{"certification":{"candidate":{"gauge":{"kind":"none"},"index_shift":0,"operator":{"coefficients":[{"denominator":["1"],"numerator":["1"]},{"denominator":["1"],"numerator":["1"]}],"shape":"derivative_polynomial"},"parent":{"first_index":1,"initial_polynomials":[["0"],["0"],["1"]],"recurrence":{"denominator":{"coeffs":[["-3"],["1"]]},"terms":[{"coeff":{"coeffs":[["-2","-8"],["1","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-1","-4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_name":"derivative of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by upward antidiagonals."},"identifying_rows":3,"identity":{"certificate":{"certificate":{"dividend_relation":{"terms":[{"coeff":{"coeffs":[["10","37","-74","24"],["-13","-33","46","-8"],["6","14","-8","0"],["-1","-2","0","0"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["10","42","-63","-8","16"],["-13","-37","34","8","0"],["6","15","-4","0","0"],["-1","-2","0","0","0"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["0","5","11","-32","16"],["0","-4","-12","16","0"],["0","1","4","0","0"]]},"deriv_order":2,"shift":0},{"coeff":{"coeffs":[["-20","34","-8"],["22","-25","4"],["-8","4","0"],["1","0","0"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["-20","34","-8"],["22","-25","4"],["-8","4","0"],["1","0","0"]]},"deriv_order":1,"shift":1}]},"exceptional_bases":[],"gauge":{"kind":"none"},"identifying_indices":[1,2,3],"index_shift":0,"input":{"first_index":1,"initial_polynomials":[["0"],["0"],["1"]],"recurrence":{"denominator":{"coeffs":[["-51","63","-12"],["41","-33","4"],["-11","4","0"],["1","0","0"]]},"terms":[{"coeff":{"coeffs":[["-30","-86","128","-32"],["28","67","-62","8"],["-9","-20","8","0"],["1","2","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-10","-27","48","-16"],["0","6","20","-16","0"],["0","-1","-4","0","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"input_valid_from_base":3,"multiplier":["-2","1"],"operator":{"coefficients":[{"denominator":["1"],"numerator":["1"]},{"denominator":["1"],"numerator":["1"]}],"shape":"derivative_polynomial"},"parent":{"first_index":1,"initial_polynomials":[["0"],["0"],["1"]],"recurrence":{"denominator":{"coeffs":[["-3"],["1"]]},"terms":[{"coeff":{"coeffs":[["-2","-8"],["1","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-1","-4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_leader_shift":1,"parent_valid_from_base":3,"quotients":[{"differentiations":[{"coeff":{"coeffs":[["0","-5","-23/2","32","-8"],["0","11/2","63/4","-24","4"],["0","-2","-7","4","0"],["0","1/4","1","0","0"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","-5","-23/2","32","-8"],["0","11/2","63/4","-24","4"],["0","-2","-7","4","0"],["0","1/4","1","0","0"]]},"deriv_order":1}],"shift":0}],"row_annihilation":{"annihilator":[{"coeff":{"coeffs":[["3"],["-7/4"],["1/4"]]},"deriv_order":0},{"coeff":{"coeffs":[["1/2","9/2"],["-1/4","-1"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","1/4","1"]]},"deriv_order":2}],"kind":"single","quotient":[{"coeff":{"coeffs":[["0","10","24","-64"],["0","-9","-28","32"],["0","2","8","0"]]},"deriv_order":0}],"row_multiplier":["0","1/4","1"],"start_base":3,"step_multiplier":{"coeffs":[["0","0","1/16","1/2","1"]]},"step_quotient":[{"coeff":{"coeffs":[["0","0","-1/16","-9/8","-6","-10"],["0","0","1/16","5/8","2","2"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","-1/16","-3/4","-3","-4"]]},"deriv_order":1}]},"shifted_parent_relation":{"terms":[{"coeff":{"coeffs":[["1","6"],["-1","-2"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","1","4"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["-2"],["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 3) and E the input rule gauged by lambda(n)/delta(n) and composed with (or conjugated through) T = phi(n, D) = 1 + (1) D; so the candidate rows satisfy the