{"certification":{"candidate":{"gauge":{"kind":"none"},"index_shift":0,"parent":{"name":"Triangle T(n, k) = coefficients of ( t(n, x) ) where t(n, x) = (1-x)^(n+1)*p(n, x)/x, p(n, x) = x*D( p(n-1, x) ), with p(1, x) = x/(1-x)^2, p(2, x) = x*(1+x)/(1-x)^3, and p(3, x) = x*(1+6*x+x^2)/(1-x)^4, read by rows.","recurrence":{"first_index":1,"initial_polynomials":[["1","6","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","1"],["0","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1","-1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"symmetry_schedule":{"first_degree":2,"first_index":1,"step":1}},"scale":"1"},"gamma_rule":"P(n) = (1 + 2nt) P(n-1) + (t - 4t^2) P'(n-1)","identifying_rows":1,"identity":{"certificate":{"candidate_leader_shift":1,"candidate_relation":{"terms":[{"coeff":{"coeffs":[["-1","-2"],["0","-2"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","-1","4"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"shift":1}]},"candidate_valid_from_base":1,"exceptional_bases":[],"gamma_relation":{"terms":[{"coeff":{"coeffs":[["-1","-2"],["0","-2"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","-1","4"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"shift":1}]},"gauge":{"kind":"none"},"identifying_indices":[1],"index_shift":0,"input":{"first_index":1,"initial_polynomials":[["1","4"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","0","0"],["0","2","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1","-4"],["0","0","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"input_valid_from_base":1,"multiplier":["1"],"parent":{"first_index":1,"initial_polynomials":[["1","6","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","1"],["0","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1","-1"]]},"deriv_order":1,"offset":1,"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) = (1 + 2nt) P(n-1) + (t - 4t^2) P'(n-1); 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 1 identifying rows gives Q_n = H_n for every n >= 1","symmetry_schedule":{"first_degree":2,"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 >= 2, and the seeds P_1..P_1 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":2,"first_index":1,"step":1}},"parent":{"gauge":{"kind":"none"},"index_shift":0,"recurrence":{"first_index":1,"initial_polynomials":[["1","6","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","1"],["0","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1","-1"]]},"deriv_order":1,"offset":1,"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":{"route":"root_domain","statement":"every parent row P_n, n >= 1, is zero or real-rooted with coefficients of one sign","status":"proved"},"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":"proved"},"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: derivatives and alternating signs are unsupported)","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":"source claim: every P_n, n >= 1, is real-rooted in (-inf, 0] (CertifiedFamilyInvariant): the completed induction for Q_n and real_mobius_equivalence (target to source; every map condition certified for all indices); deg Q_n = d_n by the degree formula, so no root of P_n is added at the image of infinity","route":"root_domain","success":true,"variant":"f_polynomial"}],"conditional_on":[],"route":"root_domain","rows_checked":10,"statement":"every parent row P_n, n >= 1, is zero or real-rooted with coefficients of one sign","status":"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_1..P_1 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":"proved: Q_n = lambda(n) Gamma_(n+0)(c z), c = 1 for every n >= 1 and every parent row P_n, n >= 1, is zero or real-rooted with coefficients of one sign; so every Q_n, n >= 1, is zero or real-rooted with roots in (-inf, 0]"},"kind":"gamma_polynomial"}