{"certification":{"candidate":{"gauge":{"denominator":["1"],"initial_value":"1/8","kind":"ratio","numerator":["1/2"]},"index_shift":2,"operator":{"coefficients":[{"denominator":["1"],"numerator":["0"]},{"denominator":["1"],"numerator":["1"]}],"shape":"derivative_polynomial"},"parent":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","0"],["0","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","2","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_name":"n-41809911e05bb4ba"},"identifying_rows":2,"identity":{"certificate":{"certificate":{"dividend_relation":{"terms":[{"coeff":{"coeffs":[["8","16","24"],["2","12","26"],["0","2","9"],["0","0","1"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["0","4","8","-12"],["0","1","6","-7"],["0","0","1","-1"]]},"deriv_order":2,"shift":0},{"coeff":{"coeffs":[["-5","-9"],["-3/2","-6"],["0","-1"]]},"deriv_order":1,"shift":1},{"coeff":{"coeffs":[["0","-3/2","3/2"],["0","-1/2","1/2"]]},"deriv_order":2,"shift":1},{"coeff":{"coeffs":[["3/4"],["1/4"]]},"deriv_order":1,"shift":2}]},"exceptional_bases":[],"gauge":{"denominator":["1"],"initial_value":"1/8","kind":"ratio","numerator":["1/2"]},"identifying_indices":[1,2],"index_shift":2,"input":{"first_index":1,"initial_polynomials":[["11","22","3"],["26","132","78","4"]],"recurrence":{"denominator":{"coeffs":[["1"],["1"]]},"terms":[{"coeff":{"coeffs":[["4","2"],["3","4"],["0","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1","-1"],["0","1","-1"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-4","0","0"],["-2","-4","-2"],["0","-2","-3"],["0","0","-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","-2","0","2"],["0","-1","-2","3"],["0","0","-1","1"]]},"deriv_order":1,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"input_valid_from_base":1,"multiplier":["1"],"operator":{"coefficients":[{"denominator":["1"],"numerator":["0"]},{"denominator":["1"],"numerator":["1"]}],"shape":"derivative_polynomial"},"parent":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","0"],["0","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","2","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_leader_shift":1,"parent_valid_from_base":1,"quotients":[{"differentiations":[{"coeff":{"coeffs":[["6"],["7/2"],["1/2"]]},"deriv_order":0},{"coeff":{"coeffs":[["-2","-6"],["-1/2","-7/2"],["0","-1/2"]]},"deriv_order":1}],"shift":0},{"differentiations":[{"coeff":{"coeffs":[["3/4"],["1/4"]]},"deriv_order":1}],"shift":1}],"shifted_parent_relation":{"terms":[{"coeff":{"coeffs":[["-2","-6"],["0","-2"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","-2","2"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"shift":1}]},"statement":"w(n) 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+2) (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) D; so the candidate rows satisfy the input rule at every base n >= 1 (the exceptional bases checked exactly), and agreement on the 2 identifying rows gives Q_n = lambda(n) T_n(P_(n+2)) for every n >= 1"},"method":"operator_division"},"status":"certified"},"input_first_index":1,"interlacing":{"from_index":2,"justification":"the parent theorem (LiuWang) proves P_(m-1) weakly interlacing P_m for every m >= 1; T = phi(n, D) = (1) D 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":["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(n0) = 1/8 != 0 and lambda(n) / lambda(n - 1) = numerator(n) / denominator(n) != 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of lambda(n) / lambda(n - 1) = numerator(n) / denominator(n) at n = 2 + u have one sign)","name":"gauge_finite_nonzero","sign_decision":{"certificate":{"condition":"nonzero","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["1/2"]},"numerator":["1/2"],"range":{"first":"2","last":null}},"decision":"holds"},"statement":"lambda(n) is finite and nonzero for every n >= n0","status":{"kind":"verified"}},{"evidence":"delta(n) != 0 for every integer n in [1, +inf) (exact; Taylor shift: the coefficients of delta(n) at n = 1 + u have one sign)","name":"operator_coefficients_defined","sign_decision":{"certificate":{"condition":"nonzero","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["1"]},"numerator":["1"],"range":{"first":"1","last":null}},"decision":"holds"},"statement":"the operator coefficients are finite at every n >= n0 (their denominators do not vanish)","status":{"kind":"verified"}}],"identity":{"method":"forward_division","status":"certified"},"input_first_index":1,"input_interlacing":{"interlacing":{"from_index":2,"justification":{"premises":["the parent theorem (LiuWang) proves P_(m-1) weakly interlacing P_m for every m >= 1; T = phi(n, D) = (1) D 