{"certification":{"candidate":{"gauge":{"kind":"none"},"index_shift":0,"parent":{"name":"Triangle of denominators in Leibniz's Harmonic Triangle a(n,k), n >= 1, 1 <= k <= n.","recurrence":{"first_index":1,"initial_polynomials":[["1"],["2","2"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-1","-2","-1"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"symmetry_schedule":{"first_degree":0,"first_index":1,"step":1}},"scale":"1"},"gamma_rule":"P(n) = 2 P(n-1) - P(n-2)","identifying_rows":2,"identity":{"certificate":{"candidate_leader_shift":2,"candidate_relation":{"terms":[{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-2"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"shift":2}]},"candidate_valid_from_base":1,"exceptional_bases":[],"gamma_relation":{"terms":[{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-2"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"shift":2}]},"gauge":{"kind":"none"},"identifying_indices":[1,2],"index_shift":0,"input":{"first_index":1,"initial_polynomials":[["1"],["2"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-1"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"input_valid_from_base":1,"multiplier":["1"],"parent":{"first_index":1,"initial_polynomials":[["1"],["2","2"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-1","-2","-1"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"quotients":[{"differentiations":[{"coeff":{"coeffs":[["1"]]},"deriv_order":0}],"shift":0}],"scale":"1","statement":"every parent row P_m, m >= 1, is symmetric at d_m; the gamma rows satisfy the transported rule P(n) = 2 P(n-1) - 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 1; with the exceptional bases checked exactly H satisfies the input rule at every base n >= 1, and agreement on the 2 identifying rows gives Q_n = H_n for every n >= 1","symmetry_schedule":{"first_degree":0,"first_index":1,"step":1}},"status":"certified"},"input_first_index":1,"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 >= 3, and the seeds P_1..P_2 are symmetric; so the reversed rows coincide with the rows by induction and every P_m, m >= 1, 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":1,"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":1,"step":1}},"parent":{"gauge":{"kind":"none"},"index_shift":0,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["2","2"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["2","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-1","-2","-1"]]},"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 >= 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, 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), 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 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":"HypothesisFailed: P_n = (a_n + b_n (x + 1)) P_(n-1) - c_n (x + 1)^2 P_(n-2) for n >= 3, with a_n = 0, b_n = 2, c_n = 1; Q_n(y) = y^(d_n) P_n(1/y - 1), d_n = deg P_1 + n - 1, satisfies Q_n = (a_n y + b_n) Q_(n-1) - c_n Q_(n-2); open hypotheses: leading_coefficient_nonzero: fails at n = 3: f(n) a_n = 0 = 0 there (exact); seeds_interlace: deg Q_2 != deg Q_1 + 1 = 1; first_coefficient_positive: not checked: the seeds do not satisfy the seed hypothesis; recurrence_coefficient_positive: fails at n = 4: the denominator of beta_n = 1 vanishes there; root_region: the region contains the pole x = -1 of the coordinate change y = 1/(x + 1)","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 >= 3, with A_n(x) = (2) + (2) x, V_n(x) = 0, W_n(x) = (-1) + (-2) x + (-1) x^2; open hypotheses: lag_degree_step_nonnegative: fails at n = 3: mu_n = -1 = -1 there (exact); multipliers_vanish_at_zero: [x^0] c_2(n, x) = -1 is not the zero polynomial; 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 = 1 (not a disproof): fails at n = 3: [x^1] W_n = -2 there (exact); derivative_or_lag_coefficient_negative: 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 = 1 (not a disproof): fails at n = 3: [x^1] W_n = -2 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-2) in the coefficient of x^(s+0) is not shown to be >= 0 for n >= 3","route":"attestation","success":false,"variant":"oeis_catalog:A003506:lean_proved"}],"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"},"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 >= 3, and the seeds P_1..P_2 are symmetric; so the reversed rows coincide with the rows by induction and every P_m, m >= 1, 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 >= 1 is certified; the lift is not proved (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))"},"kind":"gamma_polynomial"}