{ "round": "RUN-107", "gate": "src109", "DAG_D": "PASS", "D_rational": "ISOMORPHISM", "D_integral": "INDEX_11", "layers": { "D-NV1": "PASS", "D-NV2": "PASS", "D-NV3": "PASS" }, "map": { "source": "H/L = the psi^{-1}-quotient line H^1(Q, V(g)^*(psi^-1) (x) H_Gamma(-j))", "target": "A_p = the H_Gamma-line of L_p(g_E x psi)", "theorem": "Kato's explicit reciprocity law, with C1.13 and Proposition C1.1", "injective": true, "surjective": false, "status": "non-zero-divisor, non-unit (target) / unit (calibrator)" }, "one_variable_audit": { "relevant_tangent_directions": 1, "no_direction_to_choose": true, "derivative_inference_refused": true }, "local_bloch_kato_factor": { "index": 11, "owner": "p-adic reciprocity / the leading-line integral realization", "counted_times": 1, "E_may_read_not_enter": true }, "schneider_dependency": "NONE", "boundary": { "D_asks": [ "is the reciprocity map actual (a map of honest objects, not a slogan)?", "is it typed (source, target, ring, integral or rational, named theorem)?", "is it nonzero on the actual transverse quotient?" ], "E_asks": [ "the exact order of vanishing at the central point", "the leading tensor (which determinant line the leading term lives in)", "the exact leading coefficient, and its equality with the arithmetic side" ], "the_rule": "map exists and is nonzero = D; leading-order equality = E", "series_identity_is_not_E_proved": true, "what_the_series_gives_D": [ "L_E != 0 in Lambda (mu = 0)", "L_E not a unit in Lambda" ], "what_the_series_does_NOT_give": [ "the exact order of vanishing at t = 0 over Z_11", "the leading coefficient", "ord = rank", "any leading-term equality" ], "forbidden_read": "treating 'L_E(t) = 2t^2 + ... as a series fact' as E already proved", "forbidden_read_taken": false, "E_verdict_emitted": false, "F_G_touched": false }, "route": { "arrows": [ { "name": "Psi_2", "source": "H^1(Q, (1/X)V_c(f,g)^*), the 1/X-enlarged family module, rank 1 over O(V_1 x V_2 x W)", "target": "Q = H/L = the psi^{-1}-quotient line H^1(Q, V(g)^*(psi^-1) (x) H_Gamma(-j)), rank 1 over H_Gamma", "ring": "H_Gamma after the weight specialisation", "theorem": "C1.6 (the tau-quotient image is zero, so kappa_d descends into the ambient H = H^1(Q, V(f,g)^*)) + C1.7 (that lift is unique; its image in the psi^{-1}-quotient is taken); then the leading term X^n kappa-hat_d mod X", "role": "the ONE passage family -> leading term: the quotient that defines D's domain", "level": "integral over H_Gamma after the named extractions", "localised_at": [ "X^n" ], "injective": true, "surjective": false, "status_target": "non-zero-divisor, non-unit", "status_calibrator": "non-zero-divisor, non-unit", "nonzero": "X^n kappa-hat_d is well-defined and non-zero modulo X (LR, C1.7 area)", "frame": { "b_f+": "none: a quotient projection of Galois modules" }, "line_level": "class-level (quotient maps of Galois modules)", "from_log": "src99 (RUN-097) + src100 (RUN-098)" }, { "name": "Psi_3", "source": "Q = the psi-quotient line (the BF leading line)", "target": "the Kato line, the same free rank-1 H_Gamma-module (C1.1)", "ring": "H_Gamma, as written an identity in the total quotient ring", "theorem": "C1.13: kappa-hat^*_d(f_beta,g) = [C C_d log^{[r+1]} L_p(g x tau)/L_p(Ad g)] kappa(g x psi), proved scalar-first and lifted to classes by Proposition C1.1", "role": "identifies D's leading object with Kato's class", "level": "rational: fractional as written, integral after clearing L_p(Ad g)", "localised_at": [ "log^{[r+1]}", "L_p(Ad