{"certification":{"candidate":{"gauge":{"kind":"none"},"index_shift":0,"modulus":2,"parent":{"first_index":0,"initial_polynomials":[["1"],["3"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-1"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"residue":0},"identifying_rows":2,"identity":{"certificate":{"exceptional_bases":[],"gauge":{"kind":"none"},"gauged_parent_relation":{"terms":[{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-2"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"shift":2}]},"identifying_indices":[0,1],"index_shift":0,"input_leader_shift":2,"input_relation":{"terms":[{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-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":2,"parent_relation":{"terms":[{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-2"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"shift":2}]},"parent_valid_from_base":0,"quotients":[{"component":0,"differentiations":[{"coeff":{"coeffs":[["1"]]},"deriv_order":0}],"shift":0}],"residue":0,"statement":"w(n) e_0^T M = sum_nu H_nu S^nu phi(L~) with remainder zero, so w(n) (M V_(2,0)(P~))_n = 0 for every base n >= 0; with the exceptional bases checked exactly, V_(2,0)(P~) satisfies the input rule at every base n >= 0, and agreement on the 2 identifying rows gives Q_n = lambda(n) V_(2,0)(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":0},"parent":{"gauge":{"kind":"none"},"index_shift":0,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["3"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-1"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"relation":"Q_n = lambda(n) V_(2,0)(P_(n+0)) for every n >= 0"},"parent_proof":{"statement":"open: no route proved that every parent row P_n, n >= 0, is zero or real-rooted with coefficients of one sign; the first 10 parent rows pass the exact check (evidence only)","status":"open"},"root_transfer":{"input_conclusion":"every Q_n = lambda(n) V_(2,0)(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":"open"},"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: coefficient of P_(n-1) must be 1 after normalization; 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":"HypothesisFailed: P_n = (a_n x + b_n) P_(n-1) - c_n P_(n-2) for n >= 2, with a_n = 0, b_n = 2, c_n = 1; open hypotheses: leading_coefficient_nonzero: fails at n = 2: f(n) a_n = 0 = 0 there (exact); seeds_interlace: deg Q_1 != deg Q_0 + 1 = 1; first_coefficient_positive: not checked: the seeds do not satisfy the seed hypothesis; recurrence_coefficient_positive: fails at n = 3: the denominator of beta_n = 1 vanishes there; root_upper_bound: not attempted: the real-rootedness hypotheses do not all hold","route":"three_term_recurrence","success":false,"variant":"three_term_recurrence"},{"detail":"not applicable: the rule is not f(n) P_n = c_0(n, x) P_(n-1) + c_1(n, x) P'_(n-1)","route":"liu_wang","success":false,"variant":"liu_wang"},{"detail":"HypothesisFailed: P_n = A_n(x) P_(n-1) + V_n(x) P'_(n-1) + W_n(x) P_(n-2) for n >= 2, with A_n(x) = (2), V_n(x) = 0, W_n(x) = (-1); open hypotheses: seed_degree_step: degrees 0 and 0 (exact); seed_strict_interlacing: the Bezout test fails or the degrees do not differ by one (exact); degree_step_positive: fails at n = 2: lambda_n = 0 = 0 there (exact); multipliers_vanish_at_zero: [x^0] c_2(n, x) = -1 is not the zero polynomial","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":"the parent equals the flagged entry, but the flag gives real-rootedness only and one-signed coefficients are not shown: the weight of the coefficient of x^s of P_(n-2) in the coefficient of x^(s+0) is not shown to be >= 0 for n >= 2","route":"attestation","success":false,"variant":"oeis_catalog:A198954:lean_proved"},{"detail":"no lean_proved catalog entry equals the parent; 8 with the same first index and seeds differ in the rule or in later initial rows (oeis_catalog:A181371:lean_proved, oeis_catalog:A201013:lean_proved, oeis_catalog:A208509:lean_proved, oeis_catalog:A210039:lean_proved, oeis_catalog:A274498:lean_proved, oeis_catalog:A306364:lean_proved, oeis_catalog:A317497:lean_proved, oeis_catalog:A318773:lean_proved)","route":"attestation","success":false,"variant":"oeis_catalog"}],"conditional_on":[],"rows_checked":10,"statement":"open: no route proved that every parent row P_n, n >= 0, is zero or real-rooted with coefficients of one sign; the first 10 parent rows pass the exact check (evidence only)","status":"not_proved"},"rows_compared":16,"schema":"polytool.recurrence-veronese-certificate.v1","statement":"Q_n = lambda(n) V_(2,0)(P_(n+0)) for every n >= 0 is certified; the parent statement is open","verdict":"identity_certified_parent_open"},"input":{"first_index":0,"initial_polynomials":[["1"],["3"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-1"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"kind":"veronese_section"}