This paper studies concise symmetric cubic tensors of minimal border rank. It establishes a general polar-defect obstruction for tensors that are 111-abundant but not 111-sharp, and combines this obstruction with the low-dimensional geometry of cubic hypersurfaces with vanishing Hessian. As a result, every concise 111-abundant symmetric cubic in at most seven variables is proved to be 111-sharp. Consequently, ordinary tensor border rank and symmetric border rank coincide throughout the minimal-border-rank locus in these dimensions. In six variables, the paper gives a complete classification up to linear equivalence. The locus consists of twenty one-generic trace-cubic orbits arising from six-dimensional commutative Artin–Gorenstein algebras and two one-degenerate Perazzo orbits. The two Perazzo orbits are distinguished explicitly, their projective orbit dimensions are determined, and the lower-dimensional orbit is shown to be the unique concise codimension-one boundary orbit of the higher-dimensional one. Explicit symmetric degeneration families are also constructed. The accompanying computation package verifies the displayed trace cubics, the Perazzo reductions and degeneration identities, the Hessian calculations for reducible cubics, the centroid computations, and the projective stabilizer ranks. All finite calculations use exact arithmetic and include independent finite-field checks. Research methodology and AI assistance:This work was developed using the CARMA-Math research workflow, a cumulative AI-assisted mathematical research methodology using persistent research archives, literature and prior-art investigation, iterative proof exploration, and verification procedures. Generative AI (ChatGPT) was used extensively for mathematical exploration, proof development, computational reasoning, literature research, and manuscript preparation.
Akihiro Koide· Zenodo (CERN European Organi...· 0 citations
This paper studies concise symmetric cubic tensors of minimal border rank. It establishes a general polar-defect obstruction for tensors that are 111-abundant but not 111-sharp, and combines this obstruction with the low-dimensional geometry of cubic hypersurfaces with vanishing Hessian. As a result, every concise 111-abundant symmetric cubic in at most seven variables is proved to be 111-sharp. Consequently, ordinary tensor border rank and symmetric border rank coincide throughout the minimal-border-rank locus in these dimensions. In six variables, the paper gives a complete classification up to linear equivalence. The locus consists of twenty one-generic trace-cubic orbits arising from six-dimensional commutative Artin–Gorenstein algebras and two one-degenerate Perazzo orbits. The two Perazzo orbits are distinguished explicitly, their projective orbit dimensions are determined, and the lower-dimensional orbit is shown to be the unique concise codimension-one boundary orbit of the higher-dimensional one. Explicit symmetric degeneration families are also constructed. The accompanying computation package verifies the displayed trace cubics, the Perazzo reductions and degeneration identities, the Hessian calculations for reducible cubics, the centroid computations, and the projective stabilizer ranks. All finite calculations use exact arithmetic and include independent finite-field checks. Research methodology and AI assistance:This work was developed using the CARMA-Math research workflow, a cumulative AI-assisted mathematical research methodology using persistent research archives, literature and prior-art investigation, iterative proof exploration, and verification procedures. Generative AI (ChatGPT) was used extensively for mathematical exploration, proof development, computational reasoning, literature research, and manuscript preparation.
Akihiro Koide· Zenodo (CERN European Organi...· 0 citations
This work gives a complete dimension and defectivity classification for homoscedastic Gaussian moment varieties over an algebraically closed field of characteristic zero. It proves that these varieties have the expected dimension for every moment order at least four and combines this result with the known cubic classification to determine all defective cases. The paper also establishes a uniform nondefectivity theorem when the common covariance is restricted to a general positive-dimensional linear subspace, including the isotropic covariance model. In addition, it determines rational identifiability throughout the simplex range. The least rationally identifying moment order is two for one component, five for two components, and four for every model with at least three components in the simplex range. The proofs combine a degeneration of the common covariance tangent block, fat-point postulation, Waring decomposition methods, cumulant coordinates, and flat moment matrices. The three- and four-component cases are treated separately by explicit rational reconstruction and an exact reduced Gröbner-fiber argument with boundary exclusion. The accompanying computation archive provides executable exact-arithmetic verification code, fixed inputs, deterministic outputs, and a unified reproduction command. It includes independent checks of Jacobian ranks, the cubic defect classification, quartic recovery, restricted covariance models, the numerical conditions entering the fat-point argument, and the exact small-component fiber certificates. Research methodology and AI assistance:This work was developed using the CARMA-Math research workflow, a cumulative AI-assisted mathematical research methodology using persistent research archives, literature and prior-art investigation, iterative proof exploration, and verification procedures. Generative AI (ChatGPT) was used extensively for mathematical exploration, proof development, computational reasoning, literature research, and manuscript preparation.
Akihiro Koide· Zenodo (CERN European Organi...· 0 citations
This work gives a complete dimension and defectivity classification for homoscedastic Gaussian moment varieties over an algebraically closed field of characteristic zero. It proves that these varieties have the expected dimension for every moment order at least four and combines this result with the known cubic classification to determine all defective cases. The paper also establishes a uniform nondefectivity theorem when the common covariance is restricted to a general positive-dimensional linear subspace, including the isotropic covariance model. In addition, it determines rational identifiability throughout the simplex range. The least rationally identifying moment order is two for one component, five for two components, and four for every model with at least three components in the simplex range. The proofs combine a degeneration of the common covariance tangent block, fat-point postulation, Waring decomposition methods, cumulant coordinates, and flat moment matrices. The three- and four-component cases are treated separately by explicit rational reconstruction and an exact reduced Gröbner-fiber argument with boundary exclusion. The accompanying computation archive provides executable exact-arithmetic verification code, fixed inputs, deterministic outputs, and a unified reproduction command. It includes independent checks of Jacobian ranks, the cubic defect classification, quartic recovery, restricted covariance models, the numerical conditions entering the fat-point argument, and the exact small-component fiber certificates. Research methodology and AI assistance:This work was developed using the CARMA-Math research workflow, a cumulative AI-assisted mathematical research methodology using persistent research archives, literature and prior-art investigation, iterative proof exploration, and verification procedures. Generative AI (ChatGPT) was used extensively for mathematical exploration, proof development, computational reasoning, literature research, and manuscript preparation.
