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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 4 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 = phi(n, D) = 1 + (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 [1, 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 [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 (ThreeTermRecurrence) proves P_(m-1) strictly interlacing P_m for every m >= 2; T = phi(n, D) = 1 + (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 + (1) D","kind":"preserver_image","operator":{"coefficients":[{"denominator":["1"],"numerator":["1"]},{"denominator":["1"],"numerator":["1"]}],"shape":"derivative_polynomial"}},"parent":{"gauge":{"kind":"none"},"index_shift":0,"recurrence":{"first_index":1,"initial_polynomials":[["-8","-16","4"],["-60","-65","50","-5"]],"recurrence":{"denominator":{"coeffs":[["1"],["1/2"]]},"terms":[{"coeff":{"coeffs":[["3","-3/2"],["4","-1/2"],["1","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-3"],["-11/2"],["-3"],["-1/2"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"relation":"Q_n = lambda(n) T_n(P_(n+0)) with T = phi(n, D) = 1 + (1) D, for every n >= 1"},"parent_proof":{"route":"three_term_recurrence_whole_line","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","status":"proved"},"root_transfer":{"input_conclusion":"every Q_n, n >= 1, is zero or real-rooted with roots in (-inf, +inf)","input_root_region":{},"parent_hypothesis":"every parent row used is real-rooted (anywhere on the line)","parent_root_region":{},"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":1,"interlacing":"strict","interlacing_from":2,"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"},"hypotheses":[{"evidence":"f(n) = 1/2 n + 1 != 0 for every integer n in [3, +inf) (exact; Taylor shift: the coefficients of f(n) = 1/2 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":["5/2","1/2"]},"numerator":["1","1/2"],"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 = -1/2 n - 3/2 != 0 for every integer n in [3, +inf) (exact; Taylor shift: the coefficients of f(n) a_n = -1/2 n - 3/2 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":["-3","-1/2"]},"numerator":["-3/2","-1/2"],"range":{"first":"3","last":null}},"decision":"holds"},"statement":"a_n != 0 for every n >= 3","status":{"kind":"verified"}},{"evidence":"Q_1 has degree 2, Q_2 degree 3; 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 = 16 (exact)","name":"first_coefficient_positive","statement":"beta_3 = c_3 L_1 / (a_3 L_2) > 0","status":{"kind":"verified"}},{"evidence":"beta_n = (n^4 + 7 n^3 + 17 n^2 + 17 n + 6)/(n^2 + 5 n + 6) > 0 for every integer n in [4, +inf) (exact; Taylor shift: the coefficients of beta_n = (n^4 + 7 n^3 + 17 n^2 + 17 n + 6)/(n^2 + 5 n + 6) at n = 4 + u have one sign)","name":"recurrence_coefficient_positive","sign_decision":{"certificate":{"condition":"positive","denominator":["6","5","1"],"method":{"kind":"taylor_shift","shifted_denominator":["42","13","1"],"shifted_numerator":["1050","745","197","23","1"]},"numerator":["6","17","17","7","1"],"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"}}],"lean_replay":"not_attempted","normal_form":"P_n = (a_n x + b_n) P_(n-1) - c_n P_(n-2) for n >= 3, with a_n = (-n - 3)/(n + 2), b_n = (2 n^2 + 8 n + 6)/(n + 2), c_n = (n^3 + 6 n^2 + 11 n + 6)/(n + 2)","recurrence":{"first_index":1,"initial_polynomials":[["-8","-16","4"],["-60","-65","50","-5"]],"recurrence":{"denominator":{"coeffs":[["1"],["1/2"]]},"terms":[{"coeff":{"coeffs":[["3","-3/2"],["4","-1/2"],["1","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-3"],["-11/2"],["-3"],["-1/2"]]},"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","schema":"polytool.recurrence-real-rootedness-theorem.v1","status":"proved","theorem":"three_term_recurrence"},"parent_proof":{"attempts":[],"conditional_on":[],"failing_row_index":1,"rows_checked":10,"statement":"parent row P_1 is nonzero and not real-rooted with coefficients of one sign; this parent gives no proof","status":"refuted"},"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 + (1) D for every n >= 1, the side conditions hold for every n, and 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; so every Q_n, n >= 1, is zero or real-rooted with roots in (-inf, +inf)","whole_line_parent_proof":{"conclusion":{"first_index":1,"interlacing":"strict","interlacing_from":2,"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"},"hypotheses":[{"evidence":"f(n) = 1/2 n + 1 != 0 for every integer n in [3, +inf) (exact; Taylor shift: the coefficients of f(n) = 1/2 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":["5/2","1/2"]},"numerator":["1","1/2"],"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 = -1/2 n - 3/2 != 0 for every integer n in [3, +inf) (exact; Taylor shift: the coefficients of f(n) a_n = -1/2 n - 3/2 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":["-3","-1/2"]},"numerator":["-3/2","-1/2"],"range":{"first":"3","last":null}},"decision":"holds"},"statement":"a_n != 0 for every n >= 3","status":{"kind":"verified"}},{"evidence":"Q_1 has degree 2, Q_2 degree 3; 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 = 16 (exact)","name":"first_coefficient_positive","statement":"beta_3 = c_3 L_1 / (a_3 L_2) > 0","status":{"kind":"verified"}},{"evidence":"beta_n = (n^4 + 7 n^3 + 17 n^2 + 17 n + 6)/(n^2 + 5 n + 6) > 0 for every integer n in [4, +inf) (exact; Taylor shift: the coefficients of beta_n = (n^4 + 7 n^3 + 17 n^2 + 17 n + 6)/(n^2 + 5 n + 6) at n = 4 + u have one sign)","name":"recurrence_coefficient_positive","sign_decision":{"certificate":{"condition":"positive","denominator":["6","5","1"],"method":{"kind":"taylor_shift","shifted_denominator":["42","13","1"],"shifted_numerator":["1050","745","197","23","1"]},"numerator":["6","17","17","7","1"],"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"}}],"lean_replay":"not_attempted","normal_form":"P_n = (a_n x + b_n) P_(n-1) - c_n P_(n-2) for n >= 3, with a_n = (-n - 3)/(n + 2), b_n = (2 n^2 + 8 n + 6)/(n + 2), c_n = (n^3 + 6 n^2 + 11 n + 6)/(n + 2)","recurrence":{"first_index":1,"initial_polynomials":[["-8","-16","4"],["-60","-65","50","-5"]],"recurrence":{"denominator":{"coeffs":[["1"],["1/2"]]},"terms":[{"coeff":{"coeffs":[["3","-3/2"],["4","-1/2"],["1","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-3"],["-11/2"],["-3"],["-1/2"]]},"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","schema":"polytool.recurrence-real-rootedness-theorem.v1","status":"proved","theorem":"three_term_recurrence"}},"kind":"preserver_image"}