{"certification":{"candidate":{"gauge":{"kind":"none"},"index_shift":0,"operator":{"coefficients":[{"denominator":["1"],"numerator":["0"]},{"denominator":["1"],"numerator":["1"]}],"shape":"derivative_polynomial"},"parent":{"first_index":0,"initial_polynomials":[["1"],["4","4"]],"recurrence":{"denominator":{"coeffs":[["10"],["-21"],["9"]]},"terms":[{"coeff":{"coeffs":[["0","0","2"],["0","0","3"],["0","0","-9"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","-10"],["0","-24"],["0","18"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_name":"one_plus_derivative of 3n + 1 preceded by n zeros."},"identifying_rows":2,"identity":{"certificate":{"certificate":{"dividend_relation":{"terms":[{"coeff":{"coeffs":[["0","0","-4"],["0","-4","-7"],["0","-7","-3"],["0","-3","0"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["0","0"],["-1","1"],["-2","3"],["3","0"]]},"deriv_order":1,"shift":1}]},"exceptional_bases":[],"gauge":{"kind":"none"},"identifying_indices":[0,1],"index_shift":0,"input":{"first_index":0,"initial_polynomials":[["0"],["4"]],"recurrence":{"denominator":{"coeffs":[["-4","2"],["12","-5"],["-11","3"],["3","0"]]},"terms":[{"coeff":{"coeffs":[["0","0","0"],["0","-1","1"],["0","-2","3"],["0","3","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"input_valid_from_base":1,"multiplier":["1/3","1"],"operator":{"coefficients":[{"denominator":["1"],"numerator":["0"]},{"denominator":["1"],"numerator":["1"]}],"shape":"derivative_polynomial"},"parent":{"first_index":0,"initial_polynomials":[["1"],["4","4"]],"recurrence":{"denominator":{"coeffs":[["10"],["-21"],["9"]]},"terms":[{"coeff":{"coeffs":[["0","0","2"],["0","0","3"],["0","0","-9"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","-10"],["0","-24"],["0","18"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_leader_shift":2,"parent_valid_from_base":0,"quotients":[],"row_annihilation":{"kind":"system","system":{"generators":[{"annihilator":{"terms":[{"coeff":{"coeffs":[["-4","-4"],["4","-6"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","-4","3"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["1"],["3"]]},"deriv_order":0,"shift":1}]},"quotient":[{"coeff":{"coeffs":[["0","0","0","0","0","0"],["0","0","0","0","-1","1"],["0","0","0","-2","0","3"],["0","0","-1","-3","9","0"],["0","0","-2","9","0","0"],["0","0","3","0","0","0"]]},"deriv_order":1}],"step_multiplier":{"coeffs":[["0","0","0","0","1"],["0","0","0","2","6"],["0","0","1","12","9"],["0","0","6","18","0"],["0","0","9","0","0"]]},"step_quotients":[[{"coeff":{"coeffs":[["0","0","0","0","0","16"],["0","0","0","0","48","108"],["0","0","0","48","324","216"],["0","0","16","324","648","108"],["0","0","108","648","324","0"],["0","0","216","324","0","0"],["0","0","108","0","0","0"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","0","0","-16","12"],["0","0","0","0","-32","-84","81"],["0","0","0","-16","-204","-54","162"],["0","0","0","-108","-351","216","81"],["0","0","0","-216","-54","162","0"],["0","0","0","-108","81","0","0"]]},"deriv_order":1}],[{"coeff":{"coeffs":[["0","0","0","0","0","0"],["0","0","0","64","-96","36"],["0","0","64","336","-612","243"],["0","0","432","216","-1053","486"],["0","0","864","-864","-162","243"],["0","0","432","-648","243","0"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","0","-64","96","-36"],["0","0","0","-64","-336","612","-243"],["0","0","0","-432","-216","1053","-486"],["0","0","0","-864","864","162","-243"],["0","0","0","-432","648","-243","0"]]},"deriv_order":1}]]},{"annihilator":{"terms":[{"coeff":{"coeffs":[["0","0"],["1","-1"],["-1","0"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","0","1"],["0","1","0"]]},"deriv_order":1,"shift":0}]},"quotient":[{"coeff":{"coeffs":[["0","0","0","0","-4"],["0","0","0","-12","-15"],["0","0","-8","-45","-9"],["0","0","-30","-27","0"],["0","0","-18","0","0"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","0","0","0"],["0","0","0","-4","7","-3"],["0","0","-4","-1","15","-9"],["0","0","-8","30","-18","0"],["0","0","12","-9","0","0"]]},"deriv_order":1}],"step_multiplier":{"coeffs":[["0","0","0","0","1"],["0","0","0","2","6"],["0","0","1","12","9"],["0","0","6","18","0"],["0","0","9","0","0"]]},"step_quotients":[[{"coeff":{"coeffs":[["0","0","0","0","0","-4"],["0","0","0","0","-12","-31"],["0","0","0","-12","-93","-81"],["0","0","-4","-93","-243","-81"],["0","0","-31","-243","-243","-27"],["0","0","-81","-243","-81","0"],["0","0","-81","-81","0","0"],["0","0","-27","0","0","0"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","0","0","4","4"],["0","0","0","0","8","39","27"],["0","0","0","4","66","135","54"],["0","0","0","31","189","189","27"],["0","0","0","81","216","81","0"],["0","0","0","81","81","0","0"],["0","0","0","27","0","0","0"]]},"deriv_order":1}],[{"coeff":{"coeffs":[["0","0","0","0","0","28"],["0","0","0","-16","52","213"],["0","0","-16","-100","399","540"],["0","0","-124","-138","1026","513"],["0","0","-324","162","999","162"],["0","0","-324","378","324","0"],["0","0","-108","162","0","0"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","0","16","4","-12"],["0","0","0","16","128","3","-81"],["0","0","0","124","339","-108","-162"],["0","0","0","324","297","-297","-81"],["0","0","0","324","-27","-162","0"],["0","0","0","108","-81","0","0"]]},"deriv_order":1}]]}],"restart_bases":[],"row_multiplier":{"coeffs":[["0","0","0","0","3"],["0","0","0","6","0"],["0","0","3","0","0"]]},"start_base":1}},"shifted_parent_relation":{"terms":[{"coeff":{"coeffs":[["0","0","28"],["0","0","33"],["0","0","9"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","-14"],["0","-48"],["0","-18"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["4"],["15"],["9"]]},"deriv_order":0,"shift":2}]},"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+0) (valid