{"certification":{"candidate":{"gauge":{"kind":"none"},"index_shift":0,"operator":{"intercept":"0","shape":"affine","slope":"1"},"parent":{"first_index":0,"initial_polynomials":[["1","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","1","0"],["0","2","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-4","-4","-1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_name":"existing definition"},"identifying_rows":2,"identity":{"certificate":{"certificate":{"dividend_relation":{"terms":[{"coeff":{"coeffs":[["-4","-4","-1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"shift":2}]},"exceptional_bases":[],"gauge":{"kind":"none"},"identifying_indices":[0,1],"index_shift":0,"input":{"first_index":0,"initial_polynomials":[["1","1"],["2","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["4","4","1"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"input_valid_from_base":0,"multiplier":["1"],"operator":{"intercept":"0","shape":"affine","slope":"1"},"parent":{"first_index":0,"initial_polynomials":[["1","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","1","0"],["0","2","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-4","-4","-1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_leader_shift":1,"parent_valid_from_base":0,"quotients":[{"differentiations":[{"coeff":{"coeffs":[["8","28","30","13","2"],["0","8","12","6","1"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","-16","-32","-24","-8","-1"]]},"deriv_order":1}],"shift":0},{"differentiations":[{"coeff":{"coeffs":[["4","4","1"]]},"deriv_order":0}],"shift":1}],"row_annihilation":{"annihilator":[{"coeff":{"coeffs":[["0"],["1"],["1"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0"],["-4","-2"]]},"deriv_order":1},{"coeff":{"coeffs":[["4","4","1"]]},"deriv_order":2}],"kind":"single","quotient":[{"coeff":{"coeffs":[["0","0","16","32","24","8","1"]]},"deriv_order":0}],"row_multiplier":["4","4","1"],"start_base":0,"step_multiplier":{"coeffs":[["16","32","24","8","1"]]},"step_quotient":[{"coeff":{"coeffs":[["-96","-272","-320","-200","-70","-13","-1"],["0","32","80","80","40","10","1"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","-64","-192","-240","-160","-60","-12","-1"]]},"deriv_order":1}]},"shifted_parent_relation":{"terms":[{"coeff":{"coeffs":[["-2","-3","-1"],["0","-2","-1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","4","4","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 0) 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 >= 0 (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 >= 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","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","1","0"],["0","2","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-4","-4","-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 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":"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: the theorem needs two seed rows","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 theorem needs two seed rows","route":"liu_wang_three_term","success":false,"variant":"liu_wang_three_term"},{"detail":"HypothesisFailed: P_n = a_n(x) S_n(P_(n-1)) for n >= 1, with a_n(x) = (x + 2) ((1)) and S_n = A + B D + C D^2 with A = 1 + n x, B = -2 x - x^2, C = 0; open hypotheses: degree_step: delta = 2, deg P_(n-1) = 2 n - 1: fails at n = 1: Lambda(n, deg P_(n-1)) = -n + 1 = 0 there (exact); weyl_step_leading_coefficients_same_sign: fails at n = 1: A_1 B_lead = -n = -1 there (exact); weyl_step_leading_coefficients_one_signed: fails at n = 1: A_a B_(a+1) = -n = -1 there (exact)","route":"iterated_preserver","success":false,"variant":"iterated_preserver"},{"detail":"no lean_proved catalog entry equals the parent; 4 with the same first index and seeds differ in the rule or in later initial rows (oeis_catalog:A101842:lean_proved, oeis_catalog:A199478:lean_proved, oeis_catalog:A199881:lean_proved, oeis_catalog:A272471:lean_proved)","route":"attestation","success":false,"variant":"oeis_catalog"}],"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"}