this observation map to give a local criterion for non-identifiability and to construct an exact three-dimensional collision at a closed squared-length cutoff of one: two non-isometric lattices have identical complete truncated vector-length data, although their Gram determinants are different. A signed-cycle construction gives a higherdimensional family whose complete rank-one length multisets agree at every cutoff below an explicit threshold, with an exact first-shell discrepancy. We then turn to multi-vector data. The full degree-n Gram theta data determines every rank-n Euclidean lattice, by a basis-witness and covolume-index argument. On each compact subset of lattice moduli space, the same argument yields a finite cutoff; a determinant-one cusp family shows that the basis-witness cutoff supplied by this method cannot be uniform on the full moduli space. Finally, a published rank-32 theta-series coincidence, together with a minimum-shell spanning identity, is stabilized by orthogonal direct sums to every rank n ≥ 32. If k(n) denotes the least degree of full Gram theta data that universally determines rank-n Euclidean lattices, we obtain 4 ≤ k(n) ≤ n for n ≥ 32. The paper separates proved statements, cited lattice-theoretic input, and computational checks; it does not claim a solution of the exact minimum-degree problem.

math.NT — Number Theory

A congruence modulo p^2 for the p-th coefficient of e(1-z)^((1-z)/z)

Contributed by David Niedbala Giraudin

Define E_n by sum_{n>=0} E_n z^n = exp(sum_{k>=1} z^k/(k(k+1))) = e(1-z)^((1-z)/z). Equivalently, e(1+1/x)^(-x) = sum_{n>=0} E_n (x+1)^(-n), the reciprocal of the expansion of e^(-1)(1+1/x)^x used by Yang in refinements of Carleman's inequality. We prove that for every odd prime p, E_p = -p/2 (mod p^2); in particular v_p(E_p) = 1. The proof is elementary. With Phi(u) = e(1+u)^(-1/u), it combines the two expressions of E_p through the coefficients of Phi that come from the identities e(1-z)^((1-z)/z) = Phi(-z)/(1-z) and e(1-z)^((1-z)/z) = Phi(z/(1-z)), and it reduces the congruence to the differential recurrence of the E_n read modulo p. A script checks the statement and the intermediate congruences for all odd primes below 200 and a few larger ones.

math.NT — Number Theory

The exact multiplicity of the Wilson supercongruence in F_q[t]

Contributed by David Niedbala Giraudin

Let q = p^s with p an odd prime, and let P be a monic irreducible polynomial of degree d >= 2 in F_q[t]. The Wilson analogue F_d = -1 mod P holds for the product F_d of all nonzero polynomials of degree less than d, and Thakur proved that P is a Wilson prime, i.e. F_d = -1 mod P^2, exactly when all exponents of P are congruent to 0 or 1 mod p, in which case the congruence holds modulo P^(p-1). He asked whether it can hold modulo a higher power, and what the exact multiplicities are. Let r be the largest integer such that all exponents of P are congruent to 0 or 1 modulo p^r. We prove that the exact multiplicity m = v_P(F_d + 1) is 1 if r = 0, p^r - 1 if 1 <= r <= s, and q + 1 if r > s. In particular m <= q + 1, and the multiplicity of a Wilson prime in the numerator of the sum over i < d of 1/(t^(q^i) - t) is exactly m - 1. Examples over F_3, F_5 and F_9 show that all regimes occur. A script checks the theorem on all primes of small degree and on these examples.

math.NT — Number Theory

The p-adic valuations of the coefficients in Yang's expansion of (1+1/x)^x

Contributed by David Niedbala Giraudin

Yang's coefficients b_n, defined by (1+1/x)^x = e(1 - sum_{n>=1} b_n (x+1)^{-n}), appear in refinements of Carleman's inequality, and the literature on them concerns their analytic properties. We study their p-adic valuations for an odd prime p. From the identity 1 - sum_n b_n u^n = exp(-sum_{k>=1} u^k/(k(k+1))) one gets v_p(b_n) >= -M(n), where M(n) is the optimum of a knapsack problem on the compositions of n. We compute M(n) in closed form and show that, for n >= 2, the optimum is attained by a unique composition exactly when n = (p-1)m with m in an explicit set U_p defined by a digit condition. For these n the bound is an equality: v_p(b_n) = -(m + v_p(m!)). The set U_p has density 1/(p-1), so the valuation of b_n is determined on a set N_p of indices of density 1/(p-1)^2. The set N_p contains the indices p^K - 1, at which b_{p^K-1} is the unique coefficient of minimal valuation among b_1, ..., b_{p^K-1}. The same statements hold for the coefficients of e(1-u)^{1/u}, and the minimality statement for those of e(1-u)^{1/u-1}. A script checks all statements numerically in finite ranges.

math.NT — Number Theory

Torsion of elliptic curves over number fields of degree $6$, $7$, $8$ and $9$

Contributed by Filip Najman

We determine all the groups that appear as the torsion group of an elliptic curve over a number field of degree $d$ for $d=6$, $7$, $8$ and $9$. Apart from the groups that occur infinitely often, exactly $4$ groups occur in degree $6$, $3$ in degree $7$, $8$ in degree $8$ and $9$ in degree $9$, and among them the groups $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/24\mathbb{Z}$ over sextic fields, $\mathbb{Z}/3\mathbb{Z}\times\mathbb{Z}/15\mathbb{Z}$ over octic fields and $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/26\mathbb{Z}$ over nonic fields were not known to occur. For $13$ of these $24$ pairs of a degree and a group we also determine all the elliptic curves that realize them. The proofs combine the global method of Derickx, Hecke sieves, formal immersion criteria at the prime $2$ with Hecke operators, and Riemann-Roch computations on modular curves and their quotients over finite fields.

Author comments: Fixed ten references to equations (6) and (10), which appeared as "??" in v1 (compiled with TeX Live 2025). No mathematical changes.

math.NT — Number Theory

A sharper Type I/II estimate for primes in arithmetic progressions to smooth moduli, and H_m << e^{3.8043 m}

Contributed by David Niedbala Giraudin

In Stadlmann's Type I/II equidistribution estimate for x^delta-smooth moduli, one of the three conditions, 64 varpi + 20 delta + 2 sigma < 1, can be replaced by 48 varpi + 18 delta + 2 sigma < 1. This weaker condition follows from the second one, 48 varpi + 16 delta + 4 sigma < 1, whenever delta <= sigma. The proof follows Stadlmann's argument and makes two changes: the length of a block in one Taylor expansion depends on the pair of shifts, and the bounds are summed over the pairs instead of being multiplied by their number. Combined with Stadlmann's general Harman-sieve construction, this gives a minorant for the primes with exponent of distribution 1/2 + 2 varpi to smooth moduli for every varpi < 7/544, instead of varpi = 0.01265. As a consequence, liminf (p_{n+m} - p_n) << e^{cm} for every c > 3.80428, and for every c > 3.80423 if a numerically computed value of the sieve deficit is used. This improves Stadlmann's exponent 3.8075.

