File size: 6,566 Bytes
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"kernel": "run_sabrina_soft_mellin_residue_identity",
"role": "post-computation assertion only",
"schema": "ouroboros_scientific_reference_assertion_v1",
"scientific_payload": {
"claim_boundary": {
"capabilities_removed": [],
"distributional_endpoint_convention": "symmetric half-line delta weight",
"paper_leading_and_subleading_results_challenged": false,
"scope": "The source immediately applies n=0 and n=1; both are exact because n!=1.",
"smoothness_assumption": "g(omega)=omega a(omega) is smooth at omega=0",
"unqualified_n_ge_2_extension_challenged": true
},
"classification": "physical_n0_n1_exact_unqualified_all_n_extension_requires_factorial",
"exact_checks": {
"archive_hash_matches": true,
"corrected_identity_passes_n0_through_n8": true,
"fourier_annihilation_phase": true,
"fourier_creation_phase": true,
"mellin_residue_is_derivative_over_factorial": true,
"member_hash_matches": true,
"n0_source_identity_exact": true,
"n1_source_identity_exact": true,
"n2_is_exact_counterexample_to_unqualified_all_n_extension": true,
"n2_mismatch_factor_is_two": true,
"source_blocks_match": true,
"source_equation_labels_unique": true
},
"finite_exact_rows": [
{
"annihilation_coefficient": "pi",
"creation_coefficient": "pi",
"displayed_identity_matches": true,
"endpoint_derivative": "1",
"factorial_corrected_identity_matches": true,
"mellin_residue": "1",
"mellin_test_function": "a(omega)=omega^-1 exp(-omega)",
"n": 0,
"source_formula_ratio": "1"
},
{
"annihilation_coefficient": "-I*pi",
"creation_coefficient": "I*pi",
"displayed_identity_matches": true,
"endpoint_derivative": "1",
"factorial_corrected_identity_matches": true,
"mellin_residue": "1",
"mellin_test_function": "a(omega)=omega^0 exp(-omega)",
"n": 1,
"source_formula_ratio": "1"
},
{
"annihilation_coefficient": "-2*pi",
"creation_coefficient": "-2*pi",
"displayed_identity_matches": false,
"endpoint_derivative": "2",
"factorial_corrected_identity_matches": true,
"mellin_residue": "1",
"mellin_test_function": "a(omega)=omega^1 exp(-omega)",
"n": 2,
"source_formula_ratio": "2"
},
{
"annihilation_coefficient": "6*I*pi",
"creation_coefficient": "-6*I*pi",
"displayed_identity_matches": false,
"endpoint_derivative": "6",
"factorial_corrected_identity_matches": true,
"mellin_residue": "1",
"mellin_test_function": "a(omega)=omega^2 exp(-omega)",
"n": 3,
"source_formula_ratio": "6"
},
{
"annihilation_coefficient": "24*pi",
"creation_coefficient": "24*pi",
"displayed_identity_matches": false,
"endpoint_derivative": "24",
"factorial_corrected_identity_matches": true,
"mellin_residue": "1",
"mellin_test_function": "a(omega)=omega^3 exp(-omega)",
"n": 4,
"source_formula_ratio": "24"
},
{
"annihilation_coefficient": "-120*I*pi",
"creation_coefficient": "120*I*pi",
"displayed_identity_matches": false,
"endpoint_derivative": "120",
"factorial_corrected_identity_matches": true,
"mellin_residue": "1",
"mellin_test_function": "a(omega)=omega^4 exp(-omega)",
"n": 5,
"source_formula_ratio": "120"
},
{
"annihilation_coefficient": "-720*pi",
"creation_coefficient": "-720*pi",
"displayed_identity_matches": false,
"endpoint_derivative": "720",
"factorial_corrected_identity_matches": true,
"mellin_residue": "1",
"mellin_test_function": "a(omega)=omega^5 exp(-omega)",
"n": 6,
"source_formula_ratio": "720"
},
{
"annihilation_coefficient": "5040*I*pi",
"creation_coefficient": "-5040*I*pi",
"displayed_identity_matches": false,
"endpoint_derivative": "5040",
"factorial_corrected_identity_matches": true,
"mellin_residue": "1",
"mellin_test_function": "a(omega)=omega^6 exp(-omega)",
"n": 7,
"source_formula_ratio": "5040"
},
{
"annihilation_coefficient": "40320*pi",
"creation_coefficient": "40320*pi",
"displayed_identity_matches": false,
"endpoint_derivative": "40320",
"factorial_corrected_identity_matches": true,
"mellin_residue": "1",
"mellin_test_function": "a(omega)=omega^7 exp(-omega)",
"n": 8,
"source_formula_ratio": "40320"
}
],
"paper": {
"arxiv_id": "2105.09792",
"source_equations": [
"Mellin",
"mellinmode",
"eq:IDDelta1",
"eq:IDDelta2"
],
"title": "Revisiting the Conformally Soft Sector with Celestial Diamonds"
},
"result_version": "sabrina_soft_mellin_residue_scope_theorem_v1",
"source_evidence": {
"archive_sha256": "96fe817814bf7bfc432df67f49f60ea0afa72f6eeb335e93624b6d8bb54359f3",
"block_hashes": {
"Mellin": "6bceb4357552b5aa5e81879be46ef7a2bca646b1766392f58887007fa4ffb310",
"eq:IDDelta1": "4972b66bb470f919154ab6fd3324c373869538b172947c553b8508ce4fcdaddf",
"eq:IDDelta2": "c7e9d17afa4c3d9719ddce22c97340c3be4facd874f86d1bb519cc51c8902c5c",
"mellinmode": "e019183a7f11f077ce6d9e78ace905c0e421de10bd9940f9b6d1036b3f242122"
},
"checks": {
"archive_hash_matches": true,
"member_hash_matches": true,
"source_blocks_match": true,
"source_equation_labels_unique": true
}
},
"status": "complete",
"theorem": {
"corrected_annihilation_identity": "LHS=(-i)^n pi n! Res_{Delta=1-n}[a_Delta] (1+z zbar)^(-1-n).",
"corrected_creation_identity": "LHS=(+i)^n pi n! Res_{Delta=1-n}[a_Delta^dagger] (1+z zbar)^(-1-n).",
"equivalent_source_form": "Replace u_pm^n on the left by u_pm^n/n!.",
"fourier_endpoint_rule": "Integral_0^infinity g(omega) delta^(n)(c omega) d omega =(-1)^n g^(n)(0)/(2 c^(n+1)) under the symmetric endpoint convention.",
"mellin_residue_rule": "Res_{Delta=1-n} Integral_0^infinity omega^(Delta-2) g(omega) d omega = g^(n)(0)/n!."
}
},
"scientific_payload_sha256": "2bac79f473b925a796d826a6c2d4fdd4003b5c269b8843d9f37dd83ed0e4d58f"
}
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