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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":{"from_index":2,"justification":"the parent theorem (ThreeTermRecurrence) proves P_(m-1) strictly interlacing P_m for every m >= 2; T = P((1) x + (0)) is independent of n and maps every real-rooted polynomial to a real-rooted polynomial or zero, so by the Hermite--Kakeya--Obreschkoff theorem (both directions) T(P_(m-1)) and T(P_m) have weakly interlacing zeros; the nonzero gauge does not move zeros, so Q_(n-1) and Q_n interlace weakly (orientation not determined) for every n >= 2","kind":"weak","orientation":"unordered","status":"claimed"},"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":1,"input_interlacing":{"interlacing":{"from_index":2,"justification":{"premises":["the parent theorem (ThreeTermRecurrence) proves P_(m-1) strictly interlacing P_m for every m >= 2; T = P((1) x + (0)) is independent of n and maps every real-rooted polynomial to a real-rooted polynomial or zero, so by the Hermite--Kakeya--Obreschkoff theorem (both directions) T(P_(m-1)) and T(P_m) have weakly interlacing zeros; the nonzero gauge does not move zeros, so Q_(n-1) and Q_n interlace weakly (orientation not determined) for every n >= 2"],"theorem":"Hermite--Kakeya--Obreschkoff theorem (N. Obreschkoff, Verteilung und Berechnung der Nullstellen reeller Polynome, 1963; see S. Fisk, Polynomials, Roots, and Interlacing, arXiv:math/0612833): real polynomials f, g have weakly interlacing zeros iff alpha f + beta g is real-rooted or zero for all real alpha, beta; a linear T independent of n that maps every real-rooted polynomial to a real-rooted polynomial or zero therefore maps interlacing pairs to pairs with weakly interlacing zeros (orientation not determined)"},"kind":"weak","orientation":"unordered"},"status":"proved"},"lean_replay":"not_attempted","operation":{"description":"P((1) x + (0))","kind":"preserver_image","operator":{"intercept":"0","shape":"affine","slope":"1"}},"parent":{"gauge":{"kind":"none"},"index_shift":0,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["2","1"]],"recurrence":{"denominator":{"coeffs":[["1"],["-1"]]},"terms":[{"coeff":{"coeffs":[["6","3"],["-4","-2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"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+0)) with T = P((1) x + (0)), for every n >= 1"},"parent_proof":{"route":"three_term_recurrence","statement":"every parent row P_n, n >= 1, is zero or real-rooted with coefficients of one sign","status":"proved"},"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":"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_interlacing_report":{"conclusion":{"first_index":1,"interlacing":"strict","interlacing_from":2,"root_region":{"upper":{"closed":false,"value":"0"}},"simple_roots":true,"statement":"every P_n, n >= 1, is nonzero and real-rooted, with simple roots from n = 1 on; P_(n-1) strictly interlaces P_n for n >= 2; every root lies in (-inf, 0)"},"hypotheses":[{"evidence":"f(n) = -n + 1 != 0 for every integer n in [3, +inf) (exact; Taylor shift: the coefficients of f(n) = -n + 1 at n = 3 + u have one sign)","name":"left_factor_nonzero","sign_decision":{"certificate":{"condition":"nonzero","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["-2","-1"]},"numerator":["1","-1"],"range":{"first":"3","last":null}},"decision":"holds"},"statement":"the left-hand factor f(n) is nonzero for every n >= 3","status":{"kind":"verified"}},{"evidence":"f(n) a_n = -4 n + 6 != 0 for every integer n in [3, +inf) (exact; Taylor shift: the coefficients of f(n) a_n = -4 n + 6 at n = 3 + u have one sign)","name":"leading_coefficient_nonzero","sign_decision":{"certificate":{"condition":"nonzero","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["-6","-4"]},"numerator":["6","-4"],"range":{"first":"3","last":null}},"decision":"holds"},"statement":"a_n != 0 for every n >= 3","status":{"kind":"verified"}},{"evidence":"Q_1 has degree 0, Q_2 degree 1; simple real roots and strict interlacing checked exactly","name":"seeds_interlace","statement":"Q_1 and Q_2 have simple real roots, degrees e and e + 1, and Q_1 strictly interlaces Q_2","status":{"kind":"verified"}},{"evidence":"beta_3 = 1/12 (exact)","name":"first_coefficient_positive","statement":"beta_3 = c_3 L_1 / (a_3 L_2) > 0","status":{"kind":"verified"}},{"evidence":"beta_n = (1/16 n^2 - 1/4 n + 1/4)/(n^2 - 4 n + 15/4) > 0 for every integer n in [4, +inf) (exact; Taylor shift: the coefficients