{"certification":{"candidate":{"gauge":{"kind":"none"},"index_shift":0,"parent":{"name":"Square array read by antidiagonals: S(p,q) = (p+q+1)!(2p+2q+1)!/((p+1)!(2p+1)!(q+1)!(2q+1)!) (p,q>=0).","recurrence":{"first_index":0,"initial_polynomials":[["1"],["1","1"]],"recurrence":{"denominator":{"coeffs":[["6"],["7"],["2"]]},"terms":[{"coeff":{"coeffs":[["4","-2"],["13","1"],["-2","16"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","6","-6"],["0","18","-18"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["2","4","2"],["-6","-12","-6"],["4","8","4"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"symmetry_schedule":{"first_degree":0,"first_index":0,"step":1}},"scale":"1"},"gamma_rule":"(6 + 7n + 2n^2) P(n) = (4 + 13n - 2n^2 - 12t - 24nt + 36n^2t) P(n-1) + (6t + 18nt - 24t^2 - 72nt^2) P'(n-1) + (2 - 6n + 4n^2) P(n-2)","identifying_rows":2,"identity":{"certificate":{"candidate_leader_shift":2,"candidate_relation":{"terms":[{"coeff":{"coeffs":[["-6"],["-10"],["-4"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-22","-84"],["-5","-120"],["2","-36"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["0","-42","168"],["0","-18","72"]]},"deriv_order":1,"shift":1},{"coeff":{"coeffs":[["28"],["15"],["2"]]},"deriv_order":0,"shift":2}]},"candidate_valid_from_base":0,"exceptional_bases":[],"gamma_relation":{"terms":[{"coeff":{"coeffs":[["-6"],["-10"],["-4"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-22","-84"],["-5","-120"],["2","-36"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["0","-42","168"],["0","-18","72"]]},"deriv_order":1,"shift":1},{"coeff":{"coeffs":[["28"],["15"],["2"]]},"deriv_order":0,"shift":2}]},"gauge":{"kind":"none"},"identifying_indices":[0,1],"index_shift":0,"input":{"first_index":0,"initial_polynomials":[["1"],["1"]],"recurrence":{"denominator":{"coeffs":[["6"],["7"],["2"]]},"terms":[{"coeff":{"coeffs":[["4","-12","0"],["13","-24","0"],["-2","36","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","6","-24"],["0","18","-72"],["0","0","0"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["2","0","0"],["-6","0","0"],["4","0","0"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"input_valid_from_base":0,"multiplier":["28","15","2"],"parent":{"first_index":0,"initial_polynomials":[["1"],["1","1"]],"recurrence":{"denominator":{"coeffs":[["6"],["7"],["2"]]},"terms":[{"coeff":{"coeffs":[["4","-2"],["13","1"],["-2","16"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","6","-6"],["0","18","-18"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["2","4","2"],["-6","-12","-6"],["4","8","4"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"quotients":[{"differentiations":[{"coeff":{"coeffs":[["28"],["15"],["2"]]},"deriv_order":0}],"shift":0}],"scale":"1","statement":"every parent row P_m, m >= 0, is symmetric at d_m; the gamma rows satisfy the transported rule (6 + 7n + 2n^2) P(n) = (4 + 13n - 2n^2 - 12t - 24nt + 36n^2t) P(n-1) + (6t + 18nt - 24t^2 - 72nt^2) P'(n-1) + (2 - 6n + 4n^2) P(n-2); w(n) M = sum_nu H_nu S^nu L with remainder zero for the relation L of H_n = lambda(n) Gamma_(n+0)(c z), c = 1, valid from base 0; with the exceptional bases checked exactly H satisfies the input rule at every base n >= 0, and agreement on the 2 identifying rows gives Q_n = H_n for every n >= 0","symmetry_schedule":{"first_degree":0,"first_index":0,"step":1}},"status":"certified"},"input_first_index":0,"lift_summary":{"hypotheses":[{"evidence":"the reversal transport x^(d_m) P_m(1/x) of the parent rule defines the same row map, both left factors are nonzero polynomials at every target m >= 2, and the seeds P_0..P_1 are symmetric; so the reversed rows coincide with the rows by induction and every P_m, m >= 0, is symmetric at d_m","name":"parent_rows_symmetric","statement":"every