{"certification":{"candidate":{"gauge":{"kind":"none"},"index_shift":0,"modulus":2,"parent":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-1","1"],["1","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"residue":1},"identifying_rows":2,"identity":{"certificate":{"exceptional_bases":[],"gauge":{"kind":"none"},"gauged_parent_relation":{"terms":[{"coeff":{"coeffs":[["0","-1"],["-1","0"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"shift":1}]},"identifying_indices":[0,1],"index_shift":0,"input_leader_shift":2,"input_relation":{"terms":[{"coeff":{"coeffs":[["0","-1"],["0","0"],["1","0"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-1"],["-2"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"shift":2}]},"input_valid_from_base":0,"modulus":2,"multiplier":["1"],"parent_leader_shift":1,"parent_relation":{"terms":[{"coeff":{"coeffs":[["0","-1"],["-1","0"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"shift":1}]},"parent_valid_from_base":0,"quotients":[{"component":0,"differentiations":[{"coeff":{"coeffs":[["1"]]},"deriv_order":0}],"shift":0},{"component":1,"differentiations":[{"coeff":{"coeffs":[["0"],["-1"]]},"deriv_order":0}],"shift":0},{"component":1,"differentiations":[{"coeff":{"coeffs":[["1"]]},"deriv_order":0}],"shift":1}],"residue":1,"statement":"w(n) e_1^T M = sum_nu H_nu S^nu phi(L~) with remainder zero, so w(n) (M V_(2,1)(P~))_n = 0 for every base n >= 0; with the exceptional bases checked exactly, V_(2,1)(P~) satisfies the input rule at every base n >= 0, and agreement on the 2 identifying rows gives Q_n = lambda(n) V_(2,1)(P_(n+0)) for every n >= 0"},"kind":"forward_division","status":"certified"},"input_first_index":0,"lean_replay":"not_attempted","lift_summary":{"hypotheses":[{"evidence":"lambda(n) = 1","name":"gauge_finite_nonzero","statement":"lambda(n) is finite and nonzero for every n >= n0","status":{"kind":"verified"}}],"identity":{"method":"forward_division","status":"certified"},"input_first_index":0,"input_interlacing":{"reason":"the parent statement is not proved","status":"not_proved"},"lean_replay":"not_attempted","operation":{"kind":"veronese_section","modulus":2,"residue":1},"parent":{"gauge":{"kind":"none"},"index_shift":0,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-1","1"],["1","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"relation":"Q_n = lambda(n) V_(2,1)(P_(n+0)) for every n >= 0"},"parent_proof":{"labels":["oeis_catalog:A132393:lean_proved"],"statement":"every parent row P_n, n >= 0, is zero or real-rooted with coefficients of one sign (real-rootedness from the catalog flag, not proved here; one-signed coefficients proved here)","status":"attested"},"root_transfer":{"input_conclusion":"every Q_n = lambda(n) V_(2,1)(P_(n+0)), n >= 0, is zero or real-rooted with roots in (-inf, 0]","input_root_region":{"upper":{"closed":true,"value":"0"}},"parent_hypothesis":"every parent row used is zero or real-rooted with all roots in (-inf, 0] (coefficients of one sign)","parent_root_region":{"upper":{"closed":true,"value":"0"}},"theorem":"Veronese sections of Polya frequency sequences (Aissen--Schoenberg--Whitney--Edrei): if P is real-rooted with coefficients of one sign, equivalently nonzero with all roots in (-inf, 0], every section V_(r,j)(P) is zero or real-rooted with roots in (-inf, 0]"},"verdict":"conditional_on_attestation"},"notes":[],"parent_proof":{"attempts":[{"detail":"the rule is not P_(n+k) = c(x) P_n with c free of n","route":"periodic_product","success":false,"variant":"P_(n+k) = c(x) P_n"},{"detail":"no theorem matched (weighted_renewal_window: coefficients must be index-independent; fixed_delay_predecessor_window: coefficient of P_(n-1) must be 1 (+ w(n) t) after normalization)","route":"paper_theorem","success":false,"variant":"match_recurrence_theorems"},{"detail":"not applicable: the theorem needs at least two seed rows","route":"three_term_recurrence","success":false,"variant":"three_term_recurrence"},{"detail":"HypothesisFailed: P_n = a_n(x) P_(n-1) + b_n(x) P'_(n-1) for n >= 1, with a_n(x) = (n - 1) + (1) x and b_n(x) = (0) + (0) x + (0) x^2; open hypotheses: constant_term_positive: fails at n = 1: a_n(0) = n - 1 = 0 there (exact)","route":"liu_wang","success":false,"variant":"liu_wang"},{"detail":"not applicable: the theorem needs two seed rows","route":"liu_wang_derivative_lag","success":false,"variant":"liu_wang_derivative_lag"},{"detail":"not applicable: the rule is not f(n) P_n = c_0(n, x) P_(n-1) + gamma(n) (x - L)(x - U) P'_(n-1)","route":"ma_wang_interval","success":false,"variant":"ma_wang_interval"},{"detail":"every row of the bundled OEIS catalog entry A132393 is zero or real-rooted (flag lean_proved, as of 2026-10-03: a machine-checked Lean proof made outside polytool); every row has coefficients of one sign: the seeds do, the leader is positive and every combined weight is >= 0 for n >= 1, with deg P_m <= 0 + 1 m","route":"attestation","success":true,"variant":"oeis_catalog:A132393:lean_proved"}],"conditional_on":["oeis_catalog:A132393:lean_proved"],"route":"attestation","rows_checked":10,"statement":"every parent row P_n, n >= 0, is zero or real-rooted with coefficients of one sign (real-rootedness from the catalog flag, not proved here; one-signed coefficients proved here)","status":"conditional_on_attestation"},"rows_compared":16,"schema":"polytool.recurrence-veronese-certificate.v1","statement":"conditional: Q_n = lambda(n) V_(2,1)(P_(n+0)) for every n >= 0 is certified; if the attestation [\"oeis_catalog:A132393:lean_proved\"] holds, every Q_n is zero or real-rooted with nonpositive roots","verdict":"conditional_on_attestation"},"input":{"first_index":0,"initial_polynomials":[["0"],["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-3","0"],["2","0"],["0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-4","1"],["4","0"],["-1","0"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"kind":"veronese_section"}