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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 1) and E the input rule gauged by lambda(n)/delta(n) and composed with (or conjugated through) T = phi(n, D) = 1 + (1) D; 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+0)) 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":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"lift_summary":{"hypotheses":[{"evidence":"the polynomial with ascending coefficients [1, 1] is fixed; exact check: real-rooted = true, coefficients of one sign = 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here)","status":"attested"},"root_transfer":{"input_conclusion":"every Q_n, n >= 1, 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":"Laguerre's theorem on differential operators: if phi is a real polynomial with only real roots, then phi(D) maps every real-rooted polynomial to a real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n"},"verdict":"conditional_on_attestation"},"notes":["the division remainder is removed by row annihilators of the shifted parent"],"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":"no theorem matched (weighted_renewal_window: support must satisfy offset = 1 + ell * t_degree; fixed_delay_predecessor_window: only P_(n-1) and one term w(n) t P_(n-L) are supported)","route":"paper_theorem","success":false,"variant":"match_recurrence_theorems"},{"detail":"not applicable: c_2(n, x) = gamma(n) w(x) with w neither constant nor a constant times a square (x + v_0)^2","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) = (1) + (1) x, V_n(x) = 0, W_n(x) = (1) + (1) x; open hypotheses: multipliers_vanish_at_zero: [x^0] c_2(n, x) = 1 is not the zero polynomial; lag_coefficient_nonpositive: W_n is not separable with a factor of one sign on (-inf, 0), and the sufficient monomial test (-1)^k [x^k] W_n <= 0 fails for k = 0 (not a disproof): fails at n = 3: [x^0] W_n = 1 there (exact); derivative_or_lag_coefficient_negative: W_n is not separable with a factor of one sign on (-inf, 0), and the sufficient monomial test (-1)^k [x^k] W_n <= 0 fails for k = 0 (not a disproof): fails at n = 3: [x^0] W_n = 1 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":"every row of the bundled OEIS catalog entry A063967 is zero or real-rooted (flag lean_proved, as of 2026-10-03: a machine-checked Lean proof made outside polytool); every row has coefficients of one sign: the seeds do, the leader is positive and every combined weight is >= 0 for n >= 3, with deg P_m <= -1 + 1 m","route":"attestation","success":true,"variant":"oeis_catalog:A063967:lean_proved"}],"conditional_on":["oeis_catalog:A063967:lean_proved"],"route":"attestation","rows_checked":10,"statement":"every parent row P_n, n >= 1, is zero or real-rooted with coefficients of one sign (real-rootedness from the catalog flag, not proved here; one-signed coefficients proved here)","status":"conditional_on_attestation"},"rows_compared":16,"schema":"polytool.recurrence-preserver-image.v1","statement":"Q_n = lambda(n) T_n(P_(n+0)) with T = phi(n, D) = 1 + (1) D for every n >= 1 is certified; the lift is ConditionalOnAttestation (every parent row P_n, n >= 1, is zero or real-rooted with coefficients of one sign (real-rootedness from the catalog flag, not proved here; one-signed coefficients proved here))"},"kind":"preserver_image"}