{"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":[["0"],["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["8","6"],["-21","-15"],["13","9"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["4","7","3"],["-8","-14","-6"]]},"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":[["0","0","0"],["27","54","27"],["18","36","18"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-18","-12"],["-40","-24"],["-20","-12"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["2"],["5"],["2"]]},"deriv_order":0,"shift":2}]},"exceptional_bases":[0],"gauge":{"kind":"none"},"identifying_indices":[0,1],"index_shift":0,"input":{"first_index":0,"initial_polynomials":[["1"],["1"]],"recurrence":{"denominator":{"coeffs":[["0"],["-3"],["2"]]},"terms":[{"coeff":{"coeffs":[["-18","-36","-18"],["45","90","45"],["-18","-36","-18"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["18","12"],["-40","-24"],["20","12"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"input_valid_from_base":0,"multiplier":["0","2","5","4","1"],"operator":{"intercept":"0","shape":"affine","slope":"1"},"parent":{"first_index":0,"initial_polynomials":[["1"],["1"]],"recurrence":{"denominator":{"coeffs":[["0"],["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["8","6"],["-21","-15"],["13","9"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["4","7","3"],["-8","-14","-6"]]},"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"],["24","66","60","18"],["52","143","130","39"],["36","99","90","27"],["8","22","20","6"]]},"deriv_order":0},{"coeff":{"coeffs":[["-32","-112","-146","-84","-18"],["-304/3","-1064/3","-1387/3","-266","-57"],["-256/3","-896/3","-1168/3","-224","-48"],["-64/3","-224/3","-292/3","-56","-12"]]},"deriv_order":1}],"shift":0},{"differentiations":[{"coeff":{"coeffs":[["0","0","0"],["8/3","14/3","2"],["28/3","49/3","7"],["28/3","49/3","7"],["8/3","14/3","2"]]},"deriv_order":0}],"shift":1}],"row_annihilation":{"annihilator":[{"coeff":{"coeffs":[["0"],["-1"],["1"]]},"deriv_order":0},{"coeff":{"coeffs":[["7/3","2"],["-8/3","-2"]]},"deriv_order":1},{"coeff":{"coeffs":[["4/3","7/3","1"]]},"deriv_order":2}],"kind":"single","quotient":[{"coeff":{"coeffs":[["96","336","438","252","54"],["496","1736","2263","1302","279"],["864","3024","3942","2268","486"],["576","2016","2628","1512","324"],["128","448","584","336","72"]]},"deriv_order":0}],"row_multiplier":["4/3","7/3","1"],"start_base":1,"step_multiplier":{"coeffs":[["16/9","56/9","73/9","14/3","1"]]},"step_quotient":[{"coeff":{"coeffs":[["16/9","56/9","73/9","14/3","1","0"],["112/9","440/9","679/9","57","21","3"],["208/9","872/9","1453/9","401/3","55","9"]]},"deriv_order":0},{"coeff":{"coeffs":[["-64/9","-112/3","-244/3","-847/9","-61","-21","-3"],["-128/9","-224/3","-488/3","-1694/9","-122","-42","-6"]]},"deriv_order":1}]},"shifted_parent_relation":{"terms":[{"coeff":{"coeffs":[["0","0"],["-5","-3"],["-13","-9"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["4","7","3"],["8","14","6"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["0"],["1"],["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 >= 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":{"denominator":{"coeffs":[["0"],["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["8","6"],["-21","-15"],["13","9"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["4","7","3"],["-8","-14","-6"]]},"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) = ((13 n^2 - 21 n + 8)/(n^2 - n)) + ((9 n^2 - 15 n + 6)/(n^2 - n)) x and b_n(x) = ((-8 n + 4)/(n^2 - n)) + ((-14 n + 7)/(n^2 - n)) x + ((-6 n + 3)/(n^2 - n)) x^2; open hypotheses: derivative_coefficient_vanishes_at_zero: [x^0] c_1(n, x) = -8 n + 4 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 = -21/2 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) = ((13 n^2 - 21 n + 8)/(n^2 - n)) + ((9 n^2 - 15 n + 6)/(n^2 - n)) x, V_n(x) = ((-8 n + 4)/(n^2 - n)) + ((-14 n + 7)/(n^2 - n)) x + ((-6 n + 3)/(n^2 - n)) 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); multipliers_vanish_at_zero: [x^0] c_1(n, x) = -8 n + 4 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 = -21/2 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 = -21/2 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 = -4/3, U = -1, alpha_n = (9 n^2 - 15 n + 6)/(n^2 - n), beta_n = (13 n^2 - 21 n + 8)/(n^2 - n), kappa_n = (-6 n + 3)/(n^2 - n); open hypotheses: slope_positive: fails at n = 2: a_n = (-3/2 n^2 + 5/2 n - 1)/(n - 1/2) = -4/3 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 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: P_n = a_n(x) S_n(P_(n-1)) for n >= 2, with a_n(x) = (1) ((1)) and S_n = (A + B D + C D^2 with A = 8 - 21 n + 13 n^2 + 6 x - 15 n x + 9 n^2 x, B = 4 - 8 n + 7 x - 14 n x + 3 x^2 - 6 n x^2, C = 0) / (n^2 - n); open hypotheses: weyl_step_leading_coefficients_same_sign: fails at n = 2: A_1 B_lead = -54 n^3 + 117 n^2 - 81 n + 18 = -108 there (exact); weyl_step_interlacing: fails at n = 2: B_2 (B_2 A_0^2 - B_1 A_0 A_1 + B_0 A_1^2) = 144 n^6 - 504 n^5 + 684 n^4 - 450 n^3 + 144 n^2 - 18 n = 972 there (exact); weyl_step_leading_coefficients_one_signed: fails at n = 2: A_a B_(a+1) = -54 n^3 + 117 n^2 - 81 n + 18 = -108 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"}