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

A mix of human-written and AI-generated textHuman understanding: all partsmath.NT — Number Theorymath.AG — Algebraic Geometry

Contributed by Zeng Chunping ↗

Version 1 / Nov 12, 2026 / CC BY 4.0

Abstract

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⋅ωX/B=−Z2Z\cdot\omega_{\mathcal{X}/B}=-Z^{2}, the Riemann--Hurwitz intersection numbers, the invariants of the general (g0,r,b)(g_0,r,b) construction, and a Deligne--Riemann--Roch decomposition c=2b2+2b−112+(D0−T)/ω2c=\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/Nt=12/N, N≡1(mod3)N\equiv 1\pmod 3 (at the literature point (q,λ)=(5,3)(q,\lambda)=(5,3); for varying qq this is an arithmetic coincidence, not a structural barrier). A local hyperelliptic model gives T=8kT=8k, t=12t=12 as the unique transverse output; second-order torsion gives n1=n2n_1=n_2, hence Θ=T/k∈2Z\Theta=T/k\in 2\mathbb{Z}, leaving (Θ,q)=(12,27)(\Theta,q)=(12,27) as the sole lattice candidate. The unified criterion E=12(q−1)/(qNγ)E=12(q-1)/(qN\gamma) has the unique solution (N,Θ,q,γ)=(13,12,27,8/9)(N,\Theta,q,\gamma)=(13,12,27,8/9); the height is the canonical-bundle height of a fixed genus-22 curve (γ=2/5\gamma=2/5, confirmed by 12/γ=3012/\gamma=30 in the source), so (12,27)(12,27) requires genus 1313 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≪N1/3(log⁡N)3\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 η(q)=(q−1)/q\eta(q)=(q-1)/q, unify the exponent as E=tη/γE=t\eta/\gamma, and obtain Emax⁡=12q−1q⌈2q−2ℓ⌉E_{\max}=12\frac{q-1}{q}\bigl\lceil\frac{2q-2}{\ell}\bigr\rceil. A discriminant stratification sharpens this to c≤Aε(N/M5)24+εc\le A_{\varepsilon}(N/M_5)^{24+\varepsilon}; bootstrapping to exponent 11 is impossible (inf⁡θ=0\inf\theta=0); the discriminant-to-degree ratio is tower-monotone; 4/54/5 is an Artin conductor identity; and the isometry step (32) of the source contributes only O(d′)O(d'), derived unconditionally from a finite Čech complex, leaving 32=m⋅c\frac32=m\cdot c, 3636, 1212, 2424 and E(c)=12/(5−18c)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 1212 decomposes as 13⋅mρ−mc\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⁡θ=0\inf\theta=0, tower monotonicity, bridge breaking, the SS-smoothness loop, the double exclusion T/k=47/3T/k=47/3, the g0=13g_0=13 collapse, index-ideal absorption with exact slope 2/(n−1)2/(n-1), and the distributional vanishing of the discount (the multiplicity excess and θ\theta have density 1−o(1)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.

