MathExplore Notes

MathExplore Notes is a collection of AI-assisted mathematical research results, including manuscripts and reproducible verification materials.

Featured result

The minimum degree of nonnegative multiples of cyclotomic polynomials proves Conjecture 1 in Steinberger's 2012 paper for every integer n>1, with an exact minimum degree and a complete classification of the equality case.

Research index

Result and files	Areas and keywords	Main result	Version and review
Circuit orderability of cographic matroids of complete and complete bipartite graphs PDF · BibTeX · DOI	Matroid theory, cographic matroids, complete graphs, surface topology	For every integer n >= 4, M*(K_n) is orderable iff n = 4 or n = 6; for all positive integers r,s, M*(K_{r,s}) is orderable iff min(r,s) <= 2. This classifies the two specified families, not all cographic matroids.	v0.1-beta Internally checked preprint; external review pending
Classification of central graphical zonotopal algebras PDF · BibTeX · DOI	Zonotopal algebras, graphical matroids, matroid reconstruction, Cyclic flats, SPQR decomposition	Classifies central graphical zonotopal algebras of all finite connected loopless multigraphs over the real numbers by their bridge-deleted graphical matroids, without requiring grading-preserving isomorphisms. This resolves Nenashev's graphical Conjecture 8; it is not a classification of arbitrary real matrix configurations.	v0.1-beta Internally reviewed preprint; external review pending
Positive Jacobi realizations for complete-bipartite symmetric edge polytopes PDF · BibTeX · DOI	Ehrhart theory, symmetric edge polytopes, Jacobi matrices, Canonical-line conjecture, Kölbl	Proves the canonical-line property for K(4,n) and K(5,n) by positive Jacobi realizations, and positivity of the first two inverse tail coefficients for all 3 <= m <= n; does not settle the full conjecture.	v0.1-beta Internally reviewed preprint; external review pending
Real-rooted h-star polynomials of rank-two matroids with parallel classes of size at most three PDF · BibTeX · DOI	Matroid theory, Ehrhart theory, matroid base polytopes, real-rootedness, Sparse paving matroids, Sturm sequences	Proves real-rootedness for every rank-two matroid whose nonloop parallel classes have size at most three, extending the sparse-paving boundary from two to three.	v0.1-beta Internally verified public Beta; external review pending
A Walsh-convolution proof of two conjectures of Lachaume PDF · BibTeX · DOI	Polynomial convolutions, real-rootedness, root location, hyperbolic polynomials, Walsh symmetric additive convolution	Identifies Lachaume's derivative convolution exactly with a rescaled Walsh symmetric additive convolution, proving Conjectures 4 and 5 and a complex convex-hull root-location statement. Uses the classical Walsh theorem; no claim is made for Lachaume's Conjectures 1 and 3.	v0.1-beta Internally verified public Beta; external review pending
Affine stabilizers of two o-monomial value sets PDF · BibTeX · DOI	Finite geometry, finite fields, o-monomials, hyperovals, affine stabilizers, Ding--Tang, Segre and Glynn hyperovals	Proves trivial affine stabilizers for the $\overline{\mathrm{Segre}}$ and Glynn I value sets for every odd $m\ge5$, completing the two families left untreated among all currently known nontranslation o-monomials. The coverage is of the known families, not a classification of unknown o-monomials.	v0.1-beta Internally verified public Beta; external review pending
Parallel-support rank and a one-sided Merino--Welsh inequality PDF · BibTeX · DOI	Matroid theory, Tutte polynomial, parallel and series classes, basis activities, Merino--Welsh conjecture	Proves the one-sided inequality $T_M(0,2)\ge T_M(1,1)$ whenever the nontrivial parallel classes span all but at most one rank direction, with strictness, complete equality classification, dual and graphic forms, and a sharp rank threshold. The theorem concerns finite loopless, coloopless matroids; the full conjecture remains outside its scope.	v0.1-beta Internally verified public Beta; external review pending
Triangle sums, peripheral cycles, and contractions for orderable cographic matroids PDF · BibTeX · DOI	Matroid theory, cographic matroids, surface triangulations, graph embeddings, Face-adjacency orderings, peripheral cycles	Proves a rooted gluing theorem, an exact triangle-sum criterion, infinite orderable families on every nonorientable genus, a peripheral-cycle obstruction, vertex-split nonclosure, and legal-contraction descent. The surface criteria concern the prescribed face-adjacency ordering, not a classification of all cographic matroids.	v0.1-beta Internally verified public Beta; external review pending
