{"certification":{"candidate":{"gauge":{"kind":"none"},"index_shift":0,"operator":{"intercept":"0","shape":"affine","slope":"1"},"parent":{"first_index":0,"initial_polynomials":[["1"],["1"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["-5","-2"],["5","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-18","-17","-4"]]},"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":[["0","0"],["-2","-1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0"],["-1"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["0"],["1"]]},"deriv_order":0,"shift":2}]},"exceptional_bases":[],"gauge":{"kind":"none"},"identifying_indices":[0,1,2],"index_shift":0,"input":{"first_index":0,"initial_polynomials":[["1"],["1"],["5","2"]],"recurrence":{"denominator":{"coeffs":[["-2"],["1"]]},"terms":[{"coeff":{"coeffs":[["-4","-2"],["2","1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-2"],["1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"input_valid_from_base":1,"multiplier":["0","1","1"],"operator":{"intercept":"0","shape":"affine","slope":"1"},"parent":{"first_index":0,"initial_polynomials":[["1"],["1"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["-5","-2"],["5","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-18","-17","-4"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_leader_shift":1,"parent_valid_from_base":1,"quotients":[{"differentiations":[{"coeff":{"coeffs":[["0","0","0","0"],["18","26","25/2","2"],["18","26","25/2","2"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","0","0"],["-81","-153","-433/4","-34","-4"]]},"deriv_order":1}],"shift":0},{"differentiations":[{"coeff":{"coeffs":[["0","0","0"],["0","0","0"],["9/2","17/4","1"]]},"deriv_order":0}],"shift":1}],"row_annihilation":{"annihilator":[{"coeff":{"coeffs":[["0"],["-1/4"],["1/4"]]},"deriv_order":0},{"coeff":{"coeffs":[["13/4","3/2"],["-9/4","-1"]]},"deriv_order":1},{"coeff":{"coeffs":[["9/2","17/4","1"]]},"deriv_order":2}],"kind":"single","quotient":[{"coeff":{"coeffs":[["0","0","0","0","0"],["324","612","433","136","16"]]},"deriv_order":0}],"row_multiplier":["9/2","17/4","1"],"start_base":1,"step_multiplier":{"coeffs":[["81/4","153/4","433/16","17/2","1"]]},"step_quotient":[{"coeff":{"coeffs":[["-162","-387","-739/2","-705/4","-42","-4"],["405/4","927/4","3389/16","773/8","22","2"]]},"deriv_order":0},{"coeff":{"coeffs":[["-729/2","-4131/4","-9747/8","-12257/16","-1083/4","-51","-4"]]},"deriv_order":1}]},"shifted_parent_relation":{"terms":[{"coeff":{"coeffs":[["0","0"],["-5","-2"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["18","17","4"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["0"],["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 = 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"],["1"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["-5","-2"],["5","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-18","-17","-4"]]},"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":"HypothesisFailed: P_n = a_n(x) P_(n-1) + b_n(x) P'_(n-1) for n >= 2, with a_n(x) = ((5 n - 5)/(n - 1)) + ((2 n - 2)/(n - 1)) x and b_n(x) = ((-18)/(n - 1)) + ((-17)/(n - 1)) x + ((-4)/(n - 1)) x^2; open hypotheses: degree_step_positive: fails at n = 3: lambda_n = (-2 n + 6)/(n - 1) = 0 there (exact); derivative_coefficient_vanishes_at_zero: [x^0] c_1(n, x) = -18 is not the zero polynomial; 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 = -17 there (exact)","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) = ((5 n - 5)/(n - 1)) + ((2 n - 2)/(n - 1)) x, V_n(x) = ((-18)/(n - 1)) + ((-17)/(n - 1)) x + ((-4)/(n - 1)) x^2, W_n(x) = 0; open hypotheses: seed_degree_step: degrees 0 and 0 (exact); seed_strict_interlacing: the Bezout