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so the candidate rows satisfy the input rule at every base n >= 0 (the exceptional bases checked exactly), and agreement on the 1 identifying rows gives Q_n = lambda(n) T_n(P_(n+1)) for every n >= 0"},"method":"operator_division"},"status":"certified"},"input_first_index":0,"interlacing":{"reason":"the schedule d_n depends on n","status":"none"},"lean_declarations":["RealRooted.X_pow_mul_reverse_eq_zero_or_splits","RealRooted.reflect_eq_zero_or_splits"],"lean_route":"reversed","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":"the schedule d_n depends on n","status":"not_proved"},"lean_replay":"not_attempted","operation":{"description":"x^(d_n) P(1/x), d_n = 0 + 3 (n - 0)","kind":"preserver_image","operator":{"schedule":{"first_degree":0,"first_index":0,"step":3},"shape":"reversal"}},"parent":{"gauge":{"kind":"none"},"index_shift":1,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["1"],["0","1","1"]],"recurrence":{"denominator":{"coeffs":[["-2"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0","-2"],["0","0","1"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"relation":"Q_n = lambda(n) T_n(P_(n+1)) with T = x^(d_n) P(1/x), d_n = 0 + 3 (n - 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":"HypothesisFailed: B(n) P_n(x) = A(n,x) P_(n-2)(x), n >= 3; open hypotheses: earlier_seeds: P_2 is zero or not real-rooted with every root in (-inf, 0) (exact); multiplier_fixed_factor: the fixed factor fails the exact root check","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":"no unconditional all-index source claim with roots in (-inf, 0]","route":"root_domain","success":false,"variant":"identity (affine 1, 0)"},{"detail":"no affine degree schedule fits the first parent rows","route":"root_domain","success":false,"variant":"f_polynomial"},{"detail":"no affine degree schedule fits the first parent rows","route":"root_domain","success":false,"variant":"reversal"},{"detail":"no affine degree schedule fits the first parent rows","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 affine degree schedule fits the first parent rows","route":"root_domain","success":false,"variant":"f_polynomial, induction start moved 1 row(s)"},{"detail":"no affine degree schedule fits the first parent rows","route":"root_domain","success":false,"variant":"reversal, induction start moved 1 row(s)"},{"detail":"no affine degree schedule fits the first parent rows","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 affine degree schedule fits the first parent rows","route":"root_domain","success":false,"variant":"f_polynomial, induction start moved 2 row(s)"},{"detail":"no affine degree schedule fits the first parent rows","route":"root_domain","success":false,"variant":"reversal, induction start moved 2 row(s)"},{"detail":"no affine degree schedule fits the first parent rows","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 affine degree schedule fits the first parent rows","route":"root_domain","success":false,"variant":"f_polynomial, induction start moved 3 row(s)"},{"detail":"no affine degree schedule fits the first parent rows","route":"root_domain","success":false,"variant":"reversal, induction start moved 3 row(s)"},{"detail":"no affine degree schedule fits the first parent rows","route":"root_domain","success":false,"variant":"mobius (1, 0, -1, 1), induction start moved 3 row(s)"},{"detail":"not applicable: the rule is not f(n) P_n = c_1(n, x) P_(n-1) + c_2(n, x) P_(n-2) with both terms nonzero","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) = 0, V_n(x) = 0, W_n(x) = ((n - 2)/(n - 2)) x^2; open hypotheses: seed_roots_negative: P_2 is zero, not real-rooted, or has a root >= 0 (exact); seed_degree_step: degrees 0 and 2 (exact); seed_strict_interlacing: the Bezout test fails or the degrees do not differ by one (exact); constant_term_positive: fails at n = 3: A_n(0) = (0)/(n - 2) = 0 there (exact); lag_coefficient_nonpositive: W_n(x) = gamma(n) w(x) / f(n) with w(x) = -2 x^2, which is negative on (-inf, 0) (no root there, exact): fails at n = 3: gamma(n)/f(n) = (-1/2 n + 1)/(n - 2) = -1/2 there (exact); derivative_or_lag_coefficient_negative: W_n(x) = gamma(n) w(x) / f(n) with w(x) = -2 x^2, which is negative on (-inf, 0) (no root there, exact): fails at n = 3: gamma(n)/f(n) = (-1/2 n + 1)/(n - 2) = -1/2 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":"not applicable: the rule is not f(n) P_n = c_1(n, x) P_(n-1) + c_d(n, x) P'_(n-1) + c_2(n, x) P_(n-2) with a nonzero lag term and a nonzero P_(n-1) or P'_(n-1) term","route":"liu_wang_three_term","success":false,"variant":"liu_wang_three_term"},{"detail":"P_n = a_n(x) S_n(P_(n-2)) for n >= 3, with a_n(x) = (x^2) (((-2 + n)/(-2 + n))) and S_n = the identity: every P_n, n >= 0, is nonzero and real-rooted with every root in (-inf, 0]; deg P_n = deg P_(n-2) + 2 for n >= 3","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+1)) with T = x^(d_n) P(1/x), d_n = 0 + 3 (n - 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":"preserver_image"}