Mathematical research drafts
Read the research website — full manuscripts, typeset mathematics, and direct PDFs. Automatically rebuilt from this repository after every update to main.
AI-generated mathematical research drafts. Not peer reviewed or independently verified by human experts. Corrections and prior-art references welcome.
This public collection contains 57 distinct manuscripts, each with its PDF, LaTeX source, and available supporting checks. These are proposed mathematical results, not certified solutions of open problems. Internal checks by other AI agents do not establish correctness, novelty, significance, or publication readiness.
The material was prepared with OpenAI Codex and collaborating AI agents, including problem selection, proof construction, literature searches, computations, and writing. No human expert verification or journal acceptance is claimed. Hosting these files makes them accessible and citable; it is not an endorsement of their conclusions.
Positivity and counterexamples for rising-factorial Toeplitz determinants
P57: Positivity and counterexamples for rising-factorial Toeplitz determinants, by Henry Zweiman, addresses the remaining conjectures in Karp's 2012 paper. It proves Conjecture 1, gives exact counterexamples to Conjectures 2, 5, and 6, and proves a corrected coefficient-positivity theorem with precise nondegeneracy conditions for Conjecture 4. This is a separate paper from P56.
The complete manuscript, nine-page PDF, LaTeX, and exact verification include the proofs and integer-arithmetic certificates. The literature assessment distinguishes the arguments from Karp's special-case proof and leaves historical priority unverified. Prepared with OpenAI Codex; not peer reviewed.
Negative real zeros of a rising-factorial transform
P56: Negative real zeros of a rising-factorial transform, by Henry Zweiman, proves the coefficient-pairing assertion for rising factorials and the positive-endpoint form of Karp's 2012 Conjecture 3. The complete proof, PDF, and LaTeX derive the factorial-transform lemma by interlacing and account explicitly for Karp's binomial coefficient. Zero endpoints are treated separately, with a nonpositive-root limit theorem and a boundary example.
The scope assessment, proof audit, and exact checks document the argument. This is an AI-assisted research preprint, not independently reviewed; no priority determination is claimed.
A planar comparison argument for radial bound states
P55: A planar comparison argument for radial bound states, prepared for Henry Zweiman, presents a candidate proof reconstruction for planar radial bound-state uniqueness and shooting classification for all powers
Independent PDE review is pending; no first-proof claim is made. The assessment credits the prior 2011 theorem and Zhang–Zhang's 2026 planar claim. The self-audit and expert brief identify the points requiring scrutiny. This archive entry does not certify a new original theorem.
A spherical Weinstock inequality at large perimeter
P54: A spherical Weinstock inequality at large perimeter, by Henry Zweiman, gives a proposed ball-maximization theorem for all smooth convex domains in the unit sphere in every dimension at least three, throughout a uniform interval of perimeters adjacent to equatorial area. The 17-page PDF, complete Markdown, and LaTeX include the harmonic boundary center, quantitative endpoint stability, strict lune comparison, and collapsed-limit argument.
The assessment and proof review distinguish the claimed interval result from the full spherical conjecture, which remains open. The interval size is nonexplicit. Primary-source read scopes and final-text access limits are recorded; no independent verification or journal acceptance is claimed. These related results constitute one preprint.
Endpoint saddles and nearby spectral critical domains
P53: Endpoint saddles and the local classification of eigenvalue–torsion critical domains, by Henry Zweiman, gives proposed nonlinear ball saddles at the volume and perimeter stability endpoints in every dimension and classifies all nearby critical domains up to similarities. The 25-page PDF, complete Markdown and LaTeX contain the proofs, including balanced even-dimensional nondegeneracy and exact local branch counts.
The assessment compares the final August 2026 endpoint question and a September 15 preprint. The review checks analytic boundary maps, full-kernel reduction, finite PDE jets and all-dimensional nonvanishing. Established Hessians, equivariant bifurcation methods and Bessel transcendence are credited. The original global convex perimeter conjecture remains open. This is one internally assessed, unreviewed preprint.
Complete splitting by a single planar Robin hole
P52: Complete splitting of planar Robin eigenvalues by a single shrinking hole, by Henry Zweiman, gives proposed all-order asymptotics and complete splitting at a prescribed interior point. One analytic strictly convex hole shape near a disk works across every finite spectral window, for either sign of a fixed nonzero Robin parameter. The size threshold depends on the window. The 20-page PDF, complete Markdown and LaTeX contain the hand proofs.
The assessment compares the final June 2026 Robin frontier, the current Neumann splitting paper and corrected Dirichlet matrix theory. The review checks weighted remainders, signed branches, planar normalization and the one-shape construction. Existing Schur-complement and polarization-tensor methods are credited. Older full-text access limits remain disclosed. No common positive threshold for the entire infinite spectrum or independent expert verification is claimed. This is one provisionally admitted, unreviewed preprint.
