{"certification":{"candidate":{"gauge":{"kind":"none"},"index_shift":0,"operator":{"coefficients":[{"denominator":["1"],"numerator":["1"]},{"denominator":["1"],"numerator":["1"]}],"shape":"derivative_polynomial"},"parent":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","-8","0"],["0","4","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","2","-8"],["0","0","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_name":"gamma_polynomial of Triangle read by rows: Eulerian numbers of type B, T(n,k) (1 <= k <= n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (2*n - 2*k + 1)*T(n-1, k-1) + (2*k - 1)*T(n-1, k)."},"identifying_rows":2,"identity":{"certificate":{"certificate":{"dividend_relation":{"terms":[{"coeff":{"coeffs":[["-3","50","-272","544","-256"],["6","-60","264","-672","256"],["0","16","-160","192","0"],["0","0","32","0","0"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-3","48","-236","320","320","-512"],["6","-56","216","-448","-128","0"],["0","16","-144","128","0","0"],["0","0","32","0","0","0"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["0","-2","36","-224","576","-512"],["0","4","-48","224","-384","0"],["0","0","16","-64","0","0"]]},"deriv_order":2,"shift":0},{"coeff":{"coeffs":[["10","-86","184","-64"],["-8","60","-80","0"],["0","-16","0","0"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["10","-80","144"],["-8","56","-64"],["0","-16","0"]]},"deriv_order":1,"shift":1},{"coeff":{"coeffs":[["0","6","-40","64"],["0","-4","16","0"]]},"deriv_order":2,"shift":1},{"coeff":{"coeffs":[["-3","8"],["2","0"]]},"deriv_order":0,"shift":2},{"coeff":{"coeffs":[["-3","8"],["2","0"]]},"deriv_order":1,"shift":2}]},"exceptional_bases":[],"gauge":{"kind":"none"},"identifying_indices":[1,2],"index_shift":0,"input":{"first_index":1,"initial_polynomials":[["1"],["1"]],"recurrence":{"denominator":{"coeffs":[["-7","8"],["2","0"]]},"terms":[{"coeff":{"coeffs":[["15","-234","1696","-2656","768"],["-6","124","-1288","1440","-256"],["0","-16","352","-192","0"],["0","0","-32","0","0"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","10","-196","928","-1344","512"],["0","-4","112","-480","384","0"],["0","0","-16","64","0","0"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-26","270","-344","64"],["8","-124","80","0"],["0","16","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-14","72","-64"],["0","4","-16","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"input_valid_from_base":1,"multiplier":["1"],"operator":{"coefficients":[{"denominator":["1"],"numerator":["1"]},{"denominator":["1"],"numerator":["1"]}],"shape":"derivative_polynomial"},"parent":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","-8","0"],["0","4","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","2","-8"],["0","0","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_leader_shift":1,"parent_valid_from_base":1,"quotients":[{"differentiations":[{"coeff":{"coeffs":[["7","-78","184","-64"],["-18","80","-48","0"],["8","-8","0","0"]]},"deriv_order":0},{"coeff":{"coeffs":[["1","-14","56","-64"],["-2","16","-48","0"],["0","-8","0","0"]]},"deriv_order":1}],"shift":0},{"differentiations":[{"coeff":{"coeffs":[["-3","8"],["2","0"]]},"deriv_order":0},{"coeff":{"coeffs":[["-3","8"],["2","0"]]},"deriv_order":1}],"shift":1}],"shifted_parent_relation":{"terms":[{"coeff":{"coeffs":[["-1","4"],["0","-4"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","-2","8"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"shift":1}]},"statement":"w(n) E = sum_nu H_nu S^nu L' (with any remainder removed by row annihilators of the parent), where L' is the parent rule at P'_n = P_(n+0) (valid from base 1) and E the input rule gauged by lambda(n)/delta(n) and composed with (or conjugated through) T = phi(n, D) = 1 + (1) D; so the candidate rows satisfy the input rule at every base n >= 1 (the exceptional bases checked exactly), and agreement on the 2 identifying rows gives Q_n = lambda(n) T_n(P_(n+0)) for every n >= 1"},"method":"operator_division"},"status":"certified"},"input_first_index":1,"interlacing":{"reason":"no parent theorem report with an interlacing