{"certification":{"candidate":{"gauge":{"kind":"none"},"index_shift":0,"operator":{"intercept":"0","shape":"affine","slope":"1"},"parent":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["0"],["1"]]},"terms":[{"coeff":{"coeffs":[["8","2"],["-7","-1"],["3","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["60","36","4"],["-24","-12","-1"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["48","48","14","1"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_name":"existing 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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 0) 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 >= 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"]],"recurrence":{"denominator":{"coeffs":[["0"],["1"]]},"terms":[{"coeff":{"coeffs":[["8","2"],["-7","-1"],["3","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["60","36","4"],["-24","-12","-1"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["48","48","14","1"]]},"deriv_order":2,"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 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 rows","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 theorem needs two seed rows","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 >= 1, with a_n(x) = (1) ((1)) and S_n = (A + B D + C D^2 with A = n - n x + 5 n^2 x + 36 x^2 - 60 n x^2 + 24 n^2 x^2 + 96 x^3 - 144 n x^3 + 48 n^2 x^3, B = -n x + 20 x^2 - 16 n x^2 + 132 x^3 - 72 n x^3 + 192 x^4 - 96 n x^4, C = x^2 + 14 x^3 + 48 x^4 + 48 x^5) / (n); open hypotheses: degree_step: delta = 3, deg P_(n-1) = 3 n - 3: fails at n = 1: Lambda(n, deg P_(n-1)) = 192 n^2 - 288 n + 96 = 0 there (exact); weyl_step_discriminant_nonnegative: at n = 1: B^2 - 4 A C = -31202904692094748946173490180535/1298074214633706907132624082305024 < 0 at x = -16983/32768: T is not a real-rootedness preserver on the whole line (Borcea--Branden 2010, Theorem 1.3, necessary condition); weyl_step_interlacing: at n = 1: 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 = 3: A_a B_(a+1) = -4608 n^3 + 23040 n^2 - 36864 n + 18432 = -9216 there (exact)","route":"iterated_preserver","success":false,"variant":"iterated_preserver"},{"detail":"no lean_proved catalog entry equals the parent; 549 with the same first index and seeds differ in the rule or in later initial rows (oeis_catalog:A001263:lean_proved, oeis_catalog:A003982:lean_proved, oeis_catalog:A007318:lean_proved, oeis_catalog:A008299:lean_proved, oeis_catalog:A008303:lean_proved, oeis_catalog:A008306:lean_proved, oeis_catalog:A008459:lean_proved, oeis_catalog:A008517:lean_proved, oeis_catalog:A008518:lean_proved, 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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 = P((1) x + (0)) for every n >= 0 is certified; the lift is Open (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))"},"kind":"definition_equality"}