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":"phi(n, D) = (1) D","kind":"preserver_image","operator":{"coefficients":[{"denominator":["1"],"numerator":["0"]},{"denominator":["1"],"numerator":["1"]}],"shape":"derivative_polynomial"}},"parent":{"gauge":{"denominator":["1"],"initial_value":"1/8","kind":"ratio","numerator":["1/2"]},"index_shift":2,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","0"],["0","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","2","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"relation":"Q_n = lambda(n) T_n(P_(n+2)) with T = phi(n, D) = (1) D, for every n >= 1"},"parent_proof":{"route":"root_domain","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 >= 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":"proved"},"notes":[],"parent_interlacing_report":{"conclusion":{"first_index":0,"interlacing":"weak","interlacing_from":1,"root_region":{"upper":{"closed":false,"value":"0"}},"simple_roots":false,"statement":"every P_n, n >= 0, is nonzero and real-rooted with every root in (-inf, 0) (so its coefficients have one sign); deg P_n = deg P_(n-1) + 1 and P_(n-1) interlaces P_n (Liu--Wang) for n >= 1"},"hypotheses":[{"evidence":"P_0 has degree 0; checked exactly","name":"seed_roots_negative","statement":"P_0 is nonzero and real-rooted with every root < 0","status":{"kind":"verified"}},{"evidence":"lambda_n = 2 > 0 for every integer n in [1, +inf) (exact; Taylor shift: the coefficients of lambda_n = 2 at n = 1 + u have one sign)","name":"degree_step_positive","sign_decision":{"certificate":{"condition":"positive","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["2"]},"numerator":["2"],"range":{"first":"1","last":null}},"decision":"holds"},"statement":"lambda_n = [x^1] a_n + (deg P_(n-1)) [x^2] b_n > 0 for every n >= 1 (deg P_n = deg P_(n-1) + 1 with leading coefficients of one sign)","status":{"kind":"verified"}},{"evidence":"[x^0] c_1(n, x) is the zero polynomial","name":"derivative_coefficient_vanishes_at_zero","statement":"b_n(0) = 0 for every n >= 1","status":{"kind":"verified"}},{"evidence":"b_n(x) = gamma(n) w(x) / f(n) with w(x) = -2 x^2 + 2 x, which is negative on (-inf, 0) (no root there, exact): gamma(n)/f(n) = 1 >= 0 for every integer n in [1, +inf) (exact; Taylor shift: the coefficients of gamma(n)/f(n) = 1 at n = 1 + u have one sign)","name":"derivative_coefficient_nonpositive","sign_decision":{"certificate":{"condition":"nonnegative","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["1"]},"numerator":["1"],"range":{"first":"1","last":null}},"decision":"holds"},"statement":"b_n(x) <= 0 for every x < 0 and every n >= 1","status":{"kind":"verified"}},{"evidence":"a_n(0) = 2 > 0 for every integer n in [1, +inf) (exact; Taylor shift: the coefficients of a_n(0) = 2 at n = 1 + u have one sign)","name":"constant_term_positive","sign_decision":{"certificate":{"condition":"positive","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["2"]},"numerator":["2"],"range":{"first":"1","last":null}},"decision":"holds"},"statement":"a_n(0) > 0 for every n >= 1","status":{"kind":"verified"}},{"evidence":"exact","name":"requested_root_region","statement":"the requested region (-inf, 0) contains (-inf, 0), the region this theorem certifies","status":{"kind":"verified"}}],"lean_replay":"not_attempted","normal_form":"P_n = a_n(x) P_(n-1) + b_n(x) P'_(n-1) for n >= 1, with a_n(x) = (2) + (2 n) x and b_n(x) = (0) + (2) x + (-2) x^2","recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","0"],["0","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","2","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"reference":"L. L. Liu and Y. Wang, A unified approach to polynomial sequences with only real zeros, Adv. in Appl. Math. 38 (2007) 542-560, Theorem 2.1 with g = f' (as in their Corollary 2.4), with nonnegativity of the coefficients replaced by an exact induction keeping every root in (-inf, 0)","requested_root_region":{"upper":{"closed":false,"value":"0"}},"schema":"polytool.recurrence-real-rootedness-theorem.v1","status":"proved","theorem":"liu_wang"},"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: coefficients must be index-independent; fixed_delay_predecessor_window: derivatives and alternating signs are unsupported)","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":"source claim: every P_n, n >= 0, is real-rooted in (-inf, 0] (CertifiedFamilyInvariant): the completed induction for Q_n and real_mobius_equivalence (target to source; every map condition certified for all indices); deg Q_n = d_n by the degree formula, so no root of P_n is added at the image of infinity","route":"root_domain","success":true,"variant":"f_polynomial"}],"conditional_on":[],"route":"root_domain","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+2)) with T = phi(n, D) = (1) D for every n >= 1, 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 >= 1, is zero or real-rooted with roots in (-inf, 0]"},"kind":"preserver_image"}