math.NT — Number Theory

Serre modularity over real quadratic fields

Contributed by Lassina Dembélé, Luis V. Dieulefait, Nuno Freitas, Ariel Pacetti

We prove Serre's modularity conjecture, in every odd residual characteristic and with arbitrary conductor, over twenty real quadratic fields $K=\Q(\sqrt d)$. The proofs for $d=2,5,13,17,41$ use classical discriminant bounds. For $d=33$ the proof uses the zero-free region $\Re s>31/32$ of Cureton and Thorner, and for $d=6$ the region $\Re s>0.9586$, which we derive from specified analytic inputs. For the remaining thirteen fields we assume GRH for Dedekind zeta functions. The main tools are conductor reductions at an odd split prime and at two, a weight-reduction step by congruences, a scalar dyadic modularity criterion, and integral trace orders. We also use a modularity theorem for the mixed ordinary--nonordinary case and, for the seven fields in which two splits, dyadic propagation results.

math.CO — Combinatorics

Three-colour cuboids from four-colour cubes

Contributed by Jason Davies

We determine the least number of four-colour cubes that guarantees a cuboid with monochromatic faces using exactly three colours. Each source uses all four colours of a fixed palette; only proper rotations are allowed, and interior faces need not match. For integer side lengths $2\le a\le b\le c$, the sharp threshold for the corner-and-edge frame is $\max\{16a-3,4(a+b+c)-16,\lfloor(24c-1)/5\rfloor\}$, and the complete-cuboid threshold is $\max\{abc,16a-3,\lfloor(24c-1)/5\rfloor\}$. The three frame bounds arise from colour-bucket capacity, the number of positions and balanced checkerboards. The upper bound combines corner containment, a weighted graph selecting edge directions, and Hall's theorem. For cubic frames the threshold is $16n-3$, with exactly two failing collections of size $16n-4$. For $n\ge3$ we classify every failure of size at least $14n$ and count these collections and the inclusion-maximal ones. The case $n=2$ answers the threshold question in Problem 2 of Berkove, Eickemeyer and Schulman (2018). All threshold, stability and counting results are formalised in Lean.

math.NT — Number Theory

Turán's pure power sum problem: certified bounds for finite $n$ and $\limsup R_n\le 0.68637$

Contributed by Vasileios Bakeas, Ricardo Pinho

For $n\ge1$ let $R_n=\inf\max_{1\le k\le n}|z_1^k+\dots+z_n^k|$, the infimum being over all complex $z_1=1,z_2,\dots,z_n$. This is Turán's pure power sum problem (Erdős problem #519). The best published upper bound for the constant $\limsup R_n$ is $0.69368$, from a computation of Harcos reported by Biró in 2000. We prove $\limsup_{n\to\infty}R_n\le0.68637$. The construction is the Biró–Harcos linearization, with the first half of the power sums prescribed as $s_k=Te^{i\varphi(k/n)}$ for a piecewise-linear phase $\varphi$ that varies from the start. We prove a limit theorem: the quantity that controls the second half converges to an explicit functional of the solution of a linear Volterra equation whose associated operator is a contraction. The resulting inequality is certified in ball arithmetic by an a posteriori error bound, with two separately written programs that follow the same mathematical argument. We also give rigorous upper bounds for $R_n$ at 13 values of $n$ between $300$ and $10^6$, from explicit exact certificates, each checked by one or two programs written independently of the generator. In an appendix, R. Pinho applies the limit theorem to a finer profile, with sharper error bounds, and certifies $\limsup R_n\le0.6863607950$. The work was done with substantial help from AI models; it has not been refereed.

math.NT — Number Theory

Arcs of Π(1+z^p): prime races, the Gaussian core, Dirichlet L-functions and a certified solver for Subset Sum

Contributed by Antonio Mennillo

The subsets of $\{1\} \cup \{p \le N\}$ reach every integer between 0 and their total, by Bertrand's postulate, so deciding whether a target is a subset sum of the primes is trivial. Counting the solutions is not: the count carries the arithmetic of the primes. We study the generating function $F_N(z) = \Pi(1 + z^w)$ on the unit circle and follow one object, the arc at a rational $a/q$, through three settings. For the primes, the centre of the arc $a/q$ rises above the sea of the minor arcs exactly when $q = 2^k (k \ge 2)$ or $q = 2p^j$, with excess $\Lambda(q/2)/\varphi(q)$, and the neighbourhoods of all the other arcs are bounded uniformly in $q$; Theorem E gives the expansion arc by arc. The height of an arc is a race of the primes modulo $q$ with weights $\log |\cos (\pi ra/q)|$. By Landau's theorem a one-sided bound on it is equivalent to GRH for the even characters it sees, unconditionally at $q = 8$, 10 and 14, where the zero-free segments it needs are certified; at $q = 14$ the race changes sign infinitely often. The principal arc carries $\zeta$ in the centre and width of its Gaussian. For arbitrary weights, the arcs become the decision rule of a Subset Sum solver that reads them on the original weights and ends with a meet-in-the-middle filtered by a random residue class. Every answer it returns is verified. The dummy weights of HURS explain why the reading is exact: they blur the counts by a flat window, and the blur is invertible on counts, though not on reachable sets. On random weights of $b$ bits the method has rigorous guarantees, in time $2^{b/2+o(b)}$ and, with the filter, $2^{b/3+o(b)}$, resting on a local limit theorem uniform in the size of the weights and on a concentration bound for signed subset sums. Finally, the solver is read inside a general theory of constraint propagation, in which every network factors as $T = T_{\mathrm{comp}} \circ T_{\mathrm{struct}}$.

math.NT — Number Theory

On the Fontaine--Mazur conjecture for $2$-adic representations

Contributed by Luis Dieulefait

Let $F$ be a totally real field, abelian over $\Q$, in which $2$ splits completely, and let $\rho:G_F\to\GL_2(\overline\Q_2)$ be an odd irreducible representation, de Rham with distinct Hodge--Tate weights at the places above $2$, whose residual representation is reducible with distinct characters at the places above $2$. We prove that $\rho$ is modular in the following cases: if $\rho$ is non-split at every place above $2$ at which it is reducible, which includes the case where $\rho$ is irreducible at all places above $2$, and, without any condition on the splitness, if $\rho$ is of Steinberg type at some finite place not dividing $2$, using an analogue at $2$, which we prove, of Pan's finiteness theorem for fully ordinary deformation rings. Both results include the mixed case, in which $\rho$ is reducible at some places above $2$ and irreducible at others. The pro-modularity input is obtained by extending to these fields the method of \cite{OAI}, which treats representations of $G_\Q$; this is where we use that the residual characters are distinct at $2$. For $F=\Q(\sqrt q)$ with $q\equiv1\pmod8$ prime, we remove the condition at $2$ from the first result in most cases; for $F=\Q(\sqrt{17})$ this includes every case where the residual semisimplification is scalar. The first result also holds when the residual representation is irreducible with solvable image, where residual modularity follows from an odd Artin lift. For the second result we combine the method of \cite{Die26b} with Thorne's potential modularity theorem for ordinary representations of weight two and the local indecomposability theorem of Hida and Zhao.