Log-concavity of reduced composition polynomials via adjacent B-spline refinement columns PDF · BibTeX · DOI	Enumerative combinatorics, composition polynomials, discrete B-splines, total positivity, Ardila--Doker, total nonnegative minors	Proves that the coefficient sequence of the Ardila--Doker reduced composition polynomial is log-concave, and hence unimodal, for every positive integer composition. The proof uses adjacent discrete B-spline refinement columns; it does not assert real-rootedness.	v0.1-beta Internally verified public Beta; external review pending
An unbounded gap between minimum degree and the minor-connectivity ceiling PDF · BibTeX · DOI	Graph minors, vertex connectivity, minimum degree, Mader graph, graph blow-ups, Barát conjecture, branch sets	Proves $\kappa^(M_{12}[\overline K_t])=\lfloor9t/2\rfloor$ for every $t\ge2$, yielding an unbounded gap $\delta-\kappa^=\lceil t/2\rceil$ and infinitely many counterexamples to Barát's conjectured additive-one bound.	v0.1-beta Internally verified public Beta; external review pending
Every two-vertex premaniplex is the symmetry type graph of a finite abstract polytope PDF · BibTeX · DOI	Abstract polytopes, maniplexes, symmetry type graphs, voltage constructions, Two-orbit polytopes, flag graphs	Proves that every connected two-vertex premaniplex of every rank at least three is realized as the full symmetry type graph of a finite abstract polytope. The construction allows arbitrary link sets and uses a path-intersection criterion.	v0.1-beta Internally verified public Beta; external review pending
Odd-dimensional descent obstructions in the third support spectrum of binary second-order Reed--Muller codes PDF · BibTeX · DOI	Coding theory, Reed--Muller codes, higher support spectra, quadratic Boolean functions, Walsh spectra, Alternating polar forms	Classifies an exponentially widening local parity band for three-dimensional subcodes and proves an infinite family of odd-to-even support descent obstructions. This is a local classification and obstruction theorem, not the complete third support spectrum.	v0.1-beta Internally verified public Beta; external review pending
Small exact values and a dense-anchor bound for weak cube saturation PDF · BibTeX · DOI	Weak saturation, hypercubes, extremal graph theory, exhaustive enumeration, Dense-anchor constructions	Determines wsat(K8,Q3)=15, wsat(K9,Q3)=16, and wsat(K10,Q3)=18; corrects the reported order-nine value, disproves the associated 2n-1 proposal, and proves the general upper bound wsat(K_n,Q3)<=floor((7n+2)/4) for n>=9. The general bound is not asserted to be the exact asymptotic value.	v0.1-beta Internally verified public Beta; external review pending
Finite-field basis obstructions for zero link Turán density PDF · BibTeX · DOI	Hypergraph Turán theory, link Turán density, finite fields, hypergraph homomorphisms, Independent-set hosts	Proves a general finite-field basis-homomorphism obstruction giving an explicit positive link Turán density, and constructs two six-vertex three-graphs whose vertex links are all tripartite but whose link Turán densities are positive. The general obstruction does not classify all hypergraphs of zero link Turán density.	v0.1-beta Internally verified candidate proof; external review pending
Connected Peck posets with non-log-concave antichain polynomials PDF · BibTeX · DOI	Posets, Peck posets, antichain polynomials, log-concavity, Ding--Dong Conjecture C, perfect matchings	Constructs an infinite family of finite connected height-one Peck posets with non-log-concave antichain polynomials for every parameter $a\ge7$, disproving Ding--Dong Conjecture C.	v0.1-beta Internally verified candidate proof; external review pending
Exact first separation in the Thue--Morse run-length sequence PDF · BibTeX · DOI	Automatic sequences, Thue--Morse sequence, Jacobsthal representations, 2-kernels, Shallit conjecture, multiplicative orders	Determines for every $q\ge2$ the exact first separation of $d(2^{q+2}n)$ and $d(2^qn)$, extends it along arbitrary ratio-four scale distances, and gives an exact infinite multiplier family governed by multiplicative orders. The result treats these multiplier families, not all possible pairs of multipliers.	v0.1-beta Internally verified candidate proof; external review pending
