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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 = phi(n, D) = 1 + (1) D; 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":"no parent theorem report with an interlacing conclusion","status":"none"},"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"lift_summary":{"hypotheses":[{"evidence":"the polynomial with ascending coefficients [1, 1] is fixed; exact check: real-rooted = true, coefficients of one sign = 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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 theorem needs at least two seed rows","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 theorem needs two seed 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(exact); weyl_step_discriminant_nonnegative: at n = 2: B^2 - 4 A C = -3554006546718992946601903359/79228162514264337593543950336 < 0 at x = -8351/8192: 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: deg A = 1, deg B = 4: neither deg B = deg A nor deg B = deg A + 1; weyl_step_leading_coefficients_one_signed: fails at n = 4: A_a B_(a+1) = -128 n^3 + 1024 n^2 - 2688 n + 2304 = -256 there (exact)","route":"iterated_preserver","success":false,"variant":"iterated_preserver"},{"detail":"no lean_proved catalog entry equals the parent; 68 with the same first index and seeds differ in the rule or in later initial rows (oeis_catalog:A003506:lean_proved, oeis_catalog:A008277:lean_proved, oeis_catalog:A008278:lean_proved, oeis_catalog:A008288:lean_proved, oeis_catalog:A008292:lean_proved, oeis_catalog:A019538:lean_proved, oeis_catalog:A026729:lean_proved, oeis_catalog:A035607:lean_proved, 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oeis_catalog:A334218:lean_proved, oeis_catalog:A390433: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 >= 1, is zero or real-rooted with coefficients of one sign; the first 10 parent rows pass the exact check (evidence only)","status":"not_proved"},"rows_compared":16,"schema":"polytool.recurrence-preserver-image.v1","statement":"Q_n = lambda(n) T_n(P_(n+0)) with T = phi(n, D) = 1 + (1) D for every n >= 1 is certified; the lift is Open (open: no route proved that every parent row P_n, n >= 1, is zero or real-rooted with coefficients of one sign; the first 10 parent rows pass the exact check (evidence only))"},"kind":"preserver_image"}