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

Entry	Areas	Contribution	Public version	Status
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	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	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	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	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	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	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	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	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	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	Proves that the coefficient sequence of the Ardila--Doker reduced composition polynomial is log-concave, and hence unimodal, for every positive integer composition	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	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	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	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	Classifies an exponentially widening local parity band for three-dimensional subcodes and proves an infinite family of odd-to-even support descent obstructions	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	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	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	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	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	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	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	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	Constructs four pairwise nonisomorphic counterexample families for every order $n\ge8$, proves the common coefficient formula, and completely classifies the four minimum-order counterexamples	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	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 $r\ge2$	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	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	\mathrm{GL}(n,2)	$ labelled orders
Weighted recursive constructions for shattering triples with six permutations PDF · BibTeX · DOI	Extremal combinatorics, permutation shattering, weighted blow-ups, probabilistic recursion	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 $482/975$	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	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	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	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	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	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	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	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	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	Constructs an exact-parameter proper transposed sesqui array for every power of two	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	Constructs an exact-parameter proper transposed sesqui array for every odd prime power $q\equiv3\pmod4$, $q\geq11$	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	Classifies exactly which looped double paths allow the non-symmetric strong spectral property	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	Characterizes the nSSP of every fixed root-loop bidirected spider matrix by pairwise coprimality of its arm polynomials	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	Characterizes the nSSP of every fixed root-loop bidirected tree matrix by recursive sibling-subtree coprimality	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	Determines for every dimension the exact set of support sizes attained by two-dimensional subcodes of $RM_2(2,n)$	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	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	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	Gives a finite-anchor synchronization theorem and a sufficient bi-Esakia representability criterion for well-ordered rooted trees with uniformly comb-bounded finite crowns	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	Proves a mod-$8$ obstruction and sharply classifies the near-minimal boundary through $I=3N-2$, with an infinite one-point-sharp family	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	Classifies exactly which odd-squarefree-period binary words produce infinite ringsets in the Lipschitz quaternions	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	Classifies exactly which arbitrary odd-period binary words produce infinite ringsets, strictly generalizing the preceding squarefree result	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	Classifies exactly which full six-letter words of arbitrary period produce infinite ringsets, strictly containing both preceding quaternion classifications	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	Determines exactly the basic and full minorizing-measure sets, all real truncation slices, and the exposed extreme points of the full-rank basic corridor	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	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	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	Determines $\chi(F_7^*)$ exactly and characterizes regular matroids as precisely the 6-entropic 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	Proves componentwise linearity of the third symbolic power for every co-chordal graph over every field	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	Constructs a recurrent ternary word, aperiodic by projection, with exact Abelian maximal pattern complexity $p_{\alpha}^{*\mathrm{ab}}(3)=7$	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	Proves the sharp ordinary-periodicity threshold $13$, the next sharp Abelian-periodicity layer at $14$, and a frequency-denominator sufficient condition	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	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	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	Proves that a known orbit lower bound uniformly exceeds Johnson's claimed upper bound for every prime power $q$, odd prime $t\geq3$, and $z\geq2$	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	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	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	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	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	Exactly classifies a one-parameter three-runner family and gives continuous counterexamples under the conjecture's printed hypotheses	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	Proves the conjectured minimum degree and unique monic minimizer for the first two parameters outside Steinberger's general theorem	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	Gives an explicit sufficient condition for every fixed number of at least four prime factors and proves infinite families beyond Steinberger's reciprocal condition	v0.1-beta	Internally verified candidate proof; external review pending

Current results

Rank-two matroid h-star real-rootedness at parallel-class size three

For every rank-two matroid whose nonloop parallel classes have size at most three, the Ehrhart h*-polynomial of its base polytope has only real zeros and therefore has log-concave, unimodal coefficients. The result advances the known sparse-paving boundary from parallel-class size two to size three. A uniform analytic estimate handles every ground-set size n>=10; exact Sturm and finite checks settle the complete smaller boundary.

