Claimed results and mathematical scope
All entries describe claims in AI-generated drafts. They have not been independently verified by human experts. Consult the manuscripts for full hypotheses and references.
| Paper | Main advance | Material boundary or prior input |
|---|---|---|
| P01 | Complete characteristic-zero WLP/SLP classification for fully whiskered simplicial complexes, resolving the graph conjecture | The universal kernel and positive proofs supplement an explicitly credited known cokernel; positive classification is characteristic zero |
| P02 | Exact squarefree recurrence repair factors, exact Lucas repair, and optimal universal sampling exponents | Concerns Dold divisibility; sign conditions for dynamical realizability are separate |
| P03 | Prime-field-coefficient maximum-degree local permutation polynomials for every q>3 and every arity; exact recursive coefficient failures | Recursion classification concerns nonprime odd fields; characteristic-five failures are n=6,7 modulo 10; known patching and seed results credited |
| P04 | Every divisor of a product of Carlitz binomial polynomials is a subproduct up to a unit | Works in Int(Fq[t]); includes arbitrary mixed indices and powers; known individual irreducibility is credited |
| P05 | Uniform symbolic/integral defect comparison, asymptotic equivalence, and exact normalization corrections for whiskered cover ideals | Polyhedral and semigroup predecessors credited; multi-cycle formulas are consolidated here |
| P06 | Stirling repair prime support, digit valuation witnesses, and log(Fk/rad((k−1)!)) asymptotic to sqrt(k) | Resolves three full source conjectures and a valuation subcase; other cutoff and valuation subcases remain open |
| P07 | An irreducible prefix in an exceptional even-degree unicritical family remains irreducible after every allowed tail | Explicit restrictions on a, d, and tail prime support; positive-density statements use specified alphabets |
| P08 | Componentwise polymatroidal ideals whose intermediate homological shifts fail exchange in every large component degree | Negative answer to a published question and counterexample to a specified preprint theorem; equigenerated results are unaffected |
| P09 | Every finite-dimensional centerless metabelian Lie algebra over an infinite field has unique addition as a Lie ring | General faithful-action criterion is broader; older triangular proofs recover the associative-unital special case and are credited |
| P10 | Infinite counterexamples to the one-sided restricted-sumset conjecture; necessary loss coefficient 8/3; exact degree-three minimum | Optimal universal coefficient remains between 8/3 and 3; no two-sided forbidden-degree assertion |
| P11 | Seven conjectured elongated partition families and exact surviving parameter classes for the remaining pair | Patel's earlier obstruction/necessity is credited; 35 rigorously reduced finite certificates are proof dependencies |
| P12 | Sharp maximum critical-group order across all connected simple graphs; star/complete-graph conjecture and equality cases | AKLN's geometric multiplicity bound is the deep external input; simple graphs only, and stars indexed by leaves |
| P13 | Full nilperiod-implies-NI conjecture, finite unital classification, and exact Corbas power-period counts | Arbitrary associative rings for the main theorem; additive-closure precedent credited; no claim about the broader weakly-periodic-ring problem |
P14: scalar power-subsequence classification
P14 gives a finite iff criterion for scalar recurrence power subsequences, including degeneracy and transients; an exact coefficient-field exponent in the nondegenerate case; an ultimately periodic exponent spectrum; and a terminating least-multiplier computation for s>1. It concerns Dold divisibility, not positivity or journal certification. The assessment records the original Minton PDF access limitation and provisional priority.
P40: low-temperature logarithmic Sobolev boundary selection
The claimed theorem gives the sharp first-order coefficient for convex quadratic confinement around smooth bounded convex domains in every finite dimension, incorporating nonlinear extremizers and spectral limits. The unbounded exterior entropy estimate and its vanishing-energy uniformity are the principal proposed advances. Convexity, nondegenerate quadratic growth and a Laplacian bound remain hypotheses. The known spectral formula and interval asymptotic are credited; other boundary exponents and general extremizer classification remain open. See the full assessment.