Uniform magnetic Steklov transitions and geometric comparison
P51: Uniform weak-field and small-flux asymptotics for exterior magnetic Steklov operators, by Henry Zweiman, gives a proposed uniform operator expansion through integral Aharonov–Bohm flux, a lowest-eigenvalue crossover, a signed-flux asymmetry and a second-order geometric gap. Every fixed smooth simply connected nondisk is strictly below the equal-perimeter disk in an obstacle-dependent small-field, small-mass and small-flux neighborhood. No symmetry or convexity hypothesis is imposed. The 21-page PDF, complete Markdown and LaTeX contain the hand proofs.
The assessment credits the scalar capacity mechanism, circle formulas, conformal covariance and classical boundary calculus, and records the incomplete final-text access for two comparison sources. The review audits uniform radial cancellation, the singular gauge, the operator topology and the geometric coefficients. No shape-uniform threshold, all-field-range theorem or independent expert verification is claimed. This is one provisionally admitted, unreviewed preprint.
All-times heat-content maximization on metric graphs
P50: Heat-content maximization on compact metric graphs at every time, by Henry Zweiman, proposes a complete answer to the all-times question in Bifulco-Täufer's final EJP article. The equal-length Dirichlet-Neumann interval maximizes heat content at every positive time; equality at one time characterizes that interval. A stronger cumulative-rearrangement theorem covers arbitrary nonnegative L2 initial data. The twenty-four-page PDF, full Markdown, and LaTeX contain the proofs.
Revision 1.1 also proves a fixed-total-killing comparison: among graphs of fixed length with a finite nonnegative killing measure of fixed positive mass, the interval with all killing at one endpoint uniquely maximizes heat content at every positive time. The result includes delta interactions, integrable potentials and singular measures. The assessment credits prior torsion and rearrangement results and records the bounded source search. The review checks the cumulative Robin boundary condition, measure approximation and one-time rigidity. Revision 1.3 proves both sharp Kohler-Jobin comparisons for arbitrary finite nonnegative killing measures: the endpoint interval uniquely minimizes the torsion-eigenvalue product at fixed length and total killing, and the first eigenvalue at fixed torsion and total killing. A ground-state distribution estimate controls every potential term and handles singular measures by form convergence. Revision 1.4 extends both stationary comparisons to the p-Laplacian for every 1 < p < infinity, with explicit finite-strength constants and the same rigidity for arbitrary finite nonnegative killing measures. Özcan's nonlinear Dirichlet result and modified-torsion method are credited. Quantitative stability, nonlinear heat concentration, weighted edge forms and arbitrary couplings remain open. This is one internally assessed, unreviewed AI-assisted preprint.
Sharp annular stability for physical torsion
P49: Sharp stability of the annulus for physical torsion, by Henry Zweiman, proposes sharp quadratic stability for perforated planar Lipschitz domains. The deficit controls the outer domain and hole union by disks with a common center. A fixed area-ratio range also gives universal individual-asymmetry constants. The ten-page PDF, full Markdown, and LaTeX contain the plateau and calibrated-transport proofs and an analytic sharpness example.
The assessment compares the final 2026 open questions and recent primary literature; the review audits boundary regularity, jump accounting, fixed-center transport and scaling. Hensel-Laux's global calibration theorem and Guerra-Machado-Ramos's transport viewpoint are essential credited inputs. Best constants, nonlinear and higher-dimensional counterparts remain unresolved. This is one internally assessed, unreviewed preprint.
Local minimality for the generalized Neumann Hénon problem
P48: Local minimality for the generalized Neumann Hénon problem, by Henry Zweiman, proves strict Sobolev local minimality of radial minimizers below the known Steklov threshold. It covers q near p for all 2<p<n and the full subcritical range when n>=4 and p>2 is close to two. Revision 1.1 proves that the local-minimum neighborhood is uniform through the critical trace exponent and has a sharp power decay above it. The fourteen-page PDF, full Markdown, and LaTeX contain the weighted compactness and nonlinear coercivity arguments.
The assessment compares the explicit 2026 open question and close weighted Morse theory; the review checks concentration, normalization, boundary traces and the angular threshold. The threshold formula and p=2 theorem are credited prior work. Equality at the threshold and a uniform nonlinear neighborhood remain unresolved. This is one internally assessed preprint, without independent expert verification.