math.CO — Combinatorics

Off-Diagonal Commonality of a Shorter Even Cycle and a Longer Odd Cycle

Contributed by Taeyoung Kim

Let $n\ge4$ be even and let $\ell>n$ be odd. We determine the complete off-diagonal commonality region of the ordered pair $(C_n,C_\ell)$: it is the half-open interval $(0,a_{n,\ell}^{\star}]$ cut out by the unique root $a_{n,\ell}^{\star}\in(1/2,1)$ of $\frac{a^{n-1}\bigl(n+(\ell-n)a\bigr)}{\ell}=2^{1-n}.$ The upper endpoint is witnessed by the balanced disjoint union of two cliques. For the lower bound, cycle densities are expressed as traces of graphon operators. Rank-one eigenvalue majorisation separates the Perron eigenvalue from the rest of the spectrum, while the Hilbert--Schmidt norm leaves room for at most one negative eigenvalue to contribute with the wrong sign. That eigenvalue is paid for by the Perron eigenvalue, leaving the elementary inequality $x^{n}+(1-x)^{n}\ge2^{1-n}$.

math.PR — Probability

Computer-assisted bounds 0.556921 <= p_c <= 0.6463 for site percolation on the square lattice

Contributed by David Niedbala Giraudin

Let p_c be the critical probability of Bernoulli site percolation on Z^2. We prove 0.556921 <= p_c <= 0.6463. The previous best rigorous bounds known to us are p_c >= 0.5567 (Ashton, 2026, extending van den Berg and Ermakov, 1996) and p_c <= 0.666894 (Wierman and Oberly, 2024). The lower bound uses the growth-process comparison of van den Berg and Ermakov with the exact posterior on the exploration tree. We reduce this posterior to a scalar interval recursion and explore the tree generation by generation, with a prescribed order among siblings. This passes the value 0.556788... that the same scalar computation gives when the order is not used, and the value 0.556788090 reported by Ashton as the limit of his relaxation. The upper bound uses the substitution method of Wierman, in two stages with 4x4 regions, from the line graph of Z^2 (site threshold 1/2) to Z^2. Both bounds are certified by short pure-Python programs in exact rational and integer arithmetic.

math.LO — Logic

Intuitionistic Compression and Optimal Verification Trees

Contributed by Zhenyu Sun, Jilin Wang

In strict finitistic generation semantics, a formula is assertible at a generation if some stage in its verification tree forces it. We ask when assertibility agrees with intuitionistic truth on the poset of generations. The leaves of the verification trees, ordered by generation extension, form an intuitionistic Kripke model that preserves truth at every leaf, even without atomic prevalence. For a finite bundle, compression to the generations is exact if and only if each fiber of leaves has a greatest intuitionistic theory and the map selecting these theories has the back property. Compression remains exact under every restriction of the generation poset if and only if atomic prevalence holds. Conversely, consider coherent tree representations of a finite Kripke model whose atomic valuation reflects its order. Among representations with atomic prevalence, the minimum numbers of leaves and nodes over $u$ are $h(u)$ and $\delta_C+\sum_{x\preceq u}h(x)$, where $h(x)$ counts saturated paths from minimal worlds to $x$, and $\delta_C$ is $1$ if the comparability component of $u$ has no least element and $0$ otherwise. Saturated-path trees attain both minima simultaneously. Among representations attaining the node minimum at every world, the common root labels in components without a least element determine the representation up to isomorphism over the outer frame. Allowing rooted posets in place of trees can reduce the number of nodes exponentially.

math.NT — Number Theory

A certified lower bound for the essential minimum of the Zhang-Zagier height

Contributed by David Niedbala Giraudin

Let h be the absolute logarithmic Weil height. We prove that the essential minimum of the Zhang-Zagier height h(a)+h(1-a) is at least 0.2497815: for every d > 0, only finitely many algebraic numbers a satisfy h(a)+h(1-a) < 0.2497815 - d. The previous best lower bound was 0.24874 (Flammang, 2018). The proof combines Smyth's auxiliary-function method, with the polynomials of Doche and of Flammang, and the condition that the discriminant of the minimal polynomial is a nonzero integer, in the potential-theoretic form used by Smith and by Orloski, Talebizadeh Sardari and Smith for the trace problem. The infimum of the resulting auxiliary function is bounded below rigorously by monotone interval bounds and adaptive bisection, from a stored certificate. The statement concerns the essential minimum and is weaker in nature than the pointwise bounds of Doche and Flammang.

math.RA — Rings and Algebras

Cartan determinants for almost laura, quasi-laura, semilinear clannish and piecewise Bäckström algebras

Contributed by Qiyue Tang

We prove the Cartan determinant conjecture for corners of finite global dimension in almost laura, quasi-laura and (m)-quasi-laura Artin algebras, finite-dimensional piecewise Bäckström algebras, and semilinear clannish Artin algebras of finite dimension over their defining division ring.

math.CO — Combinatorics

Kernel Dimension of the Z/2 Connection Laplacian on a Finite Simple Graph (A Formalisation in Lean 4 / Mathlib v4.11.0)

Contributed by Jesus Vilela Jato

We present a Lean 4/Mathlib v4.11.0 formalization of the kernel-dimension theory of the $\mathbb{Z}/2$ connection Laplacian on finite simple graphs equipped with designated wrap edges. Each edge carries either the identity or the coordinate-swap matrix. A Hadamard–Kronecker change of basis decomposes the resulting rank-two connection Laplacian into the ordinary and signed graph Laplacians. An analogous splitting holds for the scalar Laplacian of the associated two-sheeted cover. A connected component is balanced precisely when its covering preimage has two connected components, equivalently when its sign holonomy is trivial along every closed walk. Consequently, the nullity of the signed Laplacian equals the number of balanced connected components, while the nullity of the full connection Laplacian equals the number of connected components plus the number of balanced components. For the flat connection, the nullity is twice the number of connected components. We also formalize exact characteristic polynomials and eigenvalue multisets, including multiplicities, for explicit ordinary and one-wrap signed cycle matrices for $n \geq 3$. Transfer to general connection-graph encodings remains conditional on specified matrix-reindexing identities. The contribution is a machine-checked formalization, exposition, and documented verification of results rooted in classical signed-graph spectral theory, rather than a claim of priority for those identities. The accompanying repository preserves proof artifacts, computational cross-checks, historical verification evidence, and documented corrections.