Tools used

SageMath
SageMath

References

  1. C. Soulé, On the abcabc conjecture , arXiv:2611.05217v1 (3 November 2026). Introduction (Theorems A and B, hh the canonical-bundle height of a genus-22 curve), §7.2 (the construction y2+y=x5y^{2}+y=x^{5}, F=K(m5)F=K(\sqrt[5]{m}), the constant 3030), §1.7, §3.5, §5 (Prop. 8), §6 (Prop. 9, (18)--(34), Theorem 1), §7 (Theorems A, B); §6.3 (28)--(35), §7.1, Prop. 2, Cor. 1; §1.6 (Prop. 1 proved in the source), §2.1.2 (admissible-metric existence), §6.3.5 (whole-Spec⁡\operatorname{Spec} model), §7.3.2 (constants 3030 and (51)).arXiv
  2. C. Soulé, On the Arakelov theory of arithmetic surfaces , IHÉS, 2017, Lecture 3 (the 55-fold root cover, 54⋅45=1\tfrac54\cdot\tfrac45=1).
  3. C. Soulé, Secant varieties and successive minima , J. Algebraic Geom.\ 13 (2) (2004), 323--341; arXiv:math/0110254. MR 2047701.arXiv
  4. C. Soulé, Very stable extensions on arithmetic surfaces , arXiv:1009.6191.arXiv
  5. C. Soulé, Semi-stable extensions on arithmetic surfaces , arXiv:math/0505079.arXiv
  6. L. Moret-Bailly, Hauteurs et classes de Chern sur les surfaces arithmétiques , Astérisque 183 (1990), 37--58.
  7. P. Deligne, Le déterminant de la cohomologie , in: Current Trends in Arithmetical Algebraic Geometry, Contemp.\ Math. 67, Amer.\ Math.\ Soc., Providence, RI, 1987, 93--177. MR 0902592.
  8. G. Faltings, Calculus on arithmetic surfaces , Ann.\ of Math.\ (2) 119 (1984), no. 2, 387--424. MR 0740897.
  9. H. Gillet, C. Soulé, An arithmetic Riemann--Roch theorem , Invent.\ Math.\ 110 (1992), no. 3, 473--543. MR 1189489.
  10. P. Vojta, Diophantine Approximations and Value Distribution Theory , Lecture Notes in Math.\ 1239 , Springer-Verlag, Berlin, 1987.
Show all 43 referencesShow fewer references
  1. D. W. Masser, J. Oesterlé, the abcabc conjecture (1985/1988); cf.\ SouleABC .
  2. L. Szpiro, discrinant--conductor inequalities for elliptic curves, Astérisque 183 (1990); cf.\ SouleABC .
  3. G. Frey, Links between stable elliptic curves and certain Diophantine equations , Ann.\ Univ.\ Sarav.\ 1 (1986), 1--40.
  4. A. N. Parshin, fibrations of algebraic curves and the Mordell conjecture; cf.\ SouleABC .
  5. S. Yu. Arakelov, Intersection theory of divisors on an arithmetic surface , Math.\ USSR Izv.\ 8 (1974), 1179--1192; English transl.\ ibid., 1167--1180. MR 0472815.
  6. R. C. Mason, Diophantine Equations over Function Fields , London Math.\ Soc.\ Lecture Note Ser.\ 96 , Cambridge Univ.\ Press, 1984; W. D. Stothers (1981).
  7. A. Buium, arithmetic differential operators and ΩZ1=0\Omega^{1}_{\mathbb Z}=0; cf.\ SouleABC .
  8. R. Greenberg, G. Stevens, pp-adic LL-functions and pp-adic periods of modular forms , Invent.\ Math.\ 111 (1993), no. 2, 407--447. MR 1198816.
  9. D. Eriksson, G. Freixas i Montplet, arithmetic Deligne--Riemann--Roch, arXiv:2509.05077; D. V. Osipov, Izv.\ Math.\ 88 (5), 2024.arXiv
  10. H. Pasten, Shimura curves and the abcabc conjecture , J. Number Theory 254 (2024), 214--335; arXiv:1705.09251 (2017).arXiv
  11. H. Pasten, Improved bounds for Szpiro's conjecture , arXiv:2609.17390 (2026).arXiv
  12. R. Laniewski, K. Müller, The quality plane of abcabc-triples and the decomposition into gains , arXiv:2610.03212 (2026).arXiv
  13. F. Ballay, Successive minima and asymptotic slopes in Arakelov geometry , Compositio Math.\ 157 (6) (2021), 1302--1339; arXiv:2002.06026 (2020).arXiv