parent row P_m, m >= p0, is symmetric at d_m","status":{"kind":"verified"}},{"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":0,"input_interlacing":{"reason":"no published theorem transfers interlacing of consecutive parent rows to their gamma polynomials under hypotheses polytool checks","status":"not_proved"},"lean_replay":"not_attempted","operation":{"kind":"gamma_polynomial","scale":"1","symmetry_schedule":{"first_degree":0,"first_index":0,"step":1}},"parent":{"gauge":{"kind":"none"},"index_shift":0,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["1","1"]],"recurrence":{"denominator":{"coeffs":[["6"],["7"],["2"]]},"terms":[{"coeff":{"coeffs":[["4","-2"],["13","1"],["-2","16"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","6","-6"],["0","18","-18"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["2","4","2"],["-6","-12","-6"],["4","8","4"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"relation":"Q_n = lambda(n) Gamma_(n+0)(c z) with c = 1 and P_m = sum_j Gamma_(m,j) x^j (1 + x)^(d_m - 2j), for every n >= 0"},"parent_proof":{"statement":"open: no route proved that every parent row P_n, n >= 0, 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, 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), and symmetric at its index d_m","parent_root_region":{"upper":{"closed":true,"value":"0"}},"theorem":"gamma criterion (RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff): if P = sum_j gamma_j x^j (1 + x)^(d - 2j) with deg Gamma <= d/2, then P is nonzero and real-rooted with roots in (-inf, 0] if and only if Gamma(z) = sum_j gamma_j z^j is; Gamma(c z) with c != 0 has the roots of Gamma divided by c"},"verdict":"open"},"notes":[],"parent_proof":{"attempts":[{"detail":"the leader is not a nonzero constant","route":"periodic_product","success":false,"variant":"P_(n+k) = c(x) P_n"},{"detail":"no theorem matched (weighted_renewal_window: LHS must be a nonzero constant; fixed_delay_predecessor_window: LHS must be a nonzero constant)","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: the rule is not f(n) P_n = c_1(n, x) P_(n-1) + c_2(n, x) P_(n-2) with both terms nonzero","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) = ((-n^2 + 13/2 n + 2)/(n^2 + 7/2 n + 3)) + ((8 n^2 + 1/2 n - 1)/(n^2 + 7/2 n + 3)) x, V_n(x) = ((9 n + 3)/(n^2 + 7/2 n + 3)) x + ((-9 n - 3)/(n^2 + 7/2 n + 3)) x^2, W_n(x) = ((2 n^2 - 3 n + 1)/(n^2 + 7/2 n + 3)) + ((4 n^2 - 6 n + 2)/(n^2 + 7/2 n + 3)) x + ((2 n^2 - 3 n + 1)/(n^2 + 7/2 n + 3)) x^2; open hypotheses: derivative_degree_step_nonnegative: fails at n = 7: lambda_n = (-n^2 + 13/2 n + 2)/(n^2 + 7/2 n + 3) = -1/51 there (exact); multipliers_vanish_at_zero: [x^0] c_2(n, x) = 4 n^2 - 6 n + 2 is not the zero polynomial; constant_term_positive: fails at n = 7: A_n(0) = (-n^2 + 13/2 n + 2)/(n^2 + 7/2 n + 3) = -1/51 there (exact); lag_coefficient_nonpositive: W_n is not separable with a factor of one sign on (-inf, 0), and the sufficient monomial test (-1)^k [x^k] W_n <= 0 fails for k = 0 (not a disproof): fails at n = 2: [x^0] W_n = 3/14 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":"the parent equals the flagged entry, but the flag gives real-rootedness only and one-signed coefficients are not shown: the weight of the coefficient of x^s of P_(n-1) in the coefficient of x^(s+0) is not shown to be >= 0 for n >= 2","route":"attestation","success":false,"variant":"oeis_catalog:A111910:lean_proved"},{"detail":"no lean_proved catalog entry equals the parent; 91 with the same first index and seeds differ in the rule or in later initial rows (oeis_catalog:A001263:lean_proved, oeis_catalog:A008459:lean_proved, oeis_catalog:A008518:lean_proved, oeis_catalog:A024462:lean_proved, oeis_catalog:A026374:lean_proved, oeis_catalog:A026386:lean_proved, oeis_catalog:A028262:lean_proved, oeis_catalog:A028313:lean_proved, oeis_catalog:A030528:lean_proved, oeis_catalog:A038763:lean_proved, oeis_catalog:A046741:lean_proved, oeis_catalog:A046802:lean_proved, oeis_catalog:A054142:lean_proved, oeis_catalog:A056241:lean_proved, oeis_catalog:A056939:lean_proved, oeis_catalog:A081577:lean_proved, oeis_catalog:A081578:lean_proved, oeis_catalog:A081579:lean_proved, oeis_catalog:A081580:lean_proved, oeis_catalog:A081581:lean_proved, oeis_catalog:A081582:lean_proved, oeis_catalog:A092879:lean_proved, oeis_catalog:A097806:lean_proved, oeis_catalog:A100862:lean_proved, oeis_catalog:A100898:lean_proved, oeis_catalog:A102413:lean_proved, oeis_catalog:A103450:lean_proved, oeis_catalog:A105793:lean_proved, oeis_catalog:A109955:lean_proved, oeis_catalog:A109960:lean_proved, oeis_catalog:A111049:lean_proved, oeis_catalog:A114123:lean_proved, oeis_catalog:A124376:lean_proved, oeis_catalog:A124860:lean_proved, oeis_catalog:A128540:lean_proved, oeis_catalog:A128541:lean_proved, oeis_catalog:A129718:lean_proved, oeis_catalog:A133080:lean_proved, oeis_catalog:A136158:lean_proved, oeis_catalog:A141697:lean_proved, oeis_catalog:A143007:lean_proved, oeis_catalog:A143683:lean_proved, oeis_catalog:A143685:lean_proved, oeis_catalog:A144436:lean_proved, oeis_catalog:A144438:lean_proved, oeis_catalog:A144439:lean_proved, oeis_catalog:A144440:lean_proved, oeis_catalog:A144441:lean_proved, oeis_catalog:A144442:lean_proved, oeis_catalog:A144443:lean_proved, oeis_catalog:A144445:lean_proved, oeis_catalog:A146986:lean_proved, oeis_catalog:A146987:lean_proved, oeis_catalog:A146988:lean_proved, oeis_catalog:A147565:lean_proved, oeis_catalog:A147644:lean_proved, oeis_catalog:A154227:lean_proved, oeis_catalog:A154228:lean_proved, oeis_catalog:A154233:lean_proved, oeis_catalog:A154986:lean_proved, oeis_catalog:A157012:lean_proved, oeis_catalog:A157147:lean_proved, oeis_catalog:A157148:lean_proved, oeis_catalog:A157149:lean_proved, oeis_catalog:A157150:lean_proved, oeis_catalog:A157151:lean_proved, oeis_catalog:A157169:lean_proved, oeis_catalog:A157171:lean_proved, oeis_catalog:A157637:lean_proved, oeis_catalog:A166345:lean_proved, oeis_catalog:A169803:lean_proved, oeis_catalog:A176331:lean_proved, oeis_catalog:A181543:lean_proved, oeis_catalog:A184883:lean_proved, oeis_catalog:A190152:lean_proved, oeis_catalog:A193636:lean_proved, oeis_catalog:A208513:lean_proved, oeis_catalog:A209414:lean_proved, oeis_catalog:A228955:lean_proved, oeis_catalog:A236471:lean_proved, oeis_catalog:A249247:lean_proved, oeis_catalog:A258758:lean_proved, oeis_catalog:A267482:lean_proved, oeis_catalog:A305540:lean_proved, oeis_catalog:A321966:lean_proved, oeis_catalog:A350512:lean_proved, oeis_catalog:A358628:lean_proved, oeis_catalog:A358735:lean_proved, oeis_catalog:A367873:lean_proved, oeis_catalog:A371898:lean_proved, oeis_catalog:A376827: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 >= 0, is zero or real-rooted with coefficients of one sign; the first 10 parent rows pass the exact check (evidence only)","status":"not_proved"},"parent_symmetry":{"detail":"the reversal transport x^(d_m) P_m(1/x) of the parent rule defines the same row map, both left factors are nonzero polynomials at every target m >= 2, and the seeds P_0..P_1 are symmetric; so the reversed rows coincide with the rows by induction and every P_m, m >= 0, is symmetric at d_m","status":"certified"},"rows_compared":16,"schema":"polytool.recurrence-gamma-lift.v1","statement":"Q_n = lambda(n) Gamma_(n+0)(c z), c = 1 for every n >= 0 is certified; the lift is not proved (open: no route proved that every parent row P_n, n >= 0, is zero or real-rooted with coefficients of one sign; the first 10 parent rows pass the exact check (evidence only))"},"kind":"gamma_polynomial"}