Sharp total variation from spherical rearrangement
P47: Sharp total variation bounds from spherical rearrangement, by Henry Zweiman, now proves the complete variation identity and equality classification in derivative order three at every exponent, with the measure interpretation at p=1. Revision 1.12 classifies every uniform-endpoint maximizer: its third derivative has four alternating constant pieces, and V(3,infinity)=(2-sqrt(2))/48. The proof starts with an arbitrary maximizer, establishes regularity from a conserved residual quantity, extends the positive-action comparison to the endpoint, and determines the unique one-peak profile explicitly. Together with the finite-exponent theorem and measure endpoint, this completes the order-three case. Earlier results include the all-order Hilbert case, all-order height symmetry, and local exponent intervals for variation. Revision 1.13 proves all-order uniform-endpoint regularity: every maximizer saturates its highest derivative, has finitely many simple critical points, and has exactly n+r-1 simple residual zeros when there are r critical points. For each fixed n>=3, the critical-point count is uniformly bounded over all 1<p<=infinity. The bound is implicit; the remaining all-order variation conjecture is open. Revision 1.14 derives the endpoint switching Hessian with atomic root weights and polynomial projection. It also proves that V(n,infinity) is real algebraic in every order and admits a maximizing spline with algebraic knots and piece coefficients. The proof combines finite switching with a rational semialgebraic description and credited quantifier elimination. No effective stopping count or general higher-order multiple-peak exclusion is obtained. Revision 1.15 excludes every two-critical-point endpoint variation maximizer and every midpoint-symmetric three-critical-point endpoint maximizer for n>=3. It also proves that any remaining three-point candidate has its dilation-response minimum strictly outside the two outer critical points. The proof uses an atomic affine response, positive-density approximation, separated zeros and an order-four boundary sign count. These exclusions and the response condition now hold over every 1<p<=infinity; the general higher-order conjecture remains open. The 60-page PDF, full Markdown, and LaTeX give the proofs.
The assessment compares the final survey and close primary sources; the review audits the Jacobi-Poisson representation, spherical equality cases and endpoint moment geometry. This is one internally assessed preprint. The full all-p conjecture remains unresolved; independent expert verification is not claimed.
Ground-state uniqueness on convex domains and logarithmic concavity
P46: Ground-state uniqueness on convex domains and sharp logarithmic concavity on balls, by Henry Zweiman, now proves uniqueness of the positive logarithmic Schrodinger ground state on every bounded convex domain, in every dimension and without boundary smoothness assumptions. Revision 1.2 identifies every positive log-concave solution as that ground state, answering the ground-state questions in Gallo-Mosconi-Squassina, Remark 1.6 (2026). A positive-level argument proves a Gaussian-strength logarithmic Hessian bound; a local strict transport remainder yields a sharp energy inequality with equality only for multiples of the ground state. The 20-page PDF, complete Markdown, and LaTeX contain the hand proofs.
The manuscript retains its positive radial series and sharp ball concavity results. Its convex-deformation counterexamples are now identified as ground states. Contribution and scope, internal review, and ground-state source audit distinguish the new result from classical existence, transport and logarithmic Sobolev inputs. Uniqueness among all positive solutions remains open. This is one revised, internally audited preprint, without independent expert or exhaustive priority certification.
Spectral localization and uniform reactive capacitance
P45: Spectral localization and uniform reactive capacitance of thin planar patches, by Henry Zweiman, proposes a strong operator limit and a full spectral-measure law for thin variable-width patches. Every fixed weight vanishes for thin ellipses and rhombi, while capacitance has a relative asymptotic formula uniform over all positive reactivities. The twelve-page PDF, full Markdown, and LaTeX contain the proofs.
The same contribution includes bounded remainders for two published tube conjectures, a criterion for asymptotic accuracy of the one-pole approximation, and a connected smooth example with F1>F0. The assessment compares the final Grebenkov-Maurette paper and August 2026 related work; the review audits the endpoint capacity bound, curvature comparison, and cluster splitting. This is one internally assessed preprint, not an independently certified solution. Sharp spectral corrections and singular center curves remain unresolved.
Convex Robin domains with vanishing spectral gaps
P44: Robin ground states on convex polyhedra: concavity and vanishing spectral gaps, by Henry Zweiman, contains counterexamples to the universal positive-Robin gap conjecture and to Laugesen's parameter conjectures A, B and C. Revision 4.1 adds an elementary planar result: after dividing the Robin parameter by perimeter, area-one thin triangles have first eigenvalue tending to alpha/2 for positive alpha, and tending to negative infinity for negative alpha. An explicit triangle at alpha=1 has value below 123/175 while the disk exceeds 12/13. This refutes the exact Conjecture D formulation in Laugesen's arXiv v1. The thirty-eight-page revised PDF, full Markdown, and LaTeX retain all earlier results.
The assessment records the exact preprint scope, a discrepancy with the cited final Freitas--Laugesen article, and inaccessible final journal wording. The review checks the disk lower bound, triangular slicing, base-inclusive trial integrals and negative-parameter exponential. No numerical eigenvalues are used. This remains one AI-assisted research preprint without independent expert verification or exhaustive priority certification. The gap and parameter conjectures in dimensions two and three, exceptional polygonal parameters and the unrestricted local transverse-cone question remain unresolved here.
Genus obstruction for commuting boundary operators
P43: A genus obstruction for commuting boundary operators, by Henry Zweiman, proposes a complete negative answer to both cases of Speciel's final Open Problem 1.6: a compact connected oriented smooth surface with commuting Dirichlet-to-Neumann map and boundary Laplacian must have genus zero. Together with Speciel's existing theorem, this completes the oriented classification. The nine-page PDF, full Markdown, and LaTeX contain the proof.
