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