{"input":{"first_index":0,"initial_polynomials":[["1"],["0","3"],["0","0","6"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","-3"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","0","1"]]},"deriv_order":0,"offset":3,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"kind":"hadamard_factor","lift":{"factor":{"terms":[{"kind":"inverse_factorial"}]},"factor_text":"1/k!","gauge":{"kind":"none"},"hidden_coefficients":"zero","identity":{"certificate":{"conjugated_input_relation":{"terms":[{"coeff":{"coeffs":[["0","0","0","-6"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","0","0","0","-18"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["0","0","0","0","0","-9"]]},"deriv_order":2,"shift":0},{"coeff":{"coeffs":[["0","0","0","0","0","0","-1"]]},"deriv_order":3,"shift":0},{"coeff":{"coeffs":[["0","0","6"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["0","0","0","12"]]},"deriv_order":1,"shift":1},{"coeff":{"coeffs":[["0","0","0","0","3"]]},"deriv_order":2,"shift":1},{"coeff":{"coeffs":[["0","-3"]]},"deriv_order":0,"shift":2},{"coeff":{"coeffs":[["0","0","-3"]]},"deriv_order":1,"shift":2},{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"shift":3}]},"conjugation_x_shift":0,"exceptional_bases":[],"factor":{"terms":[{"kind":"inverse_factorial"}]},"gauge":{"kind":"none"},"identifying_indices":[0,1,2],"input_leader_shift":3,"input_relation":{"terms":[{"coeff":{"coeffs":[["0","0","0","-1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","0","3"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["0","-3"]]},"deriv_order":0,"shift":2},{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"shift":3}]},"input_valid_from_base":0,"multiplier":["1"],"parent":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","3","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_leader_shift":1,"parent_relation":{"terms":[{"coeff":{"coeffs":[["0","-3"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","0","-1"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"shift":1}]},"parent_valid_from_base":0,"quotients":[{"differentiations":[{"coeff":{"coeffs":[["0","0","0","2"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","0","0","0","1"]]},"deriv_order":2}],"shift":0},{"differentiations":[{"coeff":{"coeffs":[["0","0","-2"]]},"deriv_order":1}],"shift":1},{"differentiations":[{"coeff":{"coeffs":[["1"]]},"deriv_order":0}],"shift":2}],"row_multiplier":["1"],"statement":"mu(x) w(n) M'' = sum_nu H_nu S^nu L with M'' the input relation conjugated by lambda(n) 1/k!; so the candidate rows satisfy the input rule at every base n >= 0 (the exceptional bases checked exactly), and agreement on the 3 identifying rows gives Q_n = lambda(n) sum_k u(n,k) [x^k]P_n x^k for every n >= 0"},"status":"certified"},"lift_summary":{"hypotheses":[{"evidence":"1/k!: sum_k binom(m,k) x^k / k! = L_m(-x) is real-rooted with negative zeros for every m (Branden, Example 41); second condition of Theorem 40","name":"factor_multiplier_sequence","statement":"k -> 1/k! is a nonnegative multiplier sequence for every n >= 0","status":{"kind":"verified"}}],"identity":{"method":"conjugation_division","status":"certified"},"input_first_index":0,"input_interlacing":{"reason":"the parent route iterated_preserver proves no interlacing of consecutive parent rows","status":"not_proved"},"lean_replay":"not_attempted","operation":{"factor":"1/k!","kind":"hadamard_factor"},"parent":{"index_shift":0,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","3","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"relation":"Q_n = lambda(n) sum_k 1/k! [x^k]P_n x^k 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]: a nonnegative multiplier sequence keeps real-rootedness and one coefficient sign","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":"Polya--Schur characterization of multiplier sequences (G. Polya and I. Schur, J. Reine Angew. Math. 144 (1914) 89--113; stated as Theorem 40 of P. Branden, Unimodality, log-concavity, real-rootedness and beyond, arXiv:1410.6601): a real sequence (lambda_k) is a multiplier sequence (x^k -> lambda_k x^k maps real-rooted polynomials to real-rooted polynomials or zero) iff sum_k binom(m,k) lambda_k x^k is real-rooted with zeros of one sign for every m, iff sum_k lambda_k x^k / k! (or its reflection) is C x^m e^(a x) prod (1 + alpha_j x) with a, alpha_j >= 0; a product