{"certification":{"candidate":{"gauge":{"kind":"none"},"index_shift":0,"operator":{"coefficients":[{"denominator":["1"],"numerator":["1"]},{"denominator":["1"],"numerator":["2"]},{"denominator":["1"],"numerator":["1"]}],"shape":"euler_polynomial"},"parent":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","-1"],["1","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-1","-1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_name":"shift_x_plus_one of eulerian"},"identifying_rows":2,"identity":{"certificate":{"certificate":{"dividend_relation":{"terms":[{"coeff":{"coeffs":[["4","0","0"],["12","16","-16"],["13","24","-41"],["6","16","-34"],["1","4","-9"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","-4","-36","92"],["0","4","-32","160"],["0","11","-56","1"],["0","10","-32","-110"],["0","3","-4","-43"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["0","0","-16","18","178"],["0","0","-10","96","406"],["0","0","-13","8","249"],["0","0","-4","-48","-8"],["0","0","1","-16","-29"]]},"deriv_order":2,"shift":0},{"coeff":{"coeffs":[["0","0","0","-4","40","72"],["0","0","0","6","120","178"],["0","0","0","0","80","136"],["0","0","0","-2","8","26"],["0","0","0","0","-4","-4"]]},"deriv_order":3,"shift":0},{"coeff":{"coeffs":[["0","0","0","0","0","7","7"],["0","0","0","0","2","20","18"],["0","0","0","0","1","16","15"],["0","0","0","0","0","4","4"]]},"deriv_order":4,"shift":0},{"coeff":{"coeffs":[["-6","-4"],["-13","-12"],["-9","-12"],["-2","-4"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["0","-14","20"],["0","-23","44"],["0","-19","28"],["0","-6","4"]]},"deriv_order":1,"shift":1},{"coeff":{"coeffs":[["0","0","-1","36"],["0","0","7","88"],["0","0","1","68"],["0","0","-2","16"]]},"deriv_order":2,"shift":1},{"coeff":{"coeffs":[["0","0","0","1","8"],["0","0","0","4","20"],["0","0","0","2","16"],["0","0","0","0","4"]]},"deriv_order":3,"shift":1},{"coeff":{"coeffs":[["1"],["2"],["1"]]},"deriv_order":0,"shift":2},{"coeff":{"coeffs":[["0","3"],["0","6"],["0","3"]]},"deriv_order":1,"shift":2},{"coeff":{"coeffs":[["0","0","1"],["0","0","2"],["0","0","1"]]},"deriv_order":2,"shift":2}]},"exceptional_bases":[],"gauge":{"kind":"none"},"identifying_indices":[1,2],"index_shift":0,"input":{"first_index":1,"initial_polynomials":[["1"],["2","4"]],"recurrence":{"denominator":{"coeffs":[["1"],["-2"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0","4"],["0","16","-28"],["-1","-24","53"],["2","16","-38"],["-1","-4","9"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","1","5"],["0","4","4","-32"],["0","-5","8","53"],["0","2","-12","-30"],["0","0","4","4"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","0","1","1"],["0","0","2","-4","-6"],["0","0","-1","8","9"],["0","0","0","-4","-4"]]},"deriv_order":2,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","-4"],["1","12"],["-3","-12"],["2","4"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-1","0"],["0","4","-4"],["0","-2","8"],["0","0","-4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"input_valid_from_base":1,"multiplier":["1"],"operator":{"coefficients":[{"denominator":["1"],"numerator":["1"]},{"denominator":["1"],"numerator":["2"]},{"denominator":["1"],"numerator":["1"]}],"shape":"euler_polynomial"},"parent":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","-1"],["1","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-1","-1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_leader_shift":1,"parent_valid_from_base":1,"quotients":[{"differentiations":[{"coeff":{"coeffs":[["-4"],["-8"],["-5"],["-1"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","-12","16"],["0","-16","41"],["0","-11","34"],["0","-3","9"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","0","-4","30"],["0","0","2","77"],["0","0","0","64"],["0","0","-1","17"]]},"deriv_order":2},{"coeff":{"coeffs":[["0","0","0","0","7"],["0","0","0","2","18"],["0","0","0","1","15"],["0","0","0","0","4"]]},"deriv_order":3}],"shift":0},{"differentiations":[{"coeff":{"coeffs":[["1"],["2"],["1"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","3"],["0","6"],["0","3"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","0","1"],["0","0","2"],["0","0","1"]]},"deriv_order":2}],"shift":1}],"shifted_parent_relation":{"terms":[{"coeff":{"coeffs":[["-1","0"],["-1","-1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","1","1"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["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 1) and E the input rule gauged by