P41: uniform reverse Robin Kohler-Jobin inequality
The claimed theorem answers the planar case of Buttazzo--Cito--Solombrino Q2 for all bounded open Lipschitz sets, with a common exponent for every positive Robin parameter and area. The Robin p-Laplacian extension is restricted to 1<p<=n/(n-1), where a prior pointwise Talenti theorem applies. The torsion-composition boundary correction, linear spectral control and uniform exponent are the proposed advances. Faber--Krahn, its rigidity and quantitative Saint-Venant are credited inputs. Higher-dimensional linear Robin Q2, the least exponent and forward Q1 remain unresolved here. See the assessment.
P42: Euclidean persistence for anisotropic eigenvalue-torsion products
The claimed theorem identifies the Euclidean norm as the unique optimizer on every fixed ball, for sufficiently small positive product exponents in the maximum problem and sufficiently large ones in the minimum problem. It covers all normalized symmetric seminorms, every n>=2 and p>1. The two-sided angular-deficit estimate and optimal uniform-distance power (n+1)/2 belong to this single contribution. The known quadratic-seminorm ball theorem, Euclidean eigenvalue endpoint and convex symmetrization are credited. General domains, asymmetric anisotropies, optimal q thresholds and endpoint-uniform p estimates remain unresolved here. See the assessment.
P43: genus obstruction for commuting boundary operators
The claimed theorem excludes every positive genus for compact connected oriented smooth surfaces with nonempty boundary and exactly commuting Dirichlet-to-Neumann map and boundary Laplacian. It addresses both cases of Speciel's final Open Problem 1.6; the full classification corollary uses his existing genus-zero theorem. Finite-period Hilbert-transform theory and smooth conformal sewing are credited. The new argument combines Fourier-invariant holomorphic traces with degree one and a capping Schur complement. Quantitative stability, nonorientable surfaces and rough boundaries are outside the result. See the assessment.
P44: convex Robin domains with vanishing spectral gaps
Revision 4.1 adds a complete elementary perimeter-scaled first-eigenvalue comparison in the plane. Area-one thin triangles have positive-parameter limit alpha/2, beating the disk for 0 < alpha <= 4 pi, and give unboundedness below for every negative alpha. The alpha=1 triangle witness and disk lower bound are exact. This disproves the precise preprint Conjecture D in arXiv:1905.07658v1; final journal wording was inaccessible and a citation discrepancy is disclosed. Sections 9--11 retain the fundamental-gap and separate parameter A, B and C counterexamples in dimensions at least four. No sharp positive-parameter shape optimization theorem or low-dimensional gap result is claimed. This is one internally audited AI-assisted preprint, not independent certification. See the assessment.
P45: thin reactive-patch spectral measures and uniform response
The claimed strong limit on fixed embedded C3 curves identifies transverse averaging times local width. Every fixed eigenvalue tends to the normalized maximum width; every fixed weight vanishes if the maximum set has zero length. The full input spectral measure and a separate capacity bound give relative response asymptotics uniformly over all positive reactivities. The same paper proves the regular-center-curve versions of Grebenkov-Maurette (B18)-(B19), the one-pole accuracy criterion, and existence of a connected smooth F1>F0 patch. The integral representation, variational principles and approximation are credited prior work. Sharp spectral corrections, positive-length-maximum weight limits, and singular center curves remain outside the theorem. See the assessment.
P46: ground-state uniqueness on convex domains and logarithmic concavity
Revision 1.2 proves uniqueness of the positive Dirichlet logarithmic Schrodinger ground state on every bounded convex domain in every dimension, with no boundary smoothness requirement. Every positive log-concave solution is the ground state and satisfies D2(-log U)>I. A local strict logarithmic Sobolev remainder gives a sharp energy comparison on all of H1_0 and a complete equality case, resolving the ground-state questions in Gallo-Mosconi-Squassina, Remark 1.6 (2026). Their existence theorem, Caffarelli contraction and classical Gaussian inequalities are essential credited inputs. All-positive-solution uniqueness remains open.
