{"certification":{"candidate":{"gauge":{"kind":"none"},"index_shift":0,"operator":{"coefficients":[{"denominator":["1"],"numerator":["1"]},{"denominator":["1"],"numerator":["1"]}],"shape":"derivative_polynomial"},"parent":{"first_index":0,"initial_polynomials":[["0"],["3"],["4"],["0","7"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","2","0"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","-1"]]},"deriv_order":0,"offset":4,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_name":"veronese_section_2_1 of A natural number operator."},"identifying_rows":4,"identity":{"certificate":{"certificate":{"dividend_relation":{"terms":[{"coeff":{"coeffs":[["0","12","21","16"],["0","-31","12","0"],["0","15","-4","0"],["0","-2","0","0"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","12","39","-24"],["0","-31","-28","16"],["0","15","4","0"],["0","-2","0","0"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["0","0","18","-36","-16"],["0","0","-40","8","0"],["0","0","8","0","0"]]},"deriv_order":2,"shift":0},{"coeff":{"coeffs":[["0","0","0","4","-16"],["0","0","0","-8","0"]]},"deriv_order":3,"shift":0},{"coeff":{"coeffs":[["6","-26","-32"],["-5","4","0"],["-3","4","0"],["2","0","0"]]},"deriv_order":0,"shift":2},{"coeff":{"coeffs":[["6","4","-24"],["-5","12","-16"],["-3","-4","0"],["2","0","0"]]},"deriv_order":1,"shift":2},{"coeff":{"coeffs":[["0","30","20","16"],["0","8","-8","0"],["0","-8","0","0"]]},"deriv_order":2,"shift":2},{"coeff":{"coeffs":[["0","0","12","16"],["0","0","8","0"]]},"deriv_order":3,"shift":2},{"coeff":{"coeffs":[["-3","16"]]},"deriv_order":0,"shift":4},{"coeff":{"coeffs":[["-3","16"]]},"deriv_order":1,"shift":4}]},"exceptional_bases":[0],"gauge":{"kind":"none"},"identifying_indices":[0,1,2,3],"index_shift":0,"input":{"first_index":0,"initial_polynomials":[["0"],["3"],["4"],["7","7"]],"recurrence":{"denominator":{"coeffs":[["-3","16"]]},"terms":[{"coeff":{"coeffs":[["0","-504","91","-16"],["0","247","-44","0"],["0","-39","4","0"],["0","2","0","0"]]},"deriv_order":0,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["0","0","-306","104"],["0","0","104","-16"],["0","0","-8","0"]]},"deriv_order":1,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["0","0","0","-36","16"],["0","0","0","8","0"]]},"deriv_order":2,"offset":4,"sign":"none"},{"coeff":{"coeffs":[["150","-22","32"],["-115","28","0"],["27","-4","0"],["-2","0","0"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","130","-72"],["0","-72","16"],["0","8","0"]]},"deriv_order":1,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","20","-16"],["0","0","-8","0"]]},"deriv_order":2,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"input_valid_from_base":0,"multiplier":["0","0","0","0","1/16","1/2","3/2","2","1"],"operator":{"coefficients":[{"denominator":["1"],"numerator":["1"]},{"denominator":["1"],"numerator":["1"]}],"shape":"derivative_polynomial"},"parent":{"first_index":0,"initial_polynomials":[["0"],["3"],["4"],["0","7"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","2","0"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","-1"]]},"deriv_order":0,"offset":4,"sign":"none"}]},"schema":"polytool.recurrence.v1"},"parent_leader_shift":4,"parent_valid_from_base":0,"quotients":[{"differentiations":[{"coeff":{"coeffs":[["0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","-995328","5308416"],["0","0","0","0","0","0","0","0","-7962624","42467328"],["0","0","0","0","0","0","0","0","-23887872","127401984"],["0","0","0","0","0","0","0","0","-31850496","169869312"],["0","0","0","0","0","0","0","0","-15925248","84934656"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","-995328","5308416"],["