g) (inverted)" ], "injective": true, "surjective": false, "status_target": "fractional", "status_calibrator": "fractional", "nonzero": "both generators nonzero; the scalar C C_d L_p(g x tau)/L_p(Ad g) a non-zero-divisor", "frame": { "b_f+": "C = c*_f(r) scales by u; the class identity is frame-free" }, "line_level": "scalar-first, lifted to classes by Prop C1.1", "from_log": "src99 (RUN-097) + src100 (RUN-098)" }, { "name": "Psi_4", "source": "the Kato line", "target": "A_p = the H_Gamma-line spanned by L_p(g x psi)", "ring": "H_Gamma", "theorem": "Kato's explicit reciprocity law: loc_11 then the Perrin-Riou regulator for F^- V(g)^*, landing on L_p(g x psi); the Betti-period convention audited at RUN-080/096 and frozen by C6 (RUN-099)", "role": "THE reciprocity arrow: lambda_rec itself", "level": "integral over H_Gamma after the named extractions", "localised_at": [ "L_p(g_E x psi)" ], "injective": true, "surjective": false, "status_target": "non-zero-divisor, non-unit", "status_calibrator": "unit", "nonzero": "injective on the line: the p-adic L-function is a non-zero-divisor (C1.1's proof); in-tree the series are nonzero", "frame": { "Omega^+-": "G (period frame)", "lattice index": "L (2 on both signs of the target, 2 and 1 for the calibrator)" }, "line_level": "class-level; the equality of Kato's and MTT's L_p is an equality of elements of H_Gamma", "from_log": "src99 (RUN-097) + src100 (RUN-098)" } ], "family_arrow": { "name": "Psi_1", "role": "the Coleman/Perrin-Riou computation on the FAMILY line — the source of the scalars, not of lambda_rec", "theorem": "Perrin-Riou regulator L^PR (LZ16) on loc_p; image computed by LR C1.9", "image": "c_f(k) * C_d * L_p^g", "localised_at": [ "X^n", "L_p(g_E x psi)", "L_p(Ad g) (inverted)" ], "note": "listed so the route is complete; D's map is Psi_4 after Psi_2 and Psi_3" }, "composite": { "lambda_rec": "Psi_4 o Psi_3 o (the leading-term class of Psi_2)", "source": "Q = H/L, the psi-quotient line", "target": "A_p = the H_Gamma-line of L_p(g_E x psi)", "named_extractions": { "log^{[r+1]}": "C1.8 divides kappa-hat_d by it", "X^n": "the leading term at X = 0 (C1.7 area; C1.10's n)" }, "residual": "log^{[r+1]} / X^n", "every_L_cancels_in_the_composite": [ "L_p(g x psi)", "L_p(g x tau)", "L_p(Ad g)", "C_d" ] }, "rebuilt_here": false, "read_from": [ "src99-r0-comparison-route-realization.json", "src100-r05-localization-audit.json" ], "every_arrow_typed": true, "every_arrow_injective": true, "no_arrow_surjective_for_the_target": true, "kato_scalar_lifted_by_C1_1": true }, "theorems": { "items": { "LR C1.9": { "statement": "the Coleman-map image of the Beilinson-Flach class factors with a factor c_f(k), a meromorphic function on V_1 ALONE, regular and non-vanishing at every integer weight k >= -1 EXCEPT possibly k = r, where it is still regular", "what_D_takes": "the factorisation and the partner-independence of c_f", "what_D_may_NOT_take": "the non-vanishing at our point: k = r is exactly the weight the clause excepts", "covers_k_equals_r": false, "caveat_source": "src97 (RUN-095), field c19_nonvanishing_excepts_k_equals_r", "hypotheses_discharged_at": "RUN-094/095" }, "LR C1.13": { "statement": "the critical degeneration: the specialisation of the BF class at f_beta equals [C C_d log^{[r+1]} L_p(g x tau)/L_p(Ad g)] times Kato's class, with C = c^*_f(r) in L^x", "what_D_takes": "c^*_f(r) in L^x IS the nonvanishing at our point, asserted by the paper in the construction; and X^n kappa-hat_d is well-defined and nonzero modulo X", "proof_shape": "equal regulator images, then equality of CLASSES by