Formal proof of why the current `atom_max = EWMA(WIN_bound)` design
trains successfully despite empirical violations of the "structural
minimum atom_max ≥ WIN/(1-γ)" claim made in 2026-05-30-c51-atom-
resolution-design-alternatives.md.
Key findings (proven by local smoke + cluster v8 trajectory):
1. The fixed-point bound `WIN/(1-γ)` is overstated as a structural
constraint. Empirical: smoke step 999 atom_max = 4.5× the bound,
system trains healthy (qpa=+0.969).
2. The actually-binding constraint is the dynamic bound:
`atom_max ≥ max V_target_observed ≤ WIN + γ × V_max_observed`
Self-consistent and satisfied with margin 23 at smoke step 999.
3. Both speculative alternatives (Fix F = mean_abs_pnl anchor,
Fix F-v2 = V_target_observed anchor) have closed-form instability
at γ(1+ε) ≥ 1 — for γ=0.99 only ε ≤ 0.01 stable, no resolution
benefit over current design.
4. Current design is *conservative* (atom_max ~5× larger than V_target
3σ requires) but *correct* — slow EWMA hysteresis absorbs WIN
transients and keeps Q-V Bellman self-consistent.
Includes empirical trajectory data:
- Local smoke (b=128, HEAD d57bee054, 1000 steps): atom_max 1798→1468,
qpa +0.65→+0.97, clip_rate 5-11%, dynamic bound satisfied with
margin 23 at convergence.
- Cluster v8 (b=1024, alpha-rl-vxbpq @ 6d4a962e5): popart_σ collapses
56→1.69 from step 100→16830, qpa stays >+0.93, pnl_cum $164M.
Saves pearl_atom_span_dynamic_vs_fixed_point for future sessions.
Diagnostic emit of V_target_max_observed proposed (§5) but deferred
— current signals (WIN + γ × atom_max) already provide the bound
indirectly via JSONL. Real-time V_target_max is a nice-to-have for
future calibration work, not required to validate the current design.
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