{"kind":"theorem_check","report":{"conclusion":{"first_index":2,"interlacing":"weak","interlacing_from":3,"root_region":{"upper":{"closed":true,"value":"0"}},"simple_roots":false,"statement":"every P_n, n >= 2, is nonzero and real-rooted with every root in (-inf, 0]; deg P_n = deg P_(n-1) + 1 for n >= 3; P_0, ..., P_1 are zero or real-rooted with every root in (-inf, 0] (seed rows, checked exactly); P_(n-1) interlaces P_n (weakly: P_n = a_n P_(n-1) with a_n linear adds one real root) for n >= 3"},"hypotheses":[{"evidence":"f(n) = -1/3 n + 2/3 != 0 for every integer n in [3, +inf) (exact; Taylor shift: the coefficients of f(n) = -1/3 n + 2/3 at n = 3 + u have one sign)","name":"left_factor_nonzero","sign_decision":{"certificate":{"condition":"nonzero","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["-1/3","-1/3"]},"numerator":["2/3","-1/3"],"range":{"first":"3","last":null}},"decision":"holds"},"statement":"the left-hand factor f(n) is nonzero for every n >= 3","status":{"kind":"verified"}},{"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":"P_2 has degree 2; checked exactly","name":"start_row_in_region","statement":"P_2 is nonzero and real-rooted with every root in (-inf, 0]","status":{"kind":"verified"}},{"evidence":"checked exactly for P_0, ..., P_1; P_0, P_1 zero (the conclusion starts after the last zero seed row)","name":"earlier_seeds_in_region","statement":"every nonzero seed row among P_0, ..., P_1 is real-rooted with every root in (-inf, 0]","status":{"kind":"verified"}},{"evidence":"delta = 1, deg P_(n-1) = n - 1: Lambda(n, deg P_(n-1)) = -1/3 n != 0 for every integer n in [3, +inf) (exact; Taylor shift: the coefficients of Lambda(n, deg P_(n-1)) = -1/3 n at n = 3 + u have one sign)","name":"degree_step","sign_decision":{"certificate":{"condition":"nonzero","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["-1","-1/3"]},"numerator":["0","-1/3"],"range":{"first":"3","last":null}},"decision":"holds"},"statement":"deg P_n = deg P_(n-1) + delta for every n >= 3: the coefficient Lambda(n, e) of x^(e + delta) in f(n) T_n(x^e) is nonzero at e = deg P_(n-1)","status":{"kind":"verified"}},{"evidence":"exact check of the fixed polynomial","name":"multiplier_fixed_factor_in_region","statement":"the multiplier factor g(x) = x is real-rooted with every root in (-inf, 0]","status":{"kind":"verified"}}],"iterated_preserver":{"degree_step":1,"induction_start":2,"multiplier_fixed_factor":["0","1"],"multiplier_index_factor":[{"denominator":["2/3","-1/3"],"numerator":["0","-1/3"]}],"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) (((-1/3 n)/(2/3 + -1/3 n)))","region":{"upper":{"closed":true,"value":"0"}},"start_degree":2},"lean_replay":"not_attempted","normal_form":"P_n = a_n(x) S_n(P_(n-1)) for n >= 3, with a_n(x) = (x) (((-1/3 n)/(2/3 + -1/3 n))) and S_n = the identity","recurrence":{"first_index":0,"initial_polynomials":[["0"],["0"],["0","0","1"]],"recurrence":{"denominator":{"coeffs":[["2/3"],["-1/3"]]},"terms":[{"coeff":{"coeffs":[["0","0"],["0","-1/3"]]},"deriv_order":0,"offset":1,"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","schema":"polytool.recurrence-real-rootedness-theorem.v1","status":"proved","theorem":"iterated_preserver"}}