Proposition C1.1 — scalar-first, lifted", "hypotheses_discharged_at": "RUN-094 (kappa_d exists, C1.4 with (d,6S) = 1; f_beta non-critical) and RUN-095 (g noble, ordinary, residually irreducible)" }, "LR Prop C1.1": { "statement": "the Greenberg Selmer group ker( H^1(Q, V(g)^*(chi)(n) (x) H_Gamma(-j)) -> H^1(Q_p, F^- V(g)^*(chi)(n) (x) H_Gamma(-j)) ) VANISHES", "proof_content_D_uses": "at each classical specialisation H^1(Q, V(g_l)^* (x) H_Gamma(-j)) is FREE OF RANK 1 over H_Gamma and contains Kato's Euler system as a canonical element, whose localisation at p maps under the Perrin-Riou regulator for F^- V(g_l)^* to the p-adic L-function, which is NOT A ZERO-DIVISOR", "the_statement_and_the_proof_give_different_things": "the STATEMENT gives injectivity of loc_p (its kernel is the Greenberg Selmer group, which vanishes); the PROOF gives rank one and the non-zero-divisor image, hence the regulator's nonvanishing. D-NV2 needs both.", "what_D_takes": "the regulator is injective on a rank-one module — i.e. D-NV2, and the lifting of C1.13 to classes", "what_D_may_NOT_take": "surjectivity, or an integral isomorphism", "surjectivity_claimed": false, "partner_hypothesis_is_only": [ "ordinary" ], "over_strong_additions": [], "hypotheses_discharged_at": "RUN-095 (ordinary on both curves, computed); residual irreducibility needed by C1.13's Kato class, discharged by semistability + Mazur" }, "Kato reciprocity": { "statement": "localisation at p followed by the Perrin-Riou regulator for F^- V(g)^* lands on L_p(g x psi); Kato's L_p agrees with the Mazur-Tate-Teitelbaum L_p of modular symbols up to the period frame", "what_D_takes": "the arrow Psi_4 exists and is injective", "period_convention": { "audited_at": "RUN-080 (calibrator plus = Neron) and RUN-096 C8 (indices 2, 2)", "frozen_by": "C6 = PASS / canonical, eta_pdR = 1", "reopened_here": false }, "hypotheses_discharged_at": "RUN-095 (ordinary, residually irreducible, noble, prime-to-p level, p-distinguished)" } }, "inherited_hypothesis_list_389a1": { "ordinary at p (a_p not divisible by p)": true, "unit root mod p": 7, "prime-to-p level (p does not divide N)": true, "Hecke polynomial X^2 - a_p X + p has distinct roots (disc nonzero)": true, "p-distinguished (the two roots differ mod p)": true, "noble (prime-to-p level + distinct Hecke roots), Assumption B1.1": true, "non-critical slope (ordinary means slope 0 < r_1 + 1)": true, "Hida (ordinary), as section B2 restricts to": true, "newform of its level, as section B2 assumes": true, "weight 2, trivial nebentypus": true, "minimal model (Delta squarefree, and 1728 Delta = c_4^3 - c_6^2)": true, "semistable (no additive reduction: every prime dividing Delta fails to divide c_4)": true, "residually irreducible at p = 11 (semistable + Mazur: a rational l-isogeny forces l <= 7)": true }, "all_inherited_hold": true, "all_hypotheses_point_back_to_an_actual_PASS": true, "no_new_assumption_introduced": true, "c1_9_not_stretched_over_k_equals_r": true, "c1_1_not_read_as_surjective": true, "normalization_not_reopened": true }, "domain": { "enlarged_module": { "object": "H^1(Q, (1/X)V_c(f,g)^*) — the 1/X-enlarged Beilinson-Flach family module", "ring": "O(V_1 x V_2 x W)", "rank": 1, "role": "where kappa_d(f,g) is constructed; NOT the ambient of the quotient D uses" }, "tau_quotient": { "object": "the tau-quotient (1/X)V_c(f,g)^*/V(f,g)^* = Q_p(tau^-1)(1+r) (x) V(g)^* (x) H_Gamma(-j)", "what_happens_there": "C1.6: the image of kappa_d is ZERO", "role": "the direction the class DIES in — the reason it descends