The same paper retains sharp normalized square-root-log and positive power concavity on balls, plus instability under smooth uniformly convex deformations near any ball in dimensions at least two. The counterexamples are ground states by the new theorem. The weaker small-rescaling property and the optimal ball power exponent's radius dependence remain unresolved. This is a revision, not another distinct paper; see the assessment.
P47: sharp total variation from spherical rearrangement
P47 proves V(n,2)=2C(n,2) in every order, the complete measure endpoint, and strict midpoint height maximization with unique symmetric height extremizers at every finite exponent. Revision 1.11 proved V(3,p)=2C(3,p) and the complete variation equality classification for every finite p>1. Revision 1.12 completes the uniform endpoint: every nonzero variation extremizer is a scalar multiple of the explicit one-peak function whose third derivative alternates +1,-1,+1,-1, and V(3,infinity)=(2-sqrt(2))/48. A duality argument starts with an arbitrary maximizer; a conserved residual quantity proves finite regularity without an initial finite-switch assumption. The positive action extends to q=1 with a recalculated moving-root derivative. It excludes all higher nodal counts, and exact moment equations classify the final profile. Together with the measure theorem, the order-three case is complete at every exponent. Revision 1.13 proves saturation of the highest-derivative bound for every uniform-endpoint variation maximizer in every order n>=2. It also proves endpoint residual normalization, a strict mean-parameter bound, null residual and first-derivative zero sets, and fixed endpoint controls. A separate accumulation-point argument proves finite switching, and a spline zero count proves every critical point and residual zero simple, with exactly n+r-1 residual zeros. Compactness at infinity and the prior exponent results give an implicit count bound uniform over all 1<p<=infinity for each n>=3. Exact local order-four solutions show why the extra mixed derivative terms prevent extending the order-three residual bound from the conserved quantity alone; the examples are not clamped maximizers or conjecture counterexamples. 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. Earlier finite-switch, two-switch, affine-response, nodal-length and exact weighted-obstruction results remain in the manuscript. For n>=4, nonsymmetric three-switch and higher-count configurations remain unresolved outside the established exponent intervals; the uniform endpoint also remains open there. The all-order conjecture is not claimed solved. These results form one internally audited, unreviewed paper; absolute priority and independent expert certification are not claimed. See the assessment.
P48: local minimality for the generalized Neumann Hénon problem
The claimed theorem upgrades directional second-variation positivity to strict W1p local minimality modulo scaling below the known limiting Steklov threshold, for large radial-weight exponent. A common weighted space has compact boundary trace; a nonlinear convexity estimate retains concentrated perturbation energy. Above the threshold, the angular test gives instability. The threshold formula, radial convergence and p=2 local-minimum theorem are credited prior work. Revision 1.1 proves a uniform nonlinear neighborhood through the critical trace exponent and matching power bounds on its decay above that exponent, under unit energy normalization and within the positive Steklov range. Equality at the Steklov threshold and the leading radius constant remain unresolved. See the assessment.
P49: sharp annular stability for physical torsion
The claimed theorem gives exponent-two stability for planar physical torsion on a bounded Lipschitz outer domain with finitely many separated Lipschitz holes. The joint-center constant depends only on the ratio of the outer and hole areas; individual asymmetries have a universal constant when B<=2A/3. A second-harmonic perturbation proves optimality. The global calibration theorem of Hensel-Laux and the velocity-deficit strategy of Guerra-Machado-Ramos are credited. The new argument accounts for plateau ranks and propagates a fixed center across an area window. Best constants, endpoint area-ratio limits, nonlinear equations and higher dimensions remain outside the result. See the assessment.
