Whale coalitions, jitter > 0 and very slow beta were the residual "untested
adversary variants" of open item 11. None moves a conclusion:
- Concentration does not change the deflation (suppression D-hat within noise
at every stake), and a whale coalition reproduces the sec 6.4 withholding law
D-hat -> (1-beta_adv) more cleanly than a random one: 0.9005/0.6997/0.5010
against a predicted 0.9/0.7/0.5. The "lumpier share statistic" worry points
the other way, and for a reason that is about coalition CONSTRUCTION rather
than concentration: a random coalition grows until its stake first reaches
the target, so the last node added overshoots by its own size -- a whale,
under a Pareto tail. Realised block share at a nominal beta_adv = 0.1 is
0.137 +- 0.108. Logged as item 17: the beta_adv axis is a nominal target.
- jitter up to 1 slot changes nothing under attack (notch 0.390 -> 0.410,
attacker share flat, range_ratio identically 0), as sec 6.1 found honestly.
- Slow beta shrinks the notch (0.415 -> 0.080 for beta 1 -> 0.1) at flat
attacker take, but sinks the MEAN estimate to 0.765 at beta = 0.05: the
estimator can no longer track back up during the honest half of the cycle.
Slowing beta buys the defender nothing on either axis.
Unplanned: study A blew past the memory guard, which turned out to be the
sec 6.2 fold being reached. The mechanism is sec 6.2's own -- rho_eff = rho/r,
and withholding deflates r by design, so a 50 % coalition doubles the load onto
rho_eff ~ 1.1 at the design point. Swept directly, the estimate collapses once
in 144 runs at delta_max = 8 (a concentrated 50 % coalition) and never at
delta_max = 4. That retires "not an observed dynamical trap" but is one event,
so the claim is stated as a rare tail and the rate is logged unmeasured as
item 18.
Co-Authored-By: Claude Opus 5 (1M context) <noreply@anthropic.com>