• Manuscript and verification materials

Affine stabilizers of two o-monomial value sets

For $q=2^m$ with odd $m\ge5$, the manuscript proves that the one-dimensional affine stabilizer of ${x^e+x:x\in\mathbb F_q}$ is trivial for both the $\overline{\mathrm{Segre}}$ exponent $e=q-6$ and the Glynn I exponent $e=3\cdot2^{(m+1)/2}+4$. Together with Ding--Tang's earlier cases, this covers every currently known nontranslation o-monomial family. It does not classify hypothetical unknown o-monomials and does not claim the unrestricted Conjecture 2.

• Manuscript and verification materials

Parallel-support rank and the Merino--Welsh inequality

For a finite loopless, coloopless matroid $M$, let $P(M)$ be the union of its nontrivial parallel classes. The manuscript proves

$$ r_M(P(M))\ge r(M)-1 \quad\Longrightarrow\quad T_M(0,2)\ge T_M(1,1). $$

The inequality is strict at parallel-support corank one, and equality in the full class occurs exactly for direct sums of copies of $U_{1,2}$. Dual and graphic forms are included. Explicit connected and disconnected counterexamples at support corank two show that the rank threshold is sharp without additional hypotheses.

• Manuscript and verification materials

Cographic surface operations

For closed simplicial surface triangulations, the manuscript proves a general triangle-rooted gluing theorem for compatible cographic orderings and an exact if-and-only-if criterion for triangle sums. It constructs infinite 3-connected regular non-graphic orderable cographic families on every nonorientable genus, gives a peripheral-cycle obstruction and explicit vertex-split nonclosure example, and proves descent under legal edge contraction to finitely many irreducible bases on each fixed surface.

• Manuscript and verification materials

Log-concavity of reduced composition polynomials

For every positive integer composition $c$, the coefficient sequence of the Ardila--Doker reduced composition polynomial $f_c(q)$ is log-concave and hence unimodal. The proof identifies the coefficients with a discrete B-spline refinement column, realizes the two endpoint deletions as adjacent columns of a common refinement matrix, and converts their nonnegative $2\times2$ minors into the required coefficient inequalities by an exact determinant identity.

• Manuscript and verification materials

Unbounded minor-connectivity gap

For the independent $t$-fold blow-up of Mader's 12-vertex graph, the manuscript proves the exact formula

$$ \kappa^*(M_{12}[\overline K_t])=\left\lfloor\frac{9t}{2}\right\rfloor \qquad(t\ge2). $$

Since its minimum degree is $5t$, the gap between minimum degree and the largest connectivity of a minor equals $\lceil t/2\rceil$ and is unbounded. This supplies infinitely many counterexamples to Barát's conjectured universal bound $\kappa^*(G)\ge\delta(G)-1$. The proof combines an explicit uniform minor model with a three-factor torso classification and an anticomplete-shores argument.

• Manuscript and verification materials

Two-vertex polytopality

For every rank r>=3, the manuscript proves that every connected two-vertex premaniplex is the full symmetry type graph of a finite abstract polytope. The construction treats arbitrary internal link-color sets simultaneously and proves the missing path-intersection identity; two exceptional rank-four base types are supplied by exact finite verification.

• Manuscript and exact verification

Odd-dimensional descent obstructions in the third RM2 support spectrum

For three-dimensional subcodes of the binary second-order Reed--Muller code, the manuscript classifies an exponentially widening local band around the normalized aggregate 3*2^(m-1). For every m>=6, exactly the offsets +-1,+-3,+-5 occur in odd dimension, while no offset in the same band occurs in even dimension. One explicit family starts at m=5 and proves that odd-to-even support descent fails for every m>=5.

This is a local parity classification and an infinite obstruction family, not a complete determination of the third support spectrum.

• Manuscript and verification materials

Small exact values and a dense-anchor bound for weak cube saturation

For the three-dimensional cube Q3, the manuscript proves

$$ \mathrm{wsat}(K_8,Q_3)=15, \qquad \mathrm{wsat}(K_9,Q_3)=16, \qquad \mathrm{wsat}(K_{10},Q_3)=18. $$

The order-nine value corrects a reported value of 17 and disproves the associated proposal wsat(K_n,Q3)=2n-1. A certified 28-edge graph on 11 vertices with a complete seven-vertex anchor gives a four-vertex extension of cost seven and hence

$$ \mathrm{wsat}(K_n,Q_3) \le \left\lfloor\frac{7n+2}{4}\right\rfloor \qquad(n\ge9). $$

The verification package includes complete normalized enumerations, independent graph-structural checks, and four edge-by-edge activation certificates.

