{"certification":{"candidate":{"gauge":{"kind":"none"},"index_shift":0,"operator":{"coefficients":[{"denominator":["1"],"numerator":["1"]},{"denominator":["1"],"numerator":["1"]}],"shape":"derivative_polynomial"},"parent":{"first_index":0,"initial_polynomials":[["1"],["0","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","1","1"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_name":"Triangle read by rows: triangle of repeated stepped binomial coefficients."},"identifying_rows":1,"identity":{"certificate":{"certificate":{"dividend_relation":{"terms":[{"coeff":{"coeffs":[["0","2","8","6","-4","-4"],["-2","-11","-20","-12","4","8"],["-1","-1","8","20","16","0"],["1","6","12","8","0","0"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","2","8","-2","-28","-28","-8"],["-2","-13","-34","-48","-36","-8","0"],["-1","-3","-2","4","8","0","0"],["1","6","12","8","0","0","0"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["0","0","0","-8","-24","-24","-8"],["0","-2","-14","-36","-40","-16","0"],["0","-2","-10","-16","-8","0","0"]]},"deriv_order":2,"shift":0},{"coeff":{"coeffs":[["0","-2","-6","0","4"],["1","2","8","8","0"],["1","4","4","0","0"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["0","-2","-6","0","4"],["1","2","8","8","0"],["1","4","4","0","0"]]},"deriv_order":1,"shift":1}]},"exceptional_bases":[],"gauge":{"kind":"none"},"identifying_indices":[0],"index_shift":0,"input":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["0","0","-10","-8","4"],["-1","-6","0","8","0"],["1","4","4","0","0"]]},"terms":[{"coeff":{"coeffs":[["0","-6","-24","-30","-8","12"],["-3","-9","0","28","28","-8"],["4","19","28","4","-16","0"],["-1","-6","-12","-8","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","0","-4","-12","-8","8","8"],["0","-2","-6","4","24","16","0"],["0","2","10","16","8","0","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"input_valid_from_base":0,"multiplier":["1"],"operator":{"coefficients":[{"denominator":["1"],"numerator":["1"]},{"denominator":["1"],"numerator":["1"]}],"shape":"derivative_polynomial"},"parent":{"first_index":0,"initial_polynomials":[["1"],["0","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","1","1"]]},"deriv_order":0,"offset":2,"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":[["0","-1"],["1","2"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","-2","-2"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"shift":1}]},"quotient":[{"coeff":{"coeffs":[["0","0","0","-8","-40","-56","-8","32","16"],["0","0","4","16","52","104","96","32","0"],["0","0","4","24","52","48","16","0","0"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","-8","-40","-56","-8","32","16"],["0","0","4","16","52","104","96","32","0"],["0","0","4","24","52","48","16","0","0"]]},"deriv_order":1}],"step_multiplier":{"coeffs":[["0","0","4","8","4"]]},"step_quotients":[[{"coeff":{"coeffs":[["0","0","4","12","12","4"],["0","0","4","16","20","8"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","-8","-24","-24","-8"]]},"deriv_order":1}],[{"coeff":{"coeffs":[["0","0","-4","-8","-4"]]},"deriv_order":0}]]},{"annihilator":{"terms":[{"coeff":{"coeffs":[["0","0","0"],["1","-2","-4"],["1","4","4"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","2","10","8"],["0","-4","-12","-8"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["0","0","4","8","4"]]},"deriv_order":2,"shift":0}]},"quotient":[{"coeff":{"coeffs":[["0","0","-4","-24","-36","-16"],["0","0","-8","-16","-8","0"]]},"deriv_order":0}],"step_multiplier":{"coeffs":[["0","0","0","0","16","64","96","64","16"]]},"step_quotients":[[{"coeff":{"coeffs":[["0","0","0","0","32","160","320","320","160","32","0"],["0","0","0","0","48","288","736","1024","816","352","64"],["0","0","0","0","16","128","416","704","656","320","64"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","0","0","-32","-160","-320","-320","-160","-32","0"],["0","0","0","0","0","-64","-448","-1280","-1920","-1600","-704","-128"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","0","0","0","0","0","64","384","960","1280","960","384","64"]]},"deriv_order":2}],[{"coeff":{"coeffs":[["0","0","0","0","-32","-176","-384","-416","-224","-48"],["0","0","0","0","-16","-96","-224","-256","-144","-32"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","0","0","32","160","320","320","160","32"]]},"deriv_order":1}]]}],"restart_bases":[],"row_multiplier":{"coeffs":[["0","0","4","8","4"]]},"start_base":0}},"shifted_parent_relation":{"terms":[{"coeff":{"coeffs":[["0","-1","-1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["1"]]},"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 + (1) D; so the candidate rows satisfy the input rule at every base n >= 0 (the exceptional bases checked exactly), and agreement on the 1 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 [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 [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 + (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":0,"initial_polynomials":[["1"],["0","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","1","1"]]},"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 >= 0"},"parent_proof":{"route":"periodic_product","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":"c and the seeds are nonzero, real-rooted and of one sign, so P_(n0 + k m + i) = c^m P_(n0 + i) is too","route":"periodic_product","success":true,"variant":"P_(n+k) = c(x) P_n"}],"conditional_on":[],"route":"periodic_product","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 + (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"}