  14. S.-W. Zhang, Admissible pairing on a curve , Invent.\ Math.\ 112 (1993), no. 1, 171--193. MR 1207481.
  15. E. Bombieri, J. Vaaler, On Siegel's lemma , Invent.\ Math.\ 73 (1983), 11--32; addendum ibid.\ 75 (1984), 377. MR 0707346.
  16. T. Borek, Successive minima and slopes of Hermitian vector bundles over number fields , J. Number Theory 113 (2) (2005), 380--388.
  17. S.-W. Zhang, Small points and adelic metrics , J. Algebraic Geom.\ 4 (2) (1995), 281--300. MR 1311351.
  18. X. Yuan, S.-W. Zhang, Adelic line bundles over quasi-projective varieties , Annals of Mathematics Studies, Princeton Univ.\ Press, 2026; arXiv:2105.13587.arXiv
  19. A. Moriwaki, adelic sequences and adelic structures over finitely generated fields, Kyoto Univ.\ preprint 1999-19.
  20. É. Gaudron, Géométrie des nombres adélique et lemmes de Siegel généralisés , Manuscripta Math.\ 130 (2) (2009), 159--182.
  21. H. Gillet, D. Rössler, C. Soulé, An arithmetic Riemann--Roch theorem in higher degrees , Ann.\ Inst.\ Fourier (Grenoble) 58 (6) (2008), 2169--2189. MR 2473633.
  22. G. Faltings, Lectures on the arithmetic Riemann--Roch theorem , Annals of Mathematics Studies 127 , Princeton Univ.\ Press, 1992 (notes by S.-W. Zhang). MR 1158661.
  23. C. L. Stewart, K. Yu, On the abc conjecture , Math. Ann. 321 (2001), no. 3, 705--726.
  24. C. L. Stewart, K. Yu, On the abc conjecture, II , Duke Math. J. 108 (2001), no. 1, 169--181; see also J. Théor. Nombres Bordeaux 14 (2002), 353--359. DOI 10.1215/S0012-7094-01-10815-6.DOI
  25. C. L. Stewart, R. Tijdeman, On the Oesterlé--Masser conjecture , Monatsh. Math. 102 (1986), no. 3, 251--257.
  26. L. Moret-Bailly, Hauteurs et classes de Chern sur les surfaces arithmétiques , Séminaire sur les Pinceaux de Courbes Elliptiques (Paris, 1988), Astérisque No. 183 (1990), 37--58. (= reference [16] of [1]; replaces the imprecise ``cf. ref. 16 of [1]'' in the existing item [6].)
  27. E. Bombieri, W. Gubler, Heights in Diophantine Geometry , Cambridge Tracts in Mathematics 124, Cambridge Univ. Press, 2006.
  28. J. Neukirch, Algebraic Number Theory , Springer-Verlag, 1999; Chap. III, §2, Theorem 2.11 (transitivity of the discriminant).
  29. S.-W. Zhang, Positive line bundles on arithmetic varieties , J. Amer. Math. Soc. 8 (1995), no. 1, 187--221. (Distinct from the existing item [27], Small points and adelic metrics .)
  30. M. van Frankenhuysen, The ABC conjecture implies Vojta's height inequality for curves , J. Number Theory 95 (2002), no. 2, 289--302. (Reverse implication: assumes abcabc; cited only for orientation.)
  31. X. Yuan, Arithmetic bigness and a uniform Bogomolov-type result , Annals of Mathematics 203 (2026), 15--119. (= [25] of [1]; Prop. 2.6 gives an admissible metric on any line bundle, §2.1.2 of [1].)
  32. R. Hartshorne, Algebraic Geometry , Graduate Texts in Mathematics 52, Springer-Verlag, 1983 (corr. 3rd printing). (= [9] of [1]; §6.3.6 cites III, Remark 9.3.1 and III, Cor. 12.9.)
  33. J.-P. Serre, Quelques applications du théorème de densité de Chebotarev , Inst. Hautes Études Sci. Publ. Math. 54 (1981), 323--401. (= [23] of [1]; used for the relative discriminant bound (51).)

Version history

  1. v1 · 75 pagesSubmitted by Zeng ChunpingAudit 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.CurrentNov 11, 2026