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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 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"],["0","3"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1"]]},"deriv_order":0,"offset":2,"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":{"labels":["oeis_catalog:A099097:lean_proved"],"statement":"every parent row P_n, n >= 1, is zero or real-rooted with coefficients of one sign (real-rootedness from the catalog flag, not proved here; 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every row has coefficients of one sign: the seeds do, the leader is positive and every combined weight is >= 0 for n >= 3, with deg P_m <= -1 + 1 m","route":"attestation","success":true,"variant":"oeis_catalog:A099097:lean_proved"}],"conditional_on":["oeis_catalog:A099097:lean_proved"],"route":"attestation","rows_checked":10,"statement":"every parent row P_n, n >= 1, is zero or real-rooted with coefficients of one sign (real-rootedness from the catalog flag, not proved here; one-signed coefficients proved here)","status":"conditional_on_attestation"},"rows_compared":16,"schema":"polytool.recurrence-preserver-image.v1","statement":"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 is certified; the lift is ConditionalOnAttestation (every parent row P_n, n >= 1, is zero or real-rooted with coefficients of one sign (real-rootedness from the catalog flag, not proved here; one-signed coefficients proved here))"},"kind":"preserver_image"}