{"certification":{"candidate":{"gauge":{"kind":"none"},"index_shift":0,"modulus":2,"parent":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","1"],["0","-1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"residue":0},"identifying_rows":2,"identity":{"certificate":{"exceptional_bases":[],"gauge":{"kind":"none"},"gauged_parent_relation":{"terms":[{"coeff":{"coeffs":[["0","0"],["0","1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"shift":1}]},"identifying_indices":[1,2],"index_shift":0,"input_leader_shift":2,"input_relation":{"terms":[{"coeff":{"coeffs":[["0","0"],["0","-1"],["0","-1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"shift":2}]},"input_valid_from_base":1,"modulus":2,"multiplier":["1"],"parent_leader_shift":1,"parent_relation":{"terms":[{"coeff":{"coeffs":[["0","0"],["0","1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"shift":1}]},"parent_valid_from_base":1,"quotients":[{"component":1,"differentiations":[{"coeff":{"coeffs":[["0","-1"],["0","-1"]]},"deriv_order":0}],"shift":0},{"component":0,"differentiations":[{"coeff":{"coeffs":[["1"]]},"deriv_order":0}],"shift":1}],"residue":0,"statement":"w(n) e_0^T M = sum_nu H_nu S^nu phi(L~) with remainder zero, so w(n) (M V_(2,0)(P~))_n = 0 for every base n >= 1; with the exceptional bases checked exactly, V_(2,0)(P~) satisfies the input rule at every base n >= 1, and agreement on the 2 identifying rows gives Q_n = lambda(n) V_(2,0)(P_(n+0)) for every n >= 1"},"kind":"forward_division","status":"certified"},"input_first_index":1,"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":1,"input_interlacing":{"reason":"the parent statement is not proved","status":"not_proved"},"lean_replay":"not_attempted","operation":{"kind":"veronese_section","modulus":2,"residue":0},"parent":{"gauge":{"kind":"none"},"index_shift":0,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","1"],["0","-1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"relation":"Q_n = lambda(n) V_(2,0)(P_(n+0)) for every n >= 1"},"parent_proof":{"statement":"open: no route proved that every parent row P_n, n >= 1, is zero or real-rooted with coefficients of one sign; the first 10 parent rows pass the exact check (evidence only)","status":"open"},"root_transfer":{"input_conclusion":"every Q_n = lambda(n) V_(2,0)(P_(n+0)), n >= 1, 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":"open"},"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 >= 2, with a_n(x) = (0) + (-n + 1) x and b_n(x) = (0) + (0) x + (0) x^2; open hypotheses: degree_step_positive: fails at n = 2: lambda_n = -n + 1 = -1 there (exact); constant_term_positive: fails at n = 2: 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":"no lean_proved catalog entry equals the parent; 68 with the same first index and seeds differ in the rule or in later initial rows (oeis_catalog:A003506:lean_proved, oeis_catalog:A008277:lean_proved, oeis_catalog:A008278:lean_proved, oeis_catalog:A008288:lean_proved, oeis_catalog:A008292:lean_proved, oeis_catalog:A019538:lean_proved, oeis_catalog:A026729:lean_proved, oeis_catalog:A035607:lean_proved, oeis_catalog:A039755:lean_proved, oeis_catalog:A046716:lean_proved, oeis_catalog:A048854:lean_proved, oeis_catalog:A052553:lean_proved, oeis_catalog:A056242:lean_proved, oeis_catalog:A059110:lean_proved, oeis_catalog:A060187:lean_proved, oeis_catalog:A062145:lean_proved, oeis_catalog:A063007:lean_proved, oeis_catalog:A063967:lean_proved, oeis_catalog:A071951:lean_proved, oeis_catalog:A078812:lean_proved, oeis_catalog:A085478:lean_proved, oeis_catalog:A086645:lean_proved, oeis_catalog:A089627:lean_proved, oeis_catalog:A091042:lean_proved, oeis_catalog:A098158:lean_proved, oeis_catalog:A098172:lean_proved, oeis_catalog:A099097:lean_proved, oeis_catalog:A104684:lean_proved, oeis_catalog:A105278:lean_proved, oeis_catalog:A106800:lean_proved, oeis_catalog:A111125:lean_proved, oeis_catalog:A114656:lean_proved, oeis_catalog:A122542:lean_proved, oeis_catalog:A123160:lean_proved, oeis_catalog:A127648:lean_proved, oeis_catalog:A131689:lean_proved, oeis_catalog:A134991:lean_proved, oeis_catalog:A142075:lean_proved, oeis_catalog:A142459:lean_proved, oeis_catalog:A143395:lean_proved, oeis_catalog:A154602:lean_proved, oeis_catalog:A156289:lean_proved, oeis_catalog:A158909:lean_proved, oeis_catalog:A166973:lean_proved, oeis_catalog:A173018:lean_proved, oeis_catalog:A201637:lean_proved, oeis_catalog:A201730:lean_proved, oeis_catalog:A219836:lean_proved, oeis_catalog:A225117:lean_proved, oeis_catalog:A225466:lean_proved, oeis_catalog:A225467:lean_proved, oeis_catalog:A225468:lean_proved, oeis_catalog:A225469:lean_proved, oeis_catalog:A225472:lean_proved, oeis_catalog:A225473:lean_proved, oeis_catalog:A235706:lean_proved, oeis_catalog:A253283:lean_proved, oeis_catalog:A259454:lean_proved, oeis_catalog:A269939:lean_proved, oeis_catalog:A269945:lean_proved, oeis_catalog:A271697:lean_proved, oeis_catalog:A271703:lean_proved, oeis_catalog:A271705:lean_proved, oeis_catalog:A286724:lean_proved, oeis_catalog:A292219:lean_proved, oeis_catalog:A321331:lean_proved, oeis_catalog:A334218:lean_proved, oeis_catalog:A390433:lean_proved)","route":"attestation","success":false,"variant":"oeis_catalog"}],"conditional_on":[],"rows_checked":10,"statement":"open: no route proved that every parent row P_n, n >= 1, is zero or real-rooted with coefficients of one sign; the first 10 parent rows pass the exact check (evidence only)","status":"not_proved"},"rows_compared":16,"schema":"polytool.recurrence-veronese-certificate.v1","statement":"Q_n = lambda(n) V_(2,0)(P_(n+0)) for every n >= 1 is certified; the parent statement is open","verdict":"identity_certified_parent_open"},"input":{"first_index":1,"initial_polynomials":[["1"],["0"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","2"],["0","-3"],["0","1"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"kind":"veronese_section"}