{"certification":{"candidate":{"gauge":{"denominator":["1"],"initial_value":"1/4","kind":"ratio","numerator":["-1/2"]},"index_shift":2,"operator":{"intercept":"0","shape":"affine","slope":"-2"},"parent":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["8","8"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","16"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_name":"n-be41df8499a2681b"},"identifying_rows":1,"identity":{"certificate":{"certificate":{"dividend_relation":{"terms":[{"coeff":{"coeffs":[["1","1/2"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["4","4"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["0","8"]]},"deriv_order":2,"shift":0},{"coeff":{"coeffs":[["-1/2"]]},"deriv_order":0,"shift":1}]},"exceptional_bases":[],"gauge":{"denominator":["1"],"initial_value":"1/4","kind":"ratio","numerator":["-1/2"]},"identifying_indices":[1],"index_shift":2,"input":{"first_index":1,"initial_polynomials":[["3","-6","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-1","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["2","-4"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","4"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"input_valid_from_base":1,"multiplier":["1"],"operator":{"intercept":"0","shape":"affine","slope":"-2"},"parent":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["8","8"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","16"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_leader_shift":1,"parent_valid_from_base":1,"quotients":[{"differentiations":[{"coeff":{"coeffs":[["-1/2"]]},"deriv_order":0}],"shift":0}],"shifted_parent_relation":{"terms":[{"coeff":{"coeffs":[["-2","-1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-8","-8"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["0","-16"]]},"deriv_order":2,"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+2) (valid from base 1) and E the input rule gauged by lambda(n)/delta(n) and composed with (or conjugated through) T = P((-2) x + (0)); so the candidate rows satisfy the input rule at every base n >= 1 (the exceptional bases checked exactly), and agreement on the 1 identifying rows gives Q_n = lambda(n) T_n(P_(n+2)) for every n >= 1"},"method":"operator_division"},"status":"certified"},"input_first_index":1,"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(n0) = 1/4 != 0 and lambda(n) / lambda(n - 1) = numerator(n) / denominator(n) != 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of lambda(n) / lambda(n - 1) = numerator(n) / denominator(n) at n = 2 + u have one sign)","name":"gauge_finite_nonzero","sign_decision":{"certificate":{"condition":"nonzero","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["-1/2"]},"numerator":["-1/2"],"range":{"first":"2","last":null}},"decision":"holds"},"statement":"lambda(n) is finite and nonzero for every n >= n0","status":{"kind":"verified"}}],"identity":{"method":"forward_division","status":"certified"},"input_first_index":1,"input_interlacing":{"reason":"no parent theorem report with an interlacing conclusion","status":"not_proved"},"lean_replay":"not_attempted","operation":{"description":"P((-2) x + (0))","kind":"preserver_image","operator":{"intercept":"0","shape":"affine","slope":"-2"}},"parent":{"gauge":{"denominator":["1"],"initial_value":"1/4","kind":"ratio","numerator":["-1/2"]},"index_shift":2,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["8","8"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","16"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"relation":"Q_n = lambda(n) T_n(P_(n+2)) with T = P((-2) x + (0)), for every n >= 1"},"parent_proof":{"route":"iterated_preserver","statement":"every parent row P_n, n >= 1, is zero or real-rooted with coefficients of one sign","status":"proved"},"root_transfer":{"input_conclusion":"every Q_n, n >= 1, is zero or real-rooted with roots in [0, +inf)","input_root_region":{"lower":{"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":[],"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":"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":"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)","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 >= 2, with a_n(x) = (1) ((1)) and S_n = A + B D + C D^2 with A = 2 + x, B = 8 + 8 x, C = 16 x: every P_n, n >= 1, is nonzero and real-rooted with every root in (-inf, 0]; deg P_n = deg P_(n-1) + 1 for n >= 2; P_(n-1) interlaces P_n (weakly: the step preserves the alternation of P_(N-1) and P_N, Branden's Corollary 2.5) for n >= 2","route":"iterated_preserver","success":true,"variant":"iterated_preserver"}],"conditional_on":[],"interlacing":{"from_index":2,"justification":{"premises":["every hypothesis of the theorem is verified exactly for every index: P_n = a_n(x) S_n(P_(n-1)) for n >= 2, with a_n(x) = (1) ((1)) and S_n = A + B D + C D^2 with A = 2 + x, B = 8 + 8 x, C = 16 x","every P_n, n >= 1, is nonzero and real-rooted with every root in (-inf, 0]; deg P_n = deg P_(n-1) + 1 for n >= 2; P_(n-1) interlaces P_n (weakly: the step preserves the alternation of P_(N-1) and P_N, Branden's Corollary 2.5) for n >= 2"],"theorem":"iterated_preserver: Iterated real-rootedness preservers (induction on n): if P_m is nonzero and real-rooted with every root in a closed region R and, for every n > m, P_n = a_n(x) S_n(P_(n-1)) where S_n maps every polynomial that is zero or real-rooted with every root in R to zero or a real-rooted polynomial with every root in R and a_n is real-rooted with every root in R, then every P_n is real-rooted with every root in R (the roots of a product are the roots of its factors); an exact degree step keeps every row nonzero; S_n = A + B D + C D^2 with A = 2 + x, B = 8 + 8 x, C = 16 x: Branden's Weyl step: T = A + B D + C D^2 maps every real-rooted polynomial to a real-rooted polynomial when (i) B^2 - 4 A C >= 0 on the real line, (ii) A strictly interlaces B (deg B = deg A + 1) or strictly alternates left of B (deg B = deg A), with leading coefficients of the same sign when deg A >= 1, and (iii) T(x^k) has degree deg A + k with leading coefficients of one sign for every k >= 0 (P. Branden, Linear transformations preserving the Polya frequency property, Trans. Amer. Math. Soc. 358 (2006), Corollary 3.5); (i) is necessary for a real-rootedness preserver (J. Borcea and P. Branden, Proc. London Math. Soc. 101 (2010), Theorem 1.3); with coefficients of one sign T also keeps every root in (-inf, 0]; a linear operator preserving real-rootedness preserves the alternation of two polynomials (P. Branden, Trans. Amer. Math. Soc. 358 (2006), Corollary 2.5)"},"kind":"weak","orientation":"previous_interlaces_next"},"route":"iterated_preserver","rows_checked":10,"statement":"every parent row P_n, n >= 1, 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+2)) with T = P((-2) x + (0)) for every n >= 1, the side conditions hold for every n, and every parent row P_n, n >= 1, is zero or real-rooted with coefficients of one sign; so every Q_n, n >= 1, is zero or real-rooted with roots in [0, +inf)"},"kind":"preserver_image"}