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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 1 terms","kind":"preserver_image","operator":{"denominator":["1"],"shape":"weyl_operator","terms":[{"coeff":{"coeffs":[["-2","1"],["-1","0"]]},"deriv_order":0}]}},"parent":{"gauge":{"kind":"none"},"index_shift":0,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-2","1"],["-1","0"]]},"deriv_order":0,"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 1 terms, for every n >= 0"},"parent_proof":{"statement":"parent row P_1 is nonzero and not real-rooted with coefficients of one sign; this parent gives no proof (the row is real-rooted with coefficients of both signs; real-rootedness of every parent row is not proved)","status":"open"},"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":[],"conditional_on":[],"failing_row_index":1,"rows_checked":10,"statement":"parent row P_1 is nonzero and not real-rooted with coefficients of one sign; this parent gives no proof","status":"refuted"},"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 1 terms for every n >= 0 is certified; the lift is Open (parent row P_1 is nonzero and not real-rooted with coefficients of one sign; this parent gives no proof (the row is real-rooted with coefficients of both signs; real-rootedness of every parent row is not proved))"},"parent":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-2","1"],["-1","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}},"kind":"obreschkoff_combination"}