The assessment compares the final journal problem, later author papers, classical finite-period theory, and smooth conformal welding. The review checks the holomorphic trace multiplication, degree-one map, conformal energy gluing and constant-mode coercivity. This is one internally assessed preprint; independent mathematical review and absolute priority remain unverified.
Euclidean persistence for anisotropic products
P42: Euclidean persistence and sharp stability for anisotropic eigenvalue-torsion products, by Henry Zweiman, proposes exact Euclidean optimality on balls over all normalized symmetric seminorms: maximization for sufficiently small positive torsion exponents and minimization for sufficiently large ones. The proof covers every dimension n>=2 and every energy exponent p>1. The product deficit is equivalent to an angular average and controls uniform distance with the optimal power (n+1)/2. The ten-page PDF, full Markdown, and LaTeX are included.
The assessment compares the explicit questions of Buttazzo and Fernandes Horta, their quadratic-seminorm predecessor, the published eigenvalue-only theory, and the classical symmetrization input. The review checks the cap covering argument, nonlinear flux duality, global thresholds and sharpness. General domains and optimal thresholds remain outside the theorem. This is one internally assessed preprint, not an independently certified solution.
Uniform reverse Robin Kohler-Jobin inequality
P41: Torsion transplantation and a uniform reverse Kohler-Jobin inequality for Robin problems, by Henry Zweiman, proposes an affirmative answer to the planar reverse Robin question of Buttazzo, Cito and Solombrino. Balls uniquely maximize the competing eigenvalue-torsion product for sufficiently large exponents among all bounded planar Lipschitz sets of fixed area. The exponent is uniform in the positive Robin parameter and area. A nonlinear extension and equivalent product/torsion deficits belong to the same proof. The eight-page PDF, full Markdown and LaTeX are included.
The assessment records the current primary-source comparison and the essential prior Talenti and Faber-Krahn inputs. The review audits the boundary correction, global threshold and parameter limits. The linear problem in dimensions at least three and the optimal exponent remain unresolved. P40 through P52 are the thirteen manuscripts internally assessed and provisionally admitted under the revised September 15 standard; the earlier 39 remain pending reassessment. This is not independent certification of correctness, novelty or journal suitability.
Boundary selection for logarithmic Sobolev constants
P40: Boundary selection for low-temperature logarithmic Sobolev constants, by Henry Zweiman, gives a proposed first-order sharp-constant expansion in arbitrary finite dimension for convex potentials with a smooth bounded domain of minima and quadratic normal growth. The coefficient selects among nonlinear extremizers and, at equality of the limiting constants, the first Neumann eigenspace. A smooth strictly convex example establishes the nonlinear regime. The 13-page PDF, complete Markdown and LaTeX contain the full proof.
The assessment compares Ben Nejma's explicit 2026 higher-dimensional frontier and related boundary-diffusion literature. The review records the originating assistant's proof audit and artifact checks. The spectral coefficient and interval result are credited to prior work. This is the first manuscript internally assessed under the revised September 15 program standard; the earlier 39 remain pending reassessment. That internal admission does not certify journal-level significance, independent correctness or historical priority.
Five stronger revisions and full Markdown editions
The selected collection contains five consolidated revisions (36 pages), each with complete Markdown, PDF, LaTeX, and an explicit source/significance assessment. Four contain broader structural theorems; P06 retains its Stirling conjecture resolutions and asymptotic. These are revised versions of existing papers, not five additional papers. The research frontiers state what remains unresolved.
All thirteen original papers now also have complete Markdown editions. Their earlier PDFs and LaTeX files remain unchanged.
Counterexamples to the matrix-power nullity criterion
P39: Counterexamples to a nullity criterion for matrix power maps, by Henry Zweiman, gives a proposed negative answer to Saini--Singh Question 6.1. For every k>=5, a diagonal idempotent of nullity two in size k+1 has an explicit rank-three target omitted by X^k+B Y^k, despite satisfying the proposed dimension inequality. The proof works over every field. The four-page PDF, complete Markdown, and LaTeX contain the full argument.
The assessment compares the exact open question and distinguishes P33's complementary-coefficient result. The internal review records the universal proof audit and exact diagnostics. The image is also dense but not additive over algebraically closed characteristic-zero fields. No dimension-minimality or complete replacement criterion is claimed. This is one internally checked proposed manuscript; historical priority, independent correctness and significance remain provisional.
Word fibres and all metacyclic p-groups
P38: Word fibres over finite chain rings and the Amit--Ashurst conjecture for metacyclic groups, by Henry Zweiman, proposes the conjectured lower bound for every word on every finite metacyclic p-group, including nonsplit extensions, and every elementary-abelian-by-cyclic p-group. The proof combines finite-difference support with coefficient ideals over chain rings, then descends a stronger estimate through a central cyclic cover. It permits unbounded rank and nilpotency class. The eight-page PDF, complete Markdown, and LaTeX contain the proofs.
