{ "round": "RUN-105", "gate": "src107", "B": "PASS", "sub_statuses": { "actual_arithmetic_complex": "PASS — RΓ(Z_S,T), three descriptions, one object", "canonical_fundamental_line_identification": "PASS — Φ_B: det^{-1}RΓ(Z_S,T) ≅ D_MW,11 ⊗ H_str, and H_str ≅ D_MW,11 ⊗ H^1_f(Q_11,T)^{-1}", "free_MW_factor": "PASS — D_MW,11 = ∧²(E(Q)⊗Z_11) via the Kummer map, an isomorphism in degree 1 (cokernel T_11(Sha), inherited zero)", "strict_free_factor": "PASS — H_str = det(H^2)^{-1} ≅ Sel_str(T), rank one, torsion-free (index 11^0), identified with D_MW ⊗ H^1_f(Q_11,T)^{-1} through the ordinary complex", "finite_factor": "PASS — placed, every 11-length zero (Sha numeric only); the unit line", "self_dual_placement": "PASS — actual shift 3 on the ordinary complex, relaxed dual to strict; Round 034's centre-1 profile is the ordinary complex shifted by [1]; not forced", "A_element_transported": "PASS — ε·(P∧Q)⊗R_str = ε·(P∧Q)^{⊗2}⊗g^{-1}, ε unmeasured", "numerical_finite_evaluation": "NOT REQUIRED / deferred to G (all zero at 11 anyway; Sha inherited)", "C_input": "H_str = Sel_str(T) ≅ D_MW,11 ⊗ H^1_f(Q_11,T)^{-1}: the strict/free line — C asks whether the derived height pairs it with the MW determinant's complementary direction" }, "line": "D_MW (x) H_str = D_MW^{(x)2} (x) H^1_f(Q_11,T)^{-1}", "C_input": "H_str = Sel_str(T) ≅ D_MW,11 ⊗ H^1_f(Q_11,T)^{-1}: the strict/free line — C asks whether the derived height pairs it with the MW determinant's complementary direction", "outcome_read": "B PASS: the line that carries A's element is the BSD-shaped arithmetic line at 11 — the rank-2 Mordell–Weil determinant squared over the local Bloch–Kato line — with the finite package the unit line and the strict line kept whole for C", "complexes": [ { "complex": "A", "name": "RΓ(Z_S,T) (Castella–Sano)", "ring": "Z_11", "local_at_11": "none (all of H^1(Q_11,T); the relaxed condition)", "local_at_389": "none (Z_S = Z[1/11·389]: unramified outside S is the only condition, and H^1(Q_389,T) = 0 anyway)", "degrees": "perfect, cohomology in [1, 2] (H^0 = T^{G_Q} = 0 by trivial torsion; H^3 = 0 for p odd)", "duality": "Poitou–Tate/Artin–Verdier: dual to the compactly supported complex RΓ_c(Z_S,T), H^i_c ≅ H^{3-i}(Z_S, E[11^∞])^∨ — NOT self-dual", "shift": "3 (i ↔ 3−i) against RΓ_c", "theorem": "Fukaya–Kato (perfectness); CS §2.2–2.3" }, { "complex": "B", "name": "RΓ~_{F_can}(Q,T) (Macias Castillo–Sano, Nekovář; T^+ = T, T^- = 0)", "ring": "Z_11", "local_at_11": "RΓ(Q_11, T^-) = RΓ(Q_11, 0) = 0", "local_at_389": "RΓ_{/ur}(Q_389,T), acyclic (RUN-104: H^1_ur = Z_11(1)/388 = 0, H^1(Q_389,T) = 0, H^2 = 0)", "degrees": "Cone(...)[−1]: same cohomology as A", "duality": "dual to the strict-at-11 complex (F* = strict)", "shift": "3", "theorem": "MCS Definition 2.2; RUN-104's acyclicity: B ≃ A" }, { "complex": "C", "name": "C_PT,n = [H^1_{F^n}(Q,E[11^n]) → ⊕_{l|n} H^1_{/F}(Q_l,E[11^n])] (Poitou–Tate model)", "ring": "Z/11^n", "local_at_11": "relaxed", "local_at_389": "H^1_f = 0", "degrees": "[1, 2]", "duality": "H^2(C_PT) = H^1_{F*}(Q,E[11^n]^*(1))^* (MCS Lemma 2.19(ii))", "shift": "n/a (finite model)", "theorem": "MCS