Amendment 2 to the noise-draw pre-reg: read 4 — the energy score

2026-08-05 ~22:5xZ. Amends Amendment 1 by ADDING a fourth read. Posted before any per-draw number exists (the draws-10 probe has not run; no --dump-draws npz has been opened). Reads 1–3 are unchanged.

Why

Amendment 1’s honest-limits section said the comm holdout would need “a distributional column (best-of-N or an energy-distance-style score)” if the unfair-penalty signature confirms. The standing literature slice (this session) surfaced Energy Policy (arXiv 2510.12483), which trains manipulation policies directly on the energy score — a strictly proper scoring rule for distributions. We don’t need the training objective; we need the metric, and it is computable on CPU from the exact --dump-draws npz the probe already produces. Read 2 (best-of-N) is an oracle bound — it forgives all dispersion. The energy score is the principled middle: it rewards committing to valid modes and penalizes spread, so neither mode-averaging (AR-style) nor scatter wins for free. Declaring it now, before data, keeps it a read rather than a post-hoc rescue.

Read 4 (definition frozen)

Per frame, over the same valid-element mask as reads 1–3, with $a_i$ = draw $i$’s chunk restricted to valid elements ($N=10$), $y$ = truth, $m$ = valid-element count, and $d(u,v)=\Vert u-v{\Vert }_2/\sqrt{m}$ (RMS-normalized so frames of different valid counts are comparable; positive scaling preserves propriety):

$$\mathrm{ES}=\frac{1}{N}\sum _{i}d(a_i,y)-\frac{1}{2N^2}\sum _{i,j}d(a_i,a_j)$$

• Pooled across frames weighted by $m$ (matching the report’s valid-element pooling convention).
• AR baseline: the banked AR-100k npz is deterministic (N=1) — the same formula degenerates to $d(a,y)$ with a zero interaction term. Computed on the identical probe frames via the same index-join and row-agreement asserts as reads 1–3.
• Also reported: the flow single-draw ES (each draw scored alone, averaged) — the gap between it and the 10-draw ES is the value of modeling the distribution vs sampling from it once.

Pre-declared interpretation

• Flow ES ≤ AR ES while flow single-draw MAE > AR MAE: quantified evidence that the MAE deficit is (at least partly) a scoring-rule artifact — and ES becomes the candidate distributional column for ranking flow arms on the comm holdout (feeding the limit-attribution front, per Amendment 1’s honest-limits note).
• Flow ES > AR ES too: flow loses even under a mode-fair proper score — the modeling-deficit read strengthens and the AR-recipe weighting of the attribution screens stands.
• ES is quoted with the same effect-size discipline as reads 1–3; no decision rides on ES alone tonight (it is one column of the results post, not a gate).

Implementation: a read4_energy_score addition to fontaine/scripts/draws_fairness.py, to be landed with a degenerate draws=1 validation (interaction term exactly zero; ES equals the RMS-normalized L2 of the banked predictions) before the probe npz is opened.