0","0","0","0","0","0","0","0","-7962624","42467328"],["0","0","0","0","0","0","0","0","-23887872","127401984"],["0","0","0","0","0","0","0","0","-31850496","169869312"],["0","0","0","0","0","0","0","0","-15925248","84934656"]]},"deriv_order":1}],"shift":0}],"row_annihilation":{"kind":"system","system":{"generators":[{"annihilator":{"terms":[{"coeff":{"coeffs":[["0","-8","20"],["0","-28","-4"],["0","16","-16"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","0","-8","40"],["0","0","-32","32"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["0","-7"],["0","16"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["3"]]},"deriv_order":0,"shift":3}]},"quotient":[],"step_multiplier":{"coeffs":[["0","0","0","0","0"],["0","0","0","0","0"],["0","0","0","0","192"],["0","0","0","0","768"],["0","0","0","0","768"]]},"step_quotients":[[],[{"coeff":{"coeffs":[["0","0","0","0","0","0"],["0","0","0","0","0","0"],["0","0","0","0","0","960"],["0","0","0","0","0","4864"],["0","0","0","0","0","7936"],["0","0","0","0","0","4096"]]},"deriv_order":0}],[{"coeff":{"coeffs":[["0","0","0","0","0","0","0"],["0","0","0","0","0","0","0"],["0","0","0","0","0","6272","-14976"],["0","0","0","0","0","18176","-43776"],["0","0","0","0","0","11264","-27648"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","-3840","6912"],["0","0","0","0","0","0","-10752","16896"],["0","0","0","0","0","0","-6144","6144"]]},"deriv_order":1}],[{"coeff":{"coeffs":[["0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","5760","-10368"],["0","0","0","0","0","-7680","60672","-80640"],["0","0","0","0","0","-4096","65536","-98304"],["0","0","0","0","0","38912","-131072","92160"],["0","0","0","0","0","32768","-131072","98304"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","-15360","43008","-27648"],["0","0","0","0","0","0","-43008","110592","-67584"],["0","0","0","0","0","0","-24576","49152","-24576"]]},"deriv_order":1}]]},{"annihilator":{"terms":[{"coeff":{"coeffs":[["16","-43","36"],["8","-28","-12"],["-8","32","-24"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","16","-56","72"],["0","16","-64","48"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["-4","0"],["-8","24"]]},"deriv_order":0,"shift":1},{"coeff":{"coeffs":[["3"]]},"deriv_order":0,"shift":2}]},"quotient":[{"coeff":{"coeffs":[["0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","-552960","442368"],["0","0","0","0","0","0","0","0","-4755456","3981312"],["0","0","0","0","0","0","0","0","-15704064","14155776"],["0","0","0","0","0","0","0","0","-23887872","24772608"],["0","0","0","0","0","0","0","0","-14155776","21233664"],["0","0","0","0","0","0","0","0","1769472","7077888"],["0","0","0","0","0","0","0","0","3538944","0"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","663552","-221184","884736"],["0","0","0","0","0","0","0","0","4755456","-442368","5308416"],["0","0","0","0","0","0","0","0","11169792","4866048","7077888"],["0","0","0","0","0","0","0","0","5529600","21233664","-14155776"],["0","0","0","0","0","0","0","0","-13271040","28311552","-42467328"],["0","0","0","0","0","0","0","0","-14155776","7077888","-28311552"],["0","0","0","0","0","0","0","0","1769472","-7077888","0"],["0","0","0","0","0","0","0","0","3538944","0","0"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","3317760","2211840","1769472"],["0","0","0","0","0","0","0","0","0","27426816","16809984","14155776"],["0","0","0","0","0","0","0","0","0","85819392","46006272","42467328"],["0","0","0","0","0","0","0","0","0","120324096","49545216","56623104"],["0","0","0","0","0","0","0","0","0","60162048","7077888","28311552"],["0","0","0","0","0","0","0","0","0","-14155776","-14155776