{"certification":{"candidate":{"gauge":{"kind":"none"},"index_shift":0,"operator":{"intercept":"0","shape":"affine","slope":"1"},"parent":{"first_index":0,"initial_polynomials":[["1"],["6","2"],["31","20","3"]],"recurrence":{"denominator":{"coeffs":[["-4"],["0"],["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["-18","0","2"],["-9","-6","-1"],["0","-3","-1"],["3","1","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["18","30","14","2"],["9","15","7","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_name":"existing definition"},"identifying_rows":3,"identity":{"certificate":{"certificate":{"dividend_relation":{"terms":[{"coeff":{"coeffs":[["36","24","4"],["45","30","5"],["9","6","1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-6","-2"],["-24","-8"],["-6","-2"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["0"],["3"],["1"]]},"deriv_order":0,"shift":2}]},"exceptional_bases":[1],"gauge":{"kind":"none"},"identifying_indices":[0,1,2],"index_shift":0,"input":{"first_index":0,"initial_polynomials":[["1"],["6","2"],["31","20","3"]],"recurrence":{"denominator":{"coeffs":[["-2"],["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["18","12","2"],["-9","-6","-1"],["-9","-6","-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-18","-6"],["0","0"],["6","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"input_valid_from_base":1,"multiplier":["0","-32","-12","17","19","7","1"],"operator":{"intercept":"0","shape":"affine","slope":"1"},"parent":{"first_index":0,"initial_polynomials":[["1"],["6","2"],["31","20","3"]],"recurrence":{"denominator":{"coeffs":[["-4"],["0"],["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["-18","0","2"],["-9","-6","-1"],["0","-3","-1"],["3","1","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["18","30","14","2"],["9","15","7","1"]]},"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","0","0","0","0"],["-756","-1836","-1656","-712","-148","-12"],["-1377","-3267","-2862","-1194","-241","-19"],["-837","-1890","-1548","-598","-111","-8"],["-243","-513","-378","-126","-19","-1"],["-27","-54","-36","-10","-1","0"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","0","0","0","0"],["972","3240","4212","2736","948","168","12"],["567","1890","2457","1596","553","98","7"],["81","270","351","228","79","14","1"]]},"deriv_order":1}],"shift":0},{"differentiations":[{"coeff":{"coeffs":[["0","0","0","0"],["-108","-180","-84","-12"],["-9","-15","-7","-1"],["63","105","49","7"],["45","75","35","5"],["9","15","7","1"]]},"deriv_order":0}],"shift":1}],"row_annihilation":{"annihilator":[{"coeff":{"coeffs":[["0","0"],["1","-1"],["1","1"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","6","2"],["-6","-8","-2"]]},"deriv_order":1},{"coeff":{"coeffs":[["9","15","7","1"]]},"deriv_order":2}],"kind":"single","quotient":[{"coeff":{"coeffs":[["0","0","0","0","0","0","0"],["2916","9720","12636","8208","2844","504","36"],["2673","8910","11583","7524","2607","462","33"],["810","2700","3510","2280","790","140","10"],["81","270","351","228","79","14","1"]]},"deriv_order":0}],"row_multiplier":["9","15","7","1"],"start_base":2,"step_multiplier":{"coeffs":[["81","270","351","228","79","14","1"]]},"step_quotient":[{"coeff":{"coeffs":[["243","1620","3996","5004","3570","1516","380","52","3"],["729","2187","2187","459","-675","-567","-191","-31","-2"],["729","2430","3078","1782","360","-102","-70","-14","-1"],["243","891","1323","1035","465","121","17","1","0"]]},"deriv_order":0},{"coeff":{"coeffs":[["2187","10935","23328","27864","20574","9774","3000","576","63","3"],["729","3645","7776","9288","6858","3258","1000","192","21","1"]]},"deriv_order":1}]},"shifted_parent_relation":{"terms":[{"coeff":{"coeffs":[["24","8","0"],["0","9","3"],["-9","0","1"],["-3","-1","0"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-27","-45","-21","-3"],["-9","-15","-7","-1"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["-4"],["1"],["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 2) and E the input rule gauged by lambda(n)/delta(n) and composed with (or conjugated through) T = P((1) x + (0)); so the candidate rows satisfy the input rule at every base n >= 1 (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 >= 0"},"method":"operator_division"},"status":"certified"},"input_first_index":0,"interlacing":{"reason":"no parent theorem report with an interlacing conclusion","status":"none"},"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"lift_summary":{"hypotheses":[{"evidence":"lambda(n) = 1","name":"gauge_finite_nonzero","statement":"lambda(n) is finite and nonzero for every n >= n0","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":"P((1) x + (0))","kind":"preserver_image","operator":{"intercept":"0","shape":"affine","slope":"1"}},"parent":{"gauge":{"kind":"none"},"index_shift":0,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["6","2"],["31","20","3"]],"recurrence":{"denominator":{"coeffs":[["-4"],["0"],["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["-18","0","2"],["-9","-6","-1"],["0","-3","-1"],["3","1","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["18","30","14","2"],["9","15","7","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"relation":"Q_n = lambda(n) T_n(P_(n+0)) with T = P((1) x + (0)), 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":"a real affine substitution x -> a x + b (a != 0), the reversal x^d p(1/x) (d >= deg p) and a real Mobius map (c x + e)^d p((a x + b)/(c x + e)) (a e - b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically"},"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":"not applicable: deg_x c_0 > 1 or deg_x c_1 > 2: the degree may grow by more than one","route":"liu_wang","success":false,"variant":"liu_wang"},{"detail":"not applicable: deg_x c_0 > 1, deg_x c_1 > 2 or deg_x c_2 > 2: the degree may grow by more than one","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): deg_x c_0 > 1 or deg_x c_1 != 2","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) = P_n(x + -3): 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 = -12 x + 3 n^2 x + n^3 x + 2 x^2 - n x^2 - n^2 x^2, B = -4 x^2 - 2 n x^2 + 2 x^3 + n x^3, C = 0) / (n^3 - n^2 - 4); open hypotheses: degree_step: delta = 2, deg P_(n-1) = 2 n - 4: fails at n = 3: Lambda(n, deg P_(n-1)) = n^2 - n - 6 = 0 there (exact); weyl_step_interlacing: at n = 3: A and B have leading coefficients of opposite signs; weyl_step_leading_coefficients_one_signed: fails at n = 3: A_a B_(a+1) = -n^3 - 3 n^2 + 4 = -50 there (exact)","route":"iterated_preserver","success":false,"variant":"iterated_preserver"}],"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 = P((1) x + (0)) 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":"definition_equality"}