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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 = f(n, theta) = 1 + (1) theta; so the candidate rows satisfy the input rule at every base n >= 0 (the exceptional bases checked exactly), and agreement on the 1 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":["RealRooted.isMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isMultiplierSequence_iff_preserves"],"lift_summary":{"hypotheses":[{"evidence":"the polynomial with ascending coefficients [1, 1] is fixed; exact check: real-rooted = true, coefficients of one sign = 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theorem report with an interlacing conclusion","status":"not_proved"},"lean_replay":"not_attempted","operation":{"description":"f(n, theta) = 1 + (1) theta","kind":"preserver_image","operator":{"coefficients":[{"denominator":["1"],"numerator":["1"]},{"denominator":["1"],"numerator":["1"]}],"shape":"euler_polynomial"}},"parent":{"gauge":{"kind":"none"},"index_shift":0,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","3"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"relation":"Q_n = lambda(n) T_n(P_(n+0)) with T = f(n, theta) = 1 + (1) theta, for every n >= 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). Every P_n has nonnegative coefficients and constant term 1 and is real-rooted."},"kind":"weak","orientation":"previous_interlaces_next"},"route":"paper_theorem","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-preserver-image.v1","statement":"proved: Q_n = lambda(n) T_n(P_(n+0)) with T = f(n, theta) = 1 + (1) theta for every n >= 0, the side conditions hold for every n, 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 roots in (-inf, 0]"},"kind":"preserver_image"}