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78984 bytes, as of the version from 2026-10-27 01:12 (current). Recorded here, not run.
{
"tid": "T513",
"status": "coupling_selection_unresolved_prose_only",
"parameters": {
"ell": "The nonnegative angular momentum index labels partial waves. The existing s,p,d,f prefix is a modest truncation, not a completeness claim.",
"eta": "Zero is an intrinsic free-wave degeneration. +/-1/2,+/-1,+/-2 remain conventional reference couplings; signs describe attraction/repulsion but do not select those magnitudes.",
"n": "The strictly positive radial-zero ordinal is intrinsic at fixed ell and eta. It does not make the independently chosen eta values intrinsic."
},
"physical_caveat": "At fixed physical charges and masses, eta depends on scattering energy. Fixed-eta radial zeros are not the energy spectrum of a fixed physical Coulomb Hamiltonian.",
"action": "Keep all 336 numbers, parameter keys, annotations, equalities, generators and existing sources unchanged. Clarify the scientifically unresolved coupling selection only.",
"sources": [
"https://dlmf.nist.gov/33.2",
"https://dlmf.nist.gov/33.22.E4",
"https://dlmf.nist.gov/33.5"
],
"validation": {
"all_stored_endpoint_brackets": 336,
"independent_free_case_checks": 48,
"index_proof": "For |eta|<=2, ell>=0, the coefficient recurrence bounds |a_k|<=5^k. F>0 on (0,1/16]. Positivity of G(1/16) and (G/F)prime=-1/F^2 imply G>0 there. On [1/16,infinity), q<=65 gives root separation >=pi/sqrt(65)>1/4 by integration by parts and Wirtinger. Nonzero scan signs at spacing 1/4 therefore count all simple zeros.",
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