A Cobordism–Branch–Valence Framework for the ABC Conjecture

This manuscript develops a cobordism–inspired branch–valence framework for the ABC correspondence, reframing primitive triples (a, b, c) (a, b, c) (a, b, c) with a+b=ca+b=ca+b=c through geometric, analytic, and probabilistic structures. The construction encodes coarse prime data using a height-type valence functional and studies its behavior under residue-filtered geometric flows. A three-part Spiro inequality is introduced on local patches, and a residue-filtered gluing mechanism promotes patchwise control to a global normalized potential F~FF. The real part of F~FF governs the monotonicity and central envelope Φ0₀Φ0, yielding Szpiro-type inequalities along cobordism-smoothing flows. A complementary reformulation reparameterizes triples as probability measures μa, b, c₀, ₁, ₂μa, b, c on the primes, placing ABC arithmetic inside a compact weak-∗*∗ simplex. This compactness viewpoint suggests the possibility of “ABC flows’’ on measure spaces whose expected evolution of height-type functionals exhibits ABC-like behavior. A martingale-based soft-switching heuristic is proposed to reconcile gradient dynamics with stochastic fluctuations in prime support. The paper does not claim a proof of the ABC conjecture. Instead, it proposes a unifying geometric mechanism that narrows exceptional loci and clarifies how cobordism, height theory, residue filtration, and measure-theoretic compactness may interact in future approaches to the conjecture. The author gratefully acknowledges the conceptual, analytic, and expository assistance of the OpenAI large language model ChatGPT.