{"certification":{"candidate":{"gauge":{"kind":"none"},"index_shift":0,"operator":{"a":"1","b":"0","c":"1","e":"1","schedule":{"first_degree":2,"first_index":0,"step":1},"shape":"mobius"},"parent":{"first_index":0,"initial_polynomials":[["2","3","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-1","-1"],["-3","-2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["4","6","2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_name":"reversal of Triangle of Sulanke numbers: T(n,k) = T(n,k-1) + a(n-1,k) for n+k even and a(n,k) = a(n,k-1) + 2*a(n-1,k) for n+k odd."},"identifying_rows":1,"identity":{"certificate":{"certificate":{"dividend_relation":{"terms":[{"coeff":{"coeffs":[["4","-9","3","5","-3"],["3","-7","3","3","-2"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-4","6","4","-8","0","2"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["1","-3","3","-1"]]},"deriv_order":0,"shift":1}]},"exceptional_bases":[],"gauge":{"kind":"none"},"identifying_indices":[0],"index_shift":0,"input":{"first_index":0,"initial_polynomials":[["2","7","6"]],"recurrence":{"denominator":{"coeffs":[["1","2","1"]]},"terms":[{"coeff":{"coeffs":[["-5","-26","-49","-40","-12"],["-7","-33","-57","-43","-12"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["4","26","66","82","50","12"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"input_valid_from_base":0,"multiplier":["1"],"operator":{"a":"1","b":"0","c":"1","e":"1","schedule":{"first_degree":2,"first_index":0,"step":1},"shape":"mobius"},"parent":{"first_index":0,"initial_polynomials":[["2","3","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-1","-1"],["-3","-2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["4","6","2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_leader_shift":1,"parent_valid_from_base":0,"quotients":[{"differentiations":[{"coeff":{"coeffs":[["1","-3","3","-1"]]},"deriv_order":0}],"shift":0}],"shifted_parent_relation":{"terms":[{"coeff":{"coeffs":[["4","3"],["3","2"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-4","-6","-2"]]},"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 = ((1) x + (1))^(d_n) P(((1) x + (0))/((1) x + (1))), d_n = 2 + 1 (n - 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":"the schedule d_n depends on n","status":"none"},"lean_declarations":["RealRooted.splits_fPolynomial"],"lean_route":"fTransform","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":"the schedule d_n depends on n","status":"not_proved"},"lean_replay":"not_attempted","operation":{"description":"((1) x + (1))^(d_n) P(((1) x + (0))/((1) x + (1))), d_n = 2 + 1 (n - 0)","kind":"preserver_image","operator":{"a":"1","b":"0","c":"1","e":"1","schedule":{"first_degree":2,"first_index":0,"step":1},"shape":"mobius"}},"parent":{"gauge":{"kind":"none"},"index_shift":0,"recurrence":{"first_index":0,"initial_polynomials":[["2","3","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-1","-1"],["-3","-2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["4","6","2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"relation":"Q_n = lambda(n) T_n(P_(n+0)) with T = ((1) x + (1))^(d_n) P(((1) x + (0))/((1) x + (1))), d_n = 2 + 1 (n - 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 [-1, 0]","input_root_region":{"lower":{"closed":true,"value":"-1"},"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":[],"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":"HypothesisFailed: P_n = a_n(x) P_(n-1) + b_n(x) P'_(n-1) for n >= 1, with a_n(x) = (-3 n - 1) + (-2 n - 1) x and b_n(x) = (4) + (6) x + (2) x^2; open hypotheses: derivative_coefficient_vanishes_at_zero: [x^0] c_1(n, x) = 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 = 0 (not a disproof): fails at n = 1: [x^0] b_n = 4 there (exact); constant_term_positive: fails at n = 1: a_n(0) = -3 n - 1 = -4 there (exact)","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":"HypothesisFailed: P_n = (alpha_n x + beta_n) P_(n-1) + kappa_n (x - L)(x - U) P'_(n-1) for n >= 1, with L = -2, U = -1, alpha_n = -2 n - 1, beta_n = -3 n - 1, kappa_n = 2; open hypotheses: slope_positive: fails at n = 1: a_n = -n - 1/2 = -3/2 there (exact)","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) = (1) ((1)) and S_n = A + B D + C D^2 with A = -1 - 3 n - x - 2 n x, B = 4 + 6 x + 2 x^2, C = 0; open hypotheses: weyl_step_leading_coefficients_same_sign: fails at n = 1: A_1 B_lead = -4 n - 2 = -6 there (exact); weyl_step_leading_coefficients_one_signed: fails at n = 1: A_a B_(a+1) = -4 n - 2 = -6 there (exact)","route":"iterated_preserver","success":false,"variant":"iterated_preserver"}],"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 = ((1) x + (1))^(d_n) P(((1) x + (0))/((1) x + (1))), d_n = 2 + 1 (n - 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":"preserver_image"}