The assessment compares the cyclic-maximal-subgroup predecessor and later work. The internal review records the proof audit and 745 exact word-distribution checks. The general nilpotent-group conjecture remains unresolved. This is one proposed manuscript; independent correctness, historical priority, and significance remain provisional.
Sharp uniform dominance for short Gorenstein algebras
P37: Uniform dominance of short Gorenstein algebras, by Henry Zweiman, proposes the sharp equality dx(R)=1 for every commutative graded Artinian Gorenstein algebra of Hilbert function (1,e,1), e>=3, over any field. Every nonzero stable object generates the residue field in one cone and the full category in at most three extensions. The six-page PDF, full Markdown, and LaTeX contain the proof and the bound seven for arbitrary short Gorenstein local rings.
The assessment credits Ringel's essential bristle-generation theorem and Kimura's known qualitative dominance result. The internal review covers the uniform cone construction, 312 exact matrix checks, and a concrete kernel with eight simple summands. Takahashi's global Question 6.12 remains unresolved. This is one proposed manuscript; outside correctness, historical priority, and significance remain provisional.
Short local rings in arbitrary characteristic
P36: Dominance and common deformations of short local rings, by Henry Zweiman, proposes a common-deformation theorem for every Artinian local ring with maximal-ideal cube zero and its associated graded ring. It transfers dominance and uniform dominance in both directions. Combined with Kimura's essential equicharacteristic theorem and known flat descent, it answers both clauses of Kobayashi–Takahashi Question 7.1 without characteristic or residue-field restrictions. The six-page PDF, full Markdown, and LaTeX include the proof.
The assessment distinguishes the contribution from Kimura's known theorem and records the priority limits. The internal review covers the coefficient-ring construction, regular-parameter conditions, higher-dimensional reduction, and 258 exact finite quotient checks. P36 left uniform dominance of short Gorenstein rings unresolved; the proposed P37 theorem above addresses that stronger question. This is one proposed preprint; outside correctness, priority, and significance remain provisional.
Connected quadratic curves and three-quadric intersections
P35: Rational points on connected curves cut out by quadrics, by Henry Zweiman, proposes a bound of Dq+1 for geometrically connected quadratic base-locus curves of degree D <= q+5 over Fq, q >= 3. Its sharp corollary gives 8q+1 for all complete intersections of three quadrics in projective four-space, including reducible and nonreduced intersections. The six-page PDF, full Markdown, and LaTeX include the complete proof.
The assessment identifies the precise case of Lin's Conjecture 1.5 and credits the known curve bounds and extremal construction. The internal review covers the geometric proof, source dependency repair, and exact diagnostics. The general largest-intersection conjecture remains unresolved. Outside correctness, priority, and significance remain provisional; this is one proposed manuscript.
Ordinarization counts and collision asymptotics
P34: Quasipolynomiality and collision asymptotics for ordinarization counts, by Henry Zweiman, proposes an affirmative answer to Cyrusian–Kaplan Remark 3.7 and eventual strict growth for every fixed positive ordinarization number. The complete second coefficient is explicit. Revision 1.2 adds a growing-ordinarization limit under r²/g → 0, with a Poisson collision law at the scale r³/g and bounds on every configuration outside the bulk. The thirteen-page PDF, full Markdown, and LaTeX give the complete proof.
The assessment records the primary open statement, antecedent polyhedral method, priority search, and an unaudited r=3 preprint lead. The current review covers the growing-parameter proof and exact diagnostics; earlier reviews remain available. Full monotonicity at every genus remains unresolved. Outside correctness, historical novelty, and significance are provisional.
Sharp polynomial matrix completion
P33: Sharp completion bounds for polynomial values of matrices, by Henry Zweiman, proposes a sharp all-dimension criterion for completing prescribed rows to a scalar polynomial value f(X). For a rank-r coefficient A, the image of Af(X) equals AM_n exactly when n > E(f)(r-1), where E(f) measures the minimum root multiplicity in an exceptional fiber. This gives a complete surjectivity and image-linearity classification when the coefficient column spaces form a direct sum. The six-page PDF, complete Markdown, and LaTeX include the proof.
The assessment distinguishes this regime from the unresolved overlapping-coefficient question and credits the known nilpotent obstruction and classical Kronecker decomposition. The review accompanies exact decomposition and completion checks, including all 19683 matrices over F3 of size three. This is one internally checked proposed manuscript; outside correctness, priority and significance remain provisional.
New sharp arity theorem for mixed-exponent clonoids
P32: Sharp arity bounds for clonoids from mixed-exponent groups, by Henry Zweiman, proposes a complete answer to Fioravanti--Kompatscher--Rossi Question 60. For A=Z/(p^2) direct sum (Z/p)^s, every clonoid into a module on which p is invertible is generated by its (s+2)-ary functions, and this bound is optimal in general. The six-page PDF, complete Markdown, and LaTeX give the uniform reconstruction and an explicit sharpness example.
