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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":"no parent theorem report with an interlacing conclusion","status":"none"},"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"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":0,"input_interlacing":{"reason":"no parent theorem report with an interlacing conclusion","status":"not_proved"},"lean_replay":"not_attempted","operation":{"description":"P((1) x + (0))","kind":"preserver_image","operator":{"intercept":"0","shape":"affine","slope":"1"}},"parent":{"gauge":{"kind":"none"},"index_shift":0,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-2","-2"],["2","2"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"relation":"Q_n = lambda(n) T_n(P_(n+0)) with T = P((1) x + (0)), for every n >= 0"},"parent_proof":{"statement":"open: no route proved that every parent row P_n, n >= 0, is zero or real-rooted with coefficients of one sign; the first 10 parent rows pass the exact check (evidence only)","status":"open"},"root_transfer":{"input_conclusion":"every Q_n, n >= 0, is zero or real-rooted with roots in (-inf, 0]","input_root_region":{"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":"open"},"notes":["the division remainder is removed by row annihilators of the shifted parent"],"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: coefficients must be index-independent; fixed_delay_predecessor_window: only P_(n-1) and one term w(n) t P_(n-L) are supported)","route":"paper_theorem","success":false,"variant":"match_recurrence_theorems"},{"detail":"not applicable: c_2(n, x) = gamma(n) w(x) with w neither constant nor a constant times a square (x + v_0)^2","route":"three_term_recurrence","success":false,"variant":"three_term_recurrence"},{"detail":"not applicable: the rule is not f(n) P_n = c_0(n, x) P_(n-1) + c_1(n, x) P'_(n-1)","route":"liu_wang","success":false,"variant":"liu_wang"},{"detail":"HypothesisFailed: P_n = A_n(x) P_(n-1) + V_n(x) P'_(n-1) + W_n(x) P_(n-2) for n >= 2, with A_n(x) = (1), V_n(x) = 0, W_n(x) = (2 n - 2) + (2 n - 2) x; open hypotheses: seed_degree_step: degrees 0 and 0 (exact); seed_strict_interlacing: the Bezout test fails or the degrees do not differ by one (exact); degree_step_positive: fails at n = 2: lambda_n = 0 = 0 there (exact); multipliers_vanish_at_zero: [x^0] c_2(n, x) = 2 n - 2 is not the zero polynomial; lag_coefficient_nonpositive: W_n is not separable with a factor of one sign on (-inf, 0), and the sufficient monomial test (-1)^k [x^k] W_n <= 0 fails for k = 0 (not a disproof): fails at n = 2: [x^0] W_n = 2 there (exact); derivative_or_lag_coefficient_negative: W_n is not separable with a factor of one sign on (-inf, 0), and the sufficient monomial test (-1)^k [x^k] W_n <= 0 fails for k = 0 (not a disproof): fails at n = 2: [x^0] W_n = 2 there (exact)","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":"no lean_proved catalog entry equals the parent; 54 with the same first index and seeds differ in the rule or in later initial rows (oeis_catalog:A008299:lean_proved, oeis_catalog:A011973:lean_proved, oeis_catalog:A034839:lean_proved, oeis_catalog:A039991:lean_proved, oeis_catalog:A055151:lean_proved, oeis_catalog:A059022:lean_proved, oeis_catalog:A059023:lean_proved, oeis_catalog:A059344:lean_proved, oeis_catalog:A060523:lean_proved, oeis_catalog:A093127:lean_proved, oeis_catalog:A100861:lean_proved, oeis_catalog:A101350:lean_proved, oeis_catalog:A102547:lean_proved, oeis_catalog:A109446:lean_proved, oeis_catalog:A114580:lean_proved, oeis_catalog:A118394:lean_proved, oeis_catalog:A118931:lean_proved, oeis_catalog:A118933:lean_proved, oeis_catalog:A124324:lean_proved, oeis_catalog:A128099:lean_proved, oeis_catalog:A128100:lean_proved, oeis_catalog:A129186:lean_proved, oeis_catalog:A132885:lean_proved, oeis_catalog:A133081:lean_proved, oeis_catalog:A134434:lean_proved, oeis_catalog:A136011:lean_proved, oeis_catalog:A136394:lean_proved, oeis_catalog:A144299:lean_proved, oeis_catalog:A152198:lean_proved, oeis_catalog:A152815:lean_proved, oeis_catalog:A152828:lean_proved, oeis_catalog:A152830:lean_proved, oeis_catalog:A165253:lean_proved, oeis_catalog:A177040:lean_proved, oeis_catalog:A185263:lean_proved, oeis_catalog:A188440:lean_proved, oeis_catalog:A200091:lean_proved, oeis_catalog:A201701:lean_proved, oeis_catalog:A202023:lean_proved, oeis_catalog:A211956:lean_proved, oeis_catalog:A245962:lean_proved, oeis_catalog:A245964:lean_proved, oeis_catalog:A262226:lean_proved, oeis_catalog:A263789:lean_proved, oeis_catalog:A266213:lean_proved, oeis_catalog:A275442:lean_proved, oeis_catalog:A304236:lean_proved, oeis_catalog:A304429:lean_proved, oeis_catalog:A317496:lean_proved, oeis_catalog:A318772:lean_proved, oeis_catalog:A355144:lean_proved, oeis_catalog:A367270:lean_proved, oeis_catalog:A390883:lean_proved, oeis_catalog:A391403:lean_proved)","route":"attestation","success":false,"variant":"oeis_catalog"}],"conditional_on":[],"rows_checked":10,"statement":"open: no route proved that every parent row P_n, n >= 0, is zero or real-rooted with coefficients of one sign; 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