conclusion","status":"none"},"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"lift_summary":{"hypotheses":[{"evidence":"the polynomial with ascending coefficients [1, 1] is fixed; exact check: real-rooted = true, coefficients of one sign = true","name":"derivative_polynomial_real_rooted","statement":"the polynomial in D is real-rooted for every n >= n0","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"}},{"evidence":"delta(n) != 0 for every integer n in [1, +inf) (exact; Taylor shift: the coefficients of delta(n) at n = 1 + u have one sign)","name":"operator_coefficients_defined","sign_decision":{"certificate":{"condition":"nonzero","denominator":["1"],"method":{"kind":"taylor_shift","shifted_denominator":["1"],"shifted_numerator":["1"]},"numerator":["1"],"range":{"first":"1","last":null}},"decision":"holds"},"statement":"the operator coefficients are finite at every n >= n0 (their denominators do not vanish)","status":{"kind":"verified"}}],"identity":{"method":"forward_division","status":"certified"},"input_first_index":1,"input_interlacing":{"reason":"no parent theorem report with an interlacing conclusion","status":"not_proved"},"lean_replay":"not_attempted","operation":{"description":"phi(n, D) = 1 + (1) D","kind":"preserver_image","operator":{"coefficients":[{"denominator":["1"],"numerator":["1"]},{"denominator":["1"],"numerator":["1"]}],"shape":"derivative_polynomial"}},"parent":{"gauge":{"kind":"none"},"index_shift":0,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","-8","0"],["0","4","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","2","-8"],["0","0","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"relation":"Q_n = lambda(n) T_n(P_(n+0)) with T = phi(n, D) = 1 + (1) D, 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)","parent_root_region":{"upper":{"closed":true,"value":"0"}},"theorem":"Laguerre's theorem on differential operators: if phi is a real polynomial with only real roots, then phi(D) maps every real-rooted polynomial to a real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n"},"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":"not applicable: expected one nonzero lag term","route":"product_recurrence","success":false,"variant":"product_recurrence"},{"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":"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) = (1) + (4 n - 8) x and b_n(x) = (0) + (2) x + (-8) x^2; open hypotheses: degree_step_positive: fails at n = 2: lambda_n = -4 n + 8 = 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":"HypothesisFailed: P_n = (alpha_n x + beta_n) P_(n-1) + kappa_n (x - L)(x - U) P'_(n-1) for n >= 2, with L = 0, U = 1/4, alpha_n = 4 n - 8, beta_n = 1, kappa_n = -8; open hypotheses: slope_positive: fails at n = 2: a_n = -1/2 n + 1 = 0 there (exact); upper_endpoint_threshold: no hit before n = 2; E(2) = e(2) + 0 = -1/2 < 0 with no hit in between (exact); when a_n > 0 for every n <= 2, the criterion puts a root of P_2 outside the interval","route":"ma_wang_interval","success":false,"variant":"ma_wang_interval"},{"detail":"not applicable: the theorem needs two seed rows","route":"liu_wang_three_term","success":false,"variant":"liu_wang_three_term"},{"detail":"HypothesisFailed: Q_n = x^(d_n) P_n(1/x) at the nominal degree d_n, with d_(n-L) = n - 2 (an upper bound of deg P_(n-L)): Q_n = a_n(x) S_n(Q_(n-1)) for n >= 2, with a_n(x) = (1) ((1)) and S_n = A + B D + C D^2 with A = 8 - 4 n - 3 x + 2 n x, B = 8 x - 2 x^2, C = 0; open hypotheses: weyl_step_leading_coefficients_same_sign: fails at n = 2: A_1 B_lead = -4 n + 6 = -2 there (exact); weyl_step_interlacing: fails at n = 2: B_2 (B_2 A_0^2 - B_1 A_0 A_1 + B_0 A_1^2) = -64 n^2 + 192 n - 128 = 0 there (exact); weyl_step_leading_coefficients_one_signed: fails at n = 2: A_a B_(a+1) = -4 n + 6 = -2 there (exact)","route":"iterated_preserver","success":false,"variant":"iterated_preserver"},{"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-preserver-image.v1","statement":"Q_n = lambda(n) T_n(P_(n+0)) with T = phi(n, D) = 1 + (1) D for every n >= 1 is certified; the lift is Open (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":"preserver_image"}