from base 0) 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+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":["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(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 [0, +inf) (exact; Taylor shift: the coefficients of delta(n) at n = 0 + 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":"0","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":0,"input_interlacing":{"reason":"no parent theorem report with an interlacing conclusion","status":"not_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":{"kind":"none"},"index_shift":0,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["4","4"]],"recurrence":{"denominator":{"coeffs":[["10"],["-21"],["9"]]},"terms":[{"coeff":{"coeffs":[["0","0","2"],["0","0","3"],["0","0","-9"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","-10"],["0","-24"],["0","18"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"relation":"Q_n = lambda(n) T_n(P_(n+0)) with T = phi(n, D) = (1) D, for every n >= 0"},"parent_proof":{"route":"liu_wang_three_term","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, 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":["the division remainder is removed by row annihilators of the shifted parent"],"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":"not applicable: expected one nonzero lag term","route":"product_recurrence","success":false,"variant":"product_recurrence"},{"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":"HypothesisFailed: P_n = (a_n + b_n (x + 0)) P_(n-1) - c_n (x + 0)^2 P_(n-2) for n >= 2, with a_n = (0)/(n^2 - 7/3 n + 10/9), b_n = (2 n^2 - 8/3 n - 10/9)/(n^2 - 7/3 n + 10/9), c_n = (n^2 - 1/3 n - 2/9)/(n^2 - 7/3 n + 10/9); Q_n(y) = y^(d_n) P_n(1/y + 0), d_n = deg P_0 + n - 0, satisfies Q_n = (a_n y + b_n) Q_(n-1) - c_n Q_(n-2); open hypotheses: leading_coefficient_nonzero: fails at n = 2: f(n) a_n = 0 = 0 there (exact); first_coefficient_positive: a_2 or c_2 is undefined or a_2 = 0; recurrence_coefficient_positive: fails at n = 3: the denominator of beta_n = 81 n^4 - 378 n^3 + 459 n^2 - 42 n - 80 vanishes there; root_upper_bound: not attempted: the real-rootedness hypotheses do not all hold","route":"three_term_recurrence","success":false,"variant":"three_term_recurrence"},{"detail":"not applicable: the rule is not f(n) P_n = c_0(n, x) P_(n-1) + c_1(n, x) P'_(n-1)","route":"liu_wang","success":false,"variant":"liu_wang"},{"detail":"HypothesisFailed: P_n = A_n(x) P_(n-1) + V_n(x) P'_(n-1) + W_n(x) P_(n-2) for n >= 2, with A_n(x) = ((2 n^2 - 8/3 n - 10/9)/(n^2 - 7/3 n + 10/9)) x, V_n(x) = 0, W_n(x) = ((-n^2 + 1/3 n + 2/9)/(n^2 - 7/3 n + 10/9)) x^2; open hypotheses: lag_degree_step_nonnegative: fails at n = 2: mu_n = (-n^2 + 1/3 n + 2/9)/(n^2 - 7/3 n + 10/9) = -7 there (exact); degree_step_positive: fails at n = 2: lambda_n + mu_n = (n^2 - 7/3 n - 8/9)/(n^2 - 7/3 n + 10/9) = -7/2 there (exact); constant_term_positive: fails at n = 2: A_n(0) = (0)/(n^2 - 7/3 n + 10/9) = 0 there (exact)","route":"liu_wang_derivative_lag","success":false,"variant":"liu_wang_derivative_lag"},{"detail":"not applicable: the rule is not f(n) P_n = c_0(n, x) P_(n-1) + gamma(n) (x - L)(x - U) P'_(n-1)","route":"ma_wang_interval","success":false,"variant":"ma_wang_interval"},{"detail":"induction start moved 1 row(s) past the seeds: Q_n = A_n(x) Q_(n-1) + B_n(x) Q_(n-2) for n >= 3, with A_n(x) = ((2 n^2 - 8/3 n - 10/9)/(n^2 - 7/3 n + 10/9)), B_n(x) = ((-n^2 + 1/3 n + 2/9)/(n^2 - 7/3 n + 10/9)); Q_n = x^(n - 0) P_n(1/x) for n >= 1 (the reversal at the nominal degree deg P_1 + n - 1, an upper bound for deg P_n), which satisfies the rule below: every P_n, n >= 0, is nonzero and real-rooted with every root in (-inf, 0] (so its coefficients have one sign); for n >= 2, Q_(n-1) interlaces Q_n (weakly, in the sense of Liu and Wang; no interlacing of the P_n is claimed); the rule is applied from n = 3 on in the induction (start moved 1 row(s) past the supplied seeds; P_0, ..., P_2 enter as seed rows, checked exactly)","route":"liu_wang_three_term","success":true,"variant":"liu_wang_three_term"}],"conditional_on":[],"route":"liu_wang_three_term","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 = phi(n, D) = (1) D 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, 0]"},"kind":"preserver_image"}