cs.DS — Data Structures and Algorithms

Three-machine unit-job scheduling in time O((L+2)^276)

Contributed by David Niedbala Giraudin

OpenAI recently gave a deterministic polynomial-time algorithm for P3|prec,p_j=1|C_max (problem OPEN8 of Garey and Johnson), with running time O((L+2)^150020) on a multitape Turing machine, L being the input length; the same exponent appears in the accompanying Lean statement. The exponent comes from enumerating all Boolean formulas with at most 10000 leaves, each leaf choosing its own triple of jobs. We show that every description actually used by the structural proof involves at most 21 distinct job triples and at most 3 distinct rank thresholds. Restricting the enumerated family accordingly, and using the length bound on accepting paths already proved there to limit the number of reachability passes, the same algorithm runs in time O((L+2)^276).

math.NT — Number Theory

Two improved lower bounds for the 2-adic valuation of the constant of the Prouhet-Tarry-Escott problem

Contributed by David Niedbala Giraudin

For an ideal solution of the Prouhet--Tarry--Escott problem of size $n$ the associated constant $C$ is divisible by a large power of $2$. Restricting to symmetric solutions, we prove two new lower bounds for $v_2(C)$: $v_2(C)\ge14$ at $n=15$, improving $12$, and $v_2(C)\ge19$ at $n=20$, improving $18$; both earlier bounds are those of the companion article hexagon:2611.00057. The first rests on two identities satisfied by every symmetric solution of odd size, one relating $v_2(C)$ to a difference of two products and one turning the index relation into an equality, together with an arithmetic criterion for a block of entries of equal valuation. The second rests on an identity internal to each of the two sides, from which we deduce that the vanishing of the cross invariant $R$ forces $k_0\equiv0\pmod4$ when $k_0\ge4$. Two self-contained verifiers accompany the note.

math.NT — Number Theory

The 2-adic valuation of the constant for symmetric ideal Prouhet-Tarry-Escott solutions

Contributed by David Niedbala Giraudin

For an ideal solution of the Prouhet--Tarry--Escott problem of size $n$, let $C$ denote the associated constant, and let $\overline{C_n}$ be the greatest common divisor of these constants. Filaseta and Markovich obtained $v_2(\overline{C_8})\ge6$ and determined $v_2(\overline{C_9})=9$, and asked for improved lower bounds in the range $10\le n\le12$; Bera and Nair gave bounds for $n=10$ and $n=12$. We treat the symmetric case at every size, even and odd, by $2$-adic arguments throughout. A single identity, the row law, governs both parities. For $n$ even, over $\mathbb{Z}$, it splits into two regular systems and yields $v_2(C)\ge6,8,11,12,12,16,18,18$ for $n=6,8,10,\dots,20$. The values at $n=6,8,10,12$ are attained and hence exact for the symmetric constant: in the symmetric case this closes both brackets of Bera--Nair, and gives $v_2(C'_8)=8$, for which the published information was $v_2(\overline{C_8})\ge6$. We also record that the two lower bounds of Bera--Nair, $9$ at $n=10$ and $11$ at $n=12$, are attained by $2$-adic solutions symmetric about $-1/2$, so that no argument using $2$-adic information alone can raise them. For $n$ odd, over $\mathbb{Z}_2$, it combines with a finite-difference lemma refining a congruence of Filaseta--Markovich to give $v_2(C)\ge11,12,12$ for $n=11,13,15$, against $8$, $10$ and $11$ from Kleiman's bound $(n-1)!\mid C$; the chain also reproduces the known values $4,6,9$ at $n=5,7,9$. At $n=11$ the bound is sharp: the two quadratic near-solutions of Coppersmith, Mossinghoff, Scheinerman and VanderKam are symmetric ideal solutions over $\mathbb{Z}_2$ with $v_2(C)=11$, since in both of the real quadratic fields involved the prime $2$ splits. This answers, for $n=11$ and $p=2$, two questions of Filaseta--Markovich, and exhibits for those sizes the case $p\le n$ of the question closing Rees--Smyth, who construct $p$-adic ideal solutions only for $p>n$: at the prime $2$ there is no local obstruction at these sizes. At $n=13$ we exhibit a symmetric ideal solution over $\mathbb{Z}_2$, obtained from a second object of the same authors, although its constant has valuation $17$.

math.CO — Combinatorics

Log-concavity in the Newton basis and support of rank-two scattering coefficients

Contributed by Qiyue Tang

We prove that polynomials with nonnegative, interval-supported log-concave Newton coefficients are closed under multiplication and, for integer Newton coefficients, under $D(m)\mapsto\binom{D(m)}k$ for integers $k\ge0$. Coordinatewise Newton multiplication preserves two-variable Schur positivity. We also determine the exact positive Newton support of symmetric rank-two cluster scattering coefficients.

cs.DS — Data Structures and Algorithms

Ulam Metric $k$-Center in Parameterized Linear Time

Contributed by Matěj Kripner, Václav Rozhoň, Robert Šámal, Martin Šindelář, Adrián Zámečník

Continuous Ulam Metric $k$-Center asks for at most $k$ permutations whose radius-$d$ balls cover the input rankings. For $m$ rankings of $n$ symbols, a deterministic algorithm takes $2^{O(kd\log(k+d+2))}mn$ word operations. Permanent reference rankings, short insertion lists, and small obstructions obtained by patience sorting control the search. For one center, this answers a question of Bai, Fomin, Golovach, More, and Wietheger with linear input dependence. An arbitrary-alphabet Hamming embedding with a nonexpansive retraction also gives the matching $2^{o(d\log d)}$ exclusion under the Exponential Time Hypothesis.

cs.DS — Data Structures and Algorithms

The Exact One-Half Threshold for Arbitrary-Capacity Bottleneck Multiple Subset Sum

Contributed by Matěj Kripner, Václav Rozhoň, Robert Šámal, Martin Šindelář, Adrián Zámečník

In bottleneck multiple subset sum, items with weights are packed into knapsacks with capacities, possibly leaving items unpacked, so as to maximize the minimum load. For identical capacities the best possible approximation ratio is $2/3$, and Chen et al.\ asked whether $2/3$ is also achievable for arbitrary capacities. We show that it is not: for arbitrary capacities we give a $1/2$-approximation using $O(n\log(n+1)+m\log(m+1))$ arithmetic and comparison operations for $n$ items and $m$ knapsacks, and prove that no fixed ratio above $1/2$ is achievable unless $\mathsf P=\mathsf{NP}$. We also give a short proof of the classical $1/2$ guarantee of Friesen and Langston for unit profits.

cs.CC — Computational Complexity

Universal-Clique Amplification and Hardness of Assignment-Optimal Edge Clique Covers

Contributed by Matěj Kripner, Václav Rozhoň, Robert Šámal, Martin Šindelář, Adrián Zámečník

