Derivations: a closed-form reference for the (n,2n) Legendre transfer binding (sn-n2n-transfer-binding is a sentinel)

[deOliveira-R]

The gap (archivist, #426 step 3, 2026-09-04)

derivations/ has no script for the (n,2n) Legendre stack: every number the transfer family is verified against is either an end-to-end SN solve (the Be-reflected flagship, tests/sn/verification/analytical/test_be_reflected_n2n_anisotropy.py) or a tape read (tests/data/test_n2n_yield_convention.py). The corpus pass scored the family's "derivation source" dimension 3/6 for that reason, and sn-n2n-transfer-binding (the per-ℓ (n,2n) binding equation in docs/theory/methods/sn/adjoint.rst) is a documented SENTINEL — implemented, not verified by a foundation edge.

What would close it, in cost order

1. Cheap, term tier. A SymPy derive_*() in orpheus/derivations/ proving emission_matrix(y=2) ≡ 2·emission_matrix(y=1) and, more usefully, that the per-ℓ (n,2n) source is y·Σ_{2n,ℓ}ᵀ φ_ℓ block by block — the F2 ruling as algebra — with a verifies("sn-n2n-transfer-binding") marker. Moves the label from sentinel to a real foundation edge in the V&V matrix.
2. Eigenvalue tier. An ANISOTROPIC extension of derive_2g_n2n (the existing 2-group infinite-medium (n,2n) case, orpheus/derivations/continuous/analytical/homogeneous.py): a P1 (n,2n) block in an infinite medium changes k only through the leakage-free P1 closure, so the honest reference is a P1/B1 infinite-medium solve with a scattering-order-1 (n,2n) term — the only route that reaches the eigenvalue layer with a closed form (an MMS would not: the (n,2n) anisotropy is a streaming-coupled source, and MMS verifies discretization, not the model).
3. Not worth it: an ℓ ≥ 2 closed form — the flagship's ladder converges by ℓ = 1 ([M] |Δ(ℓ=2)|/|Δ(ℓ=1)| = 3.1e-3).

Context: .claude/plans/n2n_anisotropy_kept.md §5b.3; ERR-082 in the error catalogue.