• Manuscript and verification materials

Finite-field basis obstructions for zero link Turán density

For a finite $r$-uniform hypergraph $H$, a prime power $q$, and $m\ge0$, let $B_{q,m}(r)$ be the $r$-graph of linearly independent $r$-sets in $\mathbb F_q^{r+m}$, with the zero vector isolated. The manuscript proves that the absence of a homomorphism $H\to B_{q,m}(r)$ implies

$$ \pi_\infty(H)\ge \prod_{j=m+1}^{\infty}(1-q^{-j})>0. $$

It then constructs a six-vertex three-graph and its dual for which every vertex link is tripartite, while both have link Turán density at least $\prod_{j\ge1}(1-2^{-j})\approx0.288788$. Thus local tripartite links do not suffice for zero link Turán density.

• Manuscript and verification materials

Exact Thue--Morse run separation

Let $d$ be the adjacent-run-length sequence of the Thue--Morse word. For every $q\ge2$, the manuscript proves that the first position at which $d(2^{q+2}n)$ and $d(2^qn)$ differ is

$$ \left\lfloor\frac{2^{E_q}}9\right\rfloor, \qquad E_q=\begin{cases}2^{q+1}-2,&q\text{ even},\2^{q+1}+1,&q\text{ odd}. \end{cases} $$

It also treats arbitrary distances in the same ratio-four scale chain and gives an exact infinite multiplier family characterized by a multiplicative-order condition.

• Manuscript and verification materials

Connected-set polynomial counterexamples

For every order $n\ge8$, the manuscript constructs four pairwise nonisomorphic connected graphs with at least as many edges as vertices whose connected set polynomials are not unimodal. A common rooted-block formula proves the fixed coefficient pattern $10,11,10$. A complete scan of all 11,117 connected eight-vertex graph classes shows that eight is the minimum counterexample order and that the four first family members are exactly all minimum-order counterexamples.

• Manuscript and verification materials

Higher-order Stirling cycle row log-concavity

For

$$ C_r(n,k)=[n+(r-1)k,k]_{\ge r}, $$

the manuscript proves that every row is log-concave for every $n$ if and only if $1\le r\le5$. It settles the previously open $r=3,4,5$ layers, proves failure for every $r\ge6$ already at $n=3$, and establishes a factorial-normalized log-concavity theorem for every $r\ge2$. The $r=5$ layer combines exact symbolic certificates, an exhaustive finite prefix, and an effective Fourier--Edgeworth tail estimate.

• Manuscript and verification materials

Compatible orderings of binary vector spaces

A total order on a finite vector space is compatible when the unique order-preserving bijection between any two subspaces of the same dimension is linear. For every $n\ge4$, this manuscript proves that compatible total orders on $\mathbb F_2^n$ form exactly eight orbits under $\mathrm{GL}(n,2)$, and therefore their labelled number is

$$ 8|\mathrm{GL}(n,2)| =8\prod_{i=0}^{n-1}(2^n-2^i). $$

An exhaustive, symmetry-broken four-dimensional classification supplies the only computer-assisted base lemma. A local line condition then forces an initial complete flag, and a unique-lift theorem propagates the two surviving ordered Fano types through every higher dimension.

• Manuscript and exact verification

Weighted recursion for six-permutation shattering

For any finite six-permutation template and any probability weights with at least two positive entries, the manuscript proves an exact weighted lexicographic recursion. Applied to the published 26-point template, it gives

$$ c_3\ge\frac{1288385}{2599242}=0.495677201276\ldots, $$

strictly improving the uniform-recursion value (482/975) for the same template. The included exact certificate reconstructs all 2600 triples and the rational weighted bound.