An edge clique cover has assignment cost equal to the sum of its clique sizes. Unless $P=NP$, this cost admits no polynomial-time $O(N^{1/2-\eta})$-approximation for any fixed $0<\eta<1/2$, even on connected $N$-vertex graphs. A universal-clique join transfers coloring hardness without the additive obstruction in earlier reductions. Its exact residual formula also exhibits a change of optimal cover when a universal twin is added. Explicit counterexamples refute two proposed twin-extension rules, while weighted compression of true-twin classes preserves the optimum.

math.PR — Probability

Stationary diffusion on nonnegatively curved graphs

Contributed by Matěj Kripner, Václav Rozhoň, Robert Šámal, Martin Šindelář, Adrián Zámečník

Lazy random walk on a bounded-degree graph of nonnegative Ollivier–Ricci curvature has diffusive second moment and Gaussian upper tails when started in stationarity. If the maximum degree is $D$, then $\mathbb E d(X_0,X_n)^2\le4Dn$, with optimal order of dependence on $D$. The results also hold for stationary random rooted infinite graphs. The proof combines known heat-kernel total-variation decay with a reversible energy identity for the conditional mean displacement. Its exponential form controls variation between starting vertices; martingale concentration then controls the walk itself.

math.IT — Information Theory

Least-Atom Extremizers for Resampling Noise

Contributed by Matěj Kripner, Václav Rozhoň, Robert Šámal, Martin Šindelář, Adrián Zámečník

A single least-mass atom determines the extremal behavior of resampling noise on a finite probability space. A fixed-moment comparison gives the sharp generalized Samorodnitsky inequality for every $1<q<2$ through one scalar curve, and identifies all nonconstant scalar extremizers. The same principle determines every $L^p$-to-$L^q$ resampling norm for $1<p<q<\infty$, resolving a conjecture of Cao, Fan, Han, Qiu, and Wang. The sharp hypercontractive parameter increases strictly with the least atom. The uniform-binary phase diagram is explicit, and product equality cases are classified for every nonuniform reference law and for the uniform binary law.

math.NT — Number Theory

Nested Squares and Distinct-Prime Goldbach Partitions

Contributed by Frank Vega

The distinct-prime Goldbach conjecture asserts that for every $N\geq 4$ there is $M\in A_N=[1,N-3]$ with $N-M$ and $N+M$ both prime; geometrically, nested squares of side lengths $N$ and $M$ have an L-shaped difference of semiprime area $(N-M)(N+M)$. With $C_N$ and $D_N$ the sets of $M\in A_N$ for which $N-M$, respectively $N+M$, is prime, the conjecture for $2N$ is $C_N\cap D_N\neq\varnothing$. We embed this centered condition in a straddling one: $C_N\cap D_N$ lies in the set $E_N$ of half-differences $(Q-P)/2\in A_N$ of primes $2<P<N<Q<2N$. From Mikawa's theorem on prime pairs with almost all differences we deduce $|A_N\setminus E_N|\ll_D N/(\log N)^D$ for every $D>0$, and with explicit constants we prove $|E_N|>N/760$ for every $N\geq 3275$. Consequently, for all large $N$ and all but $O_D(N/(\log N)^D)$ primes $P<N$, the number $2N-P$ becomes prime after adding the difference $P'-P$ of two primes $P'<N$; Goldbach's partition is the case $P'=P$. In contrast, $|C_N\cap D_N|<(N-3)-\log^2(2N)$ for every $N\geq 34$, and covering all but $\log^2(2N)$ values of $A_N$ for all large $N$ would imply the Cram\'er-type bound $p'-p<2\log^2(2p+2)+2$. The method also shows that $2N$ is a sum of two distinct primes for more than $X/9600$ integers $N\leq X$, for every $X\geq 10^6$. Computations up to $N=10^6$ compare the number of distinct-prime partitions with the Hardy--Littlewood prediction and measure the missing values of $E_N$ forced by prime gaps. The conjecture itself is not proved here.

math.AP — Analysis of PDEs

Energy Solutions and Harnack Estimates for Nonlocal Drift–Diffusion

Contributed by Zhenyu Sun

We consider nonlocal drift–diffusion equations of order $2s\in[1,2)$ with bounded divergence-free drift. The symmetric measurable jump kernel satisfies local coercivity, a global upper form bound, and a scale-invariant bound on the intensity of long jumps. The transport form is bounded and skew-symmetric on $H^{1/2}\times H^{1/2}$. In the fractional energy space, distributional and variational solutions coincide and satisfy nonlinear testing identities. Viscous approximations converge strongly in $C_tL^2_x\cap L^2_tH^s_x$ to the unique solution of the Cauchy problem. For sources bounded in the cylinder, supersolutions nonnegative there satisfy a weak Harnack inequality involving the time integral of the negative exterior tail. Under this source bound, locally bounded solutions whose exterior tail belongs to $L^q$ in time for some $q>1$ are locally Hölder continuous. This also gives Hölder continuity at positive times for homogeneous global solutions with $L^2$ data. Under upper jumping smoothness, control of the positive tail on an earlier time interval yields the full parabolic Harnack inequality for solutions nonnegative on the full spatial slab. A kernel supported on alternating dyadic annuli satisfies all the kernel assumptions. The local estimates depend on the drift through $R^{2s-1}\|b\|_\infty$.

math.NT — Number Theory

Non-Archimedean Riesz-Frechet Representation

Contributed by K. MAHESH KRISHNA

We derive a Riesz-Frechet representation for bounded linear functionals defined on the p-adic Hilbert spaces introduced by Kalisch [\textit{Ann. of Math. (2), 1947}]. We also notice the surprising difference between the Archimedean case and the non-Archimedean case (exact non-Archimedean version of Riesz-Frechet representation fails).

math.GN — General Topology

Torus fibrations on $3$-dimensional manifolds which are smooth functions with finitely many critical values and stable fold maps

Contributed by Naok Kitazawa

We present smooth functions which are torus fibrations with finitely many critical values on $3$-dimensional closed manifolds, from (stable) fold maps into the plane. Fold maps are higher dimensional versions of Morse functions and introduced and studied first in the 1950s by Thom and Whitney. Saeki, Sakuma, and so on, are important researchers, studying related singularity theory and differential topology mainly, since the 1990s. $3$-dimensional closed manifolds always admit ones into the plane. Surface fibrations are important tools and objects in complex geometry, symplectic geometry, and $4$-dimensional geometry. It is important to construct important examples of the fibrations. In our paper, we present a $3$-dimensional related new study. Related to this, the author has succeeded in explicit construction of explicit functions with prescribed level sets and fold maps whose singular values sets are concentric on manifolds such as connected sums of sphere bundles over spheres.

