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0"},"parent_proof":{"route":"paper_theorem","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, 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":"Laguerre's theorem on multiplier sequences: if f is a real polynomial whose roots are all real and <= 0, then (f(k))_(k >= 0) is a multiplier sequence, so x^k -> f(k) x^k maps every real-rooted polynomial to a real-rooted polynomial or zero (Laguerre; see T. Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977); Polya--Schur, Theorem 40 of P. Branden, arXiv:1410.6601: sum_k f(k) x^k / k! = f(theta) e^x = C e^x x^m prod (1 + x/a_j), a_j > 0); f(k) has one sign for k >= 0, so coefficients of one sign stay of one sign; applied to f(n, .) for each n"},"verdict":"proved"},"notes":["the division remainder is removed by row annihilators of the shifted parent"],"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":"P_n = P_(n-1) + 3 t P_(n-2); P_0 = 1, P_m = 0 for m < 0; generating function 1/(1 - x R(t x^1)) (paper_proof_not_lean_checked): For all n, s >= 0 with 1*s <= 1*1+1: P_n << P_(n+s) (oriented weak interlacing; shared and repeated zeros allowed). Every P_n has nonnegative coefficients and constant term 1 and is real-rooted.; the identifying seeds equal the canonical rows (Canonical), so every parent row is a canonical row","route":"paper_theorem","success":true,"variant":"weighted_renewal_window"},{"detail":"the parent rows are not the canonical boundary rows (Mismatch)","route":"paper_theorem","success":false,"variant":"fixed_delay_predecessor_window"}],"conditional_on":[],"interlacing":{"from_index":1,"justification":{"premises":["homogeneous, index-independent, no derivatives or sign factors; every t^j term has lag 1 + 1*j with ell = 1 >= 1 (exact_structural)","R(z) = 1 + 3 z has R(0) = 1 and nonnegative coefficients (exact_rational)","R(z) has only real zeros (exact_root_count)","window: P_n << P_(n+s) for every n >= 0 and 0 <= s <= 2 (1*s <= 1*1+1 (d*s <= d*ell+1))","the identifying parent seeds equal the canonical rows (Canonical), so every parent row is a canonical row and consecutive parent rows are consecutive canonical rows"],"theorem":"weighted_renewal_window (paper_proof_not_lean_checked; https://github.com/PerAlexandersson/RealRooted/issues/2037): For all n, s >= 0 with 1*s <= 1*1+1: P_n << P_(n+s) (oriented weak interlacing; shared and repeated zeros allowed). 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