into the ambient, not the transverse quotient" }, "H": { "object": "H^1(Q, V(f,g)^*) — the Beilinson-Flach FAMILY module itself", "ring": "O(V_1 x V_2 x W)", "rank": 1, "role": "the AMBIENT. Not the transverse quotient.", "why_the_class_is_here": "C1.6 kills the tau-quotient image, so kappa_d descends; C1.7: that lift is UNIQUE", "source": "src99's type record for L1 (RUN-097); LR C1.6, C1.7" }, "L": { "object": "ker( H -> the psi^{-1}-quotient ) — the image of H^1(Q, V(g)^* (x) H_Gamma(-j)) under the inclusion 1 -> V(f_beta)^*", "why_the_1_part_is_the_sub": "dualising RUN-093's ordering 0 -> psi -> V(f_beta) -> 1 -> 0 gives 0 -> 1 -> V(f_beta)^* -> psi^-1 -> 0, so the 1-part is a SUB of V(f,g)^* and psi^-1 is its quotient", "role": "the reference subobject L of PC-001/Round 030 — the direction the leading term is taken transverse to" }, "Q": { "object": "H/L = the psi^{-1}-quotient line H^1(Q, V(g)^*(psi^-1) (x) H_Gamma(-j))", "ring": "H_Gamma", "rank": 1, "freeness": "free of rank 1 over H_Gamma (Prop C1.1's proof)", "why_this_is_the_quotient": "C1.7: the unique lift's image in the psi^{-1}-quotient is the object that is taken", "role": "D's DOMAIN" }, "first_nonzero_projective_class": { "object": "X^n kappa-hat_d(f,g) mod X", "assertion": "well-defined and NON-ZERO modulo X", "source": "LR, the construction inside C1.13 (quoted in the tree's EXTRACTS)", "label": "CITED_THEOREM", "which_n": "the weight-jet order n of C1.10, equal to Round 030's m (RUN-098's Psi_2 localisation at X^n)", "is_the_first_nonzero_graded_class": true }, "round_030_language": "ambient H, reference subobject L, quotient Q = H/L, the first nonzero projective class", "Q_is_the_quotient_of_H_by_L": true, "the_class_survives_in": "the psi^{-1}-quotient", "the_class_dies_in": "the tau-quotient", "quotients_the_right_way_round": true, "whole_BF_family_line_called_the_transverse_quotient": false, "H_is_not_Q": true, "H_is_not_the_enlarged_module": true, "the_leading_object_sent_into_Q": { "object": "X^n kappa-hat_d(f,g) mod X", "in": "H/L = the psi^{-1}-quotient line H^1(Q, V(g)^*(psi^-1) (x) H_Gamma(-j))", "nonzero": true, "by": "LR's own assertion, lifted to a class statement by Prop C1.1 (C1.13's proof)" }, "two_variables_kept_apart": { "weight direction X": "where the leading-term extraction happens, order n", "cyclotomic direction t": "where the target A_p lives; its vanishing order is E's" } }, "nonvanishing": { "series": { "L_E": { "reduction_nonzero": true, "mu": 0, "lambda": 2, "unit_in_Lambda": false, "zero_mod_p_and_t": true, "why_mu": "some coefficient of the reduction is a unit, so the element is nonzero in Lambda", "why_not_unit": "the element reduces to 0 modulo (p, t), so its constant term is not a p-adic unit", "declared_lambda": 2, "declared_mu": 0 }, "lambda_E": { "reduction_nonzero": true, "mu": 0, "lambda": 0, "unit_in_Lambda": true, "zero_mod_p_and_t": false, "why_mu": "some coefficient of the reduction is a unit, so the element is nonzero in Lambda", "why_not_unit": "the element reduces to 0 modulo (p, t), so its constant term is not a p-adic unit" }, "L_0": { "reduction_nonzero": true, "mu": 0, "lambda": 0, "unit_in_Lambda": true, "zero_mod_p_and_t": false, "why_mu": "some coefficient of the reduction is a unit, so the element is nonzero in Lambda", "why_not_unit": "the element reduces to 0 modulo (p, t), so its constant term is not a p-adic unit" }, "lambda_0": { "reduction_nonzero": true, "mu": 0, "lambda": 0, "unit_in_Lambda": true, "zero_mod_p_and_t": false, "why_mu": "some coefficient