math.NT — Number Theory

Linear independence of even Dirichlet beta values

Contributed by twinheterodox

Let $\beta(s)=1-3^{-s}+5^{-s}-\cdots$ denote the Dirichlet beta function. For all sufficiently large even integers $s$, we prove $\dim_{\mathbb{Q}}\mathrm{span}_{\mathbb{Q}}\{\beta(2),\beta(4),\ldots,\beta(s)\}\geq 0.21\sqrt{s/\log s}$.

cs.CC — Computational Complexity

The Siriaisa Algorithm for Minimum Independent Dominating Set

Contributed by Frank Vega

The Minimum Independent Dominating Set problem (MIDS) asks for a smallest maximal independent set of an undirected graph; unless $\mathrm{P}=\mathrm{NP}$, it cannot be approximated within $n^{1-\varepsilon}$ for any $\varepsilon>0$. We present version 0.0.8 of \textsc{Siriaisa}, an ensemble of eighteen strategies that adapts the vertex cover algorithm \textsc{Salvador} and the independent set algorithm \textsc{Esperanza} to MIDS, with every exact subproblem a forest solved by a conditioned tree dynamic program. We prove that the output is an independent dominating set computed in $O((n+m)\,\alpha_{\mathrm A}(n))$ expected time, that its ratio is at most $\Delta$, that it is optimal on forests, and that its ratio is at most $\sqrt n$ on three classes of graphs. A degree-splitting strategy yields a dominating set of size at most $\lfloor\sqrt n\rfloor\,(i(G)+s(G))$ for an explicit splitting excess $s(G)$, which vanishes on bounded-degree families for large $n$ and is largest on complete graphs, where the output is optimal. More generally, splitting at $\lfloor n^{1-\varepsilon}\rfloor$ gives the bound $n^{1-\varepsilon}\,(i(G)+s_\varepsilon(G))$ for every $\varepsilon\in(0,1]$. We state the ratio $n^{3/4}$ as a conjecture whose proof would imply $\mathrm{P}=\mathrm{NP}$. Through a reduction of Chleb{\'\i}k and Chleb{\'\i}kov{\'a}, every output decodes into a vertex cover, with conditional consequences for the Unique Games Conjecture. In experiments the output is optimal on $9\,999$ of $10\,000$ graphs with an exact integer-program optimum and on all $95$ bipartite graphs whose optimum was proved. As an application, we analyse the SAT solver \texttt{satypy} 0.0.4: it answers every unsatisfiable formula correctly, every SATISFIABLE answer carries a satisfying assignment, a wrong answer requires a ratio larger than $2p$ for $p$ variables, and its construction gives a self-contained proof of Halld{\'o}rsson's inapproximability theorem.

math.GR — Group Theory

Soluble subgroup pairs in symmetric and alternating groups

Contributed by Qiyue Tang

We prove that every pair of soluble subgroups of the symmetric group $S_n$ or the alternating group $A_n$ has trivially intersecting conjugates whenever $n\ge21$. This answers the second question of Anagnostopoulou-Merkouri and Burness in Problem 21.3 of the Kourovka Notebook.

math-ph — Mathematical Physics

A Bound Helium Dianion Does Not Exist

Contributed by Alec Shelley, Daniel Tartakovsky

For a non-relativistic atom with $Z$ protons and $N$ bound electrons, Lieb proved in 1984 that $N<2Z+1$. Several improved bounds have been developed, but they all leave open the question of whether a bound helium dianion can exist. We prove that the answer is no. The ground-state energies of the four- and three-electron helium Hamiltonians coincide, $E_4(Z{=}2)=E_3(Z{=}2)$, so a helium nucleus binds at most three electrons. A weighted form of Lieb's identity reduces the problem to the nonnegativity of an explicit two-electron Schrodinger operator $m$, and Barta's inequality further reduces it to the pointwise nonnegativity of an explicit function of three variables. We prove the nonnegativity of this function by a combination of analytic estimates and a computer-assisted certification in interval arithmetic, and we publish the certificate with the paper.

math.NT — Number Theory

Recycled roots under reduction and descent in the Casas-Alvero problem

Contributed by David Niedbala Giraudin

For a Casas-Alvero polynomial $f$, the type of $f$ is one less than the minimal number of roots of $f$ needed to meet every Hasse derivative $H_1f,\dots,H_{d-1}f$. We show that the type does not increase under reduction modulo a prime and under the descent from degree $np^e$ to degree $n$ in characteristic $p$. Consequently, if no Casas-Alvero polynomial of degree $n$ and type at most $k$ exists over $\overline{\mathbb F}_p$, then none of degree $np^e$ and type at most $k$ exists in characteristic $0$, for every $e\ge 0$. For $k=2$ the hypothesis is equivalent to $p\nmid N_n$ for an explicit integer $N_n$, and we show by a finite computation that $N_n\ne0$ for $3\le n\le 22$ and $n=24$. In particular there is no counterexample of type at most $2$ in degree $24$. This gives infinite families of degrees for which the conjecture is not known and every counterexample has type at least $3$: the degrees $5\cdot 8009^e$, the degrees $6p^e$ for $21$ primes $p$ that are bad for degree $6$, the degrees $7p^e$, $8p^e$, $9p^e$ and $10p^e$ for $13$, $17$, $16$ and $103$ explicit primes $p$ that are bad for these degrees, and the degrees $20p^e$ and $24p^e$ for all primes $p$ outside a finite set (for $20p^e$, including $p=2^{31}-1$). For $k=3$ we obtain the first degrees in which every counterexample has type at least $4$: the degrees $6p^e$ for $p\in\{983,1901,51313000813080529\}$, the degrees $8\cdot397^e$, the degrees $9p^e$ for $p\in\{613,757\}$ and the degrees $10p^e$ for $13$ primes $p$.

math.CO — Combinatorics

Exact diameter of the symmetric group with LX generators

Contributed by Sergei Uskov, Alexander Chervov, Fedor Levkovich-Maslyuk, Ivan Litvak, Andrei Smolensky, Alexander Soibelman

We prove an exact formula for the diameter of the directed Cayley graph for the symmetric group $S_n$ with generators $L$ (left cyclic shift) and $X$ (exchange of the first two elements). For every $n \geq 5$, we show that the diameter equals $(3n^2-8n+12)/4$ for even $n$ and $(3n^2-8n+9)/4$ for odd $n$. This resolves a longstanding question first proposed in the 1960's and refined in more recent work. The proof is AI-assisted and is based on a variety of combinatorial arguments. We also describe the agentic system we developed that led to this proof and looks highly promising for attacking other questions of this type. An accompanying Lean 4 formalization rigorously verifies the result.

math.FA — Functional Analysis

Column rigidity in noncommutative $\L^1$-spaces and completely isometric embeddings of Hilbertian operator spaces

