{"certification":{"candidate":{"gauge":{"kind":"none"},"index_shift":0,"operator":{"a":"1","b":"0","c":"1","e":"1","schedule":{"first_degree":0,"first_index":1,"step":1},"shape":"mobius"},"parent":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["12","3"],["-9","-1"],["2","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["28","31","5"],["-8","-8","-1"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["8","16","9","1"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_name":"shift_x_plus_one of Square array of Delannoy numbers D(i,j) (i >= 0, j >= 0) read by antidiagonals."},"identifying_rows":1,"identity":{"certificate":{"certificate":{"dividend_relation":{"terms":[{"coeff":{"coeffs":[["-5","23","-40","30","-5","-5","2"],["5","-24","45","-40","15","0","-1"],["-2","10","-20","20","-10","2","0"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-20","77","-89","-10","90","-55","3","4"],["8","-32","41","-5","-30","22","-3","-1"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["-8","24","-9","-36","35","8","-19","4","1"]]},"deriv_order":2,"shift":0},{"coeff":{"coeffs":[["0","0","0","0","0","0"],["1","-5","10","-10","5","-1"]]},"deriv_order":0,"shift":1}]},"exceptional_bases":[],"gauge":{"kind":"none"},"identifying_indices":[1],"index_shift":0,"input":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["-1","-3","-3","-1"],["1","3","3","1"]]},"terms":[{"coeff":{"coeffs":[["116","777","2153","3161","2595","1130","204"],["-93","-624","-1737","-2567","-2124","-933","-170"],["18","120","334","496","414","184","34"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["76","631","2218","4286","4924","3367","1270","204"],["-24","-200","-707","-1376","-1594","-1100","-419","-68"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["8","80","345","839","1260","1198","705","235","34"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"input_valid_from_base":1,"multiplier":["0","1"],"operator":{"a":"1","b":"0","c":"1","e":"1","schedule":{"first_degree":0,"first_index":1,"step":1},"shape":"mobius"},"parent":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["12","3"],["-9","-1"],["2","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["28","31","5"],["-8","-8","-1"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["8","16","9","1"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_leader_shift":1,"parent_valid_from_base":1,"quotients":[{"differentiations":[{"coeff":{"coeffs":[["0","0","0","0","0","0"],["1","-5","10","-10","5","-1"]]},"deriv_order":0}],"shift":0}],"shifted_parent_relation":{"terms":[{"coeff":{"coeffs":[["-5","-2"],["5","1"],["-2","0"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["-20","-23","-4"],["8","8","1"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["-8","-16","-9","-1"]]},"deriv_order":2,"shift":0},{"coeff":{"coeffs":[["0"],["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 = ((1) x + (1))^(d_n) P(((1) x + (0))/((1) x + (1))), d_n = 0 + 1 (n - 1); so the candidate rows satisfy the input rule at every base n >= 1 (the exceptional bases checked exactly), and agreement on the 1 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":"the schedule d_n depends on n","status":"none"},"lean_declarations":["RealRooted.splits_fPolynomial"],"lean_route":"fTransform","lift_summary":{"hypotheses":[{"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":"the schedule d_n depends on n","status":"not_proved"},"lean_replay":"not_attempted","operation":{"description":"((1) x + (1))^(d_n) P(((1) x + (0))/((1) x + (1))), d_n = 0 + 1 (n - 1)","kind":"preserver_image","operator":{"a":"1","b":"0","c":"1","e":"1","schedule":{"first_degree":0,"first_index":1,"step":1},"shape":"mobius"}},"parent":{"gauge":{"kind":"none"},"index_shift":0,"recurrence":{"first_index":1,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["12","3"],["-9","-1"],["2","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["28","31","5"],["-8","-8","-1"]]},"deriv_order":1,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["8","16","9","1"]]},"deriv_order":2,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"relation":"Q_n = lambda(n) T_n(P_(n+0)) with T = ((1) x + (1))^(d_n) P(((1) x + (0))/((1) x + (1))), d_n = 0 + 1 (n - 1), 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 [-1, 0]","input_root_region":{"lower":{"closed":true,"value":"-1"},"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":"a real affine substitution x -> a x + b (a != 0), the reversal x^d p(1/x) (d >= deg p) and a real Mobius map (c x + e)^d p((a x + b)/(c x + e)) (a e - b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically"},"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":"not applicable: expected one nonzero lag term","route":"product_recurrence","success":false,"variant":"product_recurrence"},{"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":"not applicable: the theorem needs at least two seed rows","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":"not applicable: the theorem needs two seed rows","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":"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 = -1 + n + 4 x - 7 n x + 3 n^2 x + 40 x^2 - 36 n x^2 + 8 n^2 x^2 + 48 x^3 - 40 n x^3 + 8 n^2 x^3, B = x - n x + 23 x^2 - 10 n x^2 + 68 x^3 - 24 n x^3 + 48 x^4 - 16 n x^4, C = x^2 + 9 x^3 + 16 x^4 + 8 x^5) / (n - 1); open hypotheses: degree_step: delta = 3, deg P_(n-1) = 3 n - 6: fails at n = 2: Lambda(n, deg P_(n-1)) = 32 n^2 - 112 n + 96 = 0 there (exact); weyl_step_discriminant_nonnegative: at n = 2: B^2 - 4 A C = -3554006546718992946601903359/79228162514264337593543950336 < 0 at x = -8351/8192: T is not a real-rootedness preserver on the whole line (Borcea--Branden 2010, Theorem 1.3, necessary condition); weyl_step_interlacing: at n = 2: deg A = 1, deg B = 4: neither deg B = deg A nor deg B = deg A + 1; weyl_step_leading_coefficients_one_signed: fails at n = 4: A_a B_(a+1) = -128 n^3 + 1024 n^2 - 2688 n + 2304 = -256 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 = ((1) x + (1))^(d_n) P(((1) x + (0))/((1) x + (1))), d_n = 0 + 1 (n - 1) 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"}