of the reduction is a unit, so the element is nonzero in Lambda", "why_not_unit": "the element reduces to 0 modulo (p, t), so its constant term is not a p-adic unit" } }, "weierstrass_split": { "m": 2, "u_first_coefficients": [ 2, 2, 2, 0, 5 ], "u_is_a_unit": true, "u_times_u_inverse": [ 1, 0, 0, 0 ], "split_reassembles_exactly": true }, "non_zero_divisor": { "order_additive_on_every_trial": true, "trials": 40, "sample": [ { "ord_g": 4, "ord_product": 6, "expected": 6, "nonzero": true }, { "ord_g": 7, "ord_product": 9, "expected": 9, "nonzero": true }, { "ord_g": 3, "ord_product": 5, "expected": 5, "nonzero": true } ] }, "layers": [ { "layer": "D-NV1 — the class is nonzero", "object": "X^n kappa-hat_d mod X in Q", "status": "PASS", "by": "LR's assertion in C1.13's construction (well-defined and nonzero mod X), lifted to classes by Prop C1.1", "label": "CITED_THEOREM" }, { "layer": "D-NV2 — the map is nonzero", "object": "lambda_rec : Q -> A_p, the composite of loc_11 with the Perrin-Riou regulator for F^- V(g)^*", "status": "PASS", "by": [ "Prop C1.1 as stated: the Greenberg Selmer group — BY DEFINITION the kernel of loc_p into H^1(Q_p, F^- V(g)^*(chi)(n) (x) H_Gamma(-j)) — VANISHES. So the FIRST factor, loc_p, is injective on Q.", "Prop C1.1's proof: Q is free of rank 1 over H_Gamma, and the regulator carries Kato's canonical element to L_p(g x psi), which is NOT a zero-divisor. So the SECOND factor is nonzero on Q.", "H_Gamma is a domain (for Gamma = Z_11 it is a ring of functions analytic on the open unit disc), so multiplication by a nonzero element is injective; an injection followed by that is injective." ], "why_the_two_steps_are_not_the_same_kernel": "the vanishing Greenberg Selmer group is the kernel of loc_p, not of the composite; the composite's injectivity needs the second factor's nonvanishing as well, and that is what the non-zero-divisor clause supplies", "strength": "injective, not merely nonzero", "surjective": false, "label": "CITED_THEOREM", "rank_one_argument": "a nonzero map of free rank-one modules over a domain is injective; a nonzero map of one-dimensional spaces over a field is an isomorphism" }, { "layer": "D-NV3 — the image is nonzero", "object": "lambda_rec(z-bar) = (nonzero scalar) * L_p(g_E x psi)", "status": "PASS", "scalars": { "C = c^*_f(r)": "in L^x (C1.13)", "C_d": { "value": 26, "mod_p": 4, "unit": true }, "L_p(g x tau)": "unit (RUN-098)", "1/L_p(Ad g)": "fractional, cleared (RUN-098)", "log^{[r+1]}": "non-zero-divisor, non-unit (RUN-098)" }, "series_nonvanishing": { "mu": 0, "reduction_nonzero": true, "why": "some coefficient of the reduction is a unit, so the element is nonzero in Lambda", "label": "COMPUTED_HERE" }, "and_not_a_unit": { "unit_in_Lambda": false, "why": "the element reduces to 0 modulo (p, t), so its constant term is not a p-adic unit", "label": "COMPUTED_HERE" }, "what_is_NOT_claimed": [ "the order of vanishing at t = 0 over Z_11", "the leading coefficient", "any leading-term equality" ] } ], "declared_layers": [ "D-NV1", "D-NV2", "D-NV3" ], "annihilation": { "what_vanishes": "the VALUE of L_E at the central point is not a unit (L_E = 0 mod (11, t))", "what_does_not_vanish": "L_E itself: mu = 0, so L_E != 0 in Lambda", "the_map": "multiplication by L_E on a free rank-one H_Gamma-module", "kernel": 0, "kernel_witness": "ord(L_E * g) = ord(L_E) + ord(g) for every nonzero g tested, so no nonzero g is killed", "correct_form": "L_E = t^2 * u with u(0) = 2 != 0 — a nonzero map vanishing to positive order", "D_needs_only": "u != 0, equivalently