Contributed by Cédric ARHANCET

We study exact embedding obstructions for Hilbertian operator spaces in noncommutative $\mathrm{L}^1$-spaces. We prove a quantitative version of the column inequality, whose equality case forces the absolute values of the entries to be proportional. We deduce that an operator space embedded completely isometrically in a noncommutative $\mathrm{L}^1$-space is a row Hilbert space whenever its first-column norms at the two-by-two level are Hilbert-Schmidt norms. This gives an exact embedding criterion for sums of weighted row and column spaces, including coordinate-dependent weights which are ordered in the same direction. In particular, $\mathrm{R}_n+\mathrm{C}_n$ does not embed completely isometrically into any noncommutative $\mathrm{L}^1$-space when $n \geq 2$. For $\mathrm{OH}_n$, whose complete isometric embeddability is established in a companion paper, we retain the common-support and commutative-target obstructions. We also show that distortion one is approached by matrix models but is attained by none of them, and that the matrix size must tend to infinity as the distortion tends to one.

math.NT — Number Theory

Hecke eigenvalues in level 1 modulo powers of 2 and 3: Hatada's question (1981) and the Coleman-Stein conjecture

Contributed by David Niedbala Giraudin

Let $f$ be a normalized Hecke eigenform of level $1$ and weight $k$, with eigenvalues $a_n(f)$. In 1981 Hatada asked whether $a_p(f)\equiv 1+p \pmod{2^a}$ whenever $p\equiv\pm1\pmod{2^{a-1}}$ and $6\le a\le 13$. Rustom's congruence modulo $128$ answers this for $a\le 7$. We show that the answer is affirmative for $a\le 10$ and negative for $a=11$, $12$ and $13$, the counterexamples coming from weight $16$ and $p=12289$. The proof combines Rustom's theorem with a certificate for $T_5$ modulo $512$ on period polynomials, a $2$-adic Galois law (for instance $a_p\equiv 1+p^{k-1}\pmod{2^{10}}$ for every $p\equiv 1\pmod 8$), and a reduction to a computation in a finite group. As a consequence, $a_p(f)=0$ implies $p\equiv -1\pmod{2^{11}}$. At the prime $3$ we show that, on Hatada's lattice of period polynomials, $T_2^2\equiv 9(k-1)^2\pmod{27}$, with refinements modulo $81$. This gives an independent proof of Conjecture 9.1 of Rustom without any ramification hypothesis and of the level $1$ case of the Coleman--Stein conjecture, both of which also follow from Rustom's recent classification modulo $81$. We also prove that $a_n(f)\equiv\sigma_{k-1}(n)\pmod{81}$ for every $n\equiv 1\pmod 3$, and a $3$-adic Galois law modulo $243$ on the Galois group of $\mathbb{Q}(\sqrt{-3})$, which gives congruences modulo $243$ and $729$ for suitable split primes and a bilinear law for primes $\ell\equiv 2\pmod 3$. Beyond $729$, for every prime $\ell\equiv 2\pmod 3$ we prove $a_\ell^2\equiv\sigma_{k-1}(\ell)^2\pmod{3^{\min(2j+2,10)}}$, where $j=v_3(\ell+1)$; the exponent $2j+2$ is attained for $j\le 4$ and fails for $j=5$. Each statement is marked as proved, known, verified or open. The finite computations are certified by seven Python scripts, provided as ancillary files.

math.QA — Quantum Algebra

Haagerup symmetries of the $E_8$ theory and conformal nets at central charge eight

Contributed by Cédric ARHANCET

A Haagerup algebra is a positive strongly rational vertex operator algebra $U$ of central charge eight with $\mathrm{Rep}(U)$ braided equivalent to the Drinfeld center $Z(\cal H_3)$ of the Haagerup fusion category. We prove that $U$ has exactly three holomorphic extensions, each isomorphic to $V_{E_8}$, and that the corresponding Lagrangian algebras have module categories $\cal H_1$, $\cal H_2$ and $\cal H_3$. The extensions attached to $\cal H_2$ and $\cal H_3$ are the two lattice extensions of $V_{E_6 \oplus A_2}$. We prove a recognition theorem for $\cal H_2$ and $\cal H_3$ among unitary fusion categories of rank six with three invertible objects, a noncommutative Grothendieck ring and trivial pointed associator. For simple regular unitary vertex operator algebras with bounded localized vertex operators, we construct the transport module of two ordinary unitary modules when the second integrates in Tener's annular sense. For a unitary conformal Haagerup subalgebra of $V_{E_6 \oplus A_2}$, these results give a completely rational net with representation category $Z(\cal H_3)$, a realization of the Haagerup subfactor and the generalized orbifold of Bischoff. In particular, they apply to the strongly local explicit algebra $S$ of levels $(3,13)$, whose Haagerup property is established in \cite[Theorem 6.1]{ArhIII}. In \cite{ArhIII}, the net of $S$ is recognized from strong rationality and lattice branching, and a diagonal extension of the net is then used to identify $\mathrm{Rep}(S)$. Here an exact construction of the compatible unitary structure on the diagonal lattice algebra $V_{E_8 \oplus E_8}$, followed by passage to the twin algebra, proves complete unitarity and Condition I+ for every Haagerup algebra. This also gives the VOA--net comparison through smeared vertex operators for unitary conformal embeddings in $V_{E_6 \oplus A_2}$.

Author comments: 66 pages

math.NT — Number Theory

On Bounds for Reciprocal Sums and Residues in Collatz Orbits, and Their Consequences

Contributed by Behzad Nikzad, Karl Frinkle, Fatemeh Bagherzadeh

We prove uniform bounds for reciprocal sums and residue products along Collatz paths. Let $\mathcal F$ be a finite nonrepeating shortcut path whose input states are greater than $1$, let $\mathcal R_{\mathcal F}$ be the sum of the reciprocals of its odd inputs, and let $\rho_{\mathcal F}$ be its finite-path residue. Then $\mathcal R_{\mathcal F}<1, \qquad \rho_{\mathcal F}<\frac{4}{3}.$ For the first-arrival trajectory of a terminating positive integer $n$, these inequalities give $\mathcal R(n)<1, \qquad \rho(n)=\frac{2^{E(n)}}{3^{O(n)}n}<\frac{4}{3},$ where $E(n)$ and $O(n)$ are the division and odd-operation counts. More generally, when each odd value greater than $1$ is counted once, the corresponding reciprocal-sum and residue bounds hold for every forward orbit, without a termination assumption. The proof combines verified finite height-delay records and residue bounds, canonical-block reciprocal estimates, and an exact finite count organized by gap brackets. Gap brackets, closely related to parity vectors, record the divisions by two between successive odd inputs and organize the finite counting argument; together with a related coordinate representation, they also yield exact formulas for reconstructing finite trajectories. After taking base-$2$ logarithms, the bound $\rho_{\mathcal F}<4/3$ leaves an uncertainty of less than $1/2$ in the relation between the even- and odd-operation counts. Since these counts are integers, either count is determined exactly by the other count and the path endpoints. We obtain nearest-integer restrictions on the changes in $E$ and $O$ between adjacent integers, apart from explicit small exceptions, and prove that adjacent terminating integers of equal height have identical operation counts. Thus every equal-height streak lies in a single operation-count class. We further introduce the charge $\lambda=E-2O$ and associated height and charge slice functions to study the persistence and eventual break of equal-height streaks. We distinguish class records, which are the least starting integers attaining specified values, from delay records, which attain new extrema relative to all smaller starting integers. The resulting record theory shows that $\lambda$-delay records form a subsequence of the odd-count delay records, which form a subsequence of the common division-count and height-delay list. On the class side, the nonpositive charge-class records lie among the odd-count class records, which lie among the division-count class records, which in turn lie among the height-class records. The corresponding hierarchies also hold for first-arrival paths to a common endpoint $z$ when $z=1$ or when $z>2$ is nonperiodic under the shortcut map.