Counterexamples to unimodality of the connected set polynomial PDF · BibTeX · DOI	Graph polynomials, connected induced subgraphs, unimodality, graph enumeration, Connected set polynomials	Constructs four pairwise nonisomorphic counterexample families for every order $n\ge8$, proves the common coefficient formula, and completely classifies the four minimum-order counterexamples. The minimum counterexamples have eight vertices.	v0.1-beta Internally verified candidate proof; external review pending
A complete log-concavity classification for higher-order Stirling cycle rows PDF · BibTeX · DOI	Enumerative combinatorics, associated Stirling numbers, log-concavity, Fourier--Edgeworth estimates, Deb--Sokal, associated permutations	Proves that every higher-order Stirling cycle row is log-concave for every row index exactly for orders $1\le r\le5$, settling the previously open $r=3,4,5$ layers, and proves log-concavity of the factorial-normalized rows for every $r\ge2$. This is a log-concavity classification, not a real-rootedness or total-positivity theorem.	v0.1-beta Internally verified candidate proof; external review pending
Compatible orderings of binary vector spaces PDF · BibTeX · DOI	Finite geometry, binary vector spaces, general linear groups, ordered Fano planes, Cameron, BCC30, flag orders	Classifies every compatible total order on $\mathbb F_2^n$ for all $n$: for $n\ge4$ there are exactly eight $\mathrm{GL}(n,2)$-orbits and $8\lvert\mathrm{GL}(n,2)\rvert$ labelled orders. The classification is binary; it makes no claim for larger finite fields.	v0.1-beta Internally verified candidate proof; external review pending
Weighted recursive constructions for shattering triples with six permutations PDF · BibTeX · DOI	Extremal combinatorics, permutation shattering, weighted blow-ups, probabilistic recursion, Černá--Kielak--Volec	Proves a weighted recursion theorem for every finite six-permutation template and applies it to the published 26-point template to obtain the exact lower bound $c_3\ge1288385/2599242$, strictly improving its uniform-recursion bound $482/975$. The result improves a lower bound, without determining the optimum.	v0.1-beta Internally verified candidate proof; external review pending
Projective-plane triangulations yield orderable cographic matroids PDF · BibTeX · DOI	Matroid theory, cographic matroids, graph embeddings, projective-plane triangulations, Crenshaw--Oxley conjecture	Proves that every projective-plane triangulation yields an orderable cographic matroid and constructs infinitely many 3-connected regular non-graphic counterexamples to Crenshaw--Oxley Conjecture 4. The surfaces are finite simplicial triangulations of the real projective plane.	v0.1-beta Internally verified candidate proof; external review pending
Realizability and minimum dimension of Fano half-rank profiles over F2 PDF · BibTeX · DOI	Alternating bilinear forms, Fano plane, rank profiles, finite fields, Hilbert bases, Pfaffian parity, finite conductors	Classifies every realizable seven-point half-rank profile of a three-parameter alternating net over $\mathbb F_2$ and determines its exact minimum ambient dimension. Classifies rank-profile data, not alternating matrix spaces up to isomorphism.	v0.1-beta Internally verified candidate proof; external review pending
Exact regions and low-polar-rank recursion in the third support spectrum of binary second-order Reed--Muller codes PDF · BibTeX · DOI	Coding theory, higher support spectra, quadratic Boolean functions, alternating pencils, Low-polar-rank recursion, Walsh spectra	Determines two sharp all-dimensional spectrum gaps, the exact spectra through nine variables, the complete near-full interval, a finite low-rank recursion, and an infinite forbidden support ray. The full third support spectrum in arbitrary dimension remains unresolved.	v0.1-beta Internally verified candidate proof; external review pending
The minimum degree of nonnegative multiples of cyclotomic polynomials PDF · BibTeX · DOI	Cyclotomic polynomials, nonnegative coefficients, roots of unity, trigonometric separation, Steinberger Conjecture 1, sector separation	Proves for every $n&gt;1$ that the minimum degree is $(p-1)n/p$, where $p$ is the smallest prime divisor, with equality only for positive scalar multiples of the regular $p$-gon polynomial. The all-orders theorem subsumes the two earlier cyclotomic entries below.	v0.1-beta Internally verified candidate proof; external review pending
