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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+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":["Polynomial.Splits.comp_of_degree_le_one"],"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":"no parent theorem report with an interlacing conclusion","status":"not_proved"},"lean_replay":"not_attempted","operation":{"description":"P((1) x + (1))","kind":"preserver_image","operator":{"intercept":"1","shape":"affine","slope":"1"}},"parent":{"gauge":{"kind":"none"},"index_shift":0,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-2"],["0","2"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"relation":"Q_n = lambda(n) T_n(P_(n+0)) with T = P((1) x + (1)), for every n >= 0"},"parent_proof":{"route":"paper_theorem","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, -1]","input_root_region":{"upper":{"closed":true,"value":"-1"}},"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 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 = P((1) x + (1)) 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, -1]"},"kind":"preserver_image"}