of multiplier sequences is one (composition), and a nonzero factor lambda(n) of the row changes nothing"},"verdict":"proved"},"multiplier_sequence":"1/k!: sum_k binom(m,k) x^k / k! = L_m(-x) is real-rooted with negative zeros for every m (Branden, Example 41); second condition of Theorem 40","parent":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","3","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"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":"not applicable: expected one nonzero lag term","route":"product_recurrence","success":false,"variant":"product_recurrence"},{"detail":"no theorem matched (weighted_renewal_window: derivatives and alternating signs are unsupported; fixed_delay_predecessor_window: derivatives and alternating signs are unsupported)","route":"paper_theorem","success":false,"variant":"match_recurrence_theorems"},{"detail":"no unconditional all-index source claim with roots in (-inf, 0]","route":"root_domain","success":false,"variant":"identity (affine 1, 0)"},{"detail":"no unconditional all-index source claim with roots in (-inf, 0]","route":"root_domain","success":false,"variant":"f_polynomial"},{"detail":"no unconditional all-index source claim with roots in (-inf, 0]","route":"root_domain","success":false,"variant":"reversal"},{"detail":"no unconditional all-index source claim with roots in (-inf, 0]","route":"root_domain","success":false,"variant":"mobius (1, 0, -1, 1)"},{"detail":"no unconditional all-index source claim with roots in (-inf, 0]","route":"root_domain","success":false,"variant":"identity (affine 1, 0), induction start moved 1 row(s)"},{"detail":"no unconditional all-index source claim with roots in (-inf, 0]","route":"root_domain","success":false,"variant":"f_polynomial, induction start moved 1 row(s)"},{"detail":"no unconditional all-index source claim with roots in (-inf, 0]","route":"root_domain","success":false,"variant":"reversal, induction start moved 1 row(s)"},{"detail":"no unconditional all-index source claim with roots in (-inf, 0]","route":"root_domain","success":false,"variant":"mobius (1, 0, -1, 1), induction start moved 1 row(s)"},{"detail":"no unconditional all-index source claim with roots in (-inf, 0]","route":"root_domain","success":false,"variant":"identity (affine 1, 0), induction start moved 2 row(s)"},{"detail":"no unconditional all-index source claim with roots in (-inf, 0]","route":"root_domain","success":false,"variant":"f_polynomial, induction start moved 2 row(s)"},{"detail":"no unconditional all-index source claim with roots in (-inf, 0]","route":"root_domain","success":false,"variant":"reversal, induction start moved 2 row(s)"},{"detail":"no unconditional all-index source claim with roots in (-inf, 0]","route":"root_domain","success":false,"variant":"mobius (1, 0, -1, 1), induction start moved 2 row(s)"},{"detail":"no unconditional all-index source claim with roots in (-inf, 0]","route":"root_domain","success":false,"variant":"identity (affine 1, 0), induction start moved 3 row(s)"},{"detail":"no unconditional all-index source claim with roots in (-inf, 0]","route":"root_domain","success":false,"variant":"f_polynomial, induction start moved 3 row(s)"},{"detail":"no unconditional all-index source claim with roots in (-inf, 0]","route":"root_domain","success":false,"variant":"reversal, induction start moved 3 row(s)"},{"detail":"no unconditional all-index source claim with roots in (-inf, 0]","route":"root_domain","success":false,"variant":"mobius (1, 0, -1, 1), induction start moved 3 row(s)"},{"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) = (0) + (3) x and b_n(x) = (0) + (0) x + (1) x^2; open hypotheses: derivative_coefficient_nonpositive: b_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 = 1: gamma(n)/f(n) = 1 = 1 there (exact); constant_term_positive: fails at n = 1: a_n(0) = 0 = 0 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":"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): the factor x^2 of c_1 has no two distinct rational roots","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":"P_n = a_n(x) S_n(P_(n-1)) for n >= 1, with a_n(x) = (x) ((1)) and S_n = f(n, theta) = 3 + (1) theta: every P_n, n >= 0, is nonzero and real-rooted with every root in (-inf, 0]; deg P_n = deg P_(n-1) + 1 for n >= 1","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}}