math.NT — Number Theory

The Hankel determinants of an automatic sequence need not be automatic

Contributed by David Niedbala Giraudin

Allouche, Han and Niederreiter asked whether the sequence of Hankel determinants of a $d$-automatic sequence must be $d$-regular, or must become $d$-automatic after reduction modulo an integer. We answer the first question in the negative, and the second one for the reduction modulo $3$. Let $\alpha$ be the root of $f^4+f^2-Tf+1=0$ in $\mathbb{F}_3((1/T))$ with $|\alpha|<1$, the quartic of Mills and Robbins, and let $S$ be its coefficient sequence, which is $3$-automatic by Christol's theorem. The Hankel determinants of $S$ vanish in runs whose lengths are unbounded but of the order of the square root of their position, because the partial quotients of $\alpha$ are unbounded while its irrationality exponent is $2$; and a lemma of Schlage-Puchta forbids runs in that regime in an automatic sequence. Hence these Hankel determinants are $k$-automatic for no $k$, and the integer Hankel determinants of any lift of $S$ are $k$-regular for no $k$.

math.NT — Number Theory

The binary Collatz dynamics of the trinomials x^n+x+1: an independent proof of two conjectures of Gallardo and Rahavandrainy

Contributed by David Niedbala Giraudin

Gallardo and Rahavandrainy studied an analogue of the Collatz map on F_2[x], in which x(x+1) plays the role of 2 and x^2+x+1 that of 3: an odd polynomial A is sent to the odd part of 1+(x^2+x+1)A. They proved that every orbit reaches 1, and conjectured that the orbit of the trinomial x^n+x+1 has length 2^s+1, where s is the largest integer with 2^(s+1) < n, and that for 1 <= t <= s-1 it contains 2^t terms of the same degree. We prove both statements for every n >= 3. They were proved earlier, with a complete Lean 4 formalization, by W. Choi; the proof given here was obtained independently. The proof rests on the identity (x^2+x+1)+x(x+1)=1: it reduces the dynamics to a map from one plateau to the next, characterises the last plateau of any orbit, and shows that along the orbit of x^n+x+1 the plateaus start at the 2^j-th powers of the odd parts of the polynomials x^m(x^2+x+1)+1, the index m following an explicit binary recursion. We also note that a later restatement of the conjecture, with a non-strict inequality, fails exactly when n is a power of 2. A script checks all the statements numerically.

math.NT — Number Theory

On the largest irreducible factor of an odd perfect polynomial over $\mathbb{F}_2$

Contributed by David Niedbala Giraudin

A polynomial $A$ over $\mathbb{F}_2$ is perfect if it equals the sum $\sigma(A)$ of all its divisors, and odd if it is coprime to $x^2+x$. Canaday conjectured in 1941 that 1 is the only odd perfect polynomial. Known necessary conditions on a counterexample bound its number of distinct irreducible factors ($\omega\ge 5$, Gallardo and Rahavandrainy), its number of irreducible factors counted with multiplicity and its degree ($\Omega\ge 12$ and $\deg A>200$, Cengiz, Pollack and Trevino), and, when all exponents equal 2, the degree of its smallest irreducible factor (at least 40, Gallardo). We bound a different invariant: every odd perfect polynomial $A\ne 1$ has an irreducible factor of degree at least 20. The proof reduces the question, for each $D$, to a finite system: if all irreducible factors of $A$ have degree at most $D$, the exponents of $A$ belong to an explicit finite set, through the cyclotomic factorisation of $\sigma(P^h)$ and the multiplicative order of $P$ modulo the factors of $\sigma(P^h)$. An iterated sieve on the resulting production relations empties for every $D\le 19$. The computation was carried out twice, by two independent programs, with identical results; the second program is attached as an ancillary file. As a cross-check of the method, the same programs reproduce, by the orbit argument of Gallardo and Rahavandrainy, the exclusion of every prime of degree at most 30 as the smallest factor of a special perfect polynomial, in agreement with the published bounds.

math.CO — Combinatorics

The inverse problem for the number of connected induced subgraphs

Contributed by Lucas Almeida Almeida

For a finite graph $G$, let $c(G)$ be the number of nonempty vertex sets that induce a connected subgraph. We show that a positive integer $k$ equals $c(G)$ for some finite connected graph $G$ if and only if $k\notin\{2,4,5,8,9,16\}$. Every attainable value $k\ne25$ is attained on $\lfloor\log_2 k\rfloor+1$ vertices, the least possible number, and $25$ needs six vertices. More precisely, for every $m\ge6$ each integer from $2^{m-1}$ to $2^m-1$ is the value of $c$ on a connected graph with exactly $m$ vertices. The proofs of these results are short and use no computer; computer calculations serve only as independent checks, and for two auxiliary results: the values on six to eight vertices, and the result on all connected subgraphs described below. The value $16$ falls into a gap above the path: a connected graph on $n\ge4$ vertices other than the path $P_n$ has at least $n-3$ more connected sets than $P_n$. The values $k\ge17$ are realized by a recursive construction from cones, disjoint unions and a family of graphs that we call near-cones, starting from seven small graphs; it produces a graph with $c(G)=k$ in time polynomial in $\log k$. The question arose from a problem of Ordoñez in continuum theory, through a reduction by Camargo, Ordoñez and Ramírez, who found the values up to $255$ by a computer search. The proof is formalized in Lean 4. We also observe that a theorem of Czabarka, Székely and Wagner on subtrees reduces the corresponding problem for the number of all connected subgraphs to a finite computation, and we report its outcome. We did not find these results in the literature, but we make no claim of novelty or priority.

math.CO — Combinatorics

Hilton’s conjecture on cycle multiplicities

Contributed by Qiyue Tang

We prove Hilton’s conjecture on the number of cycles of each length in dense Hamiltonian graphs. This resolves the problem posed by A. J.W. Hilton and recorded by Sheehan in 1977.