Explicit Weak abc Inequalities and the Discrete Obstruction to Full abc in the DIC Framework
Year: 2026
Type: preprint
Abstract: Independent audit and coefficient verification of C. Soulé's preprint "On the abc conjecture", arXiv:2611.05217 (2026). This manuscript neither proves nor disproves the abc conjecture; all No-Go statements are internal to the audited construction, and all statements that use Soulé's Theorem A are explicitly tagged as conditional on it. We axiomatise the DIC (Dimension Isolation Conjecture) framework for the abc conjecture and prove three stratified groups of results on Kodaira–Parshin arithmetic surface fibrations. (i) Unconditional geometric theorems. The adjunction identity Z·ω_{X/B} = −Z², the Riemann–Hurwitz intersection numbers, the invariants of the general (g0, r, b) construction, and a Deligne–Riemann–Roch decomposition of the arithmetic coefficient c = (2b² + 2b − 1)/12 + (D0 − T)/ω². (ii) A meta-theorem about the construction itself. Integrality of torsion lengths forces the coefficient of Theorem A onto the discrete set t = 12/N with N ≡ 1 (mod 3) — at the literature point (q, λ) = (5, 3); when q varies this is an arithmetic coincidence, not a structural barrier. An explicit local hyperelliptic model yields T = 8k and t = 12 as the unique transverse output; second-order torsion gives n1 = n2, hence Θ = T/k ∈ 2Z, leaving (27, 12) as the sole lattice candidate, while a transitive Galois action fixes Θ = 8 without transversality. A unified criterion E = 12(q − 1)/(qNγ) has the unique odd integral solution (N, Θ, q, γ) = (13, 12, 27, 8/9); in the original q = 5 construction no normalisation lands on the lattice. The height is the canonical-bundle height on a fixed genus-2 curve (γ = 2/5, confirmed by the constant 12/γ = 30 of the source); since Theorem A fixes both the genus and the Kummer degree, the lattice point (12, 27) requires genus 13 and lies outside the theorem's hypotheses — the last cell is empty. (iii) Controlled weak abc. All unconditional bounds for the abc conjecture currently known are of exponential type (log c ≪ N^{1/3}(log N)³, Stewart–Yu; see Pasten, arXiv:1705.09251, §1.1); the exponent produced below is of polynomial type, and its existence is precisely what Theorem A supplies. Taking Soulé's Theorem A as the sole external input, we derive the Kummer discriminant estimate η(q) = (q − 1)/q, unify the weak abc exponent as E = tη/γ, and obtain the explicit exponent E_max = 12(q − 1)/q · ⌈(2q − 2)/ℓ⌉ via a Riemann–Roch lower bound for γ. A discriminant stratification sharpens this to the pointwise bound c ≤ A_ε (N/M5)^{24+ε}; we prove that bootstrapping this discount to exponent 1 is impossible (inf θ = 0), that the discriminant-to-degree ratio is monotone in towers of number fields, that 4/5 is an Artin conductor identity rather than an estimate constant, and that the isometry step (32) of the source contributes only O(d′) — unconditionally derived here from a finite Čech complex — leaving every main coefficient (3/2 = m·c, 36, 12, 24, and E(c) = 12/(5 − 18c)) unchanged. A verbatim check of the source (Appendix J) confirms twelve items, including the statement and scope of Theorem A; the constant 12 is completely decomposed as (1/3)·m/(ρ − mc), and Route 1 is closed: archimedean and successive-minima refinements cannot move it. The unattainability of full abc inside the audited construction is established by six independent reasons, none of which depends on the truth of Theorem A. No Diophantine conjecture enters the derivations.
Author: Zeng, Chunping
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225771913
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