{"certification":{"candidate":{"gauge":{"kind":"none"},"index_shift":0,"operator":{"a":"1","b":"0","c":"1","e":"1","schedule":{"first_degree":0,"first_index":1,"step":1},"shape":"mobius"},"parent":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["1"],["-2"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0"],["0","0"],["0","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_name":"theta_plus_one_squared of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals."},"identifying_rows":1,"identity":{"certificate":{"certificate":{"dividend_relation":{"terms":[{"coeff":{"coeffs":[["0","-1","1"],["0","-2","2"],["0","-1","1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","0"],["0","0"],["1","-1"]]},"deriv_order":0,"shift":1}]},"exceptional_bases":[],"gauge":{"kind":"none"},"identifying_indices":[1],"index_shift":0,"input":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["1","1"],["-2","-2"],["1","1"]]},"terms":[{"coeff":{"coeffs":[["0","0","0"],["0","0","0"],["0","1","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"input_valid_from_base":1,"multiplier":["0","0","1"],"operator":{"a":"1","b":"0","c":"1","e":"1","schedule":{"first_degree":0,"first_index":1,"step":1},"shape":"mobius"},"parent":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["1"],["-2"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0"],["0","0"],["0","1"]]},"deriv_order":0,"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"],["1","-1"]]},"deriv_order":0}],"shift":0}],"shifted_parent_relation":{"terms":[{"coeff":{"coeffs":[["0","-1"],["0","-2"],["0","-1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0"],["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 = ((1) x + (1))^(d_n) P(((1) x + (0))/((1) x + (1))), 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.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":1,"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 = 0 + 1 (n - 1)","kind":"preserver_image","operator":{"a":"1","b":"0","c":"1","e":"1","schedule":{"first_degree":0,"first_index":1,"step":1},"shape":"mobius"}},"parent":{"gauge":{"kind":"none"},"index_shift":0,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["1"],["-2"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0"],["0","0"],["0","1"]]},"deriv_order":0,"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 = 0 + 1 (n - 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 [-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":"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":"HypothesisFailed: B(n) P_n(x) = A(n,x) P_(n-1)(x), n >= 2; open hypotheses: multiplier_fixed_factor: the fixed factor fails the exact root check","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) = ((0)/(n^2 - 2 n + 1)) + ((n^2)/(n^2 - 2 n + 1)) x and b_n(x) = ((0)/(n^2 - 2 n + 1)) + ((0)/(n^2 - 2 n + 1)) x + ((0)/(n^2 - 2 n + 1)) x^2; open hypotheses: constant_term_positive: fails at n = 2: a_n(0) = (0)/(n^2 - 2 n + 1) = 0 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":"not applicable: the rule is not f(n) P_n = c_0(n, x) P_(n-1) + gamma(n) (x - L)(x - U) P'_(n-1)","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":"P_n = a_n(x) S_n(P_(n-1)) for n >= 2, with a_n(x) = (x) (((n^2)/(1 + -2 n + n^2))) and S_n = the identity: every P_n, n >= 1, is nonzero and real-rooted with every root in (-inf, 0]; deg P_n = deg P_(n-1) + 1 for n >= 2; P_(n-1) interlaces P_n (weakly: P_n = a_n P_(n-1) with a_n linear adds one real root) for n >= 2","route":"iterated_preserver","success":true,"variant":"iterated_preserver"}],"conditional_on":[],"interlacing":{"from_index":2,"justification":{"premises":["every hypothesis of the theorem is verified exactly for every index: P_n = a_n(x) S_n(P_(n-1)) for n >= 2, with a_n(x) = (x) (((n^2)/(1 + -2 n + n^2))) and S_n = the identity","every P_n, n >= 1, is nonzero and real-rooted with every root in (-inf, 0]; deg P_n = deg P_(n-1) + 1 for n >= 2; P_(n-1) interlaces P_n (weakly: P_n = a_n P_(n-1) with a_n linear adds one real root) for n >= 2"],"theorem":"iterated_preserver: Iterated real-rootedness preservers (induction on n): if P_m is nonzero and real-rooted with every root in a closed region R and, for every n > m, P_n = a_n(x) S_n(P_(n-1)) where S_n maps every polynomial that is zero or real-rooted with every root in R to zero or a real-rooted polynomial with every root in R and a_n is real-rooted with every root in R, then every P_n is real-rooted with every root in R (the roots of a product are the roots of its factors); an exact degree step keeps every row nonzero; here S_n is the identity"},"kind":"weak","orientation":"previous_interlaces_next"},"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 = ((1) x + (1))^(d_n) P(((1) x + (0))/((1) x + (1))), d_n = 0 + 1 (n - 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 [-1, 0]"},"kind":"preserver_image"}