README.md

MathExplore Notes

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

Research index

Entry	Areas	Contribution	Public version	Status
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
The low-component cases for total cut complexes of disconnected graphs PDF · BibTeX · DOI	Algebraic topology, graph complexes, simplicial complexes	Resolves the three low-component cases of the total-cut wedge-of-spheres problem and completes the stated range for all $k\geq d$	v0.2-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
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 of antichain polynomials for [2] x [m] x [n] PDF · BibTeX · DOI	Posets, real-rooted polynomials, Jacobi polynomials, lattice paths	Proves that the antichain generating polynomial of $[2]\times[m]\times[n]$ has only negative real zeros for all positive $m,n$	v0.1-beta	Internally verified candidate proof; external review pending

Current results

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

Low-component total cut complexes

Let $G$ be a graph with $k$ nonempty connected components and $n$ vertices. Under the component hypotheses stated in the manuscript, this result resolves the three low-component cases $k=d,d+1,d+2$ and proves

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

Together with the previously known high-component range and an independent argument for the required $d=2$ case, this completes the formula for every $d\geq2$ and $k\geq d$ under the stated hypotheses. The proof combines explicit triangular fillings, Alexander duality, joins, deformation retractions, and a nerve-lemma argument.

• 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

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

Antichain polynomials of three-dimensional boxes

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. 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

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; ultimate periodicity; bounded discrepancy; finite overlap graphs. Desarguesian spreads; partial parallelisms; field reduction; Cayley graphs; Gaussian binomial coefficients; translation nets; correction notes.