An Adèlic Height Obstruction to Coprime Solutions of the Generalized Fermat Equation: Process-Enriched Modularity, Cusp Dilations, and the Frictionless Barrier
Abstract
The Generalized Fermat Equation (Beal’s Conjecture) asserts that the Diophantine relation A^x + B^y = C^z with integer exponents x, y, z \ge 3 admits no positive integer solutions satisfying \gcd(A, B, C) = 1. In classical Diophantine geometry, resolution of this conjecture has been obstructed by three structural barriers: (i) the Higher-Rank Modularity Wall (g \ge 2), where deformation rings fail to be complete intersections; (ii) the Ineffective Height Barrier of Faltings' Theorem and the Darmon–Granville orbifold framework; and (iii) the Frictionless Continuum Fiction of first-order axiomatic set theory (\mathsf{ZFC}), which treats modular curves as ungrounded continua lacking thermodynamic and metric backpressure on cusp degenerations (\underline{\operatorname{Cost}} \equiv 0, \rho_{\text{info}} \equiv \infty). Consequently, the existence of coprime solutions resides on the Conjectural Horizon of \mathsf{ZFC} with an operational truth-value of Null.
To resolve the conjecture without category errors, we reconstruct the modular geometry of the attached Frey curve E_{A,B,C} within the enriched process category \mathbf{CostCat} over the Lawvere metric quantale \mathcal{V}_{\text{cost}} = ([0, \infty], \ge, +, 0)
Related
- The Frictionless Process Barrier: Meta-Mathematical Independence of Fundamental Conjectures in \mathsf{ZFC} and Their Generalized Operational Resolutions
- An Algebraic Proof of the Non-Modularity of Solutions to the Canonical Modular Differential Equation
- Wormholes in Vegh--dRGT massive gravity: a null-energy transparency theorem, horizon obstruction, and exact isotropic solutions
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BibTeX
@misc{tudor2026adèlic,
title = {{An Adèlic Height Obstruction to Coprime Solutions of the Generalized Fermat Equation: Process-Enriched Modularity, Cusp Dilations, and the Frictionless Barrier}},
author = {August Tudor},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23546758},
url = {https://zenodo.org/records/23546758},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - An Adèlic Height Obstruction to Coprime Solutions of the Generalized Fermat Equation: Process-Enriched Modularity, Cusp Dilations, and the Frictionless Barrier
AU - August Tudor
PY - 2026
DA - 2026/10/21
PB - Zenodo
DO - 10.5281/zenodo.23546758
UR - https://zenodo.org/records/23546758
AB - The Generalized Fermat Equation (Beal’s Conjecture) asserts that the Diophantine relation A^x + B^y = C^z with integer exponents x, y, z \ge 3 admits no positive integer solutions satisfying \gcd(A, B, C) = 1. In classical Diophantine geometry, resolution of this conjecture has been obstructed by three structural barriers: (i) the Higher-Rank Modularity Wall (g \ge 2), where deformation rings fail to be complete intersections; (ii) the Ineffective Height Barrier of Faltings' Theorem and the Darmon–Granville orbifold framework; and (iii) the Frictionless Continuum Fiction of first-order axiomatic set theory (\mathsf{ZFC}), which treats modular curves as ungrounded continua lacking thermodynamic and metric backpressure on cusp degenerations (\underline{\operatorname{Cost}} \equiv 0, \rho_{\text{info}} \equiv \infty). Consequently, the existence of coprime solutions resides on the Conjectural Horizon of \mathsf{ZFC} with an operational truth-value of Null. To resolve the conjecture without category errors, we reconstruct the modular geometry of the attached Frey curve E_{A,B,C} within the enriched process category \mathbf{CostCat} over the Lawvere metric quantale \mathcal{V}_{\text{cost}} = ([0, \infty], \ge, +, 0)
ER -