","0"],["0","0","0","0","0","0","0","0","0","-14155776","0","0"]]},"deriv_order":2},{"coeff":{"coeffs":[["0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","1327104","1769472"],["0","0","0","0","0","0","0","0","0","0","11501568","14155776"],["0","0","0","0","0","0","0","0","0","0","38928384","42467328"],["0","0","0","0","0","0","0","0","0","0","63700992","56623104"],["0","0","0","0","0","0","0","0","0","0","49545216","28311552"],["0","0","0","0","0","0","0","0","0","0","14155776","0"]]},"deriv_order":3}],"step_multiplier":{"coeffs":[["0","0","0","0","0"],["0","0","0","0","0"],["0","0","0","0","576"],["0","0","0","0","2304"],["0","0","0","0","2304"]]},"step_quotients":[[{"coeff":{"coeffs":[["0","0","0","0","0"],["0","0","0","0","0"],["0","0","0","0","576"],["0","0","0","0","2304"],["0","0","0","0","2304"]]},"deriv_order":0}],[{"coeff":{"coeffs":[["0","0","0","0","0","0"],["0","0","0","0","0","0"],["0","0","0","0","-2304","4608"],["0","0","0","0","-10752","23040"],["0","0","0","0","-15360","36864"],["0","0","0","0","-6144","18432"]]},"deriv_order":0}],[{"coeff":{"coeffs":[["0","0","0","0","0","0","0"],["0","0","0","0","0","0","0"],["0","0","0","0","-14592","61056","-72576"],["0","0","0","0","-37632","168192","-207360"],["0","0","0","0","-16896","92160","-124416"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","0","0","0","0","0"],["0","0","0","0","0","9216","-34560","34560"],["0","0","0","0","0","23040","-87552","82944"],["0","0","0","0","0","9216","-36864","27648"]]},"deriv_order":1}],[{"coeff":{"coeffs":[["0","0","0","0","0","0","0","0"],["0","0","0","0","0","-13824","51840","-51840"],["0","0","0","0","18432","-177408","476928","-400896"],["0","0","0","0","6144","-158208","525312","-483840"],["0","0","0","0","-86016","454656","-811008","442368"],["0","0","0","0","-49152","344064","-737280","442368"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","36864","-175104","276480","-138240"],["0","0","0","0","0","92160","-442368","681984","-331776"],["0","0","0","0","0","36864","-184320","258048","-110592"]]},"deriv_order":1}]]},{"annihilator":{"terms":[{"coeff":{"coeffs":[["0","3"],["-16","13"],["8","-16"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","0","6"],["0","-48","56"],["0","8","-8"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["0","0","0","0"],["0","0","-16","16"]]},"deriv_order":2,"shift":0},{"coeff":{"coeffs":[["0"],["2"],["4"]]},"deriv_order":0,"shift":1}]},"quotient":[{"coeff":{"coeffs":[["0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","-9068544","3538944","-10616832"],["0","0","0","0","0","0","0","0","-50651136","14598144","-47775744"],["0","0","0","0","0","0","0","0","-83165184","13713408","-37158912"],["0","0","0","0","0","0","0","0","-10616832","12386304","74317824"],["0","0","0","0","0","0","0","0","56623104","63700992","63700992"],["0","0","0","0","0","0","0","0","3538944","56623104","-42467328"],["0","0","0","0","0","0","0","0","-14155776","-21233664","0"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","1327104","-442368","1769472","0"],["0","0","0","0","0","0","0","0","9510912","-88473600","-39813120","-53084160"],["0","0","0","0","0","0","0","0","22339584","-530399232","-283115520","-297271296"],["0","0","0","0","0","0","0","0","11059200","-1070972928","-596312064","-509607936"],["0","0","0","0","0","0","0","0","-26542080","-670629888","-392822784","-169869312"],["0","0","0","0","0","0","0","0","-28311552","184025088","49545216","169869312"],["0","0","0","0","0","0","0","0","3538944","155713536","42467328","0"],["0","0","0","0","0","0","0","0","7077888","-21233664","0","0"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","6635520