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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 1) and E the input rule gauged by lambda(n)/delta(n) and composed with (or conjugated through) T = P((1) x + (0)); so the candidate rows satisfy the input rule at every base n >= 0 (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 >= 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 >= 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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 leader is not a nonzero constant","route":"periodic_product","success":false,"variant":"P_(n+k) = c(x) P_n"},{"detail":"not applicable: expected one nonzero lag term","route":"product_recurrence","success":false,"variant":"product_recurrence"},{"detail":"no theorem matched (weighted_renewal_window: LHS must be a nonzero constant; fixed_delay_predecessor_window: LHS must be a nonzero constant)","route":"paper_theorem","success":false,"variant":"match_recurrence_theorems"},{"detail":"not applicable: the rule is not f(n) P_n = c_1(n, x) P_(n-1) + c_2(n, x) P_(n-2) with both terms nonzero","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":"not applicable: the rule is not f(n) P_n = c_0(n, x) P_(n-1) + c_1(n, x) P'_(n-1) + c_2(n, x) P_(n-2) with a derivative or a lag term","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":"not applicable: the rule is not f(n) P_n = c_1(n, x) P_(n-1) + c_d(n, x) P'_(n-1) + c_2(n, x) P_(n-2) with a nonzero lag term and a nonzero P_(n-1) or P'_(n-1) term","route":"liu_wang_three_term","success":false,"variant":"liu_wang_three_term"},{"detail":"HypothesisFailed: Q_n = x^(d_n) P_n(1/x) at the nominal degree d_n, with d_(n-L) = n - 1 (an upper bound of deg P_(n-L)): Q_n = a_n(x) S_n(Q_(n-1)) for n >= 2, with a_n(x) = (1) ((1)) and S_n = (A + B D + C D^2 with A = -2 + 2 n - 9 n x + 9 n^2 x + 30 x^2 - 50 n x^2 + 20 n^2 x^2 + 40 x^3 - 60 n x^3 + 20 n^2 x^3, B = 2 x + 30 x^2 - 20 n x^2 + 110 x^3 - 60 n x^3 + 80 x^4 - 40 n x^4, C = -4 x^2 + 16 x^3 + 40 x^4 + 20 x^5) / (2 n - 2); open hypotheses: degree_step: delta = 3, deg P_(n-1) = 3 n - 5: fails at n = 2: Lambda(n, deg P_(n-1)) = 80 n^2 - 280 n + 240 = 0 there (exact); weyl_step_discriminant_nonnegative: at n = 2: B^2 - 4 A C = -612164681249714073581983095075350337/30948500982134506872478105600000 < 0 at x = 95937/81920: T is not a real-rootedness preserver on the whole line (Borcea--Branden 2010, Theorem 1.3, necessary condition); weyl_step_interlacing: at n = 2: A and B have leading coefficients of opposite signs; weyl_step_leading_coefficients_one_signed: fails at n = 3: A_a B_(a+1) = -800 n^3 + 4000 n^2 - 6400 n + 3200 = -1600 there (exact)","route":"iterated_preserver","success":false,"variant":"iterated_preserver"},{"detail":"no lean_proved catalog entry equals the parent; 25 with the same first index and seeds differ in the rule or in later initial rows (oeis_catalog:A029653:lean_proved, oeis_catalog:A060920:lean_proved, oeis_catalog:A084950:lean_proved, oeis_catalog:A093375:lean_proved, oeis_catalog:A104029:lean_proved, oeis_catalog:A106246:lean_proved, oeis_catalog:A113214:lean_proved, oeis_catalog:A113413:lean_proved, oeis_catalog:A117317:lean_proved, oeis_catalog:A119308:lean_proved, oeis_catalog:A119808:lean_proved, oeis_catalog:A121574:lean_proved, oeis_catalog:A126216:lean_proved, oeis_catalog:A133087:lean_proved, oeis_catalog:A134082:lean_proved, oeis_catalog:A154929:lean_proved, oeis_catalog:A156319:lean_proved, oeis_catalog:A167194:lean_proved, oeis_catalog:A193727:lean_proved, oeis_catalog:A193728:lean_proved, oeis_catalog:A193730:lean_proved, oeis_catalog:A193735:lean_proved, oeis_catalog:A202241:lean_proved, oeis_catalog:A318397:lean_proved, oeis_catalog:A394813: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; the first 10 parent rows 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