{"certification":{"candidate":{"gauge":{"kind":"none"},"index_shift":0,"modulus":2,"parent":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","0"],["0","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"residue":1},"identifying_rows":2,"identity":{"certificate":{"exceptional_bases":[],"gauge":{"kind":"none"},"gauged_parent_relation":{"terms":[{"coeff":{"coeffs":[["0","-1"],["0","-1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"shift":1}]},"identifying_indices":[0,1],"index_shift":0,"input_leader_shift":2,"input_relation":{"terms":[{"coeff":{"coeffs":[["0","-2"],["0","-3"],["0","-1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"shift":2}]},"input_valid_from_base":0,"modulus":2,"multiplier":["1"],"parent_leader_shift":1,"parent_relation":{"terms":[{"coeff":{"coeffs":[["0","-1"],["0","-1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"shift":1}]},"parent_valid_from_base":0,"quotients":[{"component":0,"differentiations":[{"coeff":{"coeffs":[["2"],["1"]]},"deriv_order":0}],"shift":0},{"component":1,"differentiations":[{"coeff":{"coeffs":[["1"]]},"deriv_order":0}],"shift":1}],"residue":1,"statement":"w(n) e_1^T M = sum_nu H_nu S^nu phi(L~) with remainder zero, so w(n) (M V_(2,1)(P~))_n = 0 for every base n >= 0; with the exceptional bases checked exactly, V_(2,1)(P~) satisfies the input rule at every base n >= 0, and agreement on the 2 identifying rows gives Q_n = lambda(n) V_(2,1)(P_(n+0)) for every n >= 0"},"kind":"forward_division","status":"certified"},"input_first_index":0,"lean_replay":"not_attempted","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":{"interlacing":{"from_index":1,"justification":{"premises":["Q_n = lambda(n) V_(2,1)(P_(n+0)) for every n >= 0 (identity certified)","every parent row P_n, n >= 0, is zero or real-rooted with coefficients of one sign (proved, route iterated_preserver)","the parent route proves P_(m-1) << P_m (weakly) for every m >= 1: iterated_preserver: 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","lambda(n) is finite and nonzero for every n >= n0 (verified: lambda(n) = 1)","each parent row is zero or real-rooted with coefficients of one sign; its negative, where needed, has nonnegative coefficients and the same zeros, and the section commutes with the sign change, so Corollary 5.6(b) applies to consecutive parent rows","the degrees of consecutive parent rows differ by at most one, hence so do those of their nonzero sections (a real-rooted polynomial with nonnegative coefficients has no internal zero coefficient, so deg V_(r,j)(P) = floor((deg P - j) / r))","a nonzero gauge lambda(n) moves no zero","hence Q_(n-1) << Q_n (weakly) for every n >= 1"],"theorem":"C. A. Athanasiadis and D. G. Wagner, Veronese sections and interlacing matrices of formal power series (arXiv:2404.12989), Corollary 5.6(b): if P and Q are real-rooted with nonnegative coefficients and the zeros of P interlace those of Q from the left (P << Q, Q carries the largest zero; the zero polynomial interlaces every such polynomial), then the k-th r-section of P interlaces that of Q in the same sense, for all 0 <= k < r"},"kind":"weak","orientation":"previous_interlaces_next"},"status":"proved"},"lean_replay":"not_attempted","operation":{"kind":"veronese_section","modulus":2,"residue":1},"parent":{"gauge":{"kind":"none"},"index_shift":0,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","0"],["0","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"relation":"Q_n = lambda(n) V_(2,1)(P_(n+0)) for every n >= 0"},"parent_proof":{"route":"iterated_preserver","statement":"every parent row P_n, n >= 0, is zero or real-rooted with coefficients of one sign","status":"proved"},"root_transfer":{"input_conclusion":"every Q_n = lambda(n) V_(2,1)(P_(n+0)), 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":"Veronese sections of Polya frequency sequences (Aissen--Schoenberg--Whitney--Edrei): if P is real-rooted with coefficients of one sign, equivalently nonzero with all roots in (-inf, 0], every section V_(r,j)(P) is zero or real-rooted with roots in (-inf, 0]"},"verdict":"proved"},"notes":[],"parent_proof":{"attempts":[{"detail":"the rule is not P_(n+k) = c(x) P_n with c free of n","route":"periodic_product","success":false,"variant":"P_(n+k) = c(x) P_n"},{"detail":"no theorem matched (weighted_renewal_window: coefficients must be index-independent; fixed_delay_predecessor_window: coefficient of P_(n-1) must be 1 (+ w(n) t) after normalization)","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":"HypothesisFailed: P_n = a_n(x) P_(n-1) + b_n(x) P'_(n-1) for n >= 1, with a_n(x) = (0) + (n) x and b_n(x) = (0) + (0) x + (0) x^2; open hypotheses: constant_term_positive: fails at n = 1: a_n(0) = 0 = 0 there (exact)","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":"P_n = a_n(x) S_n(P_(n-1)) for n >= 1, with a_n(x) = (x) ((n)) and S_n = the identity: every P_n, n >= 0, is nonzero and real-rooted with every root in (-inf, 0]; deg P_n = deg P_(n-1) + 1 for n >= 1; P_(n-1) interlaces P_n (weakly: P_n = a_n P_(n-1) with a_n linear adds one real root) for n >= 1","route":"iterated_preserver","success":true,"variant":"iterated_preserver"}],"conditional_on":[],"interlacing":{"from_index":1,"justification":{"premises":["every hypothesis of the theorem is verified exactly for every index: P_n = a_n(x) S_n(P_(n-1)) for n >= 1, with a_n(x) = (x) ((n)) and S_n = the identity","every P_n, n >= 0, is nonzero and real-rooted with every root in (-inf, 0]; deg P_n = deg P_(n-1) + 1 for n >= 1; P_(n-1) interlaces P_n (weakly: P_n = a_n P_(n-1) with a_n linear adds one real root) for n >= 1"],"theorem":"iterated_preserver: 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"},"kind":"weak","orientation":"previous_interlaces_next"},"route":"iterated_preserver","rows_checked":10,"statement":"every parent row P_n, n >= 0, is zero or real-rooted with coefficients of one sign","status":"proved"},"rows_compared":16,"schema":"polytool.recurrence-veronese-certificate.v1","statement":"proved: Q_n = lambda(n) V_(2,1)(P_(n+0)) for every n >= 0 and every parent row P_n, n >= 0, is zero or real-rooted with coefficients of one sign; so every Q_n, n >= 0, is zero or real-rooted with nonpositive roots","verdict":"proved"},"input":{"first_index":0,"initial_polynomials":[["0"],["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","0"],["0","-1"],["0","1"],["0","0"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"kind":"veronese_section"}