{"certification":{"candidate":{"gauge":{"kind":"none"},"index_shift":0,"operator":{"intercept":"0","shape":"affine","slope":"1"},"parent":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","-1"],["0","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-4","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_name":"existing definition"},"identifying_rows":2,"identity":{"certificate":{"certificate":{"dividend_relation":{"terms":[{"coeff":{"coeffs":[["-4","-2"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"shift":2}]},"exceptional_bases":[],"gauge":{"kind":"none"},"identifying_indices":[0,1],"index_shift":0,"input":{"first_index":0,"initial_polynomials":[["1"],["2"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["4","2"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"input_valid_from_base":0,"multiplier":["1"],"operator":{"intercept":"0","shape":"affine","slope":"1"},"parent":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","-1"],["0","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-4","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_leader_shift":1,"parent_valid_from_base":0,"quotients":[{"differentiations":[{"coeff":{"coeffs":[["8","12","6","1"],["0","4","4","1"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","-16","-24","-12","-2"]]},"deriv_order":1}],"shift":0},{"differentiations":[{"coeff":{"coeffs":[["4","4","1"]]},"deriv_order":0}],"shift":1}],"row_annihilation":{"annihilator":[{"coeff":{"coeffs":[["0"],["-1/4"],["1/4"]]},"deriv_order":0},{"coeff":{"coeffs":[["3","3/2"],["-2","-1"]]},"deriv_order":1},{"coeff":{"coeffs":[["4","4","1"]]},"deriv_order":2}],"kind":"single","quotient":[{"coeff":{"coeffs":[["0","0","16","16","4"]]},"deriv_order":0}],"row_multiplier":["4","4","1"],"start_base":0,"step_multiplier":{"coeffs":[["16","32","24","8","1"]]},"step_quotient":[{"coeff":{"coeffs":[["-96","-224","-208","-96","-22","-2"],["0","16","32","24","8","1"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","-64","-160","-160","-80","-20","-2"]]},"deriv_order":1}]},"shifted_parent_relation":{"terms":[{"coeff":{"coeffs":[["-2","0"],["0","-1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","4","2"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"shift":1}]},"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 2 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"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","-1"],["0","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-4","-2"]]},"deriv_order":1,"offset":1,"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: derivatives and alternating signs are unsupported)","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) = (2) + (n - 1) x and b_n(x) = (0) + (-4) x + (-2) x^2; open hypotheses: degree_step_positive: fails at n = 1: lambda_n = -n + 1 = 0 there (exact); derivative_coefficient_nonpositive: b_n is not separable with a factor of one sign on (-inf, 0), and the sufficient monomial test (-1)^k [x^k] b_n <= 0 fails for k = 1 (not a disproof): fails at n = 1: [x^1] b_n = -4 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":"HypothesisFailed: P_n = (alpha_n x + beta_n) P_(n-1) + kappa_n (x - L)(x - U) P'_(n-1) for n >= 1, with L = -2, U = 0, alpha_n = n - 1, beta_n = 2, kappa_n = -2; open hypotheses: slope_positive: fails at n = 1: a_n = -1/2 n + 1/2 = 0 there (exact); upper_endpoint_threshold: no hit before n = 1; E(1) = e(1) + 0 = -1/2 < 0 with no hit in between (exact); when a_n > 0 for every n <= 1, the criterion puts a root of P_1 outside the interval","route":"ma_wang_interval","success":false,"variant":"ma_wang_interval"},{"detail":"no lean_proved catalog entry equals the parent; 549 with the same first index and seeds differ in the rule or in later initial rows (oeis_catalog:A001263:lean_proved, oeis_catalog:A003982:lean_proved, oeis_catalog:A007318:lean_proved, 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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"}