Akihiro Koide· Zenodo (CERN European Organi...· 0 citations
This revised preprint studies the derivative Laguerre quantities associated with the Jacobi theta kernel in the Fourier representation of the Riemann xi-function. It gives exact rational certificates showing that the ninth quantity is negative throughout a nontrivial interval around the symmetry point, with the certified range extended to absolute parameter value at most one fiftieth. It also verifies positivity at the symmetry point for levels one through eight and negativity at level nine. The proof uses explicit derivative polynomials, exact rational interval arithmetic, and elementary exponential bounds. A supplementary Python verifier reproduces the decisive sign computations using integer and rational arithmetic only. Ryan Kielhorn publicly deposited an exact level-nine counterexample at the symmetry point before the original Koide deposit. Brandon Yates later registered a Lean 4 formalization of the point counterexample. This revised version makes no priority claim for the point counterexample. Its distinct contribution is the certified interval of negativity, together with an exact and independently executable reproducibility certificate. Research methodology and AI assistance:This work was developed using the CARMA-Math research workflow, a cumulative AI-assisted mathematical research methodology using persistent research archives, literature and prior-art investigation, iterative proof exploration, and verification procedures. Generative AI (ChatGPT) was used extensively for mathematical exploration, proof development, computational reasoning, literature research, and manuscript preparation.
Akihiro Koide· Zenodo (CERN European Organi...· 0 citations
This revised preprint studies the derivative Laguerre quantities associated with the Jacobi theta kernel in the Fourier representation of the Riemann xi-function. It gives exact rational certificates showing that the ninth quantity is negative throughout a nontrivial interval around the symmetry point, with the certified range extended to absolute parameter value at most one fiftieth. It also verifies positivity at the symmetry point for levels one through eight and negativity at level nine. The proof uses explicit derivative polynomials, exact rational interval arithmetic, and elementary exponential bounds. A supplementary Python verifier reproduces the decisive sign computations using integer and rational arithmetic only. Ryan Kielhorn publicly deposited an exact level-nine counterexample at the symmetry point before the original Koide deposit. Brandon Yates later registered a Lean 4 formalization of the point counterexample. This revised version makes no priority claim for the point counterexample. Its distinct contribution is the certified interval of negativity, together with an exact and independently executable reproducibility certificate. Research methodology and AI assistance:This work was developed using the CARMA-Math research workflow, a cumulative AI-assisted mathematical research methodology using persistent research archives, literature and prior-art investigation, iterative proof exploration, and verification procedures. Generative AI (ChatGPT) was used extensively for mathematical exploration, proof development, computational reasoning, literature research, and manuscript preparation.
Akihiro Koide· Zenodo (CERN European Organi...· 0 citations
This article studies one-parameter degenerations of chains of nilpotent Jordan blocks joined along their socle vectors. It gives a complete Smith-normal-form description of the associated self-extension torsion for arbitrary chain length and all nonnegative edge valuations. The result includes an explicit path-matching formula, a sharp finite reduction in the block-size parameters, a classification of the Jordan types created by zero-valued couplings, and an equality between the number of positive Smith factors and the codimension of the corresponding nilpotent-orbit degeneration. The article also identifies the precise size gaps that cause failure of the full type-A interval profile, derives exact torsion-length deficit formulas, and packages the profile through Fitting ideals and transverse-slice dimensions. Exact verification scripts and machine-readable summaries accompany the paper. Research methodology and AI assistance:This work was developed using the CARMA-Math research workflow, a cumulative AI-assisted mathematical research methodology using persistent research archives, literature and prior-art investigation, iterative proof exploration, and verification procedures. Generative AI (ChatGPT) was used extensively for mathematical exploration, proof development, computational reasoning, literature research, and manuscript preparation.
Akihiro Koide· Zenodo (CERN European Organi...· 0 citations
This article studies one-parameter degenerations of chains of nilpotent Jordan blocks joined along their socle vectors. It gives a complete Smith-normal-form description of the associated self-extension torsion for arbitrary chain length and all nonnegative edge valuations. The result includes an explicit path-matching formula, a sharp finite reduction in the block-size parameters, a classification of the Jordan types created by zero-valued couplings, and an equality between the number of positive Smith factors and the codimension of the corresponding nilpotent-orbit degeneration. The article also identifies the precise size gaps that cause failure of the full type-A interval profile, derives exact torsion-length deficit formulas, and packages the profile through Fitting ideals and transverse-slice dimensions. Exact verification scripts and machine-readable summaries accompany the paper. Research methodology and AI assistance:This work was developed using the CARMA-Math research workflow, a cumulative AI-assisted mathematical research methodology using persistent research archives, literature and prior-art investigation, iterative proof exploration, and verification procedures. Generative AI (ChatGPT) was used extensively for mathematical exploration, proof development, computational reasoning, literature research, and manuscript preparation.
Akihiro Koide· Zenodo (CERN European Organi...· 0 citations