test fails or the degrees do not differ by one (exact); degree_step_positive: fails at n = 3: lambda_n = (-2 n + 6)/(n - 1) = 0 there (exact); multipliers_vanish_at_zero: [x^0] c_1(n, x) = -18 is not the zero polynomial; derivative_coefficient_nonpositive: V_n is not separable with a factor of one sign on (-inf, 0), and the sufficient monomial test (-1)^k [x^k] V_n <= 0 fails for k = 1 (not a disproof): fails at n = 2: [x^1] V_n = -17 there (exact); derivative_or_lag_coefficient_negative: V_n is not separable with a factor of one sign on (-inf, 0), and the sufficient monomial test (-1)^k [x^k] V_n <= 0 fails for k = 1 (not a disproof): fails at n = 2: [x^1] V_n = -17 there (exact)","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 = -9/4, U = -2, alpha_n = (2 n - 2)/(n - 1), beta_n = (5 n - 5)/(n - 1), kappa_n = (-4)/(n - 1); open hypotheses: slope_positive: fails at n = 2: a_n = -1/2 n + 1/2 = -1/2 there (exact); upper_endpoint_threshold: no hit before n = 2; E(2) = e(2) + 0 = -1 < 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 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^(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 = 6 - 2 n + 29 x - 12 n x + 36 x^2 - 18 n x^2, B = 4 x + 17 x^2 + 18 x^3, C = 0) / (n - 1); open hypotheses: degree_step: delta = 2, deg P_(n-1) = 2 n - 4: fails at n = 2: Lambda(n, deg P_(n-1)) = 18 n - 36 = 0 there (exact); weyl_step_interlacing: at n = 2: deg A = 1, deg B = 3: neither deg B = deg A nor deg B = deg A + 1; weyl_step_leading_coefficients_one_signed: fails at n = 3: A_a B_(a+1) = -324 n + 648 = -324 there (exact)","route":"iterated_preserver","success":false,"variant":"iterated_preserver"},{"detail":"no lean_proved catalog entry equals the parent; 54 with the same first index and seeds differ in the rule or in later initial rows (oeis_catalog:A008299:lean_proved, oeis_catalog:A011973:lean_proved, oeis_catalog:A034839:lean_proved, oeis_catalog:A039991:lean_proved, oeis_catalog:A055151:lean_proved, oeis_catalog:A059022:lean_proved, oeis_catalog:A059023:lean_proved, oeis_catalog:A059344:lean_proved, oeis_catalog:A060523:lean_proved, oeis_catalog:A093127:lean_proved, oeis_catalog:A100861:lean_proved, oeis_catalog:A101350:lean_proved, oeis_catalog:A102547:lean_proved, oeis_catalog:A109446:lean_proved, oeis_catalog:A114580:lean_proved, oeis_catalog:A118394:lean_proved, oeis_catalog:A118931:lean_proved, oeis_catalog:A118933:lean_proved, oeis_catalog:A124324:lean_proved, oeis_catalog:A128099:lean_proved, oeis_catalog:A128100:lean_proved, oeis_catalog:A129186:lean_proved, oeis_catalog:A132885:lean_proved, oeis_catalog:A133081:lean_proved, oeis_catalog:A134434:lean_proved, oeis_catalog:A136011:lean_proved, oeis_catalog:A136394:lean_proved, oeis_catalog:A144299:lean_proved, oeis_catalog:A152198:lean_proved, oeis_catalog:A152815:lean_proved, oeis_catalog:A152828:lean_proved, oeis_catalog:A152830:lean_proved, oeis_catalog:A165253:lean_proved, oeis_catalog:A177040:lean_proved, oeis_catalog:A185263:lean_proved, oeis_catalog:A188440:lean_proved, oeis_catalog:A200091:lean_proved, oeis_catalog:A201701:lean_proved, oeis_catalog:A202023:lean_proved, oeis_catalog:A211956:lean_proved, oeis_catalog:A245962:lean_proved, oeis_catalog:A245964:lean_proved, oeis_catalog:A262226:lean_proved, oeis_catalog:A263789:lean_proved, oeis_catalog:A266213:lean_proved, oeis_catalog:A275442:lean_proved, oeis_catalog:A304236:lean_proved, oeis_catalog:A304429:lean_proved, oeis_catalog:A317496:lean_proved, oeis_catalog:A318772:lean_proved, oeis_catalog:A355144:lean_proved, oeis_catalog:A367270:lean_proved, oeis_catalog:A390883:lean_proved, oeis_catalog:A391403: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"}