L_E != 0", "exact_m_belongs_to": "E", "forbidden_inference": "L_E(0) = L_E'(0) = 0, therefore the reciprocity map annihilates the target", "forbidden_inference_taken": false }, "directional": { "relevant_tangent_directions": 1, "why": "PC-001 fixes one weight/cyclotomic direction; after the weight specialisation exactly the cyclotomic direction remains", "round_011_statement_used": "(17.1) one variable: the order is e + m exactly, with no direction to choose", "round_030_warning_not_needed": "the multivariable requirement tau_m(xi^{(x)m}) != 0 needs a direction; with one direction there is nothing to choose", "forbidden_inference": "one variable, therefore every derivative is nonzero", "forbidden_inference_taken": false, "counterexample_from_the_tree": { "series": "the tree's own L_E mod 11", "nonzero_as_a_series": true, "value_at_0": 0, "first_derivative_at_0": 0, "first_nonzero_taylor_coefficient_index": 2, "its_value": 2, "so_one_variable_does_not_imply_a_nonzero_derivative": true } }, "mod_p_data_does_not_determine": "the order of vanishing at t = 0 over Z_11; only lambda and mu are determined", "lambda_matches_declared": true, "mu_matches_declared": true }, "integral": { "D_rational": { "verdict": "ISOMORPHISM", "map": "log_omega : H^1_f(Q_11, V) -> D_dR(V)/Fil^0 = Q_11, V = T (x) Q_11", "source_dimension": 1, "target_dimension": 1, "why_source_is_one_dimensional": "11 does not divide #E~(F_11) = 16, so E(Q_11) (x)^ Z_11 = E_1(Q_11) = Z_11 (RUN-106)", "nonzero": true, "argument": "a nonzero linear map between one-dimensional spaces over a field has zero kernel and is onto — Round 011 (3.1)-(3.2)", "label": "DERIVED_FROM_VERIFIED_INPUTS (RUN-106's v_11(log_omega(g)) = 1 < infinity)" }, "D_integral": { "verdict": "INDEX_11", "map": "log_omega on the lattice: H^1_f(Q_11, T) -> Z_11", "image": "11^1 · Z_11", "index": 11, "declared": 11, "unit_isomorphism": false, "integrally_trivial": false, "label": "DERIVED_FROM_VERIFIED_INPUTS (RUN-106 computed v_11(log_omega(g)) = 1 from the formal group of the minimal model)" }, "compatibility": { "statement": "rational reciprocity = ISOMORPHISM and integral reciprocity = index 11 are compatible", "why": "an isomorphism of Q_11-lines restricted to lattices is an injection with finite cokernel; the cokernel's order is the index", "D_can_pass_with_index_not_1": true }, "C6": { "what_C6_settled": "η_pdR = 1 — the period/de Rham comparison pairing ⟨γ, ω⟩ of the normalisation route (Eichler–Shimura; Ω^± is its value on the chosen Betti basis), RUN-099", "eta_pdR": 1, "frozen": true, "does_it_overwrite_the_local_index": false, "forbidden_inference": "η_pdR = 1 ⇒ the local log normalisation is 1", "what_it_would_cost": "a factor 11: the integral image is 11^1 · Z_11, not Z_11", "index_after_the_forbidden_inference": 1 }, "ledger": { "rows": [ { "factor": "local Bloch-Kato index at 11", "value": 11, "owner": "p-adic reciprocity / the leading-line integral realization", "entered_by": "D", "entered_times": 1, "E_may": "read it", "E_may_not": "enter it again" }, { "factor": "global index [Lambda_E : ZP + ZQ]", "value": 1, "owner": "C_NT (RUN-106)", "entered_by": "C", "entered_times": 1, "E_may": "read it", "E_may_not": "enter it again" }, { "factor": "11-adic index of loc_11(E(Q) (x) Z_11)", "value": 1, "owner": "RUN-104's H^2 torsion", "entered_by": "A", "entered_times": 1, "E_may": "read it", "E_may_not": "enter it again" } ], "double_entries": [], "rule": "an index enters exactly once, through the map it belongs to, with an owner