{"certification":{"candidate":{"gauge":{"kind":"none"},"index_shift":0,"operator":{"schedule":{"first_degree":0,"first_index":1,"step":1},"shape":"reversal"},"parent":{"first_index":1,"initial_polynomials":[["2"]],"recurrence":{"denominator":{"coeffs":[["1"],["1"]]},"terms":[{"coeff":{"coeffs":[["-1","-3"],["5","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["2","0","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_name":"shift_x_plus_one of Triangle read by rows: T(n,k) is the number of Schroeder paths of length 2n and having k weak ascents (1<=k<=n). A Schroeder path of length 2n is a lattice path from (0,0) to (2n,0) consisting of U=(1,1), D=(1,-1) and H=(2,0) steps and never going below the x-axis. A weak ascent in a Schroeder path is a maximal sequence of consecutive U and H steps."},"identifying_rows":1,"identity":{"certificate":{"certificate":{"dividend_relation":{"terms":[{"coeff":{"coeffs":[["-4","0"],["-5","-3"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-2","0","2"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["2"],["1"]]},"deriv_order":0,"shift":1}]},"exceptional_bases":[],"gauge":{"kind":"none"},"identifying_indices":[1],"index_shift":0,"input":{"first_index":1,"initial_polynomials":[["2"]],"recurrence":{"denominator":{"coeffs":[["1"],["1"]]},"terms":[{"coeff":{"coeffs":[["1","-1","-4"],["1","5","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","2","0","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"input_valid_from_base":1,"multiplier":["2","1"],"operator":{"schedule":{"first_degree":0,"first_index":1,"step":1},"shape":"reversal"},"parent":{"first_index":1,"initial_polynomials":[["2"]],"recurrence":{"denominator":{"coeffs":[["1"],["1"]]},"terms":[{"coeff":{"coeffs":[["-1","-3"],["5","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["2","0","-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":[["2"],["1"]]},"deriv_order":0}],"shift":0}],"shifted_parent_relation":{"terms":[{"coeff":{"coeffs":[["-4","0"],["-5","-3"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-2","0","2"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["2"],["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 = x^(d_n) P(1/x), d_n = 0 + 1 (n - 1); so the candidate rows satisfy the input rule at every base n >= 1 (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 >= 1"},"method":"operator_division"},"status":"certified"},"input_first_index":1,"interlacing":{"reason":"the schedule d_n depends on n","status":"none"},"lean_declarations":["RealRooted.X_pow_mul_reverse_eq_zero_or_splits","RealRooted.reflect_eq_zero_or_splits"],"lean_route":"reversed","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":1,"input_interlacing":{"reason":"the schedule d_n depends on n","status":"not_proved"},"lean_replay":"not_attempted","operation":{"description":"x^(d_n) P(1/x), d_n = 0 + 1 (n - 1)","kind":"preserver_image","operator":{"schedule":{"first_degree":0,"first_index":1,"step":1},"shape":"reversal"}},"parent":{"gauge":{"kind":"none"},"index_shift":0,"recurrence":{"first_index":1,"initial_polynomials":[["2"]],"recurrence":{"denominator":{"coeffs":[["1"],["1"]]},"terms":[{"coeff":{"coeffs":[["-1","-3"],["5","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["2","0","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"relation":"Q_n = lambda(n) T_n(P_(n+0)) with T = x^(d_n) P(1/x), d_n = 0 + 1 (n - 1), for every n >= 1"},"parent_proof":{"statement":"open: no route proved that every parent row P_n, n >= 1, 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 >= 1, 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":[],"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 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 >= 2, with a_n(x) = ((5 n - 1)/(n + 1)) + ((3 n - 3)/(n + 1)) x and b_n(x) = ((2)/(n + 1)) + ((0)/(n + 1)) x + ((-2)/(n + 1)) x^2; open hypotheses: derivative_coefficient_vanishes_at_zero: [x^0] c_1(n, x) = 2 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 = 2: [x^0] b_n = 2/3 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 >= 2, with L = -1, U = 1, alpha_n = (3 n - 3)/(n + 1), beta_n = (5 n - 1)/(n + 1), kappa_n = (-2)/(n + 1); open hypotheses: slope_positive: fails at n = 2: a_n = -3/2 n + 3/2 = -3/2 there (exact); upper_endpoint_threshold: no hit before n = 2; E(2) = e(2) + 0 = -3 < 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 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 >= 2, with a_n(x) = (1) ((1)) and S_n = (A + B D + C D^2 with A = -1 + 5 n - 3 x + 3 n x, B = 2 - 2 x^2, C = 0) / (n + 1); open hypotheses: weyl_step_leading_coefficients_same_sign: fails at n = 2: A_1 B_lead = -6 n + 6 = -6 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) = 64 n^2 + 32 n - 32 = 288 there (exact); weyl_step_leading_coefficients_one_signed: fails at n = 2: A_a B_(a+1) = -6 n + 6 = -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 >= 1, 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 = x^(d_n) P(1/x), d_n = 0 + 1 (n - 1) for every n >= 1 is certified; the lift is Open (open: no route proved that every parent row P_n, n >= 1, is zero or real-rooted with coefficients of one sign; the first 10 parent rows pass the exact check (evidence only))"},"kind":"preserver_image"}