{"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":[["0"],["0"],["1"]],"recurrence":{"denominator":{"coeffs":[["-2"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0"],["1","-3"],["0","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","0"],["0","0","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_name":"derivative of Triangle of Bessel numbers read by rows: T(n,k) is the number of k-matchings of the complete graph K(n)."},"identifying_rows":5,"identity":{"certificate":{"certificate":{"dividend_relation":{"terms":[{"coeff":{"coeffs":[["-1","4"],["-1","3"],["0","-1"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["0","0","2"],["0","0","2"]]},"deriv_order":2,"shift":0},{"coeff":{"coeffs":[["-3"],["1"]]},"deriv_order":1,"shift":1}]},"exceptional_bases":[],"gauge":{"kind":"none"},"identifying_indices":[0,1,2,3,4],"index_shift":0,"input":{"first_index":0,"initial_polynomials":[["0"],["0"],["0"],["0"],["6"]],"recurrence":{"denominator":{"coeffs":[["-4"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0"],["1","-5"],["0","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","0"],["0","0","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"input_valid_from_base":4,"multiplier":["-1","1"],"operator":{"coefficients":[{"denominator":["1"],"numerator":["0"]},{"denominator":["1"],"numerator":["1"]}],"shape":"derivative_polynomial"},"parent":{"first_index":0,"initial_polynomials":[["0"],["0"],["1"]],"recurrence":{"denominator":{"coeffs":[["-2"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0"],["1","-3"],["0","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","0"],["0","0","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_leader_shift":1,"parent_valid_from_base":2,"quotients":[{"differentiations":[{"coeff":{"coeffs":[["0","0","-3"],["0","0","1"]]},"deriv_order":1}],"shift":0}],"row_annihilation":{"annihilator":[{"coeff":{"coeffs":[["3/2"],["-5/4"],["1/4"]]},"deriv_order":0},{"coeff":{"coeffs":[["-1/2","7/2"],["0","-1"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","0","1"]]},"deriv_order":2}],"kind":"single","quotient":[{"coeff":{"coeffs":[["0","0","4"],["0","0","4"]]},"deriv_order":0}],"row_multiplier":["0","0","1"],"start_base":4,"step_multiplier":{"coeffs":[["0","0","0","0","1"]]},"step_quotient":[{"coeff":{"coeffs":[["0","0","0","0","1","-4"],["0","0","0","0","1","-3"],["0","0","0","0","0","1"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","0","0","0","-2"],["0","0","0","0","0","0","-2"]]},"deriv_order":1}]},"shifted_parent_relation":{"terms":[{"coeff":{"coeffs":[["-1","2"],["-1","1"],["0","-1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","0","2"],["0","0","2"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["-1"],["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+0) (valid from base 2) 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 >= 4 (the exceptional bases checked exactly), and agreement on the 5 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":[["0"],["0"],["1"]],"recurrence":{"denominator":{"coeffs":[["-2"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0"],["1","-3"],["0","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","0"],["0","0","-2"]]},"deriv_order":1,"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":{"statement":"open: no route proved that every parent row P_n, n >= 0, is zero or real-rooted with coefficients of one sign; the first 10 parent rows pass the exact check (evidence only)","status":"open"},"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":"open"},"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":"not applicable: the rule is not f(n) P_n = c_1(n, x) P_(n-1) + c_2(n, x) P_(n-2) with both terms nonzero","route":"three_term_recurrence","success":false,"variant":"three_term_recurrence"},{"detail":"HypothesisFailed: P_n = a_n(x) P_(n-1) + b_n(x) P'_(n-1) for n >= 3, with a_n(x) = ((n)/(n - 2)) + ((n^2 - 3 n)/(n - 2)) x and b_n(x) = ((0)/(n - 2)) + ((0)/(n - 2)) x + ((-2 n)/(n - 2)) x^2; open hypotheses: earlier_seeds: P_0 is zero or not real-rooted with every root in (-inf, 0) (exact); degree_step_positive: fails at n = 3: lambda_n = (-n^2 + 3 n)/(n - 2) = 0 there (exact)","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 >= 3, with A_n(x) = ((n)/(n - 2)) + ((n^2 - 3 n)/(n - 2)) x, V_n(x) = ((-2 n)/(n - 2)) x^2, W_n(x) = 0; open hypotheses: seed_roots_negative: P_1 is zero, not real-rooted, or has a root >= 0 (exact); seed_degree_step: a seed row of the induction is zero; seed_strict_interlacing: the Bezout test fails or the degrees do not differ by one (exact); degree_step_positive: fails at n = 3: lambda_n = (-n^2 + 3 n)/(n - 2) = 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): the factor x^2 of c_1 has no two distinct rational roots","route":"ma_wang_interval","success":false,"variant":"ma_wang_interval"},{"detail":"not applicable: the rule is not f(n) P_n = c_1(n, x) P_(n-1) + c_d(n, x) P'_(n-1) + c_2(n, x) P_(n-2) with a nonzero lag term and a nonzero P_(n-1) or P'_(n-1) term","route":"liu_wang_three_term","success":false,"variant":"liu_wang_three_term"},{"detail":"HypothesisFailed: Q_n = x^(d_n) P_n(1/x) at the nominal degree d_n, with d_(n-L) = n - 3 (an upper bound of deg P_(n-L)): Q_n = a_n(x) S_n(Q_(n-1)) for n >= 3, with a_n(x) = (1) ((1)) and S_n = (A + B D + C D^2 with A = 3 n - n^2 + n x, B = 2 n x, C = 0) / (n - 2); open hypotheses: weyl_step_interlacing: fails at n = 3: A_0 B_1 - A_1 B_0 = -2 n^3 + 6 n^2 = 0 there (exact)","route":"iterated_preserver","success":false,"variant":"iterated_preserver"},{"detail":"Unresolved: row relation p(n, k) [x^k] P_n + q(n, k) [x^(k-1)] P_n = 0, n >= 2, with p = (-2) (k), q = (4) (k - 1/2 n) (k - 1/2 n + 1/2); valuation v(n) = 0; open hypotheses: multiplier_matching: no matching of the ratio factors (identity: R(n, j) = -q(n, j + v(n)) / p(n, j + v(n))) with a base binom(M, j), complements, pairs and reciprocal rising factorials holds for every n >= 2 + delay, delay <= 4","route":"hypergeometric_row","success":false,"variant":"hypergeometric_row"}],"conditional_on":[],"rows_checked":10,"statement":"open: no route proved that every parent row P_n, n >= 0, is zero or real-rooted with coefficients of one sign; the first 10 parent rows pass the exact check (evidence only)","status":"not_proved"},"rows_compared":16,"schema":"polytool.recurrence-preserver-image.v1","statement":"Q_n = lambda(n) T_n(P_(n+0)) with T = phi(n, D) = (1) D for every n >= 0 is certified; the lift is Open (open: no route proved that every parent row P_n, n >= 0, is zero or real-rooted with coefficients of one sign; the first 10 parent rows pass the exact check (evidence only))"},"kind":"preserver_image"}