named" }, "index_matches_declared": true, "owner_named": true }, "independence": { "dependencies": [ "LR Proposition C1.1 (rank one, Kato element, regulator injective)", "LR Theorem C1.13 (the degeneration, C = c^*_f(r) in L^x)", "LR Corollary C1.9 (the factorisation and the partner-independence of c_f) — NOT its non-vanishing clause", "Kato's explicit reciprocity law for g", "the nonvanishing of L_p(g_E x psi) as an element of Lambda (COMPUTED_HERE, mu = 0)", "RUN-106's v_11(log_omega(g)) = 1, for the integral verdict only" ], "schneider": { "statement": "nondegeneracy of the p-adic height at k = 1 (Schneider's conjecture)", "status_in_the_tree": "OPEN; C_p PARTIAL at RUN-106", "in_D's_dependency_set": false, "why_not": "explicit reciprocity asks whether a map of lines is nonzero; the p-adic height asks whether a pairing is nondegenerate. No arrow of D's route is a height pairing.", "D_fails_if_Schneider_unproved": false }, "normalization": { "certificate": "PC-001 C1-C10, CLOSED as architecture (RUN-102), frozen", "scalars_D_does_not_ask_for": [ "C_pi", "q_f", "G_b", "Pi_i", "kappa_det", "eta_pdR", "epsilon" ], "a_scalar_reentered": false, "what_D_does_record": "the actual arithmetic coefficients: C_d, L_p(Ad g), the lambda-series, log^{[r+1]}", "nonzero_divisors_suffice": true, "units_required": false }, "nonzero_divisor_vs_unit": { "rule": "a nonzero divisor is not a unit (RUN-098's rule, continued)", "L_E": { "status": "non-zero-divisor, non-unit", "truth": "non-zero-divisor, non-unit", "mislabelled": false }, "log^{[r+1]}": "non-zero-divisor, non-unit", "C_d": "unit", "L_p(g x tau)": "unit" }, "target_and_calibrator": { "rows": { "target": { "curve": "389.a1", "global class nonzero": "CITED (C1.1: Kato's element is a canonical generator)", "localisation nonzero": "CITED (C1.1's proof: loc_p of the Kato element has nonzero regulator image)", "reciprocity map nonzero": "PASS (C1.1, injective)", "analytic image nonzero": "PASS", "analytic image a unit": false, "integral index of the local log": 11, "label": "COMPUTED_HERE for the analytic row" }, "calibrator": { "curve": "19.a1", "global class nonzero": "CITED (C1.1: Kato's element is a canonical generator)", "localisation nonzero": "CITED (C1.1's proof: loc_p of the Kato element has nonzero regulator image)", "reciprocity map nonzero": "PASS (C1.1, injective)", "analytic image nonzero": "PASS", "analytic image a unit": true, "integral index of the local log": "not computed this round", "label": "COMPUTED_HERE for the analytic row" } }, "calibrator_status": "unit", "declared_calibrator_status": "unit", "calibrator_used_to_prove_the_target": false, "why_it_cannot": "the target's nonvanishing comes from the target's own reduction mod 11; the calibrator is a different curve with a different series, and its being a unit says nothing about 389.a1", "the_difference_is_the_point": "the calibrator's reciprocity is an integral isomorphism, the target's is injective with a nontrivial cokernel — exactly RUN-098's unit / non-unit split" }, "schneider_absent": true, "calibrator_status_matches": true, "labels_honest": true }, "controls": { "A": { "module": "R = F_11[[t]] (truncated), free of rank 1", "map": "multiplication by t^2 * u with u(0) = 2", "map_nonzero": true, "vanishing_order": 2, "value_at_zero": 0, "annihilates_anything": false, "D_style_nonannihilation": "PASS", "E_style_leading_coefficient": "NOT CERTIFIED — this gate emits no leading-coefficient equality", "requirement_enforced": true, "ok": true }, "B": { "module": "Z_11, free of rank 1", "map": "multiplication