{"certification":{"candidate":{"gauge":{"kind":"none"},"index_shift":0,"parent":{"name":"Square the entries of Pascal's triangle.","recurrence":{"first_index":0,"initial_polynomials":[["1"],["1","1"]],"recurrence":{"denominator":{"coeffs":[["0"],["1"]]},"terms":[{"coeff":{"coeffs":[["-1","-1"],["2","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["1","-2","1"],["-1","2","-1"]]},"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":"(n) P(n) = (-1 + 2n) P(n-1) + (1 - n - 4t + 4nt) P(n-2)","identifying_rows":2,"identity":{"certificate":{"candidate_leader_shift":2,"candidate_relation":{"terms":[{"coeff":{"coeffs":[["1","-4"],["1","-4"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-3"],["-2"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["2"],["1"]]},"deriv_order":0,"shift":2}]},"candidate_valid_from_base":0,"exceptional_bases":[],"gamma_relation":{"terms":[{"coeff":{"coeffs":[["1","-4"],["1","-4"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-3"],["-2"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["2"],["1"]]},"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":[["0"],["-1"]]},"terms":[{"coeff":{"coeffs":[["1","0"],["-2","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-1","4"],["1","-4"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"input_valid_from_base":0,"multiplier":["2","1"],"parent":{"first_index":0,"initial_polynomials":[["1"],["1","1"]],"recurrence":{"denominator":{"coeffs":[["0"],["1"]]},"terms":[{"coeff":{"coeffs":[["-1","-1"],["2","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["1","-2","1"],["-1","2","-1"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"quotients":[{"differentiations":[{"coeff":{"coeffs":[["-2"],["-1"]]},"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 (n) P(n) = (-1 + 2n) P(n-1) + (1 - n - 4t + 4nt) 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":[["0"],["1"]]},"terms":[{"coeff":{"coeffs":[["-1","-1"],["2","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["1","-2","1"],["-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 >= 0"},"parent_proof":{"route":"three_term_recurrence","statement":"every parent row P_n, n >= 0, is zero or real-rooted with coefficients of one sign","status":"proved"},"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":"proved"},"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":"P_n = (a_n + b_n (x - 1)) P_(n-1) - c_n (x - 1)^2 P_(n-2) for n >= 2, with a_n = (4 n - 2)/(n), b_n = (2 n - 1)/(n), c_n = (n - 1)/(n); Q_n(y) = y^(d_n) P_n(1/y + 1), d_n = deg P_0 + n - 0, satisfies Q_n = (a_n y + b_n) Q_(n-1) - c_n Q_(n-2): every P_n, n >= 0, is nonzero and real-rooted, with simple roots from n = 0 on; P_(n-1) strictly interlaces P_n for n >= 1; every root lies in (-inf, 0)","route":"three_term_recurrence","success":true,"variant":"three_term_recurrence"}],"conditional_on":[],"interlacing":{"from_index":1,"justification":{"premises":["every hypothesis of the theorem is verified exactly for every index: P_n = (a_n + b_n (x - 1)) P_(n-1) - c_n (x - 1)^2 P_(n-2) for n >= 2, with a_n = (4 n - 2)/(n), b_n = (2 n - 1)/(n), c_n = (n - 1)/(n); Q_n(y) = y^(d_n) P_n(1/y + 1), d_n = deg P_0 + n - 0, satisfies Q_n = (a_n y + b_n) Q_(n-1) - c_n Q_(n-2)","every P_n, n >= 0, is nonzero and real-rooted, with simple roots from n = 0 on; P_(n-1) strictly interlaces P_n for n >= 1; every root lies in (-inf, 0)"],"theorem":"three_term_recurrence: three-term recurrence Q_n = (a_n y + b_n) Q_(n-1) - c_n Q_(n-2) with a_n != 0 and beta_n = c_n / (a_n a_(n-1)) > 0: simple real roots and strict interlacing of consecutive rows (zero separation for orthogonal polynomials: G. Szeg\u0151, Orthogonal Polynomials, \u00a73.3; Favard, C. R. Acad. Sci. Paris 200 (1935)); root regions by a positive-ratio (chain-sequence) induction"},"kind":"strict","orientation":"previous_interlaces_next"},"route":"three_term_recurrence","rows_checked":10,"statement":"every parent row P_n, n >= 0, 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_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":"proved: Q_n = lambda(n) Gamma_(n+0)(c z), c = 1 for every n >= 0 and every parent row P_n, n >= 0, is zero or real-rooted with coefficients of one sign; so every Q_n, n >= 0, is zero or real-rooted with roots in (-inf, 0]"},"kind":"gamma_polynomial"}