{"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":[["-2"],["3"]]},"terms":[{"coeff":{"coeffs":[["6","-2"],["2","6"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["1","-2","-3"]]},"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":[["28","56","28"],["33","66","33"],["9","18","9"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-14","-14"],["-48","-48"],["-18","-18"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["4"],["15"],["9"]]},"deriv_order":0,"shift":2}]},"exceptional_bases":[],"gauge":{"kind":"none"},"identifying_indices":[0,1],"index_shift":0,"input":{"first_index":0,"initial_polynomials":[["1"],["8","4"]],"recurrence":{"denominator":{"coeffs":[["10"],["-21"],["9"]]},"terms":[{"coeff":{"coeffs":[["2","4","2"],["3","6","3"],["-9","-18","-9"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-10","-10"],["-24","-24"],["18","18"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"input_valid_from_base":0,"multiplier":["4","15","9"],"operator":{"intercept":"0","shape":"affine","slope":"1"},"parent":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["-2"],["3"]]},"terms":[{"coeff":{"coeffs":[["6","-2"],["2","6"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["1","-2","-3"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_leader_shift":1,"parent_valid_from_base":0,"quotients":[{"differentiations":[{"coeff":{"coeffs":[["-16","-48","-48","-16"],["-76","-212","-196","-60"],["-96","-228","-168","-36"],["-36","-72","-36","0"]]},"deriv_order":0},{"coeff":{"coeffs":[["4","0","-24","-32","-12"],["15","0","-90","-120","-45"],["9","0","-54","-72","-27"]]},"deriv_order":1}],"shift":0},{"differentiations":[{"coeff":{"coeffs":[["4","8","4"],["27","54","27"],["54","108","54"],["27","54","27"]]},"deriv_order":0}],"shift":1}],"row_annihilation":{"annihilator":[{"coeff":{"coeffs":[["0"],["-1"],["1"]]},"deriv_order":0},{"coeff":{"coeffs":[["2","2"],["-2","-2"]]},"deriv_order":1},{"coeff":{"coeffs":[["1","2","1"]]},"deriv_order":2}],"kind":"single","quotient":[{"coeff":{"coeffs":[["4","-16","-8","48","36"],["15","-60","-30","180","135"],["9","-36","-18","108","81"]]},"deriv_order":0}],"row_multiplier":["1","2","1"],"start_base":0,"step_multiplier":{"coeffs":[["1","4","6","4","1"]]},"step_quotient":[{"coeff":{"coeffs":[["0","4","16","24","16","4"],["2","14","36","44","26","6"]]},"deriv_order":0},{"coeff":{"coeffs":[["1","2","-5","-20","-25","-14","-3"]]},"deriv_order":1}]},"shifted_parent_relation":{"terms":[{"coeff":{"coeffs":[["-8","-4"],["-2","-6"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-1","2","3"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["1"],["3"]]},"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":[["-2"],["3"]]},"terms":[{"coeff":{"coeffs":[["6","-2"],["2","6"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["1","-2","-3"]]},"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":"hypergeometric_row","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":"HypothesisFailed: P_n = a_n(x) P_(n-1) + b_n(x) P'_(n-1) for n >= 1, with a_n(x) = ((2/3 n + 2)/(n - 2/3)) + ((2 n - 2/3)/(n - 2/3)) x and b_n(x) = ((1/3)/(n - 2/3)) + ((-2/3)/(n - 2/3)) x + ((-1)/(n - 2/3)) x^2; open hypotheses: derivative_coefficient_vanishes_at_zero: [x^0] c_1(n, x) = 1 is not the zero polynomial; 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 = 0 (not a disproof): fails at n = 1: [x^0] 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 = 1/3, alpha_n = (2 n - 2/3)/(n - 2/3), beta_n = (2/3 n + 2)/(n - 2/3), kappa_n = (-1)/(n - 2/3); open hypotheses: slope_positive: fails at n = 1: a_n = -2 n + 2/3 = -4/3 there (exact); upper_endpoint_threshold: no hit before n = 1; E(1) = e(1) + 0 = -7/3 < 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":"HypothesisFailed: P_n = a_n(x) S_n(P_(n-1)) for n >= 1, with a_n(x) = (1) ((1)) and S_n = (A + B D + C D^2 with A = 6 + 2 n - 2 x + 6 n x, B = 1 - 2 x - 3 x^2, C = 0) / (3 n - 2); open hypotheses: weyl_step_leading_coefficients_same_sign: fails at n = 1: A_1 B_lead = -18 n + 6 = -12 there (exact); weyl_step_interlacing: fails at n = 1: B_2 (B_2 A_0^2 - B_1 A_0 A_1 + B_0 A_1^2) = -144 n^2 + 96 n + 384 = 336 there (exact); weyl_step_leading_coefficients_one_signed: fails at n = 1: A_a B_(a+1) = -18 n + 6 = -12 there (exact)","route":"iterated_preserver","success":false,"variant":"iterated_preserver"},{"detail":"row relation p(n, k) [x^k] P_n + q(n, k) [x^(k-1)] P_n = 0, n >= 0, with p = (1) (k - n - 2) (k), q = (1) (k - n - 1) (k - n - 1); valuation v(n) = 0, degree d(n) = n: every P_n, n >= 0, is nonzero and real-rooted with every root in (-inf, 0): for n >= 0 the coefficients U_j (U_j = [x^(d(n) - j)] P_n, d(n) = n) satisfy U_j = U_0(n) prod_(t <= j) R(n, t), R(n, j) = (-1) (j + 1) (j - n - 1) / ((j) (j)), so sum_j U_j y^j is U_0(n) times the image of (1 + y)^(n) under a nonnegative multiplier sequence, and U_0(n) != 0 for n >= 0 by its scalar recurrence","route":"hypergeometric_row","success":true,"variant":"hypergeometric_row"}],"conditional_on":[],"route":"hypergeometric_row","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"}