Proper transposed sesqui arrays at all Sylvester--Hadamard powers PDF · BibTeX · DOI	Combinatorial designs, finite fields, finite geometry, algebraic curves, Sylvester--Hadamard designs, two-fibre orderings, Hasse--Weil bounds	Constructs an exact-parameter proper transposed sesqui array for every $t=2^k$ with $k\ge2$. Uses the classical column-design skeleton; the contribution is a compatible proper ordering.	v0.3-beta Internally verified candidate proof; external review pending
Paley two-edge switching for proper transposed sesqui arrays PDF · BibTeX · DOI	Combinatorial designs, finite fields, elliptic curves, Paley designs, quadratic characters, edge switching	Constructs an exact-parameter proper transposed sesqui array for every odd prime power $q\equiv3\pmod4$, $q\geq11$. Uses the classical Paley skeleton with a local two-edge switching construction.	v0.3-beta Internally verified candidate proof; external review pending
Spectral separation and the nSSP for looped double paths PDF · BibTeX · DOI	Inverse eigenvalue problems, graph patterns, Jacobi matrices, Non-symmetric strong spectral property, tridiagonal matrices	Classifies exactly which looped double paths allow the non-symmetric strong spectral property. Retains a pattern-level allowability classification, distinct from the fixed-matrix tree criterion below.	v0.2-beta Internally verified candidate proof; external review pending
An exact non-symmetric strong spectral criterion for root-loop spider matrices PDF · BibTeX · DOI	Inverse eigenvalue problems, generalized stars, matrix patterns, Arm characteristic polynomials, spectral separation	Characterizes the nSSP of every fixed root-loop bidirected spider matrix by pairwise coprimality of its arm polynomials. The criterion is over the real numbers and allows nonreal arm eigenvalues.	v0.2-beta Internally verified precursor; strictly generalized by the rooted-tree entry
A recursive non-symmetric strong spectral criterion for root-loop tree matrices PDF · BibTeX · DOI	Inverse eigenvalue problems, rooted trees, matrix patterns, Non-symmetric strong spectral property, spectral collisions	Characterizes the nSSP of every fixed root-loop bidirected tree matrix by recursive sibling-subtree coprimality. This is a fixed-matrix criterion, not an allow/require classification of all loop patterns.	v0.2-beta Internally verified candidate proof; external review pending
Attainable second support weights of binary second-order Reed--Muller codes PDF · BibTeX · DOI	Coding theory, quadratic Boolean functions, Walsh spectra, Quadratic pencils, common-zero counts, generalized Hamming weights	Determines for every dimension the exact set of support sizes attained by two-dimensional subcodes of $RM_2(2,n)$. Uses a quadratic-pencil and Walsh-spectrum compatibility analysis.	v0.2-beta Internally verified candidate proof; external review pending
Acyclic component hypotheses for total cut complexes of disconnected graphs PDF · BibTeX · DOI	Algebraic topology, graph complexes, simplicial complexes, Carnero Bravo Question 30, Alexander duality, polyhedral joins	Proves the wedge-of-spheres formula for every $k\geq d$ when all relevant component total-cut complexes are void or integer-acyclic, and answers Question 30 for this class. The hypotheses are on component complexes; this does not resolve the question for all disconnected graphs.	v0.3-beta Internally verified candidate proof; external review pending
Comb-bounded crowns and bi-Esakia representability of well-ordered rooted trees PDF · BibTeX · DOI	Order theory, Esakia duality, bi-Heyting algebras, Finite crowns, bi-p-morphisms, better partial orders	Gives a finite-anchor synchronization theorem and a sufficient bi-Esakia representability criterion for well-ordered rooted trees with uniformly comb-bounded finite crowns. Allows arbitrary width and unbounded crown size; it is a sufficient criterion, not a complete classification.	v0.1-beta Internally verified candidate proof; external review pending
Near-minimal Ehrhart data on the primitive-triangle boundary of denominator-two polygons PDF · BibTeX · DOI	Ehrhart theory, rational polygons, discrete geometry, Half-integral polygons, primitive lattice triangles, parity obstructions	Proves a mod-$8$ obstruction and sharply classifies the near-minimal boundary through $I=3N-2$, with an infinite one-point-sharp family. Here $N=i(P)$ and $I=i(2P)$, with $b(P)=0$ and $b(2P)=3$; no full Ehrhart-data classification is claimed.	v0.1-beta Internally verified candidate proof; external review pending