• Manuscript and verification materials

Orderable cographic matroids from the projective plane

For every finite simplicial triangulation T of the real projective plane, with 1-skeleton G, the manuscript proves that M*(G) is orderable. Repeated stellar subdivisions of the six-vertex triangulation produce infinitely many 3-connected, regular, non-graphic, orderable binary matroids, disproving Crenshaw--Oxley Conjecture 4.

• Manuscript and verification materials

Sylvester--Hadamard powers

For every $t=2^k$, $k\geq2$, the first manuscript constructs

$$ SA^{\mathsf T}(4t-2,t,-,t-1,t:(2t-1)\times2t). $$

It combines the classical Sylvester--Hadamard column design with a compatible proper ordering based on finite-field maps, finite geometry, algebraic curves, and exact certificates.

• Manuscript and verification materials

Paley two-edge switching

For every odd prime power $q\equiv3\pmod4$, $q\geq11$, the second manuscript constructs

$$ SA^{\mathsf T}!\left( 2q,\frac{q+1}{2},-,\frac{q-1}{2},\frac{q+1}{2}:q\times(q+1) \right). $$

It converts a classical Paley matching into a proper array by a two-edge switch, with existence proved through exact character counts and the Hasse bound for elliptic curves.

• Manuscript and verification materials

Spectral separation for looped double paths

For the double path $P_{n,L}$ with loops at the vertices in $L$, the third manuscript proves

$$ P_{n,L}\text{ allows the nSSP} \quad\Longleftrightarrow\quad L\ne\varnothing\text{ and }(n\text{ is even or }L\text{ contains an odd vertex}). $$

Its main structural tool is a spectral-separation criterion for a symmetric irreducible tridiagonal matrix with one nonzero diagonal entry: nSSP is equivalent to disjoint spectra of the two Jacobi arms.

• Manuscript and verification materials

Root-loop spider matrices

For a spider with at least three arms, consider an arbitrary real matrix whose nonzero off-diagonal positions are exactly both directed entries on every tree edge, with a single nonzero diagonal entry at the root. If $P_j$ is the characteristic polynomial of arm $j$ after deleting the root, the fourth manuscript proves

$$ A\text{ has the nSSP} \quad\Longleftrightarrow\quad P_1,\ldots,P_m\text{ are pairwise coprime}. $$

This fixed-matrix criterion allows arbitrary non-symmetric real weights, negative edge products, repeated roots within an arm, and nonreal shared-root witnesses. It extends the two-arm spectral mechanism while leaving the earlier arbitrary-loop double-path classification independent.

• Manuscript and verification materials

Recursive root-loop tree matrices

For an arbitrary finite rooted tree, consider a real matrix whose nonzero off-diagonal positions are exactly both directed entries on every tree edge, with a single nonzero diagonal entry at the root. For each child of every vertex, form the characteristic polynomial of its complete descendant subtree. The fifth manuscript proves

$$ A\text{ has the nSSP} \quad\Longleftrightarrow\quad \text{the child-subtree polynomials are pairwise coprime at every vertex}. $$

This fixed-matrix theorem allows arbitrary non-symmetric real weights, negative edge products, nonreal or repeated subtree roots, and arbitrarily many nested branch vertices. It strictly extends the preceding fixed-matrix double-path and spider criteria while leaving the earlier arbitrary-loop double-path pattern classification independent.

• Manuscript and verification materials

Second support weights of binary second-order Reed--Muller codes

For the binary Reed--Muller code $RM_2(2,n)$, this manuscript determines for every $n$ the exact set

$$ \mathcal S_n ={\mathrm{wt}(D):D\le RM_2(2,n),\ \dim D=2}. $$

The even-dimensional answer is given by an explicit finite compatibility system for the zero-frequency Walsh coefficients and polar ranks of the three nonzero members of a quadratic pencil. A finite-atom construction realizes every candidate value, and the odd-dimensional spectra satisfy $\mathcal S_{2m+1}=2\mathcal S_{2m}$. The result determines support positions, not their multiplicities.

• Manuscript and verification materials

Third support regions of binary second-order Reed--Muller codes

For three-dimensional subcodes of $RM_2(2,n)$, this manuscript determines two sharp all-dimensional spectrum gaps, the exact spectra for $2\le n\le9$, and the complete near-full interval below $2^{n-5}$ common zeros. It also gives a finite sound-and-complete canonical-pencil recursion when one nonzero direction has nonzero zero-frequency Walsh coefficient and polar rank two or four.

