{"certification":{"candidate":{"basis":{"family":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"name":"A013609"},"coefficient_family":{"first_index":1,"initial_polynomials":[["1"],["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"}},"coefficient_roots":{"parent_proof":{"route":"basis_map_coefficient_roots(paper_theorem)","statement":"every coefficient row R_n, n >= 1, is zero or real-rooted with every root in (-inf, 0]","status":"proved"},"source_interval":{"upper":{"closed":true,"value":{"lower_bound":"1","radicand":"1","rational_part":"1","sqrt_coefficient":"0","text":"1","upper_bound":"1"}}},"target_interval":{"upper":{"closed":true,"value":"0"}},"upper_bound":{"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) + 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 (CanonicalAfterRelabel), 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":2,"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 + 1 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 (CanonicalAfterRelabel), 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 >= 1, is zero or real-rooted with coefficients of one sign","status":"proved"}},"conclusion":{"hypotheses":["initial_row_nonzero","initial_row_real_rooted","initial_row_nonpositive_roots"],"input_interval":{"upper":{"closed":true,"value":{"lower_bound":"1","radicand":"1","rational_part":"1","sqrt_coefficient":"0","text":"1","upper_bound":"1"}}},"nonnegative_coefficients":true,"output_interval":{"upper":{"closed":true,"value":{"lower_bound":"0","radicand":"1","rational_part":"0","sqrt_coefficient":"0","text":"0","upper_bound":"0"}}},"sharp":true,"statement":"an input root r <= 1 gives the root (r - a)/d <= 0; a larger root gives a positive root"},"identity":{"certificate":{"basis":{"family":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"name":"A013609"},"coefficient_family":{"first_index":1,"initial_polynomials":[["1"],["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"direction":"input_by_transported","exceptional_bases":[],"input":{"first_index":1,"initial_polynomials":[["1"],["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","2"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"multiplier":["1"],"quotients":[{"row":[[{"coeff":{"coeffs":[["1"]]},"deriv_order":0}]],"shift":0}]},"status":"certified"},"input":{"first_index":1,"initial_polynomials":[["1"],["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","2"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"lift_summary":{"hypotheses":[{"evidence":"exact","name":"initial_row_nonzero","statement":"g = P_0 is nonzero","status":{"kind":"verified"}},{"evidence":"exact root isolation","name":"initial_row_real_rooted","statement":"g is real-rooted","status":{"kind":"verified"}},{"evidence":"exact root isolation","name":"initial_row_nonpositive_roots","statement":"every root of g is <= 0","status":{"kind":"verified"}}],"identity":{"method":"basis_map_division_input_by_transported","status":"certified"},"input_first_index":1,"input_interlacing":{"interlacing":{"from_index":2,"justification":{"premises":["Q_n = T_P(R_n), T_P(u^k) = P_(0 + k), P the basis family A013609 (identity certified)","every coefficient row R_n, n >= 1, is zero or real-rooted with every root in (-inf, 0] (proved, route basis_map_coefficient_roots(paper_theorem))","the parent route proves P_(m-1) << P_m (weakly) for every m >= 2: 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.","g = P_0 is nonzero (verified: exact)","g is real-rooted (verified: exact root isolation)","every root of g is <= 0 (verified: exact root isolation)","T_P (T_P(u^k) = P_(0 + k), P = A013609) does not depend on n and maps every real-rooted polynomial to a real-rooted polynomial or zero (map theorem: geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary))","every alpha P_(m-1) + beta P_m is real-rooted or zero (forward direction), so is its image, and the converse direction gives weakly interlacing zeros of the images; a nonzero gauge moves no zero","hence Q_(n-1) and Q_n interlace (orientation not determined) (weakly) for every n >= 2"],"theorem":"Hermite--Kakeya--Obreschkoff theorem (N. Obreschkoff, Verteilung und Berechnung der Nullstellen reeller Polynome, 1963; see S. Fisk, Polynomials, Roots, and Interlacing, arXiv:math/0612833): real polynomials f, g have weakly interlacing zeros iff alpha f + beta g is real-rooted or zero for all real alpha, beta; a linear T independent of n that maps every real-rooted polynomial to a real-rooted polynomial or zero therefore maps interlacing pairs to pairs with weakly interlacing zeros (orientation not determined)"},"kind":"weak","orientation":"unordered"},"status":"proved"},"lean_replay":"not_attempted","operation":{"basis":"A013609","basis_family":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"kind":"basis_map","source_interval":"(-inf, 1]","target_interval":"(-inf, 0]"},"parent":{"index_shift":0,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"relation":"Q_n = T_P(R_n), T_P(u^k) = P_(0 + k), P the basis family A013609"},"parent_proof":{"route":"basis_map_coefficient_roots(paper_theorem)","statement":"every coefficient row R_n, n >= 1, is zero or real-rooted with every root in (-inf, 0]","status":"proved"},"root_transfer":{"input_conclusion":"every input row Q_n, n >= 1, is zero or real-rooted with every root in (-inf, 0] (nonnegative coefficients when the coefficient row has a positive leading coefficient)","input_root_region":{"upper":{"closed":true,"value":"0"}},"parent_hypothesis":"every coefficient row R_n is zero or real-rooted with every root in the source interval of the map theorem","parent_root_region":{"upper":{"closed":true,"value":"0"}},"theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)"},"verdict":"proved"},"map_theorem":{"basis":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"basis_name":"A013609","conclusions":[{"hypotheses":["initial_row_nonzero","initial_row_real_rooted"],"input_interval":{},"nonnegative_coefficients":false,"output_interval":{},"sharp":true,"statement":"every real-rooted input has a real-rooted image (an affine substitution times g)"},{"hypotheses":["initial_row_nonzero","initial_row_real_rooted","initial_row_nonpositive_roots"],"input_interval":{"upper":{"closed":true,"value":{"lower_bound":"1","radicand":"1","rational_part":"1","sqrt_coefficient":"0","text":"1","upper_bound":"1"}}},"nonnegative_coefficients":true,"output_interval":{"upper":{"closed":true,"value":{"lower_bound":"0","radicand":"1","rational_part":"0","sqrt_coefficient":"0","text":"0","upper_bound":"0"}}},"sharp":true,"statement":"an input root r <= 1 gives the root (r - a)/d <= 0; a larger root gives a positive root"}],"hypotheses":[{"evidence":"exact","name":"initial_row_nonzero","statement":"g = P_0 is nonzero","status":{"kind":"verified"}},{"evidence":"exact root isolation","name":"initial_row_real_rooted","statement":"g is real-rooted","status":{"kind":"verified"}},{"evidence":"exact root isolation","name":"initial_row_nonpositive_roots","statement":"every root of g is <= 0","status":{"kind":"verified"}}],"lean_replay":"not_attempted","schema":"polytool.recurrence-basis-map-theorem.v1","shape":{"constant":"1","initial_row":["1"],"kind":"geometric","slope":"2"},"status":"proved","theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)"},"schema":"polytool.recurrence-basis-map-lift.v1"},"kind":"basis_map"}