{"certification":{"candidate":{"gauge":{"kind":"none"},"index_shift":0,"operator":{"intercept":"0","shape":"affine","slope":"1"},"parent":{"first_index":0,"initial_polynomials":[["1"],["1","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-1","-1"],["1","1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_name":"existing definition"},"identifying_rows":1,"identity":{"certificate":{"certificate":{"dividend_relation":{"terms":[{"coeff":{"coeffs":[["-1","-1"],["1","0"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-2","-2"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"shift":1}]},"exceptional_bases":[],"gauge":{"kind":"none"},"identifying_indices":[0],"index_shift":0,"input":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","1"],["-1","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["2","2"]]},"deriv_order":1,"offset":1,"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","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-1","-1"],["1","1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1","1"]]},"deriv_order":0,"offset":1,"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":[["-1","-1"],["1","0"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-2","-2"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"shift":1}]},"quotient":[{"coeff":{"coeffs":[["1"]]},"deriv_order":0}],"step_multiplier":{"coeffs":[["4","8","4"]]},"step_quotients":[[{"coeff":{"coeffs":[["4","8","4"],["4","8","4"]]},"deriv_order":0},{"coeff":{"coeffs":[["-8","-24","-24","-8"]]},"deriv_order":1}],[{"coeff":{"coeffs":[["-4","-8","-4"]]},"deriv_order":0}]]},{"annihilator":{"terms":[{"coeff":{"coeffs":[["0","0"],["-1","-2"],["1","0"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["4","6","2"],["-4","-4","0"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["4","8","4"]]},"deriv_order":2,"shift":0}]},"quotient":[],"step_multiplier":{"coeffs":[["16","64","96","64","16"]]},"step_quotients":[[{"coeff":{"coeffs":[["0","-32","-128","-192","-128","-32"],["16","32","-32","-128","-112","-32"],["16","64","96","64","16","0"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","32","160","320","320","160","32"],["-64","-320","-640","-640","-320","-64","0"]]},"deriv_order":1},{"coeff":{"coeffs":[["64","384","960","1280","960","384","64"]]},"deriv_order":2}],[{"coeff":{"coeffs":[["-16","-48","-32","32","48","16"],["-16","-64","-96","-64","-16","0"]]},"deriv_order":0},{"coeff":{"coeffs":[["32","160","320","320","160","32"]]},"deriv_order":1}]]}],"restart_bases":[],"row_multiplier":{"coeffs":[["1"]]},"start_base":0}},"shifted_parent_relation":{"terms":[{"coeff":{"coeffs":[["-1","-1"],["-1","-1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-1","-1"]]},"deriv_order":0,"shift":1},{"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 = 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 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":["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","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-1","-1"],["1","1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1","1"]]},"deriv_order":0,"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":"no theorem matched (weighted_renewal_window: coefficients must be index-independent; fixed_delay_predecessor_window: only P_(n-1) and one term w(n) t P_(n-L) are supported)","route":"paper_theorem","success":false,"variant":"match_recurrence_theorems"},{"detail":"not applicable: c_2(n, x) = gamma(n) w(x) with w neither constant nor a constant times a square (x + v_0)^2","route":"three_term_recurrence","success":false,"variant":"three_term_recurrence"},{"detail":"not applicable: the rule is not f(n) P_n = c_0(n, x) P_(n-1) + c_1(n, x) P'_(n-1)","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 >= 2, with A_n(x) = (1) + (1) x, V_n(x) = 0, W_n(x) = (n - 1) + (n - 1) x; open hypotheses: multipliers_vanish_at_zero: [x^0] c_2(n, x) = n - 1 is not