","4423680","3538944","0"],["0","0","0","0","0","0","0","0","0","54853632","-53968896","-61931520","-21233664"],["0","0","0","0","0","0","0","0","0","171638784","-475987968","-445906944","-127401984"],["0","0","0","0","0","0","0","0","0","240648192","-1196163072","-905969664","-254803968"],["0","0","0","0","0","0","0","0","0","120324096","-1132462080","-537919488","-169869312"],["0","0","0","0","0","0","0","0","0","-28311552","-240648192","84934656","0"],["0","0","0","0","0","0","0","0","0","-28311552","84934656","0","0"]]},"deriv_order":2},{"coeff":{"coeffs":[["0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","2654208","3538944","0"],["0","0","0","0","0","0","0","0","0","0","23003136","12386304","-21233664"],["0","0","0","0","0","0","0","0","0","0","77856768","-21233664","-127401984"],["0","0","0","0","0","0","0","0","0","0","127401984","-141557760","-254803968"],["0","0","0","0","0","0","0","0","0","0","99090432","-198180864","-169869312"],["0","0","0","0","0","0","0","0","0","0","28311552","-84934656","0"]]},"deriv_order":3}],"step_multiplier":{"coeffs":[["0","0","0","0","0","0","0"],["0","0","0","0","0","0","0"],["0","0","0","0","0","0","0"],["0","0","0","0","0","0","1536"],["0","0","0","0","0","0","9216"],["0","0","0","0","0","0","18432"],["0","0","0","0","0","0","12288"]]},"step_quotients":[[],[{"coeff":{"coeffs":[["0","0","0","0","0","0","0"],["0","0","0","0","0","0","0"],["0","0","0","0","0","0","0"],["0","0","0","0","0","0","3072"],["0","0","0","0","0","0","23552"],["0","0","0","0","0","0","69632"],["0","0","0","0","0","0","98304"],["0","0","0","0","0","0","65536"],["0","0","0","0","0","0","16384"]]},"deriv_order":0}],[{"coeff":{"coeffs":[["0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","22528","-51456"],["0","0","0","0","0","0","120832","-279552"],["0","0","0","0","0","0","221184","-525312"],["0","0","0","0","0","0","155648","-393216"],["0","0","0","0","0","0","32768","-98304"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","-30720","41472"],["0","0","0","0","0","0","0","-147456","196608"],["0","0","0","0","0","0","0","-215040","282624"],["0","0","0","0","0","0","0","-73728","98304"],["0","0","0","0","0","0","0","24576","-24576"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","-12288","12288"],["0","0","0","0","0","0","0","0","-61440","61440"],["0","0","0","0","0","0","0","0","-98304","98304"],["0","0","0","0","0","0","0","0","-49152","49152"]]},"deriv_order":2}],[{"coeff":{"coeffs":[["0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","46080","-62208"],["0","0","0","0","0","0","-24576","522240","-635904"],["0","0","0","0","0","0","-57344","1605632","-1944576"],["0","0","0","0","0","0","163840","1286144","-1892352"],["0","0","0","0","0","0","589824","-1228800","491520"],["0","0","0","0","0","0","524288","-1900544","1376256"],["0","0","0","0","0","0","131072","-524288","393216"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","18432","-18432"],["0","0","0","0","0","0","0","-147456","552960","-405504"],["0","0","0","0","0","0","0","-737280","2433024","-1695744"],["0","0","0","0","0","0","0","-1179648","3686400","-2506752"],["0","0","0","0","0","0","0","-589824","1769472","-1179648"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","-49152","98304","-49152"],["0","0","0","0","0","0","0","0","-245760","491520","-245760"],["0","0","0","0","0","0","0","0","-393216","786432","-393216"],["0","0","0","0","0","0","0","0","-196608","393216","-196608"]]},"deriv_order":2}]]},{"annihilator":{"terms":[{"coeff":{"coeffs":[["2"],["-3"],["1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","10"],["0","-4"]]},"deriv_order":1,"shift":0},{"coeff":{"coeffs":