{ "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, "agrees": true }