Theorem 2.20: C_PT ≃ C_Nek/11^n under Hyp 2.11 + a core vertex (RUN-104 verified both at n = 1)" }, { "complex": "D", "name": "RΓ~_ord(Q,T) (the ordinary Greenberg datum T^- = T_11(Ẽ), Example 2.1) — the self-dual hub", "ring": "Z_11", "local_at_11": "im(H^1(Q_11,T^+) → H^1(Q_11,T)) = H^1_f(Q_11,T) (ordinary; good ordinary since a_11 = -4, 11 ∤ 16)", "local_at_389": "RΓ_{/ur}(Q_389,T), acyclic", "degrees": "[1, 3] a priori; here H~^3 ≅ H~^0(Q,E[11^∞])^∨ ⊆ H^0(Q,E[11^∞])^∨ = 0, so [1, 2]", "duality": "self-dual: H~^2 ≅ H^1_{ord*}(Q,E[11^∞])^∨ = Sel_{11^∞}(E/Q)^∨ (Lemma 2.14(ii); Hyp 2.11 by Remark 2.12: 11 ∤ 1·1·16)", "shift": "3 (i ↔ 3−i): degrees 1 and 2 are dual, 0 and 3 are dual", "theorem": "MCS Definition 2.2 and Lemma 2.14; Nekovář's duality (Selmer complexes §6.3; the shift is the standard one and is not re-read this round)" } ], "triangle": "RΓ~_ord(Q,T) → RΓ(Z_S,T) → RΓ(Q_11,T^-) ⊕ RΓ_{/ur}(Q_389,T) → (MCS Definition 2.2, by the definition of the cone)", "ranks": { "global": { "H1": 2, "H2": 1 }, "ordinary": { "H1": 2, "H2": 2 }, "local_at_11": { "H0(Q11,T)": 0, "H2(Q11,T)": 0, "H1(Q11,T)": 2, "H1_f(Q11,T)": 1, "H1_s(Q11,T)": 1, "H0(Q11,T-)": 0, "H2(Q11,T-)": 0, "H1(Q11,T-)": 1 }, "triangle_sequence": { "terms": [ "H^1_ord", "H^1(Z_S,T)", "H^1(Q_11,T^-)", "H^2_ord", "H^2(Z_S,T)" ], "ranks": [ 2, 2, 1, 2, 1 ], "alternating_sum": 0, "first_map_is_zero": true, "reading": "0 → H^1(Q_11,T^-) → H^2_ord → H^2(Z_S,T) → 0 is exact with ranks 1, 2, 1: the ordinary H^2 (≅ Sel^∨, rank 2) is an extension of the relaxed H^2 (≅ Sel_str^∨, rank 1) by the local singular line at 11" }, "mod_11": { "dims": [ 2, 2, 1, 2, 1 ], "alternating_sum": 0, "rho": 1 } }, "kummer_audit": { "actual_map": "the Kummer map E(Q)⊗Z_11 → H^1(Z_S,T), the inverse limit of the connecting maps of 0 → E[11^n] → E → E → 0; as a map of complexes Λ_{E,11}[−1] → τ_{≤1}RΓ(Z_S,T) → RΓ(Z_S,T)", "theorem_source": "Kummer theory; RUN-104: H^1(Z_S,T) = Sel(T) (the quotient H^1/Sel injects into the torsion-free H^1_s(Q_11,T) with rank 0)", "integral": true, "injective": "yes (E(Q)[11^∞] = 0 and the free part injects)", "cokernel": "T_11(Sha(E/Q)) — zero under the inherited input; NUMERIC ONLY", "cohomological_degree": 1, "rational_quasi_isomorphism_in_degree_1": true, "rational_quasi_isomorphism_of_complexes": false, "why_not": "H^2(Z_S,T) ⊗ Q ≠ 0 (rank 1): Round 032's concentration hypothesis (6.1) fails; Cone(ι) ≃ H^2(Z_S,T)[−2] is a free rank-one line, not a finite module", "consequence": "det H^1 = D_MW,11 is a factor of the line, not the line; the line is D_MW,11 ⊗ det(H^2)^{-1} — H^2 is kept, never killed to fit Round 032" }, "strict_formula": { "exact_sequence": "0 → Sel_str(T) → Sel(T) → H^1_f(Q_11,T) = E(Q_11)⊗̂Z_11 → 0 (onto since the index is 11^0)", "index_exponent_k": 0, "candidate_generator_mod_11": "u_Q·P − u_P·Q = 6P + 2Q", "loc_11_of_candidate_vanishes": true, "lies_on_the_strict_line": true, "strict_line": "F_11 (P + 4 Q)" }, "second_placement": "H_str ≅ ∧²Sel(T) ⊗ H^1_f(Q_11,T)^{-1} = D_MW,11 ⊗ H^1_f(Q_11,T)^{-1}, hence 𝒜(C_arith) ≅ D_MW,11^{⊗2} ⊗ H^1_f(Q_11,T)^{-1}", "H2": { "exact_object": "H^2(Z_S,T) ≅ H^1_{F_can*}(Q, T^∨(1))^∨ = Sel_str(E[11^∞])^∨ — the Pontryagin dual of the strict-at-11 Selmer group of the divisible module", "theorem": "MCS Lemma 2.14(ii) for the relaxed datum (Hyp 2.11 verified in RUN-104); independently CS (2.13) with every local H^2 vanishing (RUN-104)", "free_rank": 1, "free_rank_conditional_on": "INHERITED - Sha(E/Q)[11] = 0 carried by the programme's documents, never verified by this line; NUMERIC ONLY (unconditionally: corank Sel_str = corank Sel_{11^∞}(E/Q) − 1)", "torsion": "Z/11^0 = 0: the torsion of Sel_str^∨ is the cokernel of loc_11 on E(Q)⊗Z_11 → E(Q_11)⊗̂Z_11, of order 11^k with k = 0 (16P, 16Q at depth 1), plus Sha[11^∞]^∨-type torsion (inherited zero)", "dual_object": "Sel_str(E[11^∞]) = ker(Sel_{11^∞}(E/Q) → E(Q_11) ⊗ Q_11/Z_11); its Tate module Sel_str(T) = ker(Sel(T) → E(Q_11)⊗̂Z_11) = Z_11 · R_str", "local_condition_provenance": "strict at 11 (the dual of relaxed); at 389 the finite condition H^1_f = 0 = H^1 (no condition to impose); unramified elsewhere", "splitting": "no splitting claimed: H^2 is torsion-free here (k = 0, inherited Sha zero), so H^2 = H^2_free and det(H^2)^{-1} needs no choice", "H_str": "H_str := det(H^2)^{-1} = Hom(H^2, Z_11) ≅ T_11(Sel_str(E[11^∞])) = Sel_str(T) = Z_11 · R_str, R_str ≡ P + 4Q (mod 11)", "C_input": "H_str, a rank-one lattice inside Sel(T) — the strict/free line; trivialising it is C's derived-height question, not B's" }, "finite_ledger": [ { "factor": "torsion", "module": "E(Q)[11^∞] (rational torsion)", "sequence": "0 → E(Q)[11^∞] → H^1(Z_S,T)_tors: the image of H^0(Z_S,E[11^∞]) (Kummer); its dual in H~^3_ord(Q,T) ≅ H~^0(Q,E[11^∞])^∨ ⊆ H^0(Q,E[11^∞])^∨", "degree": 1, "exponent": 1, "source": "RUN-011 (torsion trivial)", "destination": "F_fin,11 (torsion endpoint)", "length_11": 0, "label": "COMPUTED_HERE" }, { "factor": "sha", "module": "Sha(E/Q)[11^∞]", "sequence": "0 → E(Q)⊗Q_11/Z_11 → Sel_{11^∞}(E/Q) → Sha[11^∞] → 0; its dual is the torsion of H~^2_ord ≅ Sel^∨ and of H^2(Z_S,T) ≅ Sel_str^∨", "degree": 2, "exponent": -1, "source": "the programme's documents", "destination": "F_fin,11 (middle module)", "length_11": 0, "label": "INHERITED — NUMERIC ONLY" }, { "factor": "tamagawa-389", "module": "Tamagawa c_389", "sequence": "H^1(Q_389,T)/H^1_ur(Q_389,T) and H^2(Q_389,T) = E(Q_389)[11^∞]^∨ inside RΓ_{/ur}(Q_389,T) — acyclic here", "degree": 2, "exponent": -1, "source": "RUN-067 (Kodaira I_1, c_389 = 1)", "destination": "F_fin,11 (local piece at 389)", "length_11": 0, "label": "COMPUTED_HERE" }, { "factor": "local-11-torsion", "module": "E(Q_11)[11^∞] and Ẽ(F_11)[11^∞]", "sequence": "H^0(Q_11,T^-⊗Q/Z) = H^2(Q_11,T^+)^∨ and H^0(Q_11,E[11^∞]) = H^2(Q_11,T)^∨ — the local H^2 terms of CS (2.13) and of the ordinary local condition", "degree": 2, "exponent": -1, "source": "RUN-104 (11 ∤ 16)", "destination": "F_fin,11 (local piece at 11)", "length_11": 0, "label": "COMPUTED_HERE" }, { "factor": "index-at-11", "module": "coker(loc_11: E(Q)⊗Z_11 → E(Q_11)⊗̂Z_11) = Z/11^k", "sequence": "the torsion of H^2(Z_S,T) ≅ Sel_str^∨ (degree 2) — and NOT again in the local line H^1_f(Q_11,T), which is torsion-free (index enters once)", "degree": 2, "exponent": -1, "source": "RUN-104 (depth of 16P, 16Q)", "destination": "H_str (its torsion part)", "length_11": 0, "label": "COMPUTED_HERE" }, { "factor": "unramified-389", "module": "H^1_ur(Q_389,T) and H^1(Q_389,T)", "sequence": "RΓ_ur(Q_389,T) → RΓ(Q_389,T) → RΓ_{/ur}(Q_389,T): all acyclic (Z_11(1)/388 = 0; E(Q_389)[11^∞] = 0)", "degree": 1, "exponent": 1, "source": "RUN-104", "destination": "F_fin,11 (local piece at 389)", "length_11": 0, "label": "COMPUTED_HERE" } ], "finite_package_at_11": "the unit line: every finite module that could enter has 11-length 0 (torsion trivial, c_389 = 1, 11 ∤ #Ẽ(F_11), index 11^0; Sha[11^∞] = 0 inherited, numeric only)", "duality": { "relaxed_complex": "RΓ(Z_S,T) is dual to RΓ_c(Z_S,T) (H^i_c ≅ H^{3-i}(Z_S,E[11^∞])^∨): not self-dual; its self-dual partner in the picture is the ordinary complex", "ordinary_complex": "RΓ~_ord(Q,T) ≅ RHom(RΓ~_ord(Q,T^∨(1)), Z_11)[−3], degree reflection i ↔ 3−i: H~^1 ↔ H~^2 (Lemma 2.14(ii): H~^2 ≅ Sel_{11^∞}^∨), H~^0 ↔ H~^3", "actual_shift": 3, "is_two": false, "determinant_of_the_ordinary_complex": "det(H~^1) ⊗ det(H~^2)^{-1} = ∧²Sel(T) ⊗ det(Sel_{11^∞}(E/Q)^∨)^{-1} = ∧²Sel(T) ⊗ ∧²Sel(T) = D_MW,11^{⊗2} (Sel_{11^∞} = E(Q)⊗Q_11/Z_11 under the inherited input, so Sel^∨ = Sel(T)^*)", "where_the_square_comes_from": "the duality H~^2 ≅ (H~^1)^∨ between degrees 1 and 2 — a free-part square, the one the height pairing trivialises (Round 034 §17's second mechanism); not a torsion square", "sign_bookkeeping_in_the_det^{-1}_convention": { "degree_1 (torsion E(Q)[11^∞])": "+length", "degree_2 (Sha^∨ and the local pieces)": "−length", "degree_3 (E(Q)[11^∞]^∨)": "+length", "pattern": "𝒜 ∼ #tors² / #Sha — the reciprocal of BSD's #Sha/#tors²; det (not det^{-1}) carries the BSD orientation; the same parity either way" }, "round_034_profile": { "their_profile": "H^0 = T, H^1 = M, H^2 = T^∨, centre 1, χ^𝒜 = ℓ(M) − 2ℓ(T)", "actual_profile": "H~^1 ⊇ T, H~^2 ⊇ M, H~^3 = T^∨, centre 3/2, χ^𝒜 = 2ℓ(T) − ℓ(M)", "relation": "the actual complex shifted by [1] has Round 034's profile; det(C[1]) = det(C)^{-1}, so the exponent flips sign with the shift — the whole-complex shift of Round 034 §82–83, applied globally, not per factor", "toy_check": { "T": 1, "M": 3, "chi_actual_degrees_1_2_3": -1, "chi_round_034_degrees_0_1_2": 1, "one_is_minus_the_other": true } }, "forced": false, "reading": "the actual shift is 3, not 2; nothing is forced — the BSD exponent pattern is recovered by the global shift [1], with the orientation recorded" }, "transport": { "source": "𝔷_{Q∞}(1) ∈ det^{-1}_{Z_11}RΓ(Z_S,T), a basis (RUN-104)", "first_form": "𝔷_{Q∞}(1) = ε · (P∧Q) ⊗ R_str with Δ_B := P∧Q a generator of D_MW,11, σ_str := ε·R_str a generator of H_str = Sel_str(T), φ_fin := 1 (the finite package is the unit line)", "second_form": "under H_str ≅ D_MW,11 ⊗ H^1_f(Q_11,T)^{-1}: 𝔷_{Q∞}(1) = ε · (P∧Q)^{⊗2} ⊗ g^{-1}, g a generator of H^1_f(Q_11,T) = E(Q_11)⊗̂Z_11, with (P∧Q)⊗g^{-1} ↦ a_Q·P − a_P·Q = R_str (mod 11: 6P − 9Q)", "Delta_B_nonzero": true, "sigma_str_nonzero": true, "finite_element": "1 — a determinant class