{"certification":{"candidate":{"gauge":{"kind":"none"},"index_shift":0,"operator":{"intercept":"0","shape":"affine","slope":"1"},"parent":{"first_index":0,"initial_polynomials":[["1"],["0","4"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","8"],["0","-4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","1"],["0","0","4"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_name":"existing definition"},"identifying_rows":1,"identity":{"certificate":{"certificate":{"dividend_relation":{"terms":[{"coeff":{"coeffs":[["0","-4","-4"],["0","-4","-4"],["0","-1","-1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["1","1"],["2","2"],["1","1"]]},"deriv_order":0,"shift":1}]},"exceptional_bases":[],"gauge":{"kind":"none"},"identifying_indices":[0],"index_shift":0,"input":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["0","0"],["0","0"],["1","1"]]},"terms":[{"coeff":{"coeffs":[["0","1","1"],["0","2","2"],["0","1","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"input_valid_from_base":0,"multiplier":["1","2","1"],"operator":{"intercept":"0","shape":"affine","slope":"1"},"parent":{"first_index":0,"initial_polynomials":[["1"],["0","4"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","8"],["0","-4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","1"],["0","0","4"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_leader_shift":2,"parent_valid_from_base":0,"quotients":[],"row_annihilation":{"kind":"system","system":{"generators":[{"annihilator":{"terms":[{"coeff":{"coeffs":[["0","-4"],["0","-4"],["0","-1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["1"],["2"],["1"]]},"deriv_order":0,"shift":1}]},"quotient":[{"coeff":{"coeffs":[["1","1"],["2","2"],["1","1"]]},"deriv_order":0}],"step_multiplier":{"coeffs":[["1"],["2"],["1"]]},"step_quotients":[[{"coeff":{"coeffs":[["0","-9"],["0","-22"],["0","-17"],["0","-4"]]},"deriv_order":0}],[]]},{"annihilator":{"terms":[{"coeff":{"coeffs":[["0"],["-1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","1"]]},"deriv_order":1,"shift":0}]},"quotient":[],"step_multiplier":{"coeffs":[["0","1"],["0","4"],["0","6"],["0","4"],["0","1"]]},"step_quotients":[[{"coeff":{"coeffs":[["0","-1"],["0","-3"],["0","-3"],["0","-1"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","1"],["0","0","2"],["0","0","1"]]},"deriv_order":1}],[{"coeff":{"coeffs":[["0","0","4"],["0","0","12"],["0","0","13"],["0","0","6"],["0","0","1"]]},"deriv_order":0}]]}],"restart_bases":[],"row_multiplier":{"coeffs":[["1"]]},"start_base":0}},"shifted_parent_relation":{"terms":[{"coeff":{"coeffs":[["0","0","-9"],["0","0","-4"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","0"],["0","4"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"shift":2}]},"statement":"w(n) E = sum_nu H_nu S^nu L' (with any remainder removed by row annihilators of the parent), where L' is the parent rule at P'_n = P_(n+0) (valid from base 0) and E the input rule gauged by lambda(n)/delta(n) and composed with (or conjugated through) T = P((1) x + (0)); so the candidate rows satisfy the input rule at every base n >= 0 (the exceptional bases checked exactly), and agreement on the 1 identifying rows gives Q_n = lambda(n) T_n(P_(n+0)) for every n >= 0"},"method":"operator_division"},"status":"certified"},"input_first_index":0,"interlacing":{"reason":"no parent theorem report with an interlacing conclusion","status":"none"},"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"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":"no parent theorem report with an interlacing conclusion","status":"not_proved"},"lean_replay":"not_attempted","operation":{"description":"P((1) x + (0))","kind":"preserver_image","operator":{"intercept":"0","shape":"affine","slope":"1"}},"parent":{"gauge":{"kind":"none"},"index_shift":0,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["0","4"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","8"],["0","-4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","1"],["0","0","4"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"relation":"Q_n = lambda(n) T_n(P_(n+0)) with T = P((1) x + (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, 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":"a real affine substitution x -> a x + b (a != 0), the reversal x^d p(1/x) (d >= deg p) and a real Mobius map (c x + e)^d p((a x + b)/(c x + e)) (a e - b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically"},"verdict":"open"},"notes":["the division remainder is removed by row annihilators of the shifted parent"],"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":"HypothesisFailed: P_n = (a_n + b_n (x + 0)) P_(n-1) - c_n (x + 0)^2 P_(n-2) for n >= 2, with a_n = 0, b_n = -4 n + 8, c_n = -4 n - 1; Q_n(y) = y^(d_n) P_n(1/y + 0), d_n = deg P_0 + n - 0, satisfies Q_n = (a_n y + b_n) Q_(n-1) - c_n Q_(n-2); 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 = -4 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) = (-4 n + 8) x, V_n(x) = 0, W_n(x) = (4 n + 1) x^2; open hypotheses: seed_roots_negative: P_1 is zero, not real-rooted, or has a root >= 0 (exact); derivative_degree_step_nonnegative: fails at n = 3: lambda_n = -4 n + 8 = -4 there (exact); constant_term_positive: fails at n = 2: A_n(0) = 0 = 0 there (exact); lag_coefficient_nonpositive: W_n(x) = gamma(n) w(x) / f(n) with w(x) = x^2, which is positive on (-inf, 0) (no root there, exact): fails at n = 2: gamma(n)/f(n) = 4 n + 1 = 9 there (exact); derivative_or_lag_coefficient_negative: W_n(x) = gamma(n) w(x) / f(n) with w(x) = x^2, which is positive on (-inf, 0) (no root there, exact): fails at n = 2: gamma(n)/f(n) = 4 n + 1 = 9 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-1) in the coefficient of x^(s+1) is not shown to be >= 0 for n >= 2","route":"attestation","success":false,"variant":"oeis_catalog:A127733:lean_proved"},{"detail":"no lean_proved catalog entry equals the parent; 1 with the same first index and seeds differ in the rule or in later initial rows (oeis_catalog:A131431: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-preserver-image.v1","statement":"Q_n = lambda(n) T_n(P_(n+0)) with T = P((1) x + (0)) for every n >= 0 is certified; the lift is Open (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))"},"kind":"definition_equality"}