{"certification":{"candidate":{"gauge":{"kind":"none"},"index_shift":0,"modulus":2,"parent":{"first_index":1,"initial_polynomials":[["1"],["2","2"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-1","-2","-1"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"residue":1},"identifying_rows":4,"identity":{"certificate":{"exceptional_bases":[],"gauge":{"kind":"none"},"gauged_parent_relation":{"terms":[{"coeff":{"coeffs":[["1","2","1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-2","-2"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"shift":2}]},"identifying_indices":[1,2,3,4],"index_shift":0,"input_leader_shift":4,"input_relation":{"terms":[{"coeff":{"coeffs":[["1","-2","1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-4","4"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["6","-2"]]},"deriv_order":0,"shift":2},{"coeff":{"coeffs":[["-4"]]},"deriv_order":0,"shift":3},{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"shift":4}]},"input_valid_from_base":1,"modulus":2,"multiplier":["1"],"parent_leader_shift":2,"parent_relation":{"terms":[{"coeff":{"coeffs":[["1","2","1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-2","-2"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"shift":2}]},"parent_valid_from_base":1,"quotients":[{"component":0,"differentiations":[{"coeff":{"coeffs":[["-2"]]},"deriv_order":0}],"shift":0},{"component":1,"differentiations":[{"coeff":{"coeffs":[["1","1"]]},"deriv_order":0}],"shift":0},{"component":0,"differentiations":[{"coeff":{"coeffs":[["2"]]},"deriv_order":0}],"shift":1},{"component":1,"differentiations":[{"coeff":{"coeffs":[["-2"]]},"deriv_order":0}],"shift":1},{"component":1,"differentiations":[{"coeff":{"coeffs":[["1"]]},"deriv_order":0}],"shift":2}],"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 >= 1; with the exceptional bases checked exactly, V_(2,1)(P~) satisfies the input rule at every base n >= 1, and agreement on the 4 identifying rows gives Q_n = lambda(n) V_(2,1)(P_(n+0)) for every n >= 1"},"kind":"forward_division","status":"certified"},"input_first_index":1,"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":1,"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":1,"initial_polynomials":[["1"],["2","2"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-1","-2","-1"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"relation":"Q_n = lambda(n) V_(2,1)(P_(n+0)) for every n >= 1"},"parent_proof":{"statement":"open: no route proved that every parent row P_n, n >= 1, 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_(2,1)(P_(n+0)), n >= 1, 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 + b_n (x + 1)) P_(n-1) - c_n (x + 1)^2 P_(n-2) for n >= 3, with a_n = 0, b_n = 2, c_n = 1; Q_n(y) = y^(d_n) P_n(1/y - 1), d_n = deg P_1 + n - 1, satisfies Q_n = (a_n y + b_n) Q_(n-1) - c_n Q_(n-2); open hypotheses: leading_coefficient_nonzero: fails at n = 3: f(n) a_n = 0 = 0 there (exact); seeds_interlace: deg Q_2 != deg Q_1 + 1 = 1; first_coefficient_positive: not checked: the seeds do not satisfy the seed hypothesis; recurrence_coefficient_positive: fails at n = 4: the denominator of beta_n = 1 vanishes there; root_region: the region contains the pole x = -1 of the coordinate change y = 1/(x + 1)","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 >= 3, with A_n(x) = (2) + (2) x, V_n(x) = 0, W_n(x) = (-1) + (-2) x + (-1) x^2; open hypotheses: lag_degree_step_nonnegative: fails at n = 3: mu_n = -1 = -1 there (exact); multipliers_vanish_at_zero: [x^0] c_2(n, x) = -1 is not the zero polynomial; lag_coefficient_nonpositive: W_n is not separable with a factor of one sign on (-inf, 0), and the sufficient monomial test (-1)^k [x^k] W_n <= 0 fails for k = 1 (not a disproof): fails at n = 3: [x^1] W_n = -2 there (exact); derivative_or_lag_coefficient_negative: W_n is not separable with a factor of one sign on (-inf, 0), and the sufficient monomial test (-1)^k [x^k] W_n <= 0 fails for k = 1 (not a disproof): fails at n = 3: [x^1] W_n = -2 there (exact)","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 >= 3","route":"attestation","success":false,"variant":"oeis_catalog:A003506:lean_proved"}],"conditional_on":[],"rows_checked":10,"statement":"open: no route proved that every parent row P_n, n >= 1, 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_(2,1)(P_(n+0)) for every n >= 1 is certified; the parent statement is open","verdict":"identity_certified_parent_open"},"input":{"first_index":1,"initial_polynomials":[["0"],["2"],["6"],["12","4"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["4","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-6","2","0"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["4","-4","0"]]},"deriv_order":0,"offset":3,"sign":"none"},{"coeff":{"coeffs":[["-1","2","-1"]]},"deriv_order":0,"offset":4,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"kind":"veronese_section"}