What question did this study set out to answer?

The study aims to elucidate the symmetry and analytic structure of singularities near critical points on the Brody statistical manifold.

April 4, 2026Open Access

The Duality Web of the Brody Statistical Manifold

Key Points

The study aims to elucidate the symmetry and analytic structure of singularities near critical points on the Brody statistical manifold.
Introduced the Conjugation-Singularity Lemma to identify mechanisms of singularities.
Analyzed the connection profile's behavior through Taylor coefficients.
Utilized the alternating series remainder theorem to optimize grid evaluations.
Demonstrated factorial amplification through the even-derivative sequence of the connection.
Archived five reproducible proof scripts for validation and exploration.
Identified that critical regions on the Brody manifold are easier to analyze than previously thought.
Reduced required grid points from 83 to 26 in delicate regions through new analytic insights.
Discovered that the even-derivative sequence demonstrates log-convex behavior.
Established connections between information geometry and classical duality in spectral sequences.

Abstract

In computer-assisted proofs, the region near a critical point is usually the hardest — cancellations are tightest, grid points most needed. This paper shows that for the orbit-universal connection on the Brody statistical manifold, that region is actually the easiest, once you see the symmetry hiding inside the singularity structure. The Conjugation-Singularity Lemma identifies the mechanism: the spectral duality β ↦ 2/β acts as complex conjugation on the circle |β| = √2, forcing the dominant singularity of the connection profile onto the imaginary axis and locking its Taylor coefficients into strict sign alternation. The alternating series remainder theorem then covers the entire near-fixed-point zone with zero grid evaluation — replacing 83 grid points with 26, the most delicate region handled by analytic structure rather than sampling. Along the way, a factorial amplification mechanism is discovered: the even-derivative sequence of α* (β) at the self-dual fixed point is log-convex (a Turán inequality for all n ≥ 2), despite being built from log-concave polygamma functions. The amplification reverses log-concavity through the nonlinear Dₘ/H₊ composition chain, connecting information geometry to the classical duality between Pólya frequency and Stieltjes moment sequences. Companion to "Dual Symmetries of the Brody Statistical Manifold" (DOI: 10. 5281/zenodo. 19239285), which established the Z₂×Z₂ symmetry group and the global minimum theorem (Lemma U). '- This paper reveals the analytic engine behind that theorem. Five reproducible proof scripts (mpmath, interval arithmetic) are archived as supplementary material. Part of the ICP series on the information geometry of chaos: Paper Role DOI The Instability Compression Principle ICP empirical foundation: β → compression scaling across 30 chaotic systems 10. 5281/zenodo. 18099118 The Compressibility of Chaos (Ordo ab Chao) ICP theoretical derivation: scaling coefficient α from information geometry 10. 5281/zenodo. 18834609 Variance Excess ε (β) formula, one-point/two-point divide at βc = π 10. 5281/zenodo. 18650473 Information Geometry of the Brody Distribution Fisher metric, spectral duality theorem, effective dimension 10. 5281/zenodo. 18879754 The α-Connection Structure of the Brody Manifold Amari–Chentsov tensor, orbit-universal connection 10. 5281/zenodo. 19151206 Dual Symmetries of the Brody Statistical Manifold Z₂×Z₂ symmetry group, GOE=GUE orbit-equivalence 10. 5281/zenodo. 19239285 The Duality Web of the Brody Statistical Manifold Turán structure, Conjugation-Singularity Lemma, improved Lemma U proof this record

The Duality Web of the Brody Statistical Manifold

Key Points

Abstract

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