{"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":[["2","1"],["-2","-1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","2","2"]]},"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","-1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["2"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["1"]]},"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":{"terms":[{"coeff":{"coeffs":[["0","1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"input_valid_from_base":0,"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":[["2","1"],["-2","-1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","2","2"]]},"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","-1","-1"],["0","0","-1","-1"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","2","4","2"]]},"deriv_order":1}],"shift":0},{"differentiations":[{"coeff":{"coeffs":[["0","0","0"],["0","1","1"]]},"deriv_order":0}],"shift":1}],"row_annihilation":{"annihilator":[{"coeff":{"coeffs":[["0"],["-1/4"],["1/4"]]},"deriv_order":0},{"coeff":{"coeffs":[["1","3/2"],["-1","-1"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","1","1"]]},"deriv_order":2}],"kind":"single","quotient":[{"coeff":{"coeffs":[["0","0","4","8","4"]]},"deriv_order":0}],"row_multiplier":["0","1","1"],"start_base":1,"step_multiplier":{"coeffs":[["0","0","1","2","1"]]},"step_quotient":[{"coeff":{"coeffs":[["0","0","0","2","4","2"],["0","0","-2","-5","-4","-1"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","2","6","6","2"]]},"deriv_order":1}]},"shifted_parent_relation":{"terms":[{"coeff":{"coeffs":[["0","0"],["2","1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","-2","-2"]]},"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 >= 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":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["2","1"],["-2","-1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","2","2"]]},"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) = ((-2 n + 2)/(n - 1)) + ((-n + 1)/(n - 1)) x and b_n(x) = ((0)/(n - 1)) + ((2)/(n - 1)) x + ((2)/(n - 1)) x^2; open hypotheses: degree_step_positive: fails at n = 2: lambda_n = (n - 3)/(n - 1) = -1 there (exact); 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 = 2 (not a disproof): fails at n = 2: [x^2] b_n = 2 there (exact); constant_term_positive: fails at n = 2: a_n(0) = (-2 n + 2)/(n - 1) = -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) = ((-2 n + 2)/(n - 1)) + ((-n + 1)/(n - 1)) x, V_n(x) = ((2)/(n - 1)) x + ((2)/(n - 1)) x^2, W_n(x) = 0; open hypotheses: seed_degree_step: degrees 0 and 0, leading coefficients of opposite signs (exact); seed_strict_interlacing: the Bezout test fails or the degrees do not differ by one (exact); degree_step_positive: fails at n = 2: lambda_n = (n - 3)/(n - 1) = -1 there (exact); constant_term_positive: fails at n = 2: A_n(0) = (-2 n + 2)/(n - 1) = -2 there (exact); 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 = 2 (not a disproof): fails at n = 2: [x^2] V_n = 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 = 2 (not a disproof): fails at n = 2: [x^2] V_n = 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 = -1, U = 0, alpha_n = (-n + 1)/(n - 1), beta_n = (-2 n + 2)/(n - 1), kappa_n = (2)/(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 = -3 + n - 2 x, B = -2 x - 2 x^2, C = 0) / (n - 1); open hypotheses: weyl_step_interlacing: fails at n = 3: B_2 (B_2 A_0^2 - B_1 A_0 A_1 + B_0 A_1^2) = 4 n^2 - 16 n + 12 = 0 there (exact)","route":"iterated_preserver","success":false,"variant":"iterated_preserver"},{"detail":"Unresolved: row relation p(n, k) [x^k] P_n + q(n, k) [x^(k-1)] P_n = 0, n >= 1, with p = (4) (k - n) (k), q = (4) (k - 1/2 n - 1) (k - 1/2 n - 1/2); valuation v(n) = 0; open hypotheses: multiplier_matching: no matching of the ratio factors (identity: R(n, j) = -q(n, j + v(n)) / p(n, j + v(n))) with a base binom(M, j), complements, pairs and reciprocal rising factorials holds for every n >= 1 + delay, delay <= 4","route":"hypergeometric_row","success":false,"variant":"hypergeometric_row"}],"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"}