lambda(n)/delta(n) and composed with (or conjugated through) T = f(n, theta) = 1 + (2) theta + (1) theta^2; 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 >= 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.isMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isMultiplierSequence_iff_preserves"],"lift_summary":{"hypotheses":[{"evidence":"the polynomial with ascending coefficients [1, 2, 1] is fixed; exact check: real-rooted = true, coefficients of one sign = true","name":"euler_polynomial_roots_nonpositive","statement":"the polynomial in theta is real-rooted with every root <= 0 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":"f(n, theta) = 1 + (2) theta + (1) theta^2","kind":"preserver_image","operator":{"coefficients":[{"denominator":["1"],"numerator":["1"]},{"denominator":["1"],"numerator":["2"]},{"denominator":["1"],"numerator":["1"]}],"shape":"euler_polynomial"}},"parent":{"gauge":{"kind":"none"},"index_shift":0,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","-1"],["1","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-1","-1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"relation":"Q_n = lambda(n) T_n(P_(n+0)) with T = f(n, theta) = 1 + (2) theta + (1) theta^2, for every n >= 1"},"parent_proof":{"route":"iterated_preserver","statement":"every parent row P_n, n >= 1, is zero or real-rooted with coefficients of one sign","status":"proved"},"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 multiplier sequences: if f is a real polynomial whose roots are all real and <= 0, then (f(k))_(k >= 0) is a multiplier sequence, so x^k -> f(k) x^k maps every real-rooted polynomial to a real-rooted polynomial or zero (Laguerre; see T. Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977); Polya--Schur, Theorem 40 of P. Branden, arXiv:1410.6601: sum_k f(k) x^k / k! = f(theta) e^x = C e^x x^m prod (1 + x/a_j), a_j > 0); f(k) has one sign for k >= 0, so coefficients of one sign stay of one sign; applied to f(n, .) for each n"},"verdict":"proved"},"notes":[],"parent_proof":{"attempts":[{"detail":"the rule is not P_(n+k) = c(x) P_n with c free of n","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: coefficients must be index-independent; fixed_delay_predecessor_window: derivatives and alternating signs are unsupported)","route":"paper_theorem","success":false,"variant":"match_recurrence_theorems"},{"detail":"not applicable: the theorem needs at least two seed rows","route":"three_term_recurrence","success":false,"variant":"three_term_recurrence"},{"detail":"Unresolved: P_n = a_n(x) P_(n-1) + b_n(x) P'_(n-1) for n >= 2, with a_n(x) = (n) + (n - 1) x and b_n(x) = (0) + (-1) x + (-1) x^2; open hypotheses: 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 = 2: [x^1] b_n = -1 there (exact)","route":"liu_wang","success":false,"variant":"liu_wang"},{"detail":"not applicable: the theorem needs two seed rows","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 >= 2, with L = -1, U = 0, alpha_n = n - 1, beta_n = n, kappa_n = -1; open hypotheses: slope_positive: fails at n = 2: a_n = -n + 1 = -1 there (exact); upper_endpoint_threshold: no hit before n = 2; E(2) = e(2) + 0 = -2 < 0 with no hit in between (exact); when a_n > 0 for every n <= 2, the criterion puts a root of P_2 outside the interval","route":"ma_wang_interval","success":false,"variant":"ma_wang_interval"},{"detail":"not applicable: the theorem needs two seed rows","route":"liu_wang_three_term","success":false,"variant":"liu_wang_three_term"},{"detail":"Q_n = x^(d_n) P_n(1/x) at the nominal degree d_n, with d_(n-L) = n - 2 (an upper bound of deg P_(n-L)): Q_n = a_n(x) S_n(Q_(n-1)) for n >= 2, with a_n(x) = (1) ((1)) and S_n = A + B D + C D^2 with A = 1 + 2 x, B = x + x^2, C = 0: every P_n, n >= 1, is nonzero and real-rooted with every root in (-inf, 0]; deg Q_n = deg Q_(n-1) + 1 for n >= 2 (Q_n = x^(d_n) P_n(1/x) at the nominal degree d_n, with d_(n-L) = n - 2 (an upper bound of deg P_(n-L)))","route":"iterated_preserver","success":true,"variant":"iterated_preserver"}],"conditional_on":[],"route":"iterated_preserver","rows_checked":10,"statement":"every parent row P_n, n >= 1, 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 = f(n, theta) = 1 + (2) theta + (1) theta^2 for every n >= 1, the side conditions hold for every n, and every parent row P_n, n >= 1, is zero or real-rooted with coefficients of one sign; so every Q_n, n >= 1, is zero or real-rooted with roots in (-inf, 0]"},"kind":"preserver_image"}