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so the candidate rows satisfy the input rule at every base n >= 2 (the exceptional bases checked exactly), and agreement on the 3 identifying rows gives Q_n = lambda(n) T_n(P_(n+0)) for every n >= 0"},"method":"operator_division"},"status":"certified"},"input_first_index":0,"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 [0, 1] is fixed; exact check: real-rooted = true, coefficients of one sign = true","name":"derivative_polynomial_real_rooted","statement":"the polynomial in D is real-rooted for every n >= n0","status":{"kind":"verified"}},{"evidence":"lambda(n) = 1","name":"gauge_finite_nonzero","statement":"lambda(n) is finite and nonzero for every n >= n0","status":{"kind":"verified"}},{"evidence":"delta(n) != 0 for every integer n in [0, +inf) (exact; 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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":"proved"},"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":"Q_0 = 1, Q_n = Q_(n-1) + w(n) t Q_(max(0, n-2)), w(n) = -2 + 2n (paper_proof_not_lean_checked): For every n >= 1: Q_n has nonnegative coefficients and constant term 1, is real-rooted, and Q_i << Q_n (oriented weak interlacing) for p(n) <= i <= n. Hence Q_A << Q_B whenever 0 <= A <= B and B - A <= 2.; the identifying seeds equal the canonical rows (Canonical), so every parent row is a canonical row","route":"paper_theorem","success":true,"variant":"fixed_delay_predecessor_window"}],"conditional_on":[],"interlacing":{"from_index":1,"justification":{"premises":["p(n) = max(0, n-2) with 2 >= 1 is nondecreasing and 0 <= p(n) < n for n >= 1 (exact_structural)","w(n) >= 0 for every integer n >= 1: w(1+m) has nonnegative coefficients [0, 2] in m (exact_symbolic_certificate)","window: Q_A << Q_B for 0 <= A <= B with B - A <= 2 (B - A <= 2)","the identifying parent seeds equal the canonical rows (Canonical), so every parent row is a canonical row and consecutive parent rows are consecutive canonical rows"],"theorem":"fixed_delay_predecessor_window (paper_proof_not_lean_checked; https://github.com/PerAlexandersson/RealRooted/issues/2037): For every n >= 1: Q_n has nonnegative coefficients and constant term 1, is real-rooted, and Q_i << Q_n (oriented weak interlacing) for p(n) <= i <= n. Hence Q_A << Q_B whenever 0 <= A <= B and B - A <= 2."},"kind":"weak","orientation":"previous_interlaces_next"},"route":"paper_theorem","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 = phi(n, D) = (1) D 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"}