{ "kernel": "run_sabrina_celestial_recursion_pde_generating_function", "role": "post-computation assertion only", "schema": "ouroboros_scientific_reference_assertion_v1", "scientific_payload": { "certificate_witnesses": { "generating_function": "1/((1-a*x)*(1-b*x))", "rational_kinematics": [ { "a": "1/2", "all_exact": true, "b": "2/3", "coefficients": [ "1", "-7/10", "37/100", "-7/40", "781/10000", "-3367/100000", "14197/1000000", "-2359/400000", "242461/100000000", "-989527/1000000000", "4017157/10000000000", "-649831/4000000000", "65514541/1000000000000" ], "coincident": false, "coincident_residuals": [], "divided_difference_residuals": [ "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0" ], "generating_function_residuals": [ "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0" ], "max_p": 12, "omega": "3/5", "recurrence_residuals": [ "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0" ] }, { "a": "-1/3", "all_exact": true, "b": "4/5", "coefficients": [ "1", "-49/60", "5341/3600", "-405769/216000", "35585221/12960000", "-2936636689/777600000", "248515747501/46656000000", "-20810983493209/2799360000000", "1750374488820181/167961600000000", "-146952641422223329/10077696000000000", "12346780426820275261/604661760000000000", "-1037033006695530075049/36279705600000000000", "87114151782932582944741/2176782336000000000000" ], "coincident": false, "coincident_residuals": [], "divided_difference_residuals": [ "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0" ], "generating_function_residuals": [ "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0" ], "max_p": 12, "omega": "7/4", "recurrence_residuals": [ "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0" ] }, { "a": "2", "all_exact": true, "b": "2", "coefficients": [ "1", "-4/3", "4/3", "-32/27", "80/81", "-64/81", "448/729", "-1024/2187", "256/729", "-5120/19683", "11264/59049", "-8192/59049", "53248/531441" ], "coincident": true, "coincident_residuals": [ "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0" ], "divided_difference_residuals": [ "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0" ], "generating_function_residuals": [ "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0" ], "max_p": 12, "omega": "1/3", "recurrence_residuals": [ "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0" ] }, { "a": "0", "all_exact": true, "b": "0", "coefficients": [ "1", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0" ], "coincident": true, "coincident_residuals": [ "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0" ], "divided_difference_residuals": [ "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0" ], "generating_function_residuals": [ "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0" ], "max_p": 12, "omega": "5/7", "recurrence_residuals": [ "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0" ] }, { "a": "3/4", "all_exact": true, "b": "3/4", "coefficients": [ "1", "3/5", "27/100", "27/250", "81/2000", "729/50000", "5103/1000000", "2187/1250000", "59049/100000000", "19683/100000000", "649539/10000000000", "531441/25000000000", "6908733/1000000000000" ], "coincident": true, "coincident_residuals": [ "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0" ], "divided_difference_residuals": [ "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0" ], "generating_function_residuals": [ "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0" ], "max_p": 12, "omega": "-2/5", "recurrence_residuals": [ "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0" ] } ], "resummed_pde_kernel": "1/((1+Omega*a*x)*(1+Omega*b*x))", "symbolic_recurrence_residuals_p0_to_p5": [ "0", "0", "0", "0", "0", "0" ] }, "claim_boundary": { "capabilities_removed": [], "convergence_beyond_formal_series_proved": false, "free_operator_commutativity_asserted": false, "non_mhv_sector_proved": false }, "exact_checks": { "archive_hash_matches": true, "coincident_limits_exact": true, "complete_homogeneous_sum_present": true, "distinct_divided_differences_exact": true, "exponential_shift_seed_present": true, "member_hash_matches": true, "p_indexed_tower_present": true, "rational_generating_function_exact": true, "rational_witnesses_exact_through_p12": true, "source_blocks_match": true, "symbolic_second_order_recurrence_exact": true, "taylor_seed_present": true }, "paper": { "arxiv_id": "2208.11635", "title": "Celestial Recursion" }, "result_version": "sabrina_celestial_recursion_pde_generating_function_v1", "source_evidence": { "archive_sha256": "bb41217149756d1b5a56ca91fd7a5c99c511d0be8b8e54f15c900914c712bd77", "block_hashes": { "pde_tower": "651b7d2e3f252acd15cde8eb3cf9cb048ac454c58bfc0fcb493f3bafbe9197ea", "shift_taylor_seed": "cf5e7223ab84469191344fea116685d7c771dfc6ac5f485d811224cd54a25f5c" }, "checks": { "archive_hash_matches": true, "complete_homogeneous_sum_present": true, "exponential_shift_seed_present": true, "member_hash_matches": true, "p_indexed_tower_present": true, "source_blocks_match": true, "taylor_seed_present": true }, "member_sha256": "2774400bc31506d0693cc7c189a7618ee8d7ae8983b96d8fb8cb4d4f427dd650" }, "status": "complete", "theorem": { "coefficient_closed_form": "S_p(a,b)=sum_(l=0)^p a^(p-l)b^l=(a^(p+1)-b^(p+1))/(a-b)", "coincident_limit": "the removable a=b limit is S_p(a,a)=(p+1)a^p", "generating_function": "sum_(p>=0) S_p x^p=1/((1-ax)(1-bx))", "pde_resummation": "the termwise Taylor identities resum on the amplitude to exp(xD)A=A/((1+Omega*a*x)(1+Omega*b*x))", "recurrence": "S_(p+2)=(a+b)S_(p+1)-ab S_p" } }, "scientific_payload_sha256": "94cf744ae42b66c5db61c4d853a7867562d928a7eb80e7cc3297657eb8fcf1cd" }