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so the candidate rows satisfy the input rule at every base n >= 1 (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":"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 [1, +inf) (exact; Taylor shift: the coefficients of delta(n) at n = 1 + 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":"1","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":1,"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","2"],["0","1"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","1","-1"]]},"deriv_order":1}]}},"parent":{"gauge":{"kind":"none"},"index_shift":0,"recurrence":{"first_index":1,"initial_polynomials":[["1","8","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","1"],["0","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 >= 1"},"parent_proof":{"route":"liu_wang","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, +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_interlacing_report":{"conclusion":{"first_index":1,"interlacing":"weak","interlacing_from":2,"root_region":{"upper":{"closed":false,"value":"0"}},"simple_roots":false,"statement":"every P_n, n >= 1, is nonzero and real-rooted with every root in (-inf, 0) (so its coefficients have one sign); deg P_n = deg P_(n-1) + 1 and P_(n-1) interlaces P_n (Liu--Wang) for n >= 2"},"hypotheses":[{"evidence":"P_1 has degree 2; checked exactly","name":"seed_roots_negative","statement":"P_1 is nonzero and real-rooted with every root < 0","status":{"kind":"verified"}},{"evidence":"lambda_n = 1 > 0 for every integer n in [2, +inf) (exact; Taylor shift: the coefficients of lambda_n = 1 at n = 2 + u have one sign)","name":"degree_step_positive","sign_decision":{"certificate":{"condition":"positive","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["1"]},"numerator":["1"],"range":{"first":"2","last":null}},"decision":"holds"},"statement":"lambda_n = [x^1] a_n + (deg P_(n-1)) [x^2] b_n > 0 for every n >= 2 (deg P_n = deg P_(n-1) + 1 with leading coefficients of one sign)","status":{"kind":"verified"}},{"evidence":"[x^0] c_1(n, x) is the zero polynomial","name":"derivative_coefficient_vanishes_at_zero","statement":"b_n(0) = 0 for every n >= 2","status":{"kind":"verified"}},{"evidence":"b_n(x) = gamma(n) w(x) / f(n) with w(x) = -x^2 + x, which is negative on (-inf, 0) (no root there, exact): gamma(n)/f(n) = 1 >= 0 for every integer n in [2, +inf) (exact; 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L. Liu and Y. Wang, A unified approach to polynomial sequences with only real zeros, Adv. in Appl. 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