The assessment credits the known cyclic and vector-space cases and Kovacs's matrix-algebra theorem. The review audits congruence characters, the torsion correction, coefficient descent and the lower-bound functional. Exact checks include all 262144 matrices over Z/4 of size three. This is one internally checked proposed manuscript; the general finite-module conjecture and outside correctness, priority and significance review remain outstanding.
New complete primitive-cubic classification
P31: A complete classification of a primitive cubic family, by Henry Zweiman, proposes a complete classification of the surviving cases of Awasthi--Sharma Conjecture 9.1. A primitive polynomial X^3+X^2+X+lambda over F_(q^2), with primitive lambda, exists precisely when the characteristic is not three, for every prime power q. The seven-page PDF, complete Markdown, and LaTeX also give distinct odd- and even-characteristic counting estimates.
The assessment credits the known q=3 counterexample and the Fu--Wan character theorem. The review records the exact parametrization, geometric hypothesis checks, sieve, and 261 independently verified finite certificates. The infinite cutoff is proved; no field range is left unchecked. This is one internally checked proposed manuscript, with outside correctness, historical priority, and significance provisional.
New counterexamples to Gao's zero-sum invariant conjecture
P30: Counterexamples to Gao's zero-sum invariant conjecture, by Henry Zweiman, proposes an infinite counterexample family to the conjecture that nu(G)=d(G)-1. The four-page PDF, complete Markdown, and LaTeX give the exact value nu(C2^3+C2n)=2n+2 for every odd n >= 3. A uniform six-hole configuration excludes every coset that does not contain zero.
The assessment checks the definition and conjecture in two primary sources, including the August 2026 Geroldinger--Yang paper. The internal review accompanies 50 exact family checks and a separate base-case enumeration. The Davenport formula is classical and credited; even n and the full classification remain outside scope. This is one internally checked proposed manuscript, with outside correctness and historical priority still provisional.
New exponent-eight maximal-elasticity theorem
P29: Maximal-elasticity length sets for families of two-groups of exponent eight, by Henry Zweiman, proposes a solution of the signed-support conjecture for two infinite families of finite abelian two-groups. The six-page PDF, complete Markdown, and LaTeX prove the result when the sum of the order-two and order-eight ranks is not 1 modulo 3, or when the order-eight rank is one. For arbitrary exponent-eight groups, the paper reduces the possible minimum distances to 1 and 3.
The source assessment checks the standing conjecture and distinguishes the new scope from known cases. The internal review records the universal hand arguments and 4,756 finite witness-atom checks. The general conjecture remains open; priority and outside mathematical verification are provisional. This is one consolidated manuscript.
New counterexample for weakly C-monoids
P28: Failure of the length-set structure theorem for weakly C-monoids, by Henry Zweiman, gives an internally checked proposed negative answer to Geroldinger-Kim-Loper, arXiv:2502.21020v3, Problem 20. Its six-page PDF, complete Markdown, and LaTeX construct a weakly C-monoid with factorial complete integral closure and distance set exactly {1,2} whose length sets fail the standard structure theorem. The proof gives all length sets and a quantitative obstruction for every bounded choice of differences.
The source assessment checks the exact weak-C definition and the global residue condition in an AAMP. The internal review records the hand proofs and independent finite checks. Stronger variants with additional hypotheses are not claimed. This is one paper; historical priority and significance remain provisional.
New classification of conductor ideals
P27: Conductor ideals of C-monoids with prescribed Krull closure, by Henry Zweiman, gives an internally checked proposed answer to Geroldinger--Kim--Loper, arXiv:2502.21020v3, Problem 19. The six-page PDF, complete Markdown, and LaTeX characterize realizable ideals by finiteness of their prime colon profiles, equivalently by finitely many truncated valuation sums. The proof covers infinite divisor rank and arbitrary unit groups, constructs a realization, and bounds its reduced class semigroup.
The source assessment and internal proof review state the classical dependencies and scope. Thirty exact monomial-ideal identities and twelve complete finite quotient checks accompany the universal hand proofs. This is one paper; ring conductors and general recognition of intermediate Mori monoids remain outside its claims. Historical priority and significance are provisional.
New conductor reconstruction theorem
P26: Reconstruction of Krull monoids from conductor submonoids, by Henry Zweiman, gives an internally checked proposed answer to the isomorphism question following Proposition 4.3 of Geroldinger--Yan--Zhong. The four-page PDF, complete Markdown, and LaTeX recover the original reduced Krull monoid and divisor theory from the associated conductor monoid. Automorphism and recognition consequences are included in this same note.
The assessment credits the classical tools and explicitly notes that the proof is short. Publication of this focused result does not certify its novelty or significance, or count it automatically toward 45 significant solutions. The internal review accompanies 6,311 finite generation checks and 182 permutation checks; the universal results have hand proofs.
