{"certification":{"candidate":{"basis":{"family":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"name":"A007318"},"coefficient_family":{"first_index":0,"initial_polynomials":[["1"],["-3","3"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-3","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-3","3"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"}},"coefficient_roots":{"parent_proof":{"labels":["oeis_catalog:A099093:lean_proved"],"statement":"every coefficient row R_n, n >= 0, is zero or real-rooted with every root in (-inf, 1]","status":"attested"},"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":"1"}},"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":"no theorem matched (weighted_renewal_window: coefficient of P_(n-1) must be 1 after normalization; 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":"no unconditional all-index source claim with roots in (-inf, 0]","route":"root_domain","success":false,"variant":"identity (affine 1, 0)"},{"detail":"no unconditional all-index source claim with roots in (-inf, 0]","route":"root_domain","success":false,"variant":"f_polynomial"},{"detail":"no unconditional all-index source claim with roots in (-inf, 0]","route":"root_domain","success":false,"variant":"reversal"},{"detail":"no unconditional all-index source claim with roots in (-inf, 0]","route":"root_domain","success":false,"variant":"mobius (1, 0, -1, 1)"},{"detail":"no unconditional all-index source claim with roots in (-inf, 0]","route":"root_domain","success":false,"variant":"identity (affine 1, 0), induction start moved 1 row(s)"},{"detail":"no unconditional all-index source claim with roots in (-inf, 0]","route":"root_domain","success":false,"variant":"f_polynomial, induction start moved 1 row(s)"},{"detail":"no unconditional all-index source claim with roots in (-inf, 0]","route":"root_domain","success":false,"variant":"reversal, induction start moved 1 row(s)"},{"detail":"no unconditional all-index source claim with roots in (-inf, 0]","route":"root_domain","success":false,"variant":"mobius (1, 0, -1, 1), induction start moved 1 row(s)"},{"detail":"no unconditional all-index source claim with roots in (-inf, 0]","route":"root_domain","success":false,"variant":"identity (affine 1, 0), induction start moved 2 row(s)"},{"detail":"no unconditional all-index source claim with roots in (-inf, 0]","route":"root_domain","success":false,"variant":"f_polynomial, induction start moved 2 row(s)"},{"detail":"no unconditional all-index source claim with roots in (-inf, 0]","route":"root_domain","success":false,"variant":"reversal, induction start moved 2 row(s)"},{"detail":"no unconditional all-index source claim with roots in (-inf, 0]","route":"root_domain","success":false,"variant":"mobius (1, 0, -1, 1), induction start moved 2 row(s)"},{"detail":"no unconditional all-index source claim with roots in (-inf, 0]","route":"root_domain","success":false,"variant":"identity (affine 1, 0), induction start moved 3 row(s)"},{"detail":"no unconditional all-index source claim with roots in (-inf, 0]","route":"root_domain","success":false,"variant":"f_polynomial, induction start moved 3 row(s)"},{"detail":"no unconditional all-index source claim with roots in (-inf, 0]","route":"root_domain","success":false,"variant":"reversal, induction start moved 3 row(s)"},{"detail":"no unconditional all-index source claim with roots in (-inf, 0]","route":"root_domain","success":false,"variant":"mobius (1, 0, -1, 1), induction start moved 3 row(s)"},{"detail":"not applicable: c_2(n, x) = gamma(n) w(x) with w neither constant nor a constant times a square (x + v_0)^2","route":"three_term_recurrence","success":false,"variant":"three_term_recurrence"},{"detail":"not applicable: the rule is not f(n) P_n = c_0(n, x) P_(n-1) + c_1(n, x) P'_(n-1)","route":"liu_wang","success":false,"variant":"liu_wang"},{"detail":"HypothesisFailed: P_n = A_n(x) P_(n-1) + V_n(x) P'_(n-1) + W_n(x) P_(n-2) for n >= 2, with A_n(x) = (3) x, V_n(x) = 0, W_n(x) = (3) x; open hypotheses: seed_roots_negative: P_1 is zero, not real-rooted, or has a root >= 0 (exact); constant_term_positive: fails at n = 2: A_n(0) = 0 = 0 there (exact)","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":"every row of the bundled OEIS catalog entry A099093 is zero or real-rooted (flag lean_proved, as of 2026-08-26: a machine-checked Lean proof made outside polytool); every row has coefficients of one sign: the seeds do, the leader is positive and every combined weight is >= 0 for n >= 2, with deg P_m <= 0 + 1 m","route":"attestation","success":true,"variant":"oeis_catalog:A099093:lean_proved"}],"conditional_on":["oeis_catalog:A099093:lean_proved"],"route":"attestation","rows_checked":10,"statement":"every parent row P_n, n >= 0, is zero or real-rooted with coefficients of one sign (real-rootedness from the catalog flag, not proved here; one-signed coefficients proved here)","status":"conditional_on_attestation"}},"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","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"name":"A007318"},"coefficient_family":{"first_index":0,"initial_polynomials":[["1"],["-3","3"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-3","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-3","3"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"direction":"input_by_transported","exceptional_bases":[],"input":{"first_index":0,"initial_polynomials":[["1"],["0","3"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","3"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"multiplier":["1"],"quotients":[{"row":[[{"coeff":{"coeffs":[["1"]]},"deriv_order":0}]],"shift":0}]},"status":"certified"},"input":{"first_index":0,"initial_polynomials":[["1"],["0","3"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","3"]]},"deriv_order":0,"offset":2,"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":0,"input_interlacing":{"reason":"the parent statement is not proved (open, refuted or attested)","status":"not_proved"},"lean_replay":"not_attempted","operation":{"basis":"A007318","basis_family":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","1"]]},"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":0,"initial_polynomials":[["1"],["-3","3"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-3","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-3","3"]]},"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 A007318"},"parent_proof":{"labels":["oeis_catalog:A099093:lean_proved"],"statement":"every coefficient row R_n, n >= 0, is zero or real-rooted with every root in (-inf, 1]","status":"attested"},"root_transfer":{"input_conclusion":"every input row Q_n, n >= 0, 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":"1"}},"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":"conditional_on_attestation"},"map_theorem":{"basis":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"basis_name":"A007318","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":"1"},"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"}