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	Combinatorial designs, finite fields, finite geometry, algebraic curves	Constructs an exact-parameter proper transposed sesqui array for every power of two	v0.2-beta	Internally verified candidate proof; external review pending
Paley two-edge switching for proper transposed sesqui arrays	Combinatorial designs, finite fields, elliptic curves	Constructs an exact-parameter proper transposed sesqui array for every odd prime power $q\equiv3\pmod4$, $q\geq11$	v0.2-beta	Internally verified candidate proof; external review pending
Spectral separation and the nSSP for looped double paths	Inverse eigenvalue problems, graph patterns, Jacobi matrices	Classifies exactly which looped double paths allow the non-symmetric strong spectral property	v0.1-beta	Internally verified candidate proof; external review pending
An exact nSSP criterion for root-loop spider matrices	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	v0.1-beta	Internally verified candidate proof; external review pending
A recursive nSSP criterion for root-loop tree matrices	Inverse eigenvalue problems, rooted trees, matrix patterns	Characterizes the nSSP of every fixed root-loop bidirected tree matrix by recursive sibling-subtree coprimality	v0.1-beta	Internally verified candidate proof; external review pending
Attainable second support weights of binary second-order Reed--Muller codes	Coding theory, quadratic Boolean functions, Walsh spectra	Determines for every dimension the exact set of support sizes attained by two-dimensional subcodes of $RM_2(2,n)$	v0.1-beta	Internally verified candidate proof; external review pending
The low-component cases for total cut complexes of disconnected graphs	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 $k\geq d$	v0.1-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

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.

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.