[["0","0","4"]]},"deriv_order":2,"shift":0}]},"quotient":[{"coeff":{"coeffs":[["0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","-1990656","663552","-2654208","0"],["0","0","0","0","0","0","0","0","-11612160","5087232","-15040512","0"],["0","0","0","0","0","0","0","0","-57729024","417595392","118554624","254803968"],["0","0","0","0","0","0","0","0","-199065600","2435678208","957284352","1380188160"],["0","0","0","0","0","0","0","0","-244187136","4777574400","2245459968","2187067392"],["0","0","0","0","0","0","0","0","132710400","2848849920","1819017216","382205952"],["0","0","0","0","0","0","0","0","350355456","-707788800","-198180864","-934281216"],["0","0","0","0","0","0","0","0","0","-410517504","-566231040","169869312"],["0","0","0","0","0","0","0","0","-84934656","56623104","84934656","0"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","-9953280","-6635520","-5308416","0"],["0","0","0","0","0","0","0","0","5308416","-119218176","-61046784","-67239936","0"],["0","0","0","0","0","0","0","0","38043648","-939589632","578617344","378667008","382205952"],["0","0","0","0","0","0","0","0","89358336","-3615916032","4807655424","3468165120","2208301056"],["0","0","0","0","0","0","0","0","44236800","-6168379392","11510415360","7594573824","4076863488"],["0","0","0","0","0","0","0","0","-106168320","-3485859840","10170925056","5195169792","2038431744"],["0","0","0","0","0","0","0","0","-113246208","1075838976","1387266048","-198180864","-679477248"],["0","0","0","0","0","0","0","0","14155776","835190784","-1132462080","-169869312","0"],["0","0","0","0","0","0","0","0","28311552","-113246208","84934656","0","0"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","-3981312","-5308416","0","0"],["0","0","0","0","0","0","0","0","0","26542080","-67239936","-77856768","-14155776","0"],["0","0","0","0","0","0","0","0","0","219414528","-775028736","-244187136","417595392","84934656"],["0","0","0","0","0","0","0","0","0","686555136","-3588489216","332660736","2802843648","509607936"],["0","0","0","0","0","0","0","0","0","962592768","-7297302528","2859466752","5662310400","1019215872"],["0","0","0","0","0","0","0","0","0","481296384","-6200229888","4756340736","3510632448","679477248"],["0","0","0","0","0","0","0","0","0","-113246208","-1189085184","2321547264","-339738624","0"],["0","0","0","0","0","0","0","0","0","-113246208","452984832","-339738624","0","0"]]},"deriv_order":2},{"coeff":{"coeffs":[["0","0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0","10616832","3538944","-14155776","0"],["0","0","0","0","0","0","0","0","0","0","92012544","-42467328","-134479872","84934656"],["0","0","0","0","0","0","0","0","0","0","311427072","-396361728","-424673280","509607936"],["0","0","0","0","0","0","0","0","0","0","509607936","-1075838976","-452984832","1019215872"],["0","0","0","0","0","0","0","0","0","0","396361728","-1189085184","113246208","679477248"],["0","0","0","0","0","0","0","0","0","0","113246208","-452984832","339738624","0"]]},"deriv_order":3}],"step_multiplier":{"coeffs":[["0","0","0","0","0","0","0"],["0","0","0","0","0","0","0"],["0","0","0","0","0","0","0"],["0","0","0","0","0","0","512"],["0","0","0","0","0","0","3072"],["0","0","0","0","0","0","6144"],["0","0","0","0","0","0","4096"]]},"step_quotients":[[],[],[{"coeff":{"coeffs":[["0","0","0","0","0","0","0"],["0","0","0","0","0","0","0"],["0","0","0","0","0","0","0"],["0","0","0","0","0","0","-256"],["0","0","0","0","0","0","-768"],["0","0","0","0","0","0","0"],["0","0","0","0","0","0","1024"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","1536"],["0","0","0","0","0","0","0","5120"],["0","0","0","0","0","0","0","2048