{"certification":{"candidate":{"gauge":{"kind":"none"},"index_shift":0,"operator":{"intercept":"0","shape":"affine","slope":"1"},"parent":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["0"],["2"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","1"],["2","2"],["4","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-1","-1"],["0","-2","-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":[["0","0","0"],["0","0","5"],["0","0","2"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-30","-18"],["-32","-16"],["-8","-4"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["12"],["11"],["2"]]},"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","2"]],"recurrence":{"denominator":{"coeffs":[["-2"],["3"],["2"]]},"terms":[{"coeff":{"coeffs":[["0","0","2"],["0","0","3"],["0","0","-2"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-2","2"],["0","0"],["8","4"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"input_valid_from_base":0,"multiplier":["24","50","35","10","1"],"operator":{"intercept":"0","shape":"affine","slope":"1"},"parent":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["0"],["2"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","1"],["2","2"],["4","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-1","-1"],["0","-2","-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":[["0","0","60","60"],["0","0","119","119"],["0","0","78","78"],["0","0","21","21"],["0","0","2","2"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","-60","-120","-60"],["0","0","-79","-158","-79"],["0","0","-32","-64","-32"],["0","0","-4","-8","-4"]]},"deriv_order":1}],"shift":0},{"differentiations":[{"coeff":{"coeffs":[["0","36","36"],["0","81","81"],["0","62","62"],["0","19","19"],["0","2","2"]]},"deriv_order":0}],"shift":1}],"row_annihilation":{"annihilator":[{"coeff":{"coeffs":[["0"],["2"],["1"]]},"deriv_order":0},{"coeff":{"coeffs":[["-1","-1"],["-2","-2"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","1","1"]]},"deriv_order":2}],"kind":"single","quotient":[{"coeff":{"coeffs":[["0","0","180","360","180"],["0","0","357","714","357"],["0","0","254","508","254"],["0","0","76","152","76"],["0","0","8","16","8"]]},"deriv_order":0}],"row_multiplier":["0","1","1"],"start_base":0,"step_multiplier":{"coeffs":[["0","0","1","2","1"]]},"step_quotient":[{"coeff":{"coeffs":[["0","0","9","24","21","6"],["0","0","12","32","28","8"],["0","0","4","11","10","3"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","-3","-9","-9","-3"],["0","0","0","-2","-6","-6","-2"]]},"deriv_order":1}]},"shifted_parent_relation":{"terms":[{"coeff":{"coeffs":[["-6","-6"],["-10","-8"],["-4","-3"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","3","3"],["0","2","2"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["3"],["4"],["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":{"denominator":{"coeffs":[["0"],["2"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","1"],["2","2"],["4","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-1","-1"],["0","-2","-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":{"route":"iterated_preserver","statement":"every parent row P_n, n >= 0, is zero or real-rooted with coefficients of one sign","status":"proved"},"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":"proved"},"notes":["the division remainder is removed by row annihilators of the shifted parent"],"parent_proof":{"attempts":[{"detail":"the leader is not a nonzero constant","route":"periodic_product","success":false,"variant":"P_(n+k) = c(x) P_n"},{"detail":"not applicable: expected one nonzero lag term","route":"product_recurrence","success":false,"variant":"product_recurrence"},{"detail":"no theorem matched (weighted_renewal_window: LHS must be a nonzero constant; fixed_delay_predecessor_window: LHS must be a nonzero constant)","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":"Unresolved: P_n = a_n(x) P_(n-1) + b_n(x) P'_(n-1) for n >= 1, with a_n(x) = ((4 n^2 + 2 n)/(n^2 + 2 n)) + ((3 n^2 + 2 n + 1)/(n^2 + 2 n)) x and b_n(x) = ((0)/(n^2 + 2 n)) + ((-2 n - 1)/(n^2 + 2 n)) x + ((-2 n - 1)/(n^2 + 2 n)) x^2; open hypotheses: 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 = -1 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 = -1, U = 0, alpha_n = (3 n^2 + 2 n + 1)/(n^2 + 2 n), beta_n = (4 n^2 + 2 n)/(n^2 + 2 n), kappa_n = (-2 n - 1)/(n^2 + 2 n); open hypotheses: slope_positive: fails at n = 1: a_n = (-3/2 n^2 - n - 1/2)/(n + 1/2) = -2 there (exact); upper_endpoint_threshold: no hit before n = 1; E(1) = e(1) + 0 = -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":"not applicable: the theorem needs two seed rows","route":"liu_wang_three_term","success":false,"variant":"liu_wang_three_term"},{"detail":"induction start moved 1 row(s) past the seeds: Q_n = x^(d_n) P_n(1/x) at the nominal degree d_n, with d_(n-L) = n - 1 (an upper bound of deg P_(n-L)): Q_n = a_n(x) S_n(Q_(n-1)) for n >= 2, with a_n(x) = (1) ((1)) and S_n = (A + B D + C D^2 with A = 2 + 3 n + n^2 + x + 3 n x + 2 n^2 x, B = x + 2 n x + x^2 + 2 n x^2, C = 0) / (n^2 + 2 n): every P_n, n >= 0, is nonzero and real-rooted with every root in (-inf, 0]; deg Q_n = deg Q_(n-1) + 1 for n >= 2 (Q_n = x^(d_n) P_n(1/x) at the nominal degree d_n, with d_(n-L) = n - 1 (an upper bound of deg P_(n-L))); the rule is applied from n = 2 on in the induction (start moved 1 row(s) past the supplied seeds; P_0, ..., P_1 enter as seed rows, checked exactly)","route":"iterated_preserver","success":true,"variant":"iterated_preserver"}],"conditional_on":[],"route":"iterated_preserver","rows_checked":10,"statement":"every parent row P_n, n >= 0, is zero or real-rooted with coefficients of one sign","status":"proved"},"rows_compared":16,"schema":"polytool.recurrence-preserver-image.v1","statement":"proved: Q_n = lambda(n) T_n(P_(n+0)) with T = P((1) x + (0)) for every n >= 0, the side conditions hold for every n, and every parent row P_n, n >= 0, is zero or real-rooted with coefficients of one sign; so every Q_n, n >= 0, is zero or real-rooted with roots in (-inf, 0]"},"kind":"definition_equality"}