New Koszul idempotent counterexample
P25: Primitive idempotents that cannot be homogenized in Koszul algebras, by Henry Zweiman, gives a proposed negative answer to Dramburg's Question 6.2, including over the complex numbers. The six-page PDF, complete Markdown, and LaTeX construct a Koszul matrix algebra over a two-dimensional quadratic complete-intersection domain.
In that fixed algebra, primitive idempotents can agree with a homogeneous idempotent to arbitrarily high finite order while no algebra automorphism makes them homogeneous. They become conjugate after completion. The assessment credits the classical Picard method and prior quaternion and Laurent examples; the review records the proof and exact integer checks. This is one proposed paper, with outside correctness, priority and significance assessment still outstanding.
New unmixed height-three weighting theorem
P24: The Mendez–Pinto–Villarreal conjecture in unmixed height three, by Henry Zweiman, proposes a complete resolution of the named conjecture in the class with irreducible primary components all of height three. The seven-page PDF, complete Markdown, and LaTeX prove uniform primary-component exponents for arbitrary positive exponents and any number of variables.
The proof combines a classical dimension-one complete-intersection criterion with a five-variable support reduction and an explicit polyhedral obstruction. The assessment distinguishes the result from the prior height-two and generator-support-three cases; the review records the hand proof and exact checks. Mixed heights, higher heights, and squarefree Simis classification remain unresolved here. This is one internally checked proposed manuscript; independent correctness, priority, and significance remain provisional.
New effective joint Fourier decay theorem
P23: Effective joint Fourier decay at independent Pisot and Salem scales, by Henry Zweiman, gives a proposed effective bound for products at independent algebraic parameters. The five-page PDF, complete Markdown, and LaTeX establish the rate exp(-c log log u / log log log u), allow algebraic rescalings, and cover convolutions of homogeneous self-similar measures with algebraic translations.
The qualitative Pisot result is classical. This quantitative target was formulated in the research program; the assessment records that its exact prior open status has not been independently established. The proof review records the hand arguments, 5,120 exact projection identities, and interval checks. This is one proposed manuscript, not an addition to a certified count of solved significant open problems.
New critical-scale congruence theorem
P22: Critical-scale growth bounds for congruence-preserving functions over finite fields, by Henry Zweiman, proposes a growth-scale improvement in the Bell–Nguyen problem. Its seven-page PDF, complete Markdown, and LaTeX replace the sufficient bound q^n/(27qn) by a fixed positive multiple of q^n for arbitrary functions preserving every prime congruence on F_q[t].
The paper also gives the sharp threshold for F_q-linear maps and an explicit boundary example. These results are one manuscript. The full sharp arbitrary-function conjecture remains unresolved. Assessment records classical dependencies and the unread final-journal-text limitation; proof review records the hand arguments and exact finite checks. Priority and significance remain provisional.
New explicit Lie-polynomial image construction
P21: Explicit Lie polynomials with prescribed images over finite fields, by Henry Zweiman, gives a proposed complete answer in rank one to Kishnani–Singh Question 1.2. Its five-page PDF, complete Markdown, and LaTeX give a two-variable formula of degree at most 6q-5 for every conjugation-invariant zero-containing subset of sl_2(F_q), for every odd prime power q.
The paper proves exact fiber counts and a linear degree lower bound. The general existence classification is prior work; the contribution is explicit realization with quantitative control. Assessment and proof review record the comparison, generic identities, and 162,568 exact bracket evaluations. Higher rank remains outside scope, and priority and significance remain provisional.
New average unit-circle zero theorem
P20: The average number of unit-circle zeros in Stanley's reciprocal polynomial family, by Henry Zweiman, gives a proposed answer to Stanley's average-count question. Its seven-page PDF, complete Markdown, and LaTeX prove a limiting expected fraction of 1/sqrt(3) and convergence of the mean angular measure.
The proof uses an exact sign representation, a zero-law symmetrization, Nguyen–Vu universality, and deterministic angular discrepancy. The assessment and proof review document the existing inputs and 1,023 exact checks. The constant is classical; the contribution concerns this constrained integer family. The broader cyclotomic classification remains outside scope. Priority and significance are provisional.
New finite-field positivity obstruction
P19: Paley cliques and nonpolynomial counts of totally positive matrices, by Henry Zweiman, gives a proposed counterexample to Ayyer–Prasad's Conjecture 5.12 (2026). The six-page PDF, full Markdown, and LaTeX determine the two-by-three count over every finite field and show that no polynomial branch exists in any characteristic congruent to one modulo four, even along an arbitrary progression of extension degrees.
The proof uses the known Dawsey–McCarthy four-clique formula and identifies its surviving Gaussian terms. The assessment and audit document the contribution and 29 finite-field checks. This is one paper, with six entries proved minimal; the separate semidefinite conjecture remains unaddressed. Historical priority and significance remain provisional.
