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The research aims to establish a unified theory for organizing arithmetic complexities related to polynomial continued fractions based on their associated Sakai surface types.

June 17, 2026Open Access

The Sakai Stratification of PCF Transcendence

Key Points

The research aims to establish a unified theory for organizing arithmetic complexities related to polynomial continued fractions based on their associated Sakai surface types.
Developed a two-tier master conjecture relating connection coefficients to the balanced discriminant of degree-2 polynomial continued fractions.
Established structural findings across 30 families of constants in degree-2 cases, while conjecturing in degree-3 cases.
Integrated multiple mathematical frameworks, including Borel–Galois analysis and isomonodromy theory.
For degree-2 PCFs, identified that the sign of the balanced discriminant determines whether the connection coefficient is elementary or a genuine transcendental period.
Stronger conjectures for degree-3 and higher cases are proposed but pending analysis, suggesting an extension of the established structural results.
Outlined three significant open problems regarding explicit representations and functoriality related to the connection coefficient.

Abstract

We propose and partially establish a unified theory — the Sakai stratification of PCF transcendence — organizing the arithmetic complexity of constants arising from polynomial continued fractions (PCFs) according to the Sakai surface type of their isomonodromic deformation. The central object is the connection coefficient C of the Borel-transformed generating function G (s) = Σ Qₙ sⁿ/ (dn) !, a transcendence candidate that the companion Edge–Borel Radius (EBR) series identifies as the dominant singularity amplitude. Our main claim is a two-tier master conjecture: for degree-2 PCFs, the sign of the balanced discriminant Δ = β₁² − 4β₂β₀ determines whether C is elementary or a genuine transcendental period of the governing Sakai connection, a dichotomy now established at the STRUCTURAL level across 30 families; for degree d ≥ 3, the same surface-theoretic prediction holds at the CONJECTURED level pending the higher-degree connection analysis. The theory unifies a corpus of approximately 30 deposits spanning Lean 4 / Mathlib formalization, Borel–Galois analysis, isomonodromy theory, and arithmetic computation into a single conceptual framework. Three principal open problems — an explicit period-integral representation of C, the d ≥ 3 connection problem, and a functoriality theorem for the bridge map — define the frontier.

The Sakai Stratification of PCF Transcendence

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