math.NT — Number Theory

Explicit Weak abc Inequalities and the Discrete Obstruction to Full abc in the DIC Framework

Contributed by Zeng Chunping

Independent audit and coefficient verification of C. Soulé's preprint "On the abc conjecture" (arXiv:2611.05217, 2026). This work neither proves nor disproves the abc conjecture; all No-Go statements are internal to the audited construction, and every statement that uses Soulé's Theorem A is explicitly tagged as conditional on it. We axiomatise the DIC framework and prove three stratified groups of results on Kodaira--Parshin arithmetic surface fibrations. (i) Unconditional geometric theorems: the adjunction identity $Z\cdot\omega_{\mathcal{X}/B}=-Z^{2}$, the Riemann--Hurwitz intersection numbers, the invariants of the general $(g_0,r,b)$ construction, and a Deligne--Riemann--Roch decomposition $c=\frac{2b^{2}+2b-1}{12}+(D_0-T)/\omega^{2}$ of the arithmetic coefficient. (ii) A meta-theorem: integrality of torsion lengths forces the coefficient of Theorem A onto the discrete set $t=12/N$, $N\equiv 1\pmod 3$ (at the literature point $(q,\lambda)=(5,3)$; for varying $q$ this is an arithmetic coincidence, not a structural barrier). A local hyperelliptic model gives $T=8k$, $t=12$ as the unique transverse output; second-order torsion gives $n_1=n_2$, hence $\Theta=T/k\in 2\mathbb{Z}$, leaving $(\Theta,q)=(12,27)$ as the sole lattice candidate. The unified criterion $E=12(q-1)/(qN\gamma)$ has the unique solution $(N,\Theta,q,\gamma)=(13,12,27,8/9)$; the height is the canonical-bundle height of a fixed genus-$2$ curve ($\gamma=2/5$, confirmed by $12/\gamma=30$ in the source), so $(12,27)$ requires genus $13$ and lies outside the theorem's hypotheses: the last cell is empty. (iii) Controlled weak abc: all unconditional bounds known for abc are of exponential type ($\log c\ll N^{1/3}(\log N)^{3}$; Stewart--Yu); the exponent below is of polynomial type, and its existence is what Theorem A supplies. We derive $\eta(q)=(q-1)/q$, unify the exponent as $E=t\eta/\gamma$, and obtain $E_{\max}=12\frac{q-1}{q}\bigl\lceil\frac{2q-2}{\ell}\bigr\rceil$. A discriminant stratification sharpens this to $c\le A_{\varepsilon}(N/M_5)^{24+\varepsilon}$; bootstrapping to exponent $1$ is impossible ($\inf\theta=0$); the discriminant-to-degree ratio is tower-monotone; $4/5$ is an Artin conductor identity; and the isometry step (32) of the source contributes only $O(d')$, derived unconditionally from a finite Čech complex, leaving $\frac32=m\cdot c$, $36$, $12$, $24$ and $E(c)=12/(5-18c)$ unchanged. A verbatim check of the source confirms twelve items, including the statement and scope of Theorem A; the constant $12$ decomposes as $\frac13\cdot\frac{m}{\rho-mc}$; and Route 1 is closed. Full abc is unattainable inside the audited construction for nine independent reasons, none depending on the truth of Theorem A: hypothesis exclusion, $\inf\theta=0$, tower monotonicity, bridge breaking, the $S$-smoothness loop, the double exclusion $T/k=47/3$, the $g_0=13$ collapse, index-ideal absorption with exact slope $2/(n-1)$, and the distributional vanishing of the discount (the multiplicity excess and $\theta$ have density $1-o(1)$, with explicit large-deviation rates).

Author comments: Audit of arXiv:2611.05217 (Soulé, 2026). Neither proves nor disclaims abc. Weak-abc statements are conditional on Soulé's Theorem A (tagged Th.A*); all nine No-Go statements are internal to the audited construction and independent of Theorem A's truth. This version incorporates nine audit rounds and three external review reports, including an independent re-derivation of the adelic metric layer (Appendix G), an unconditional O(d')-form of the source's step (32) from a finite Čech complex (Appendix I), a verbatim check of the source (Appendix J), a higher-genus recalibration closing the g0=13 route (Appendix K), a quantitative index-ideal absorption analysis (Appendix L), and a distributional analysis of the discount (Appendix M). Numerical cross-checks were performed in exact rational arithmetic (SymPy); the manuscript was drafted and typeset with AI assistance under the author's direction and line-by-line review.