P50: all-times metric-graph heat content, killing measures and rigidity
The claimed theorem answers the all-times question in Bifulco-Täufer's final Remark 2.9 for finite compact connected metric graphs with a nonempty Dirichlet set and Kirchhoff conditions elsewhere. There is no arithmetic restriction on edge lengths. A stronger theorem compares cumulative decreasing rearrangements of nonnegative L2 heat evolutions; equality of constant-load heat content at one time characterizes the Dirichlet-Neumann interval. Revision 1.1 extends concentration comparison to finite nonnegative killing measures of fixed total mass on graphs of fixed length, with the endpoint-killed Robin interval as unique maximizer at every time. The atomic torsion consequence is prior work of Özcan-Täufer and is credited. The proof includes form convergence for singular measures and equality at one time. Classical graph energy rearrangement and elliptic-to-parabolic concentration iteration are credited inputs. Revision 1.3 proves the sharp Kohler-Jobin product and fixed-torsion eigenvalue comparisons for arbitrary finite nonnegative killing measures, with equality only for the endpoint-killed interval. A composition test in the weak ground-state equation absorbs the potential energy, and a scalar distribution comparison yields the sharp bound. This settles both general-measure questions left open in revision 1.2. Revision 1.4 proves the analogous sharp stationary comparisons for the p-Laplacian for every 1 < p < infinity, including arbitrary finite nonnegative killing measures, exact Robin-strength constants and endpoint-interval equality. The nonlinear Dirichlet theorem of Özcan and its modified-torsion method are credited. Weighted graphs, signed killing, arbitrary couplings, quantitative stability, nonlinear heat concentration and heat-trace bounds remain outside the claims. See the assessment for the current source checks, failed search requests and journal-significance limits.
P51: uniform exterior magnetic Steklov transition
The claimed results give a uniform L2 bounded-difference operator expansion for a fixed smooth planar obstacle as nonnegative constant field and scalar mass vanish. The effective operator handles signed flux through an integer. The leading ground-state crossover, third inverse-logarithmic signed asymmetry and second-order geometric gap are derived from it. For every fixed smooth simply connected nondisk, a positive quantitative gap from the equal-perimeter disk holds throughout a sufficiently small field, mass and flux neighborhood, with no symmetry or convexity assumption. The thresholds depend on the obstacle. Scalar capacity asymptotics, conformal covariance and classical circle formulas are prior work. Arbitrary variable fields, higher dimensions, independent multi-pole fluxes, a geometric-class-uniform threshold and the full field range remain unresolved. See the assessment for the precise source-access and admission limits.
P52: complete fixed-center planar Robin splitting
For a bounded connected planar Lipschitz domain, a prescribed interior point and a fixed nonzero real Robin parameter on both boundaries, the claimed theorem constructs one area-preserving analytic strictly convex hole shape near a disk that completely splits every fixed limiting eigenvalue cluster for sufficiently small hole size. Every positive nodal order is identified through an exterior energy matrix and its successive Schur complements. The nonnodal direction has a separate signed first-order scale. Quantitative correctors and second coefficients belong to the same paper. The size threshold depends on the spectral window; no common positive threshold for the entire infinite spectrum, mixed nonnodal Neumann theorem or parameter-uniform statement is proved. Corrected Dirichlet matrix theory, generic simplicity and shape differentiation are credited prior work. See the assessment for source-access and internal-admission limits.
P53: eigenvalue–torsion endpoints and local classification
P53 gives proposed endpoint saddles and a full local classification of critical domains for both normalized spectral products in every dimension. Classical Hessians, elliptic theory, symmetry reduction and Bessel transcendence are prior inputs. The original global convex perimeter inequality and a universal balanced-branch sign remain unresolved. The paper is internally assessed, with older full-text comparison limits disclosed; no independent certification is claimed.
P54: spherical Steklov comparison near maximal perimeter
P54 claims a uniform large-perimeter comparison for smooth proper convex domains in every dimension at least three. It does not solve the full spherical fixed-perimeter conjecture. The quantitative endpoint result, weighted center and treatment of singular maximal-perimeter limits form one connected contribution. See the paper's ASSESSMENT.md and source-audit.json for the provisional significance judgment and retained final-text access limits.