"],["0","0","0","0","0","0","0","-4096"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","1024"],["0","0","0","0","0","0","0","0","4096"],["0","0","0","0","0","0","0","0","4096"]]},"deriv_order":2}],[{"coeff":{"coeffs":[["0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","-2304"],["0","0","0","0","0","0","0","-20736"],["0","0","0","0","0","0","-2048","-51200"],["0","0","0","0","0","0","-8192","-29696"],["0","0","0","0","0","0","-8192","16384"]]},"deriv_order":0},{"coeff":{"coeffs":[["0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","-1536"],["0","0","0","0","0","0","0","8192","-24576"],["0","0","0","0","0","0","0","30720","-77824"],["0","0","0","0","0","0","0","24576","-65536"],["0","0","0","0","0","0","0","-8192","8192"]]},"deriv_order":1},{"coeff":{"coeffs":[["0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","0","0"],["0","0","0","0","0","0","0","0","4096","-4096"],["0","0","0","0","0","0","0","0","16384","-16384"],["0","0","0","0","0","0","0","0","16384","-16384"]]},"deriv_order":2}]]}],"restart_bases":[0],"row_multiplier":{"coeffs":[["0","0","0","0","0","0","0","0","5308416"]]},"start_base":0}},"shifted_parent_relation":{"terms":[{"coeff":{"coeffs":[["0","0","1"]]},"deriv_order":0,"shift":0},{"coeff":{"coeffs":[["0","-2"]]},"deriv_order":0,"shift":2},{"coeff":{"coeffs":[["1"]]},"deriv_order":0,"shift":4}]},"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 0) 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 >= 0 (the exceptional bases checked exactly), and agreement on the 4 identifying rows gives Q_n = lambda(n) T_n(P_(n+0)) for every n >= 0"},"method":"operator_division"},"status":"certified"},"input_first_index":0,"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 [0, +inf) (exact; Taylor shift: the coefficients of delta(n) at n = 0 + 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":"0","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":0,"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":0,"initial_polynomials":[["0"],["3"],["4"],["0","7"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","2","0"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["0","0","-1"]]},"deriv_order":0,"offset":4,"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 >= 0"},"parent_proof":{"statement":"open: no route proved that every parent row P_n, n >= 0, 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 >= 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)","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":["the division remainder is removed by row annihilators of the shifted parent"],"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: 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":"not applicable: the rule is not f(n) P_n = c_1(n, x) P_(n-1) + c_2(n, x) P_(n-2) with both terms nonzero","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 rule is not f(n) P_n = c_0(n, x) P_(n-1) + c_1(n, x) P'_(n-1) + c_2(n, x) P_(n-2) with a derivative or a lag term","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 rule is not f(n) P_n = c_1(n, x) P_(n-1) + c_d(n, x) P'_(n-1) + c_2(n, x) P_(n-2) with a nonzero lag term and a nonzero P_(n-1) or P'_(n-1) term","route":"liu_wang_three_term","success":false,"variant":"liu_wang_three_term"}],"conditional_on":[],"rows_checked":10,"statement":"open: no route proved that every parent row P_n, n >= 0, 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 >= 0 is certified; the lift is Open (open: no route proved that every parent row P_n, n >= 0, is zero or real-rooted with coefficients of one sign; the first 10 parent rows pass the exact check (evidence only))"},"kind":"preserver_image"}