Periodic infinite ringsets in the Lipschitz quaternions PDF · BibTeX · DOI	Integer-valued polynomials, quaternion algebras, noncommutative algebra, Lipschitz quaternions, ringsets, null ideals	Classifies exactly which odd-squarefree-period binary words produce infinite ringsets in the Lipschitz quaternions. The alphabet is the two quaternion units i and j; the six-letter theorem below includes this case.	v0.1-beta Internally verified candidate proof; external review pending
Periodic two-letter ringsets in the Lipschitz quaternions: the general odd-period classification PDF · BibTeX · DOI	Integer-valued polynomials, quaternion algebras, noncommutative algebra, Lipschitz quaternions, prime-power position classes	Classifies exactly which arbitrary odd-period binary words produce infinite ringsets, strictly generalizing the preceding squarefree result. Uses fixed-divisor obstructions and prime-power fibres; the six-letter theorem below includes this case.	v0.1-beta Internally verified candidate proof; external review pending
Periodic ringsets in the Lipschitz quaternions: the complete six-letter classification PDF · BibTeX · DOI	Integer-valued polynomials, quaternion algebras, noncommutative algebra, Lipschitz quaternions, local-global criteria, antipodal fibres	Classifies exactly which full six-letter words of arbitrary period produce infinite ringsets, strictly containing both preceding quaternion classifications. Gives necessary and sufficient conditions, including the ramified prime 2 and antipodal-fibre obstructions.	v0.1-beta Internally verified candidate proof; external review pending
Minorizing measures and truncations of measurable lattice-path matroids PDF · BibTeX · DOI	Measurable matroids, infinite-dimensional convexity, continuous optimization, Minorizing measures, weak-star topology, purification	Determines exactly the basic and full minorizing-measure sets, all real truncation slices, and the exposed extreme points of the full-rank basic corridor. The result holds for arbitrary closed time sets; it does not classify all measurable matroids.	v0.1-beta Internally verified candidate proof; external review pending
The face lattice of the irreducible-Ferrers polytope is a product of triangles PDF · BibTeX · DOI	Ferrers diagrams, polyhedral combinatorics, coding theory, Face lattices, products of simplices	Proves that the irreducible-Ferrers polytope is combinatorially equivalent to a product of triangles for every $d\geq3$, resolving Conjecture 5.28 of arXiv:2604.27868. The product has $d-2$ triangles. The equivalence is combinatorial, not affine or unimodular; this is not the Etzion--Silberstein conjecture.	v0.1-beta Internally verified candidate proof; external review pending
Partition-representation degrees of the dual Fano matroid PDF · BibTeX · DOI	Matroid theory, information theory, combinatorial designs, Entropic matroids, orthogonal Latin squares, excluded minors	Determines $\chi(F_7^*)$ exactly and characterizes regular matroids as precisely the 6-entropic matroids. The degree set is exactly the positive powers of two; six is the unique alphabet size yielding precisely the regular matroids.	v0.1.1-beta Internally verified candidate proof; external review pending
The third symbolic power of a co-chordal edge ideal is componentwise linear PDF · BibTeX · DOI	Commutative algebra, monomial ideals, chordal graphs, Symbolic powers, marked-clique capacities	Proves componentwise linearity of the third symbolic power for every co-chordal graph over every field. Handles the degree-five boundary; it does not claim the analogous statement for all symbolic powers.	v0.1-beta Internally verified candidate proof; external review pending
A ternary word attaining the Abelian maximal pattern complexity bound at pattern size three PDF · BibTeX · DOI	Combinatorics on words, Abelian complexity, substitution sequences, Parikh vectors, automatic sequences, finite-state verification	Constructs a recurrent ternary word, aperiodic by projection, with exact Abelian maximal pattern complexity $p_{\alpha}^{*\mathrm{ab}}(3)=7$. Attains the lower bound at alphabet size and pattern size three, not for every pair of parameters.	v0.1-beta Internally verified candidate proof; external review pending
