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0"},"parent_proof":{"route":"liu_wang_three_term","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 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leading_coefficient_nonzero: fails at n = 3: f(n) a_n = 0 = 0 there (exact); first_coefficient_positive: a_3 or c_3 is undefined or a_3 = 0; recurrence_coefficient_positive: fails at n = 4: the denominator of beta_n = 81 n^8 - 864 n^7 + 3618 n^6 - 7440 n^5 + 7489 n^4 - 2736 n^3 - 628 n^2 + 480 n vanishes there; earlier_seeds: P_0 is zero or not real-rooted with every root in (-inf, 0) (exact); root_upper_bound: not attempted: the real-rootedness hypotheses do not all hold","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":"HypothesisFailed: P_n = A_n(x) P_(n-1) + V_n(x) P'_(n-1) + W_n(x) P_(n-2) for n >= 3, with A_n(x) = ((2 n^4 - 20/3 n^3 + 38/9 n^2 + 20/9 n)/(n^4 - 16/3 n^3 + 91/9 n^2 - 8 n + 20/9)) x, V_n(x) = 0, W_n(x) = ((-n^4 + 4/3 n^3 - 1/9 n^2 - 2/9 n)/(n^4 - 16/3 n^3 + 91/9 n^2 - 8 n + 20/9)) x^2; open hypotheses: lag_degree_step_nonnegative: fails at n = 3: mu_n = (-n^4 + 4/3 n^3 - 1/9 n^2 - 2/9 n)/(n^4 - 16/3 n^3 + 91/9 n^2 - 8 n + 20/9) = -15/2 there (exact); degree_step_positive: fails at n = 3: lambda_n + mu_n = (n^4 - 16/3 n^3 + 37/9 n^2 + 2 n)/(n^4 - 16/3 n^3 + 91/9 n^2 - 8 n + 20/9) = -45/14 there (exact); constant_term_positive: fails at n = 3: A_n(0) = (0)/(n^4 - 16/3 n^3 + 91/9 n^2 - 8 n + 20/9) = 0 there (exact)","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":"induction start moved 1 row(s) past the seeds: Q_n = A_n(x) Q_(n-1) + B_n(x) Q_(n-2) for n >= 4, with A_n(x) = ((2 n^4 - 20/3 n^3 + 38/9 n^2 + 20/9 n)/(n^4 - 16/3 n^3 + 91/9 n^2 - 8 n + 20/9)), B_n(x) = ((-n^4 + 4/3 n^3 - 1/9 n^2 - 2/9 n)/(n^4 - 16/3 n^3 + 91/9 n^2 - 8 n + 20/9)); Q_n = x^(n - 1) P_n(1/x) for n >= 2 (the reversal at the nominal degree deg P_2 + n - 2, an upper bound for deg P_n), which satisfies the rule below: P_0 is zero; every other P_n, n >= 0, is nonzero and real-rooted with every root in (-inf, 0] (so its coefficients have one sign); for n >= 3, Q_(n-1) interlaces Q_n (weakly, in the sense of Liu and Wang; no interlacing of the P_n is claimed); the rule is applied from n = 4 on in the induction (start moved 1 row(s) past the supplied seeds; P_0, ..., P_3 enter as seed rows, checked exactly)","route":"liu_wang_three_term","success":true,"variant":"liu_wang_three_term"}],"conditional_on":[],"route":"liu_wang_three_term","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 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