{"kind":"theorem_check","report":{"conclusion":{"first_index":0,"root_region":{"upper":{"closed":true,"value":"0"}},"simple_roots":false,"statement":"every P_n, n >= 0, is nonzero and real-rooted with every root in (-inf, 0]; deg P_n = deg P_(n-2) + 2 for n >= 2"},"hypotheses":[{"evidence":"q(n) delta_S(n) sum_d c_d(n, x) D^d = f(n) (g H)(n, x) o S_num holds exactly in the Ore algebra (S_n = the identity)","name":"operator_identity","statement":"the step operator T_n of the rule equals a_n(x) S_n","status":{"kind":"verified"}},{"evidence":"checked exactly for P_0, ..., P_1","name":"start_rows_in_region","statement":"the class starts P_0, ..., P_1 are nonzero and real-rooted with every root in (-inf, 0]","status":{"kind":"verified"}},{"evidence":"delta = 2, n = 0 + 2 t, deg P_(n-2) = 2 t: Lambda(n(t), e(t)) in t = 1 != 0 for every integer n in [1, +inf) (exact; Taylor shift: the coefficients of Lambda(n(t), e(t)) in t = 1 at t = 1 + u have one sign)","name":"degree_step_class_0","sign_decision":{"certificate":{"condition":"nonzero","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["1"]},"numerator":["1"],"range":{"first":"1","last":null}},"decision":"holds"},"statement":"deg P_n = deg P_(n-2) + delta for every n = 0 + 2 t, t >= 1: the coefficient Lambda(n, e) of x^(e + delta) in f(n) T_n(x^e) is nonzero at e = deg P_(n-2) = deg P_0 + delta (t - 1)","status":{"kind":"verified"}},{"evidence":"delta = 2, n = 1 + 2 t, deg P_(n-2) = 2 t: Lambda(n(t), e(t)) in t = 1 != 0 for every integer n in [1, +inf) (exact; Taylor shift: the coefficients of Lambda(n(t), e(t)) in t = 1 at t = 1 + u have one sign)","name":"degree_step_class_1","sign_decision":{"certificate":{"condition":"nonzero","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["1"]},"numerator":["1"],"range":{"first":"1","last":null}},"decision":"holds"},"statement":"deg P_n = deg P_(n-2) + delta for every n = 1 + 2 t, t >= 1: the coefficient Lambda(n, e) of x^(e + delta) in f(n) T_n(x^e) is nonzero at e = deg P_(n-2) = deg P_1 + delta (t - 1)","status":{"kind":"verified"}},{"evidence":"exact check of the fixed polynomial","name":"multiplier_fixed_factor_in_region","statement":"the multiplier factor g(x) = x^2 + 2 x + 1 is real-rooted with every root in (-inf, 0]","status":{"kind":"verified"}}],"iterated_preserver":{"degree_step":2,"induction_start":1,"lag":2,"multiplier_fixed_factor":["1","2","1"],"multiplier_index_factor":[{"denominator":["1"],"numerator":["1"]}],"operator_identity":"f(n) T_n = sum_d c_d(n, x) D^d equals f(n) a_n(x) S_n with a_n(x) = (x^2 + 2 x + 1) ((1))","region":{"upper":{"closed":true,"value":"0"}},"start_degree":2},"lean_replay":"not_attempted","normal_form":"P_n = a_n(x) S_n(P_(n-2)) for n >= 2, with a_n(x) = (x^2 + 2 x + 1) ((1)) and S_n = the identity","recurrence":{"first_index":0,"initial_polynomials":[["0","1","1"],["1","2","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","2","1"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"reference":"Iterated real-rootedness preservers (induction on n): if P_m is nonzero and real-rooted with every root in a closed region R and, for every n > m, P_n = a_n(x) S_n(P_(n-1)) where S_n maps every polynomial that is zero or real-rooted with every root in R to zero or a real-rooted polynomial with every root in R and a_n is real-rooted with every root in R, then every P_n is real-rooted with every root in R (the roots of a product are the roots of its factors); an exact degree step keeps every row nonzero; here S_n is the identity; with lag L = 2, the induction runs in each residue class of n modulo L from its start row","schema":"polytool.recurrence-real-rootedness-theorem.v1","status":"proved","theorem":"iterated_preserver"}}