Together with arithmetic forcing, the recursion yields the infinite forbidden support ray

$$ 493\cdot2^{n-9}\notin\mathcal S_n^{(3)},\qquad n\ge12. $$

The result is not a complete classification of the third support spectrum; the Walsh-zero low-rank boundary and the all-high-rank layer remain open.

• Manuscript and verification materials

Acyclic component hypotheses for total cut complexes

Let $G=G_1\sqcup\cdots\sqcup G_k$ have $k\geq d$ nonempty connected components and $n$ vertices. If every relevant component total-cut complex $\Delta_r^t(G_i)$, $2\leq r\leq d$, is void or integer-acyclic, then

$$ \Delta_d^t(G)\simeq \bigvee_{\binom{k-1}{d-1}} S^{n-d-1}. $$

This successor theorem subsumes the previously published three low-component cases, removes the component clique-complex simple-connectivity condition, and answers Question 30 for the stated component class. Its proof combines Alexander duality, an acyclic composition diagram, a homology-colimit spectral sequence, direct connectivity arguments, and a weak-composition cover with van Kampen at the boundary $k=d$.

• Manuscript materials

Comb-bounded crowns and bi-Esakia representability

Let $X$ be a well-ordered rooted tree. For every point $a$ covering the root, write $Q_a=\uparrow a$. This manuscript proves that $X$ admits a bi-Esakia topology if every $Q_a$ is finite and the order duals $Q_a^{\mathrm{op}}$ uniformly omit one fixed finite co-tree comb.

The proof combines a finite-anchor synchronization theorem for finite posets amalgamated over a common finite downset with a better-partial-order theorem for comb-bounded finite co-trees. The criterion is sufficient rather than a classification, and allows arbitrary root width and unbounded crown sizes.

• Manuscript materials

Near-minimal denominator-two Ehrhart data

Let $P\subset\mathbb R^2$ be a convex polygon of denominator two satisfying $b(P)=0$ and $b(2P)=3$, and set $N=i(P)$ and $I=i(2P)$. This manuscript proves

$$ 2I+1\equiv4N\pm1\pmod8 $$

and gives a sharp classification near the lower boundary: all values through $I\le3N-3$, the unique next-boundary case $(N,I)=(10,28)$ at $I=3N-2$, and an infinite family with $N=4k+5$ and $I=3N-1$ showing one-point sharpness. The result is not a classification of all denominator-two Ehrhart data.

• Manuscript and verification materials

Periodic infinite quaternion ringsets

Let $m$ be an odd squarefree positive integer and let $w:\mathbb Z\to{\mathbf i,\mathbf j}$ be periodic with prescribed period $m$. For

$$ S_w={a+w(a):a\in\mathbb Z} $$

inside the Lipschitz quaternions, this manuscript proves that $S_w$ is a ringset exactly when $w$ is nonconstant and, for every prime $p\mid m$ with $p\equiv3\pmod4$, every residue class modulo $p$ contains both letters. The proof combines split matrix-ring idempotents, Frobenius obstruction polynomials, periodic congruence orbits, and a Chinese-remainder local-global argument.

• Manuscript materials

General odd-period quaternion ringsets

For an arbitrary prescribed odd period $m$, the new manuscript proves that

$$ S_w={a+w(a):a\in\mathbb Z} $$

is a ringset in the Lipschitz quaternions exactly when $w$ is nonconstant and, for every prime $p\mid m$ with $p\equiv3\pmod4$, every position class modulo $p^{v_p(m)}$ contains both letters. This strictly generalizes the preceding odd-squarefree-period theorem. The new ingredient is an explicit higher-prime-power fixed-divisor obstruction for monochromatic position classes.

