Does the energy model use real interactions beyond the second neighbour shell?

E220 — v5's four pair shells read by the second sampler, pre-registered (2026-09-21 18:3x)

Replaces the runs/v5_shells pyeCE read, whose positive-α shells cannot be decoded. The second sampler (scripts/ordering/v5_metropolis.py, now measuring α₁–α₄ on the 8/6/12/24 bcc shells at 2.85/3.29/4.66/5.46 Å; a cell narrower than twice a shell's radius is refused) runs v5 and the exact NN Ising on the same 6³ cell at 3000 and 1550 K true (150 + 350 sweeps, one core, runs/e220_v5_four_shells, runs/e220_ising_control). Predictions: (1) the NN Ising's α₃, α₄ at 3000 K are induced only, |α| < 0.01 (sampling noise on 432 sites ≈ 0.01); (2) v5 carries real third- and fourth-shell couplings, so |α₃| or |α₄| > 0.03 (three times the noise) at 3000 K; (3) v5's α₂/α₁ at 3000 K reproduces the −0.50 of runs/e216_v5_metropolis to ±0.05. If (2) fails, v5 is a two-shell Hamiltonian and the "not NN" statement narrows to "second shell only".

E220 RESULT (2026-09-21 18:5x): v5 is a two-shell Hamiltonian; prediction (2) FAILS as written. Both legs, 6³ cell, 150 + 350 sweeps, one core, 100 s per temperature:

(cooling / heating.) Scored against the pre-registration: (1) the NN Ising's induced α₃, α₄ at 3000 K are 0.0115 and −0.0057 — "< 0.01" holds for α₄ and is missed by α₃ by 0.0015; (3) v5's α₂/α₁ = −0.50 reproduces runs/e216_v5_metropolis exactly; (2) fails: v5's α₃ = 0.017 and α₄ = −0.010 are both under the 0.03 bar. The bar was set on an assumed sampling noise of 0.01; the two legs agree to 0.0002–0.002, so the shells are resolved eight times over — but resolved at 1.5× (α₃) and 1.8× (α₄) the values an NN model induces, against 2.7× for α₂. At 1550 K the third- and fourth-shell ratios of v5 and the NN Ising are identical (−0.42 vs −0.42, +0.25 vs +0.29). So what distinguishes v5 from a nearest-neighbour model is the second shell, and only the second shell; the "not NN" statement in the record narrows to "NN plus a real second-shell coupling", as the pre-registration said it would. The pyeCE four-shell read (runs/v5_shells_chain.sh) is superseded and its chain stopped.