{"certification":{"candidate":{"gauge":{"kind":"none"},"index_shift":0,"operator":{"intercept":"1","shape":"affine","slope":"1"},"parent":{"first_index":1,"initial_polynomials":[["0"],["1"]],"recurrence":{"denominator":{"coeffs":[["6"],["-7"],["2"]]},"terms":[{"coeff":{"coeffs":[["0","0"],["-1","-1"],["2","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-4","-1"],["0","4","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_name":"derivative of Triangle read by rows: T(k,s) = ((2*k+1)/(2*s+1))*binomial(k+s,2*s), 0 <= s <= k."},"identifying_rows":2,"identity":{"certificate":{"certificate":{"dividend_relation":{"terms":[{"coeff":{"coeffs":[["-1","0"],["-3","-1"],["-2","-1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","0","0"],["0","-4","-1"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["1"],["-3"],["2"]]},"deriv_order":0,"shift":1}]},"exceptional_bases":[],"gauge":{"kind":"none"},"identifying_indices":[1,2],"index_shift":0,"input":{"first_index":1,"initial_polynomials":[["0"],["1"]],"recurrence":{"denominator":{"coeffs":[["6"],["-7"],["2"]]},"terms":[{"coeff":{"coeffs":[["0","0"],["-2","-1"],["3","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-5","-6","-1"],["5","6","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"input_valid_from_base":2,"multiplier":["1","-3","2"],"operator":{"intercept":"1","shape":"affine","slope":"1"},"parent":{"first_index":1,"initial_polynomials":[["0"],["1"]],"recurrence":{"denominator":{"coeffs":[["6"],["-7"],["2"]]},"terms":[{"coeff":{"coeffs":[["0","0"],["-1","-1"],["2","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-4","-1"],["0","4","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_leader_shift":1,"parent_valid_from_base":2,"quotients":[{"differentiations":[{"coeff":{"coeffs":[["1"],["-3"],["2"]]},"deriv_order":0}],"shift":0}],"shifted_parent_relation":{"terms":[{"coeff":{"coeffs":[["-1","0"],["-3","-1"],["-2","-1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","0","0"],["0","-4","-1"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["1"],["-3"],["2"]]},"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 2) and E the input rule gauged by lambda(n)/delta(n) and composed with (or conjugated through) T = P((1) x + (1)); so the candidate rows satisfy the input rule at every base n >= 2 (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 >= 1"},"method":"operator_division"},"status":"certified"},"input_first_index":1,"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":1,"input_interlacing":{"reason":"no parent theorem report with an interlacing conclusion","status":"not_proved"},"lean_replay":"not_attempted","operation":{"description":"P((1) x + (1))","kind":"preserver_image","operator":{"intercept":"1","shape":"affine","slope":"1"}},"parent":{"gauge":{"kind":"none"},"index_shift":0,"recurrence":{"first_index":1,"initial_polynomials":[["0"],["1"]],"recurrence":{"denominator":{"coeffs":[["6"],["-7"],["2"]]},"terms":[{"coeff":{"coeffs":[["0","0"],["-1","-1"],["2","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-4","-1"],["0","4","1"]]},"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 + (1)), for every n >= 1"},"parent_proof":{"route":"iterated_preserver","statement":"every parent row P_n, n >= 1, is zero or real-rooted with coefficients of one sign","status":"proved"},"root_transfer":{"input_conclusion":"every Q_n, n >= 1, is zero or real-rooted with roots in (-inf, -1]","input_root_region":{"upper":{"closed":true,"value":"-1"}},"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":"proved"},"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 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 >= 3, with a_n(x) = ((n^2 - 1/2 n)/(n^2 - 7/2 n + 3)) + ((1/2 n^2 - 1/2 n)/(n^2 - 7/2 n + 3)) x and b_n(x) = ((0)/(n^2 - 7/2 n + 3)) + ((2 n - 2)/(n^2 - 7/2 n + 3)) x + ((1/2 n - 1/2)/(n^2 - 7/2 n + 3)) x^2; open hypotheses: earlier_seeds: P_1 is zero or not real-rooted with every root in (-inf, 0) (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 = 3: [x^2] b_n = 2/3 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 >= 3, with A_n(x) = ((n^2 - 1/2 n)/(n^2 - 7/2 n + 3)) + ((1/2 n^2 - 1/2 n)/(n^2 - 7/2 n + 3)) x, V_n(x) = ((2 n - 2)/(n^2 - 7/2 n + 3)) x + ((1/2 n - 1/2)/(n^2 - 7/2 n + 3)) x^2, W_n(x) = 0; open hypotheses: seed_roots_negative: P_1 is zero, not real-rooted, or has a root >= 0 (exact); seed_degree_step: a seed row of the induction is zero; seed_strict_interlacing: the Bezout test fails or the degrees do not differ by one (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 = 3: [x^2] V_n = 2/3 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 = 3: [x^2] V_n = 2/3 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 >= 3, with L = -4, U = 0, alpha_n = (1/2 n^2 - 1/2 n)/(n^2 - 7/2 n + 3), beta_n = (n^2 - 1/2 n)/(n^2 - 7/2 n + 3), kappa_n = (1/2 n - 1/2)/(n^2 - 7/2 n + 3); open hypotheses: earlier_seeds: P_1 is zero, not real-rooted, or has a root outside [-4, 0] (exact)","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":"P_n = a_n(x) S_n(P_(n-1)) for n >= 3, with a_n(x) = (1) ((1)) and S_n = (A + B D + C D^2 with A = -n + 2 n^2 - n x + n^2 x, B = -4 x + 4 n x - x^2 + n x^2, C = 0) / (2 n^2 - 7 n + 6): every P_n, n >= 2, is nonzero and real-rooted with every root in (-inf, 0]; deg P_n = deg P_(n-1) + 1 for n >= 3; P_1, ..., P_1 are zero or real-rooted with every root in (-inf, 0] (seed rows, checked exactly)","route":"iterated_preserver","success":true,"variant":"iterated_preserver"}],"conditional_on":[],"route":"iterated_preserver","rows_checked":10,"statement":"every parent row P_n, n >= 1, is zero or real-rooted with coefficients of one sign","status":"proved"},"rows_compared":16,"schema":"polytool.recurrence-preserver-image.v1","statement":"proved: Q_n = lambda(n) T_n(P_(n+0)) with T = P((1) x + (1)) for every n >= 1, the side conditions hold for every n, and every parent row P_n, n >= 1, is zero or real-rooted with coefficients of one sign; so every Q_n, n >= 1, is zero or real-rooted with roots in (-inf, -1]"},"kind":"preserver_image"}