Sharp initial thresholds for Abelian-bordered binary infinite words PDF · BibTeX · DOI	Combinatorics on words, Abelian borders, ultimate periodicity, Abelian-unbordered factors, finite overlap graphs	Proves the sharp ordinary-periodicity threshold $13$, the next sharp Abelian-periodicity layer at $14$, and a frequency-denominator sufficient condition. Retains sharp small-threshold results; the separate counterexample below addresses the unrestricted periodicity question.	v0.1-beta Internally verified candidate proof; external review pending
A sparse-defect counterexample to Abelian-border periodicity PDF · BibTeX · DOI	Combinatorics on words, Abelian borders, sparse constructions, Fici--Puzynina Problem 42, bounded discrepancy	Constructs a binary non-Abelian-ultimately-periodic word whose every factor of length at least 141 is Abelian bordered, giving a negative answer to Question 1 / Problem 42. Uses a sparse-defect bounded-height walk; 141 is not claimed to be the smallest possible threshold.	v0.1-beta Internally verified candidate proof; external review pending
On a claimed upper bound for partial Desarguesian parallelisms PDF · BibTeX · DOI	Finite geometry, Desarguesian spreads, partial parallelisms, Field reduction, Zhang--Zhou orbit bound	Proves that a known orbit lower bound uniformly exceeds Johnson's claimed upper bound for every prime power $q$, odd prime $t\geq3$, and $z\geq2$. This is a uniform correction using known constructions, not a claim to the first counterexample.	v0.1-beta Internally verified correction note; external review pending
Real-rootedness, palindromicity, and gamma-positivity of antichain polynomials for [2] x [m] x [n] PDF · BibTeX · Concept DOI	Posets, real-rooted polynomials, palindromicity, gamma-positivity, Jacobi polynomials, Ding--Dong conjectures, Jacobi interlacing	Proves negative real-rootedness for all positive $m,n$, classifies palindromicity by $m=n+1$ or $n=m+1$, and proves strict gamma-positivity in exactly those adjacent cases. Treats boxes with one side equal to two; it does not assert real-rootedness for all three-dimensional boxes.	v0.2-beta Internally verified candidate proof; external review pending
Palindromicity of antichain polynomials of three-dimensional boxes PDF · BibTeX · DOI	Posets, antichain polynomials, palindromicity, minuscule posets, last-passage matrices, Ding--Dong Conjecture B	Classifies all palindromic three-dimensional boxes: after sorting, exactly $(1,r,r)$ and $(2,r,r+1)$; this closes the rectangular minuscule branch of Ding--Dong Conjecture B. This is the full box palindromicity classification, not a general real-rootedness theorem.	v0.1-beta Internally verified candidate proof; external review pending
A continuous family of counterexamples to the middle runner conjecture PDF · BibTeX · DOI	Dynamical algebraic combinatorics, circular runners, order statistics, Middle runner conjecture, piecewise integration	Exactly classifies a one-parameter three-runner family and gives continuous counterexamples under the conjecture's printed hypotheses. The family has velocities 1, 2, 3 and initial positions 0, 0, a; it does not refute a common-starting-point variant.	v0.1-beta Internally verified candidate proof; external review pending
Exact certificates for the first two cases not covered by Steinberger's cyclotomic minimum-degree theorem PDF · BibTeX · DOI	Cyclotomic polynomials, nonnegative coefficients, CRT tensor arrays, exact certificates, Steinberger conjecture	Proves the conjectured minimum degree and unique monic minimizer for the first two parameters outside Steinberger's general theorem. The parameters are 46189 and 96577. The later all-orders theorem above subsumes these cases; the finite certificates remain available.	v0.1-beta Internally verified candidate proof; external review pending
Tail-projection certificates for cyclotomic minimum-degree polynomials beyond the reciprocal condition PDF · BibTeX · DOI	Cyclotomic polynomials, nonnegative coefficients, CRT tensor projections, prime tuples, Steinberger reciprocal condition	Gives an explicit sufficient condition for every fixed number of at least four prime factors and proves infinite families beyond Steinberger's reciprocal condition. The construction concerns odd squarefree orders; the later all-orders theorem above subsumes its existence conclusions.	v0.1-beta Internally verified candidate proof; external review pending

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