input rule at every base n >= 3 (the exceptional bases checked exactly), and agreement on the 3 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":{"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 [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":{"reason":"no parent theorem report with an interlacing conclusion","status":"not_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":[["0"],["0"],["1"]],"recurrence":{"denominator":{"coeffs":[["-3"],["1"]]},"terms":[{"coeff":{"coeffs":[["-2","-8"],["1","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-1","-4"]]},"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 + (1) D, for every n >= 1"},"parent_proof":{"statement":"open: no route proved that every parent row P_n, n >= 1, 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 >= 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":"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 >= 4, with a_n(x) = ((n - 2)/(n - 3)) + ((2 n - 8)/(n - 3)) x and b_n(x) = ((0)/(n - 3)) + ((-1)/(n - 3)) x + ((-4)/(n - 3)) x^2; open hypotheses: earlier_seeds: P_1 is zero or not real-rooted with every root in (-inf, 0) (exact); degree_step_positive: fails at n = 4: lambda_n = (-2 n + 8)/(n - 3) = 0 there (exact); derivative_coefficient_nonpositive: b_n is not separable with a factor of one sign on (-inf, 0), and the sufficient monomial test (-1)^k [x^k] b_n <= 0 fails for k = 1 (not a disproof): fails at n = 4: [x^1] b_n = -1 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 >= 4, with A_n(x) = ((n - 2)/(n - 3)) + ((2 n - 8)/(n - 3)) x, V_n(x) = ((-1)/(n - 3)) x + ((-4)/(n - 3)) x^2, W_n(x) = 0; open hypotheses: seed_roots_negative: P_2 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 = 4: lambda_n = (-2 n + 8)/(n - 3) = 0 there (exact); derivative_coefficient_nonpositive: V_n is not separable with a factor of one sign on (-inf, 0), and the sufficient monomial test (-1)^k [x^k] V_n <= 0 fails for k = 1 (not a disproof): fails at n = 4: [x^1] V_n = -1 there (exact); derivative_or_lag_coefficient_negative: V_n is not separable with a factor of one sign on (-inf, 0), and the sufficient monomial test (-1)^k [x^k] V_n <= 0 fails for k = 1 (not a disproof): fails at n = 4: [x^1] V_n = -1 there (exact)","route":"liu_wang_derivative_lag","success":false,"variant":"liu_wang_derivative_lag"},{"detail":"HypothesisFailed: P_n = (alpha_n x + beta_n) P_(n-1) + kappa_n (x - L)(x - U) P'_(n-1) for n >= 4, with L = -1/4, U = 0, alpha_n = (2 n - 8)/(n - 3), beta_n = (n - 2)/(n - 3), kappa_n = (-4)/(n - 3); open hypotheses: earlier_seeds: P_1 is zero, not real-rooted, or has a root outside [-1/4, 0] (exact); slope_positive: fails at n = 4: a_n = -1/2 n + 2 = 0 there (exact); upper_endpoint_threshold: no hit before n = 4; E(4) = e(4) + 0 = -2 < 0 with no hit in between (exact); when a_n > 0 for every n <= 4, the criterion puts a root of P_4 outside the interval","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 - 4 (an upper bound of deg P_(n-L)): Q_n = a_n(x) S_n(Q_(n-1)) for n >= 4, with a_n(x) = (1) ((1)) and S_n = (A + B D + C D^2 with A = 8 - 2 n + 2 x, B = 4 x + x^2, C = 0) / (n - 3); open hypotheses: weyl_step_interlacing: fails at n = 4: B_2 (B_2 A_0^2 - B_1 A_0 A_1 + B_0 A_1^2) = 4 n^2 - 16 n = 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 >= 3, with p = (1) (k) (k - n + 1), q = (4) (k - 1/2 n + 1/2) (k - 1/2 n + 1); 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 >= 3 + 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 >= 1, 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 + (1) D for every n >= 1 is certified; the lift is Open (open: no route proved that every parent row P_n, n >= 1, is zero or real-rooted with coefficients of one sign; the first 10 parent rows pass the exact check (evidence only))"},"kind":"preserver_image"}