of beta_n = (1/16 n^2 - 1/4 n + 1/4)/(n^2 - 4 n + 15/4) at n = 4 + u have one sign)","name":"recurrence_coefficient_positive","sign_decision":{"certificate":{"condition":"positive","denominator":["15/4","-4","1"],"method":{"kind":"taylor_shift","shifted_denominator":["15/4","4","1"],"shifted_numerator":["1/4","1/4","1/16"]},"numerator":["1/4","-1/4","1/16"],"range":{"first":"4","last":null}},"decision":"holds"},"statement":"beta_n = c_n / (a_n a_(n-1)) > 0 for every n >= 4","status":{"kind":"verified"}},{"evidence":"seeds checked exactly; positive-ratio induction at t = 0 with g(n) = (1/4 n - 1/4)/(n - 3/2) (the exact ratio r_n): g(n) > 0 for n >= 3, r_3 >= g(3) exactly, and (t - alpha_n) - beta_n/g(n-1) - g(n) >= 0 for every integer n in [4, +inf) (exact; Taylor shift: the coefficients of (t - alpha_n) - beta_n/g(n-1) - g(n) at n = 4 + u have one sign)","name":"root_upper_bound","sign_decision":{"certificate":{"condition":"nonnegative","denominator":["-135/8","783/16","-225/4","32","-9","1"],"method":{"kind":"taylor_shift","shifted_denominator":["375/8","1775/16","415/4","48","11","1"],"shifted_numerator":[]},"numerator":[],"range":{"first":"4","last":null}},"decision":"holds"},"statement":"every root of Q_n is < 0 for every n >= 1 (region (-inf, 0) for P)","status":{"kind":"verified"}}],"lean_replay":"not_attempted","normal_form":"P_n = (a_n + b_n (x + 0)) P_(n-1) - c_n (x + 0)^2 P_(n-2) for n >= 3, with a_n = (4 n - 6)/(n - 1), b_n = (2 n - 3)/(n - 1), c_n = (n - 2)/(n - 1); Q_n(y) = y^(d_n) P_n(1/y + 0), d_n = deg P_1 + n - 1, satisfies Q_n = (a_n y + b_n) Q_(n-1) - c_n Q_(n-2)","recurrence":{"first_index":1,"initial_polynomials":[["1"],["2","1"]],"recurrence":{"denominator":{"coeffs":[["1"],["-1"]]},"terms":[{"coeff":{"coeffs":[["6","3"],["-4","-2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","-2"],["0","0","1"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"reference":"three-term recurrence Q_n = (a_n y + b_n) Q_(n-1) - c_n Q_(n-2) with a_n != 0 and beta_n = c_n / (a_n a_(n-1)) > 0: simple real roots and strict interlacing of consecutive rows (zero separation for orthogonal polynomials: G. Szeg\u0151, Orthogonal Polynomials, \u00a73.3; Favard, C. R. Acad. Sci. Paris 200 (1935)); root regions by a positive-ratio (chain-sequence) induction","requested_root_region":{"upper":{"closed":false,"value":"0"}},"schema":"polytool.recurrence-real-rootedness-theorem.v1","status":"proved","theorem":"three_term_recurrence"},"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":"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":"P_n = (a_n + b_n (x + 0)) P_(n-1) - c_n (x + 0)^2 P_(n-2) for n >= 3, with a_n = (4 n - 6)/(n - 1), b_n = (2 n - 3)/(n - 1), c_n = (n - 2)/(n - 1); Q_n(y) = y^(d_n) P_n(1/y + 0), d_n = deg P_1 + n - 1, satisfies Q_n = (a_n y + b_n) Q_(n-1) - c_n Q_(n-2): every P_n, n >= 1, is nonzero and real-rooted, with simple roots from n = 1 on; P_(n-1) strictly interlaces P_n for n >= 2; every root lies in (-inf, 0)","route":"three_term_recurrence","success":true,"variant":"three_term_recurrence"}],"conditional_on":[],"interlacing":{"from_index":2,"justification":{"premises":["every hypothesis of the theorem is verified exactly for every index: P_n = (a_n + b_n (x + 0)) P_(n-1) - c_n (x + 0)^2 P_(n-2) for n >= 3, with a_n = (4 n - 6)/(n - 1), b_n = (2 n - 3)/(n - 1), c_n = (n - 2)/(n - 1); Q_n(y) = y^(d_n) P_n(1/y + 0), d_n = deg P_1 + n - 1, satisfies Q_n = (a_n y + b_n) Q_(n-1) - c_n Q_(n-2)","every P_n, n >= 1, is nonzero and real-rooted, with simple roots from n = 1 on; P_(n-1) strictly interlaces P_n for n >= 2; every root lies in (-inf, 0)"],"theorem":"three_term_recurrence: three-term recurrence Q_n = (a_n y + b_n) Q_(n-1) - c_n Q_(n-2) with a_n != 0 and beta_n = c_n / (a_n a_(n-1)) > 0: simple real roots and strict interlacing of consecutive rows (zero separation for orthogonal polynomials: G. Szeg\u0151, Orthogonal Polynomials, \u00a73.3; Favard, C. R. Acad. Sci. Paris 200 (1935)); root regions by a positive-ratio (chain-sequence) induction"},"kind":"strict","orientation":"previous_interlaces_next"},"route":"three_term_recurrence","rows_checked":10,"statement":"every parent row P_n, n >= 1, 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 >= 1, the side conditions hold for every n, and every parent row P_n, n >= 1, is zero or real-rooted with coefficients of one sign; so every Q_n, n >= 1, is zero or real-rooted with roots in (-inf, 0]"},"kind":"definition_equality"}