of the unit line", "orientation": "P∧Q versus Q∧P is a sign, absorbed into ε; not an obstruction", "sign_treated_as_obstruction": false, "scalar": "ε ∈ Z_11^× UNMEASURED — not scalarised; no height is applied to σ_str (that is C)", "line_versus_element": "the line isomorphism Φ_B is an isomorphism of rank-one lattices; it is not an equality of scalars, and ε is what it does not determine", "line_iso_read_as_scalar": false, "transported": true }, "controls": { "A": { "complex": "[Z^2 --0--> Z^1] in degrees [1, 2]", "factors": [ { "degree": 1, "rank": 2, "exponent": 1 }, { "degree": 2, "rank": 1, "exponent": -1 } ], "reading": "𝒜 = det(Z^2) ⊗ det(Z)^{-1}: two factors; a gate that compresses to det H^1 alone is wrong", "accepted": true }, "B": { "profile": "H^0 = T (length 1), H^1 = M (length 3), H^2 = T^∨ (length 1)", "chi_A": 1, "expected_l(M)-2l(T)": 1, "accepted": true }, "agrees": true }, "not_claimed": [ "the unit ε (𝔷_{Q∞}(1) = ε·(P∧Q)⊗R_str) — not computed; no scalar equality", "any trivialisation of H_str — the height / regulator bridge is C", "Sha(E/Q)[11] = 0 verified here (inherited; every number that needs it is NUMERIC ONLY)", "Nekovář's duality theorem re-read this round (its shift is cited as standard; the degree bookkeeping used is Lemma 2.14(ii))", "any finite VALUE beyond the 11-lengths that this tree computed (torsion, c_389, #Ẽ(F_11), the index): G's business", "nodes C, D, E, F, G", "BSD" ], "claim_labels": { "COMPUTED_HERE": [ "the rank ledger of the triangle over Z_11 and mod 11 (alternating sums 0)", "the local Euler characteristics at 11", "the strict-line formula checked mod 11 (u_Q P − u_P Q on the strict line)", "the finite ledger's 11-lengths (torsion, c_389, #Ẽ(F_11), the index)", "the sign bookkeeping on toy complexes (controls A, B; the shift flips the exponent)" ], "CITED_THEOREM": [ "MCS Definition 2.2 (the triangle), Lemma 2.14(ii) (H~^2 ≅ (H^1_{F*})^∨), Remark 2.12 (Hyp 2.11 for the ordinary datum)", "CS (2.13) and the convention det^{-1}RΓ = det H^1 ⊗ det^{-1} H^2", "Nekovář's duality shift [3] (standard; not re-read)", "Tate local duality and the self-duality of the ordinary filtration" ], "DERIVED_FROM_VERIFIED_INPUTS": [ "RUN-104's ranks (2, 1), u_P = 9, u_Q = 6, index 11^0, strict line P + 4Q", "RUN-011's torsion, RUN-067's c_389" ], "INHERITED": [ "Sha(E/Q)[11] = 0 — numeric only: the ranks 2, 1 and the cokernel of the Kummer map" ], "UNMEASURED": [ "the unit ε (𝔷_{Q∞}(1) = ε·(P∧Q)⊗R_str) — not computed; no scalar equality", "any trivialisation of H_str — the height / regulator bridge is C", "Sha(E/Q)[11] = 0 verified here (inherited; every number that needs it is NUMERIC ONLY)", "Nekovář's duality theorem re-read this round (its shift is cited as standard; the degree bookkeeping used is Lemma 2.14(ii))", "any finite VALUE beyond the 11-lengths that this tree computed (torsion, c_389, #Ẽ(F_11), the index): G's business", "nodes C, D, E, F, G", "BSD" ], "FAILED": [] }, "status": { "normalization_certificate": "CLOSED (architecture), frozen", "DAG": { "A": "PASS (cited-theorem route)", "B": "PASS", "C-G": "untouched" }, "BSD": "OPEN" } }