• Manuscript materials

Complete arbitrary-period six-letter quaternion ringsets

Let $w:\mathbb Z\to{\pm\mathbf i,\pm\mathbf j,\pm\mathbf k}$ have an arbitrary prescribed period $m$. This manuscript gives a complete necessary-and-sufficient criterion for

$$ S_w={a+w(a):a\in\mathbb Z} $$

to be a ringset in the Lipschitz quaternions. Ramified and nonsplit highest prime-power position fibres must contain at least two letters, while at split primes the unique additional obstruction is a square-root-separated pair of singleton antipodal fibres. The theorem covers even periods and the full six-letter alphabet, strictly containing both preceding quaternion entries.

• Manuscript and verification materials

Measurable lattice-path matroids

For a measurable lattice-path matroid specified by a closed time set $K$ and lower and upper cumulative paths $a\leq b$, this manuscript proves that its basic minorizing measures form exactly the fixed-mass path corridor and that all minorizing measures form exactly the interval relaxation

$$ F(t)-F(s)\leq b(t)-a(s)\qquad(s\leq t,\ s,t\in K). $$

Every real truncation is obtained by one mass slice of this relaxation. For the full-rank basic corridor, the paper also gives the contact-set criterion for extreme points and proves that every such extreme point is weak-star exposed. The statements allow arbitrary closed time sets, including infinitely many gaps and Cantor-type contact sets.

• Manuscript materials

Irreducible-Ferrers polytopes

For every integer $d\geq3$, this manuscript proves that the polytope $\mathfrak P_d$ associated with irreducible Ferrers diagrams is combinatorially equivalent to a Cartesian product of $d-2$ triangles:

$$ \mathfrak P_d\simeq_{\mathrm{comb}}(\Delta_2)^{d-2}. $$

This resolves Conjecture 5.28 of Beeloo-Sauerbier Couvée and Neri. The proof uses their nonnegative affine-slice model and complete vertex-support classification, together with a general block-support lemma that recovers the entire face lattice. It claims combinatorial equivalence only, not affine or unimodular equivalence, and does not resolve the Etzion--Silberstein conjecture itself.

• Manuscript materials

Dual Fano partition degrees and six-symbol regularity

For the dual Fano matroid, this manuscript determines the complete partition-representation degree spectrum:

$$ \chi(F_7^*)={2^k:k\geq1}. $$

The proof propagates an $M(K_4)$ group slice through the remaining circuits and forces the group to be elementary abelian. Combined with the excluded minors for regular matroids and the orthogonal-Latin-square existence theorem, it proves that a finite matroid is 6-entropic exactly when it is regular, and that six is the unique alphabet size with this property.

• Manuscript and finite calibration

Third symbolic powers of co-chordal edge ideals

For every finite co-chordal graph $G$ over an arbitrary field, this manuscript proves that

$$ I(G)^{(3)} $$

is componentwise linear. Existing results settle the second power, give counterexamples from the fourth power onward, and leave only the degree-five component undecided for the third power. A marked-clique capacity theorem on chordal graphs proves that this remaining component has a five-linear resolution.

• Manuscript and finite proof-identity calibration

Ternary Abelian maximal pattern complexity at pattern size three

The fixed point beginning in 0 of the primitive substitution

0 -> 001,  1 -> 020,  2 -> 000

is recurrent and aperiodic by projection and satisfies

$$ p_{\alpha}^{*\mathrm{ab}}(3)=7. $$

Thus the general lower bound (r-1)k+1 is attained at r=3, k=3. The global upper bound is certified by an exact 938-state base-three patch automaton, while the pattern {0,2,9} supplies sharpness.

• Manuscript and exact verification

Sharp Abelian-border periodicity thresholds

For a binary infinite word $x$, let $\mu_{\mathrm{ab}}(x)$ be the maximum length of an Abelian-unbordered factor, when finite. This manuscript proves that $\mu_{\mathrm{ab}}(x)\leq13$ forces ordinary ultimate periodicity with sharp period bound $13$, while $\mu_{\mathrm{ab}}(x)\leq14$ forces Abelian ultimate periodicity with sharp Abelian-period bound $14$. Hence the minimum finite value among non-ordinarily-periodic words is exactly $14$.

More generally, if the frequency of $1$ is the reduced fraction $p/q$ and $\mu_{\mathrm{ab}}(x)<2q$, then the tail splits into consecutive length-$q$ blocks of weight $p$. Exact finite overlap graphs and independent Python/C++ implementations certify the two initial threshold layers.