the zero polynomial; lag_coefficient_nonpositive: W_n is not separable with a factor of one sign on (-inf, 0), and the sufficient monomial test (-1)^k [x^k] W_n <= 0 fails for k = 0 (not a disproof): fails at n = 2: [x^0] W_n = 1 there (exact); derivative_or_lag_coefficient_negative: W_n is not separable with a factor of one sign on (-inf, 0), and the sufficient monomial test (-1)^k [x^k] W_n <= 0 fails for k = 0 (not a disproof): fails at n = 2: [x^0] W_n = 1 there (exact)","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)","route":"ma_wang_interval","success":false,"variant":"ma_wang_interval"},{"detail":"no lean_proved catalog entry equals the parent; 92 with the same first index and seeds differ in the rule or in later initial rows (oeis_catalog:A001263:lean_proved, oeis_catalog:A008459:lean_proved, oeis_catalog:A008518:lean_proved, oeis_catalog:A024462:lean_proved, oeis_catalog:A026374:lean_proved, oeis_catalog:A026386:lean_proved, oeis_catalog:A028262:lean_proved, oeis_catalog:A028313:lean_proved, oeis_catalog:A030528:lean_proved, oeis_catalog:A038763:lean_proved, oeis_catalog:A046741:lean_proved, oeis_catalog:A046802:lean_proved, oeis_catalog:A054142:lean_proved, oeis_catalog:A056241:lean_proved, oeis_catalog:A056939:lean_proved, oeis_catalog:A081577:lean_proved, oeis_catalog:A081578:lean_proved, oeis_catalog:A081579:lean_proved, oeis_catalog:A081580:lean_proved, oeis_catalog:A081581:lean_proved, oeis_catalog:A081582:lean_proved, oeis_catalog:A092879:lean_proved, oeis_catalog:A097806:lean_proved, oeis_catalog:A100862:lean_proved, oeis_catalog:A100898:lean_proved, oeis_catalog:A102413:lean_proved, oeis_catalog:A103450:lean_proved, oeis_catalog:A105793:lean_proved, oeis_catalog:A109955:lean_proved, oeis_catalog:A109960:lean_proved, oeis_catalog:A111049:lean_proved, oeis_catalog:A111910:lean_proved, oeis_catalog:A114123:lean_proved, oeis_catalog:A124376:lean_proved, oeis_catalog:A124860:lean_proved, oeis_catalog:A128540:lean_proved, oeis_catalog:A128541:lean_proved, oeis_catalog:A129718:lean_proved, oeis_catalog:A133080:lean_proved, oeis_catalog:A136158:lean_proved, oeis_catalog:A141697:lean_proved, oeis_catalog:A143007:lean_proved, oeis_catalog:A143683:lean_proved, oeis_catalog:A143685:lean_proved, oeis_catalog:A144436:lean_proved, oeis_catalog:A144438:lean_proved, oeis_catalog:A144439:lean_proved, oeis_catalog:A144440:lean_proved, oeis_catalog:A144441:lean_proved, oeis_catalog:A144442:lean_proved, oeis_catalog:A144443:lean_proved, oeis_catalog:A144445:lean_proved, oeis_catalog:A146986:lean_proved, oeis_catalog:A146987:lean_proved, oeis_catalog:A146988:lean_proved, oeis_catalog:A147565:lean_proved, oeis_catalog:A147644:lean_proved, oeis_catalog:A154227:lean_proved, oeis_catalog:A154228:lean_proved, oeis_catalog:A154233:lean_proved, oeis_catalog:A154986:lean_proved, oeis_catalog:A157012:lean_proved, oeis_catalog:A157147:lean_proved, oeis_catalog:A157148:lean_proved, oeis_catalog:A157149:lean_proved, oeis_catalog:A157150:lean_proved, oeis_catalog:A157151:lean_proved, oeis_catalog:A157169:lean_proved, oeis_catalog:A157171:lean_proved, oeis_catalog:A157637:lean_proved, oeis_catalog:A166345:lean_proved, oeis_catalog:A169803:lean_proved, oeis_catalog:A176331:lean_proved, oeis_catalog:A181543:lean_proved, oeis_catalog:A184883:lean_proved, oeis_catalog:A190152:lean_proved, oeis_catalog:A193636:lean_proved, oeis_catalog:A208513:lean_proved, oeis_catalog:A209414:lean_proved, oeis_catalog:A228955:lean_proved, oeis_catalog:A236471:lean_proved, oeis_catalog:A249247:lean_proved, oeis_catalog:A258758:lean_proved, oeis_catalog:A267482:lean_proved, oeis_catalog:A305540:lean_proved, oeis_catalog:A321966:lean_proved, oeis_catalog:A350512:lean_proved, oeis_catalog:A358628:lean_proved, oeis_catalog:A358735:lean_proved, oeis_catalog:A367873:lean_proved, oeis_catalog:A371898:lean_proved, oeis_catalog:A376827: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"}