by 11", "rational_isomorphism": true, "integral_image": "11^1 * Z_11", "integral_index": 11, "integral_unit_isomorphism": false, "requirement_enforced": true, "ok": true }, "what_A_separates": "a nonzero map with positive vanishing order from an annihilating map", "what_B_separates": "a rational isomorphism from an integral unit isomorphism", "switching_a_requirement_off_is_itself_a_defect": true, "agrees": true }, "E_input": [ "the one relevant tangent direction (the cyclotomic one), with no directional choice left", "the local Bloch-Kato factor index 11 at 11, owner named, already counted once — read it, do not multiply by it again", "lambda(L_E) = 2 and mu(L_E) = 0 from mod-11 data; the EXACT order of vanishing at t = 0 over Z_11 is UNMEASURED and is E's to establish", "C = c^*_f(r) in L^x, C_d = 26 a unit, L_p(g x tau) a unit, log^{[r+1]} a non-zero-divisor and not a unit", "D-rational = ISOMORPHISM over V; D-integral = index 11 over T — E must not conflate them", "the calibrator's series is a Lambda-unit and the target's is not; the calibrator cannot carry the target's leading-term claim" ], "not_claimed": [ "the order of vanishing of L_p(g_E x psi) at t = 0 over Z_11", "the leading coefficient, the leading tensor, or any leading-term equality", "ord = rank", "nondegeneracy of any p-adic height (Schneider, open)", "the complex BSD comparison", "epsilon", "E, F, G", "BSD" ], "claim_labels": { "COMPUTED_HERE": [ "L_p(g_E x psi) != 0 in Lambda: its reduction mod 11 is nonzero, so mu = 0", "L_p(g_E x psi) is not a unit in Lambda: it reduces to 0 modulo (11, t)", "lambda(L_E) = 2 as the order of the reduction, with the Weierstrass split L_E = t^2 u and u^-1 exhibited", "L_E is a non-zero-divisor: ord(L_E g) = 2 + ord(g) on every trial, so nothing is annihilated", "the directional counterexample on the tree's own series: nonzero while the value and the first derivative at 0 both vanish", "the calibrator's L_p(g_0 x psi) IS a Lambda-unit (constant term 9 mod 11), and the two twisted series are units", "C_d = 26 is a unit mod 11" ], "CITED_THEOREM": [ "LR Proposition C1.1: rank one over H_Gamma, Kato's element canonical, regulator image not a zero-divisor, Greenberg Selmer group zero — D-NV2", "LR Theorem C1.13: the degeneration with C = c^*_f(r) in L^x, and X^n kappa-hat_d nonzero mod X — D-NV1", "LR Corollary C1.9: the factorisation and the partner-independence of c_f (NOT its non-vanishing clause, which excepts k = r)", "LR C1.6 / C1.7: the tau-quotient image is zero, the lift is unique and lands in the psi-quotient — the definition of Q", "Kato's explicit reciprocity law for g" ], "DERIVED_FROM_VERIFIED_INPUTS": [ "D-rational = ISOMORPHISM, from RUN-106's v_11(log_omega(g)) = 1 and both sides one-dimensional", "D-integral = index 11, from RUN-106's log_omega(H^1_f(Q_11,T)) = 11 Z_11", "the arrow-by-arrow route, read from RUN-097's chain and RUN-098's localisation audit", "the inherited LR hypothesis list, discharged at RUN-094/095" ], "INHERITED": [ "log^{[r+1]} a non-zero-divisor and not a unit; L_p(g x tau) a unit; the C6 period frame" ], "UNMEASURED": [ "the exact order of vanishing of L_p(g_E x psi) at t = 0 over Z_11", "the leading coefficient and the leading tensor (E)", "nondegeneracy of the p-adic height (Schneider)", "epsilon" ], "FAILED": [] }, "status": { "normalization_certificate": "CLOSED (architecture), frozen", "DAG": { "A": "PASS (cited-theorem route)", "B": "PASS", "C": "PASS (C_NT PASS, C_p PARTIAL)", "D": "PASS", "E-G": "untouched" }, "BSD": "OPEN" } }