• Manuscript and exact verification

A sparse-defect counterexample to Abelian-border periodicity

This manuscript constructs a binary infinite word that is not Abelian ultimately periodic but whose every factor of length at least 141 is Abelian bordered. It therefore gives a negative answer to Question 1 of Charlier, Harju, Puzynina and Zamboni and to Problem 42 in the Fici--Puzynina survey.

The construction starts from a four-height periodic nearest-neighbor path with a complete local endpoint-sum palette. Sparse replacement walks destroy every constant arithmetic-progression tail, while five translated witnesses preserve the local Abelian-border certificates. The exact periodic palette, 28 replacement blocks and a finite sparse-prefix positive control are independently reproducible.

• Manuscript and exact verification

Correction to a Desarguesian partial-parallelism bound

For every prime power $q$, odd prime $t\geq3$, and $z\geq2$, set $n=zt$. This note combines the Zhang--Zhou orbit construction with an elementary comparison to prove

$$ D(n,t,q)> \frac{q^t-1}{q^n-1} \left[\begin{matrix}n-1\t-1\end{matrix}\right]_q

\frac{q^{n-1}-1}{q^{t-1}-1}. $$

Thus the upper bound claimed by Johnson in 2010 fails throughout its nontrivial odd-prime parameter range. Earlier $PG(5,2)$ work already implies a complete finite counterexample, so the note claims a uniform correction, not the first counterexample. A compact certificate independently verifies 18 pairwise disjoint Desarguesian $3$-spreads in $V(6,2)$.

• Correction note and verification certificate

Two-layer antichain polynomials

For all positive integers $m,n$, this manuscript proves that the antichain generating polynomial

$$ N_{[2]\times[m]\times[n]}(x) $$

has only negative real zeros, proving Ding and Dong's Conjecture 4.3. It is palindromic if and only if $|m-n|=1$, and in precisely these adjacent cases all gamma coefficients are strictly positive. This proves Conjecture 4.5 and the general gamma-positivity conjecture for the complete subfamily $k=2$, $P=[m]\times[n]$.

A canonical two-gap tail-switching bijection first establishes the previously unproved $2\times2$ determinant for the associated Zhang--Zhang polynomial. A Möbius transformation then factors it into controlled combinations of adjacent Jacobi polynomials, whose zeros are located by interlacing and endpoint signs.

• Manuscript and exact verification

Complete palindromicity classification for three-dimensional boxes

For positive integers $a\leq b\leq c$, the antichain generating polynomial of $[a]\times[b]\times[c]$ is palindromic exactly for

$$ (a,b,c)=(1,r,r)\quad\text{or}\quad(2,r,r+1). $$

The new higher-layer theorem excludes every box with $a,b,c\geq3$ by comparing the first and next-to-leading coefficients through nonnegative last-passage matrices, path complementation and minimum separators. Together with the one- and two-layer cases, this closes the rectangular minuscule branch of Ding--Dong Conjecture B.

• Manuscript and exact verification

Middle runner counterexamples

For velocities $(1,2,3)$ and initial positions $(0,0,a)$, this note evaluates the average median position exactly over one common period. Modulo one, the conjectured value $1/2$ occurs only for $a=0$ and $a=1/2$; every other offset is a counterexample to the 2015 AIM middle runner conjecture under its printed hypotheses. Thus the failure is an uncountable continuous family rather than an isolated numerical example.

The proof is a complete piecewise integration, supported by an exact rational verifier covering the explicit example, controls, phase invariance, and 775 rational parameter representatives.

• Manuscript and exact verification

Status and review policy

Entries are research records, not automatically peer-reviewed publications. Each directory states its own claim and status. The labels used here mean:

• Exploration: an investigated direction without a retained theorem claim.
• Candidate result: a precise claim with supporting argument or evidence, still undergoing internal checks.
• Internally verified: the written proof and supplied computations have passed the repository's internal checks; independent mathematical review may still be pending.
• Externally reviewed: outside mathematical feedback has been received and its disposition is recorded by the entry.

