The coefficient supplied to this internal helper for key "i,j,k,l" is the FULL WEIGHTED prefactor of this operator string and is applied verbatim — the weighted core folds in the ½ of the two-body Hamiltonian term ½·Σ g_pqrs a†_p a†_q a_r a_s; no additional ½ is applied here. The raw-physicist primary builders reach this helper via the internal raw→weighted mapping (½ and r↔s swap), so a raw ⟨pq|rs⟩ factory is supplied to computeHamiltonianWith directly. The Fcidump adapters produce a conforming raw factory.