Three Axioms for the Motive: Constructive Reverse Mathematics Meets Grothendieck's Universal Cohomology (Paper 50, Lean 4 Formalization)
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2026
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| _version_ | 1866901655526572032 |
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| author | Lee, Paul Chun-Kit |
| author_facet | Lee, Paul Chun-Kit |
| contents | <p>Five major conjectures in arithmetic geometry — Weight-Monodromy, Tate, Fontaine-Mazur, Birch–Swinnerton-Dyer, and Hodge — all encode the same logical pattern: continuous data over a complete field (where zero-testing requires the Limited Principle of Omniscience) secretly descends to discrete algebraic data (decidable in Bishop-style constructive mathematics). We call this pattern "de-omniscientizing descent."</p> <p>Calibrating all five conjectures against the constructive hierarchy BISH ⊂ BISH+MP ⊂ BISH+LPO ⊂ CLASS reveals that Grothendieck's conjectural category of numerical motives is logically specified by exactly three axioms: (1) decidable equality on morphism spaces — equivalently, Standard Conjecture D; (2) algebraic spectrum — endomorphisms satisfy monic Z-polynomials; (3) Archimedean polarization — a positive-definite inner product on the real fiber, which exists over R but is obstructed over Q_p by the u-invariant u(Q_p) = 4. The motive category is the initial Decidable Polarized Tannakian (DPT) category equipped with a Weil cohomology functor.</p> <p>From these three axioms: the Weil Riemann Hypothesis reduces to a single cancellation step (Theorem A, one-line Lean proof); Standard Conjecture D is exactly the axiom that collapses LPO-dependent homological equality to BISH-decidable numerical equality (Theorem C); and the subcategory of CM elliptic motives — all 13 rational CM j-invariants from the 13 imaginary quadratic orders of class number 1 — is unconditionally BISH-decidable, with four bridge lemmas (Lieberman, Damerell, Shimura-Taniyama, Lefschetz (1,1)) bypassing every obstruction (Theorem E). The failure boundary is sharp: CM abelian varieties of dimension ≥ 4 require open conjectures.</p> <p>The Lean 4 formalization over Mathlib comprises 8 files (2214 build targets), compiling with 0 errors and 2 principled sorries, using 46 custom axioms (25 True-valued placeholders for upstream results). Theorems A, C, and E carry zero sorries. This is Paper 50 in the Constructive Reverse Mathematics series.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_18687415 |
| institution | Zenodo |
| language | eng |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Three Axioms for the Motive: Constructive Reverse Mathematics Meets Grothendieck's Universal Cohomology (Paper 50, Lean 4 Formalization) Lee, Paul Chun-Kit Constructive reverse mathematics Motives Grothendieck conjectures Standard Conjecture D Weil Riemann Hypothesis Tannakian categories Lean 4 Mathlib Formal verification Arithmetic geometry De-omniscientizing descent Limited Principle of Omniscience <p>Five major conjectures in arithmetic geometry — Weight-Monodromy, Tate, Fontaine-Mazur, Birch–Swinnerton-Dyer, and Hodge — all encode the same logical pattern: continuous data over a complete field (where zero-testing requires the Limited Principle of Omniscience) secretly descends to discrete algebraic data (decidable in Bishop-style constructive mathematics). We call this pattern "de-omniscientizing descent."</p> <p>Calibrating all five conjectures against the constructive hierarchy BISH ⊂ BISH+MP ⊂ BISH+LPO ⊂ CLASS reveals that Grothendieck's conjectural category of numerical motives is logically specified by exactly three axioms: (1) decidable equality on morphism spaces — equivalently, Standard Conjecture D; (2) algebraic spectrum — endomorphisms satisfy monic Z-polynomials; (3) Archimedean polarization — a positive-definite inner product on the real fiber, which exists over R but is obstructed over Q_p by the u-invariant u(Q_p) = 4. The motive category is the initial Decidable Polarized Tannakian (DPT) category equipped with a Weil cohomology functor.</p> <p>From these three axioms: the Weil Riemann Hypothesis reduces to a single cancellation step (Theorem A, one-line Lean proof); Standard Conjecture D is exactly the axiom that collapses LPO-dependent homological equality to BISH-decidable numerical equality (Theorem C); and the subcategory of CM elliptic motives — all 13 rational CM j-invariants from the 13 imaginary quadratic orders of class number 1 — is unconditionally BISH-decidable, with four bridge lemmas (Lieberman, Damerell, Shimura-Taniyama, Lefschetz (1,1)) bypassing every obstruction (Theorem E). The failure boundary is sharp: CM abelian varieties of dimension ≥ 4 require open conjectures.</p> <p>The Lean 4 formalization over Mathlib comprises 8 files (2214 build targets), compiling with 0 errors and 2 principled sorries, using 46 custom axioms (25 True-valued placeholders for upstream results). Theorems A, C, and E carry zero sorries. This is Paper 50 in the Constructive Reverse Mathematics series.</p> |
| title | Three Axioms for the Motive: Constructive Reverse Mathematics Meets Grothendieck's Universal Cohomology (Paper 50, Lean 4 Formalization) |
| topic | Constructive reverse mathematics Motives Grothendieck conjectures Standard Conjecture D Weil Riemann Hypothesis Tannakian categories Lean 4 Mathlib Formal verification Arithmetic geometry De-omniscientizing descent Limited Principle of Omniscience |
| url | https://doi.org/10.5281/zenodo.18687415 |