{"certificate":{"first_index":0,"identity":{"candidate":{"gauge":{"kind":"none"},"index_shift":0,"operator":{"denominator":["1"],"shape":"weyl_operator","terms":[{"coeff":{"coeffs":[["2","1"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","1","1"]]},"deriv_order":1}]},"parent":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_name":"obreschkoff"},"identifying_rows":1,"identity":{"certificate":{"certificate":{"dividend_relation":{"terms":[{"coeff":{"coeffs":[["-2","-4","-2"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","-4","-8","-4"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["0","0","-1","-2","-1"]]},"deriv_order":2,"shift":0},{"coeff":{"coeffs":[["2","1"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["0","1","1"]]},"deriv_order":1,"shift":1}]},"exceptional_bases":[],"gauge":{"kind":"none"},"identifying_indices":[0],"index_shift":0,"input":{"first_index":0,"initial_polynomials":[["2","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"input_valid_from_base":0,"multiplier":["1"],"operator":{"denominator":["1"],"shape":"weyl_operator","terms":[{"coeff":{"coeffs":[["2","1"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","1","1"]]},"deriv_order":1}]},"parent":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_leader_shift":1,"parent_valid_from_base":0,"quotients":[{"differentiations":[{"coeff":{"coeffs":[["2","1"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","1","1"]]},"deriv_order":1}],"shift":0}],"shifted_parent_relation":{"terms":[{"coeff":{"coeffs":[["-1","-1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","-1","-1"]]},"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 = a Weyl-algebra operator with 2 terms; 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":"a general Weyl operator has no preservation theorem","status":"none"},"lift_summary":{"hypotheses":[{"evidence":"no transfer theorem is recorded for a general Weyl-algebra operator; finite row checks are not a proof","name":"operator_preserves_real_rootedness","statement":"the operator maps every parent row to a real-rooted polynomial or zero","status":{"kind":"unresolved"}},{"evidence":"lambda(n) = 1","name":"gauge_finite_nonzero","statement":"lambda(n) is finite and nonzero for every n >= n0","status":{"kind":"verified"}},{"evidence":"delta(n) != 0 for every integer n in [0, +inf) (exact; Taylor shift: the coefficients of delta(n) at n = 0 + u have one sign)","name":"operator_coefficients_defined","sign_decision":{"certificate":{"condition":"nonzero","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["1"]},"numerator":["1"],"range":{"first":"0","last":null}},"decision":"holds"},"statement":"the operator coefficients are finite at every n >= n0 (their denominators do not vanish)","status":{"kind":"verified"}}],"identity":{"method":"forward_division","status":"certified"},"input_first_index":0,"input_interlacing":{"reason":"a general Weyl operator has no preservation theorem","status":"not_proved"},"lean_replay":"not_attempted","operation":{"description":"a Weyl-algebra operator with 2 terms","kind":"preserver_image","operator":{"denominator":["1"],"shape":"weyl_operator","terms":[{"coeff":{"coeffs":[["2","1"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","1","1"]]},"deriv_order":1}]}},"parent":{"gauge":{"kind":"none"},"index_shift":0,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"relation":"Q_n = lambda(n) T_n(P_(n+0)) with T = a Weyl-algebra operator with 2 terms, for every n >= 0"},"parent_proof":{"route":"ma_wang_interval","statement":"every parent row P_n, n >= 0, is zero or real-rooted with coefficients of one sign","status":"proved"},"root_transfer":{"input_conclusion":"every Q_n, n >= 0, is zero or real-rooted with roots in (-inf, +inf)","input_root_region":{},"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":"none: a general Weyl-algebra operator need not preserve real-rootedness"},"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":"no theorem matched (weighted_renewal_window: derivatives and alternating signs are unsupported; 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":"Unresolved: P_n = a_n(x) P_(n-1) + b_n(x) P'_(n-1) for n >= 1, with a_n(x) = (1) + (1) x and b_n(x) = (0) + (1) x + (1) x^2; open hypotheses: 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 = 1: [x^2] b_n = 1 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":"P_n = (alpha_n x + beta_n) P_(n-1) + kappa_n (x - L)(x - U) P'_(n-1) for n >= 1, with L = -1, U = 0, alpha_n = 1, beta_n = 1, kappa_n = 1: every P_n, n >= 0, is nonzero and real-rooted with every root in [-1, 0); deg P_n = deg P_(n-1) + 1 and P_(n-1) interlaces P_n (Ma--Wang) for n >= 1","route":"ma_wang_interval","success":true,"variant":"ma_wang_interval"}],"conditional_on":[],"interlacing":{"from_index":1,"justification":{"premises":["every hypothesis of the theorem is verified exactly for every index: P_n = (alpha_n x + beta_n) P_(n-1) + kappa_n (x - L)(x - U) P'_(n-1) for n >= 1, with L = -1, U = 0, alpha_n = 1, beta_n = 1, kappa_n = 1","every P_n, n >= 0, is nonzero and real-rooted with every root in [-1, 0); deg P_n = deg P_(n-1) + 1 and P_(n-1) interlaces P_n (Ma--Wang) for n >= 1"],"theorem":"ma_wang_interval: S.-M. Ma and Y. Wang, q-Eulerian polynomials and polynomials with only real zeros, Electron. J. Combin. 15 (2008) R17, Theorem 2 and Corollary 3 (arXiv:math/0611826), in the closed two-endpoint form of the RealRooted library (https://github.com/PerAlexandersson/RealRooted, RealRooted.affineEndpointStep_closed), after the change of variable x = U + (U - L) y, with the endpoint multiplicities tracked exactly through the induction"},"kind":"weak","orientation":"previous_interlaces_next"},"route":"ma_wang_interval","rows_checked":10,"statement":"every parent row P_n, n >= 0, is zero or real-rooted with coefficients of one sign","status":"proved"},"rows_compared":16,"schema":"polytool.recurrence-preserver-image.v1","statement":"Q_n = lambda(n) T_n(P_(n+0)) with T = a Weyl-algebra operator with 2 terms for every n >= 0 is certified; the lift is Open (every parent row P_n, n >= 0, is zero or real-rooted with coefficients of one sign)"},"parent":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}},"kind":"obreschkoff_combination"}