Profile: results-envelope/1, results-claims/1, results-pairs/1, results-figures/1
| ID | Claim |
|---|---|
| curve-and-generators | 389.a1 (a-invariants [0, 1, 1, -2, 0], discriminant 389) has trivial torsion; P = (0,0) and Q = (1,0) lie on it and are the generators the programme uses |
| good-ordinary-at-11 | #E(F_11) = 16, a_11 = -4: good ordinary, and E(Q_11)[11^∞] = 0 (11 ∤ 16, ramification index 1) |
| kurihara-primes | #E(F_397) = 374 and #E(F_991) = 1045, each with v_11 = 1 and cyclic 11-part, so 397 and 991 satisfy the Kurihara condition a_ℓ ≡ ℓ + 1 (mod 11) and E(Q_ℓ)[11] ≅ Z/11 |
| localization-matrix | the localization matrix of (P, Q) at (397, 991) on E(Q)/11 is [[1, 2], [1, 4]] with determinant 2 ≠ 0 mod 11, so localization is injective on E(Q)/11 |
| selmer-cube | the kernel lines of loc_397 and loc_991 on E(Q)/11 are Q - 2P and Q - 4P, and their intersection is 0 (dimensions {'empty': 2, '397': 1, '991': 1, 'both': 0}) |
| kurihara-number | the Kurihara number δ̃_(397·991) recomputed from the modular-symbol sum (392040 terms) is 5 mod 11 in the disclosed normalisation; every unit is reachable by the choice of primitive roots |
| bockstein-sweep | det B_N = 2·XY in F_11[C_11×C_11]/I³; under 100 generator changes the line F_11·XY is invariant and the coefficient reaches every unit of F_11 (True) |
| certificate-canonical-values | the certificate's canonical values, each recomputed on exact rational instances: η_pdR = 1 (C6, with q = [γ_Kato : γ_tree] a lattice index filed in bucket q_D), κ_det = 1 (C7, one route), transport-graph holonomy with every cycle product 1 (C9), ρ = 1 with a 15-row choice table and no surviving choice (C10) |
| b-n-typed | P5 v1.3's B_N is the canonical localization map Sel_11(E/Q) → ⊕ E(F_ℓ)⊗F_11 dressed with chosen augmentation generators — a hand-built finite matrix, not the connecting morphism of any complex; it is not the Burns–Kurihara–Sano Bockstein regulator, and BKS Theorem 1.3's hypothesis (d) fails at 397 and 991 |
| split-at-389 | a_389 = 1 (split multiplicative), so the S-Euler factor is 𝒫_389 = 1 - 389^{-1} gamma_389, equal to 388/389 at the trivial character — an 11-adic unit |
| local-vanishing | E(Q_11)[11^∞] = 0 and E(Q_389)[11^∞] = 0 (c_389 = 1, 11 ∤ 388), so every local H² in the Poitou–Tate sequence vanishes at the trivial character and H²(Z_S,T) ≅ Sel_str(E[11^∞])^∨ unconditionally |
| formal-group-depth | 16P and 16Q lie in E_1(Q_11) ∖ E_2(Q_11) with classes 9 and 6 in E_1/E_2 ≅ F_11 (additive, checked on P+Q, P−Q, 2P): localization at 11 is onto E(Q_11)⊗̂Z_11 with index 11^0, and the strict-Selmer line mod 11 is F_11 (P + 4 Q) |
| ranks-at-trivial-character | rank H¹(Z_S,T) = 2 and rank H²(Z_S,T) = 1 at the trivial character, by two routes, so det⁻¹RΓ(Z_S,T) = Z_11·(P∧Q) ⊗ Sel_str(T) — the rank-2 determinant tensored with the strict-Selmer line; the distinguished element does not descend to H¹ |
| core-rank | the core rank of the relaxed (Kato) Selmer structure is 1 by Macias Castillo–Sano's Euler-characteristic formula and 1 by their Definition 2.16 at the tree's own core vertex 397·991 on E[11] (dim H¹_(F^n) = 3); the ordinary structure has core rank 0 |
| hypothesis-2-11 | Macias Castillo–Sano's Hypothesis 2.11 (i)–(iv) holds for the relaxed structure at T = T_11(389.a1), RΓ_/ur(Q_389,T) is acyclic, and 397, 991 lie in the prime set 𝒫_1 — so their Theorem 3.4 applies at n = 1 with every hypothesis instantiated |
| fundamental-line | det⁻¹RΓ(Z_S,T) ≅ D_MW,11 ⊗ 𝓗_str with 𝓗_str = det(H²)⁻¹ ≅ Sel_str(T) ≅ D_MW,11 ⊗ H¹_f(Q_11,T)⁻¹ — the rank-2 Mordell–Weil determinant squared over the local Bloch–Kato line at 11; the triangle RΓ̃_ord → RΓ(Z_S,T) → RΓ(Q_11,T⁻) ⊕ RΓ_/ur(Q_389,T) has rank ledger (2, 2, 1, 2, 1) over Z_11 and mod 11; the finite package at 11 is the unit line; the actual duality shift is 3 |
| global-primitive-basis | [E(Q)/tors : ZP + ZQ] = 1 — P = (0,0) and Q = (1,0) are a global primitive basis — proved twice: by Hermite's rank-2 cap (0.2254) plus an exhaustive search of the resulting box (27 points, smallest canonical height 0.3270), and independently, citing nothing, by an ℓ-sieve giving for every ℓ ≤ 19 and every projective class mod ℓ a witness prime q with (aP+bQ) ∉ ℓ·E(F_q) |
| regulator-and-convention | the Néron–Tate pairing in the convention ½(ĥ(P+Q) − ĥ(P) − ĥ(Q)), agreeing with the parallelogram route to 2.8e-9, gives det(h♭) = Reg_NT = 0.152460139 (residual 3.9e-8); the convention is discriminated by Ω·Reg·∏c_v/#tors² against L''(E,1)/2, which a doubled or halved pairing would move to 0.93 or 1.02 |
| local-bloch-kato-line | the local line at 11 is NOT integrally trivial: the formal-group logarithm of the minimal model has v₁₁(log_ω(g)) = 1, so log_ω carries H¹_f(Q₁₁,T) = E(Q₁₁)⊗̂Z₁₁ onto 11·Z₁₁ — index 11; C6's η_pdR = 1 concerns a different normalisation and does not transfer |
| mutation-drill | every gate of the tree is drilled: each planted defect must be caught by the check named for it, and the controls must disturb nothing (figures in gates.mutation_drill, rendered as pairs) |
What this line does not establish, stated by the line itself — rendered on the same screen as the claims above, not behind a toggle.
| Figure / against | What it measures |
|---|---|
| 687 / 692 | planted defects caught by the check named for them, of those planted |
| 0 / 179 | controls that disturbed a check, of those run |
| 10 / 10 | certificate cells closed as architecture, of C1–C10 |
| 4 / 7 | DAG nodes passed, of A–G |
| Gate | Result |
|---|---|
| Mutation drill | true |
Rendered 2026-10-25 from agent/bsd-verification-zhuiheng at build time — not vendored into this site. Full reports and gate logs: the private repository amral-research-trees, folder bsd-verification-zhuiheng.