Corrections, counterexamples, equivalent prior results, and independent reproductions are welcome. A status label is a record of the review stage, not a substitute for reading the proof.

Repository organization

Each research topic lives under research/ in its own directory. An entry may contain a focused README, manuscript and source, verification programs, generated evidence, checksums, and entry-specific licensing.

Citation

Use the CITATION.bib file in an entry directory when citing a specific result. The repository-level CITATION.cff describes the collection as a whole and enables GitHub's citation interface.

Licensing and attribution

Licenses are declared per research entry. Manuscripts and documentation typically use CC BY 4.0, while Python verification code typically uses the MIT License. Some entries separately place specified JSON results, certificates, or manifests under CC0 1.0; the declaration inside each entry is authoritative.

Unless an entry states otherwise, the responsible author is Carptopus. Contact: carptopus@163.com.

Repository-level keywords

AI-assisted mathematics; mathematical exploration; experimental mathematics; computer-assisted mathematics; reproducible mathematics; exact computation; constructive mathematics; combinatorics; combinatorial designs; row-column designs; transposed sesqui arrays; finite geometry; finite fields; Paley designs; quadratic characters; edge switching; algebraic combinatorics; algebraic curves over finite fields; elliptic curves; Hasse and Hasse--Weil bounds; cross-disciplinary mathematical methods; inverse eigenvalue problems; graph patterns; non-symmetric strong spectral property; Jacobi matrices; tridiagonal matrices; spectral separation; root-loop spiders; spider matrices; generalized stars; bidirected trees; arm characteristic polynomials; rooted tree matrices; recursive subtree polynomials; pairwise coprimality; spectral collisions; centralizer witnesses; Reed--Muller codes; higher weight spectra; second support weights; quadratic Boolean functions; Walsh spectra; quadratic pencils; common-zero counts; alternating polar forms; generalized Hamming weights; total cut complexes; bounded-independence complexes; simplicial complexes; Alexander duality; polyhedral joins; nerve lemma; wedges of spheres; disconnected graphs; graph complexes; algebraic topology; order theory; bi-Esakia spaces; bi-Heyting algebras; well-ordered rooted trees; finite crowns; finite combs; bi-p-morphisms; inverse limits; better partial orders; Priestley duality; Esakia duality. Ehrhart theory; rational polygons; half-integral polygons; denominator-two polygons; lattice polygons; primitive lattice triangles; parity obstructions; internal hulls; outer hulls; lattice width; discrete geometry; computational geometry. Integer-valued polynomials; Lipschitz quaternions; quaternion algebras; noncommutative algebra; ringsets; null ideals; periodic words; local-global principles; matrix rings; Chinese remainder theorem; prime-power position classes; fixed-divisor obstructions; ramified primes; split primes; antipodal fibres. Measurable matroids; lattice-path matroids; minorizing measures; basic minorizing measures; weak-star topology; interval relaxations; real truncations; extreme points; exposed points; continuous linear programming; purification. Ferrers diagrams; Ferrers polytopes; face lattices; products of triangles; products of simplices; polyhedral combinatorics; combinatorial equivalence; rank-metric codes; Etzion--Silberstein conjecture. Entropic matroids; partition representations; variable-strength orthogonal arrays; dual Fano matroid; regular matroids; excluded minors; orthogonal Latin squares; finite-group representations. Combinatorics on words; Abelian maximal pattern complexity; recurrent words; aperiodicity by projection; primitive substitutions; automatic sequences; Parikh vectors; sparse factors; finite-state verification; Abelian borders; Abelian-unbordered factors; Abelian periodicity; ultimate periodicity; sparse defects; counterexamples; bounded discrepancy; finite overlap graphs. Desarguesian spreads; partial parallelisms; field reduction; Cayley graphs; Gaussian binomial coefficients; translation nets; correction notes. Middle runner conjecture; dynamical algebraic combinatorics; circular runners; order statistics; periodic trajectories; continuous counterexamples; piecewise integration. Compatible orderings; binary vector spaces; ordered Fano planes; general linear groups; finite-geometry classification; flag orders. Abstract polytopes; maniplexes; premaniplexes; symmetry type graphs; voltage constructions; flag graphs; two-orbit polytopes.