New algebraic extrapolation classification
P18: Integer-polynomial lifting and the classification of algebraic extrapolation sets, by Henry Zweiman, gives a proposed all-degree resolution of the linked extrapolation questions in Fenner–Green–Homer (2026). Its nine-page PDF, full Markdown, and LaTeX give an exact arithmetic membership criterion, the strong-PV discreteness classification, a model-set description, and finite generation of the internal cube.
The source assessment and proof audit distinguish the lifting argument from classical approximation and known special cases. Finite-generation statements exclude zero and one. These related results form one paper; historical priority and significance remain provisional.
New rigid nonassociative algebra
P17: A rigid nonassociative shift-associative algebra in dimension five, by Henry Zweiman, gives a proposed affirmative answer to the explicit rigidity question in Abdelwahab–Kaygorodov–Sartayev (2025). Its six-page PDF, full Markdown, and LaTeX give a nilpotent example in the smallest possible dimension and extend rigidity to all higher dimensions by adjoining scalar factors.
An explicit integer minor certifies the full tangent calculation; an independent symbolic implementation confirms it. The source assessment and proof audit record scope and limits. This is one consolidated paper, with no full five-dimensional classification claimed. Historical priority and significance remain provisional.
New restricted-sumset density theorem
P16: Complement transfer and density bounds for one-sided restricted sumsets, by Henry Zweiman, gives a proposed affirmative answer to the existence-of-an-absolute-linear-coefficient question preceding Ouyang's Conjecture 1.13. Its six-page PDF, full Markdown, and LaTeX include an explicit positive-density bound and a companion logarithmic bound.
The source assessment and internal proof audit state the exact contribution and dependencies. The full sharp conjecture remains unresolved; the explicit constant is extremely small. This is one paper, distinct from P10's integer counterexample problem. Priority and significance remain provisional.
New countable semiring partition theorem
P15: Countable semiring partitions of real function fields, by Henry Zweiman, gives a proposed complete affirmative answer to the countable-decomposition question in Kiss–Somlai–Terpai, Journal of Algebra 664 (2025), Section 7. Its five-page PDF, full Markdown, and LaTeX include a direct endpoint construction and an extension to all finitely generated transcendental subfields of the real numbers.
The source assessment and internal proof audit accompany exact rational-function checks. This is one consolidated paper. The finite-partition conjecture remains unresolved here. The originating agent performed the review; historical priority and scholarly significance are not independently certified.
New scalar recurrence classification
P14: Scalar recurrence sequences with finitely repairable power subsequences, by Henry Zweiman, is a new nine-page manuscript with complete Markdown, PDF, and LaTeX. It gives a finite scalar criterion, the exact coefficient-field exponent in the nondegenerate case, eventual periodicity of admissible exponents, and a terminating least-multiplier procedure for sampling powers greater than one.
Its source assessment, internal proof audit, and artifact manifest state the evidence and limits. The originating agent performed the audit; no separate-agent or human review is claimed. Historical novelty and journal-level significance remain provisional. It is one consolidated contribution, not several papers counted by theorem. The original collection manifest continues to describe the earlier thirteen drafts; P14 has its own manifest.
The original research target was 45 algebra/number-theory papers. The revised target is 20 qualifying papers across mathematics and applied mathematics. Historical archive size does not establish completion, and revisions do not increase the distinct-paper count.
Original thirteen drafts
| ID | Individual paper page | Claimed contribution | |
|---|---|---|---|
| P01 | Lefschetz properties of fully whiskered simplicial complexes | Complete characteristic-zero WLP/SLP classification | |
| P02 | Squarefree repair factors and optimal exponents for linear recurrences | Sharp factors, Lucas repair, optimal universal exponents | |
| P03 | Maximum-degree local permutation polynomials over the prime field | Every q>3 and every arity; exact recursion failures over nonprime odd fields | |
| P04 | Divisors and unique factorizations of Carlitz binomial polynomials | Complete divisor and mixed-product factorization classification | |
| P05 | Symbolic defect and normalization of whiskered cover ideals | Uniform comparison and exact normalization corrections | |
| P06 | Prime support and growth of Stirling repair factors | Conjecture resolutions, digit valuations, growth asymptotic | |
| P07 | Irreducible compositions in an exceptional unicritical family | Irreducible prefixes and explicit positive densities | |
| P08 | Componentwise polymatroidality is not preserved by homological shifts | Uniform counterexamples in arbitrarily many homological positions | |
| P09 | Unique addition in centerless metabelian Lie algebras | Full centerless metabelian class theorem over infinite fields | |
| P10 | Counterexamples to a conjecture on one-sided restricted sumsets | Infinite counterexamples and exact degree-three minimum | |
| P11 | Finite parameter tests for partition congruences and the surviving classes of an elongated partition conjecture | Seven families and exact surviving classes for the remaining pair | |
| P12 | Sharp orders of arithmetical critical groups via canonical lattice simplices | Sharp all-simple-graph bound and the star/complete-graph conjecture | |
| P13 | Nilperiod rings and additive periods of power maps | Full nilperiod conjecture, finite classification, corrected twisted-ring counts |
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