{"kind":"theorem_check","report":{"conclusion":{"first_index":1,"interlacing":"strict","interlacing_from":3,"root_region":{"lower":{"closed":false,"value":"0"},"upper":{"closed":false,"value":"4"}},"simple_roots":true,"statement":"every P_n, n >= 1, is nonzero and real-rooted, with simple roots from n = 2 on; P_(n-1) strictly interlaces P_n for n >= 3; every root lies in (0, 4)"},"hypotheses":[{"evidence":"f(n) = n - 3 != 0 for every integer n in [4, +inf) (exact; Taylor shift: the coefficients of f(n) = n - 3 at n = 4 + u have one sign)","name":"left_factor_nonzero","sign_decision":{"certificate":{"condition":"nonzero","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["1","1"]},"numerator":["-3","1"],"range":{"first":"4","last":null}},"decision":"holds"},"statement":"the left-hand factor f(n) is nonzero for every n >= 4","status":{"kind":"verified"}},{"evidence":"f(n) a_n = n - 3 != 0 for every integer n in [4, +inf) (exact; Taylor shift: the coefficients of f(n) a_n = n - 3 at n = 4 + u have one sign)","name":"leading_coefficient_nonzero","sign_decision":{"certificate":{"condition":"nonzero","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["1","1"]},"numerator":["-3","1"],"range":{"first":"4","last":null}},"decision":"holds"},"statement":"a_n != 0 for every n >= 4","status":{"kind":"verified"}},{"evidence":"Q_2 has degree 1, Q_3 degree 2; simple real roots and strict interlacing checked exactly","name":"seeds_interlace","statement":"Q_2 and Q_3 have simple real roots, degrees e and e + 1, and Q_2 strictly interlaces Q_3","status":{"kind":"verified"}},{"evidence":"beta_4 = 1 (exact)","name":"first_coefficient_positive","statement":"beta_4 = c_4 L_2 / (a_4 L_3) > 0","status":{"kind":"verified"}},{"evidence":"beta_n = (n^2 - 7 n + 12)/(n^2 - 7 n + 12) > 0 for every integer n in [5, +inf) (exact; Taylor shift: the coefficients of beta_n = (n^2 - 7 n + 12)/(n^2 - 7 n + 12) at n = 5 + u have one sign)","name":"recurrence_coefficient_positive","sign_decision":{"certificate":{"condition":"positive","denominator":["12","-7","1"],"method":{"kind":"taylor_shift","shifted_denominator":["2","3","1"],"shifted_numerator":["2","3","1"]},"numerator":["12","-7","1"],"range":{"first":"5","last":null}},"decision":"holds"},"statement":"beta_n = c_n / (a_n a_(n-1)) > 0 for every n >= 5","status":{"kind":"verified"}},{"evidence":"checked exactly for P_1, ..., P_1","name":"earlier_seeds","statement":"the seed rows P_1, ..., P_1 are nonzero and real-rooted with every root in (0, 4)","status":{"kind":"verified"}},{"evidence":"seeds checked exactly; positive-ratio induction at t = 4 with g(n) = 1 (the exact ratio r_n): g(n) > 0 for n >= 4, r_4 >= g(4) exactly, and (t - alpha_n) - beta_n/g(n-1) - g(n) >= 0 for every integer n in [5, +inf) (exact; Taylor shift: the coefficients of (t - alpha_n) - beta_n/g(n-1) - g(n) at n = 5 + u have one sign)","name":"root_upper_bound","sign_decision":{"certificate":{"condition":"nonnegative","denominator":["-36","33","-10","1"],"method":{"kind":"taylor_shift","shifted_denominator":["4","8","5","1"],"shifted_numerator":[]},"numerator":[],"range":{"first":"5","last":null}},"decision":"holds"},"statement":"every root of P_n is < 4 for every n >= 2 (region (0, 4) for P)","status":{"kind":"verified"}},{"evidence":"on the reflected rows (-1)^(deg) p_n(-y): seeds checked exactly; positive-ratio induction at t = 0 with g(n) = 1 (the exact ratio r_n): g(n) > 0 for n >= 4, r_4 >= g(4) exactly, and (t - alpha_n) - beta_n/g(n-1) - g(n) >= 0 for every integer n in [5, +inf) (exact; Taylor shift: the coefficients of (t - alpha_n) - beta_n/g(n-1) - g(n) at n = 5 + u have one sign)","name":"root_lower_bound","sign_decision":{"certificate":{"condition":"nonnegative","denominator":["-36","33","-10","1"],"method":{"kind":"taylor_shift","shifted_denominator":["4","8","5","1"],"shifted_numerator":[]},"numerator":[],"range":{"first":"5","last":null}},"decision":"holds"},"statement":"every root of P_n is > 0 for every n >= 2 (region (0, 4) for P)","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 >= 4, with a_n = (n - 3)/(n - 3), b_n = (-2 n + 6)/(n - 3), c_n = (n - 3)/(n - 3)","recurrence":{"first_index":1,"initial_polynomials":[["1"],["-2","1"],["2","-4","1"]],"recurrence":{"denominator":{"coeffs":[["-3"],["1"]]},"terms":[{"coeff":{"coeffs":[["6","-3"],["-2","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["3"],["-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":{"lower":{"closed":false,"value":"0"},"upper":{"closed":false,"value":"4"}},"schema":"polytool.recurrence-real-rootedness-theorem.v1","status":"proved","theorem":"three_term_recurrence"}}