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Three engine defects, found while building the multi-coalition study §6.9 flags
as open. The first is the serious one.
1. COALITION SIZING (engine._adversary_mask, `random` selection — the default).
The coalition was the smallest random prefix whose stake reached the target.
Under a Pareto tail a whale straddling the cut carries it far past its label:
over 60 replicates, a nominal adversary_frac of 0.4 realised a MAJORITY in
~10% of them and reached 0.97, and 0.2 reached 0.90. The median was always
on-label, which is why it hid — it distorts the tail, not the centre.
Both other places in the code that size a set by stake had already rejected
this rule: the `whale` arm uses fit-then-close, and _churn_inactive_mask
documents the identical failure ("a 30% label realising up to ~53%"). The
`random` arm kept it. Now fit-then-close in random order, and a draw where
the tail leaves no subset near the label warns instead of silently running a
different attacker. Realised stake is now within 0.1% of its label.
Re-ran the load-bearing studies. §8.4 capstone (2 of 8 replicates
contaminated, one a 61% majority): spec rule 0.994 -> 0.995, p_ref 0.936 ->
0.937. The parent-anchored variant is far more sensitive — 0.974 -> 0.990,
p_ref 0.875 -> 0.923 — because a tighter window and a larger suppressing
coalition compound, so §8.4's argument for the W = 12 pairing rested on
0.021 of cost that is really 0.006. The pairing itself survives re-measurement
and is now better supported: p_ref reaches parity at W = 12 too, not at 15.
§6.8's uncle-margin sweep and §6.5's random-arm variants are flagged as
needing re-measurement (§8.3 item 20), not silently carried.
2. SM1 NEVER TERMINATED under a forking honest network. Textbook SM1 waits while
it leads, assuming the lead returns to zero. But honest blocks fork against
each other, so the public chain's HEIGHT grows at ~(1-a)*f*(1-fork) while a
coalition sharing one view extends privately at the full a*f; past a fork rate
of ~1 - a/(1-a) the private chain outruns the public one and `wait` never
fires. The lead ran to thousands and every block was stranded at the epoch
boundary — 98% of adversarial blocks at alpha=0.4, delta_max=8 — scoring an
attacker that WON the race as having earned nothing. selfish_lead_cap
(default: the finality depth k) publishes a lead that can no longer be caught.
Inert unless `wait` stops terminating; pinned paired.
3. RIVAL COALITIONS (adversary_coalitions = K) for the §6.9 study: K private
chains, each invisible to the others by the same arrival sentinel that hides
them from honest nodes, so they orphan each other as well as the honest chain.
Stake-balanced partition (LPT), K=1 bit-identical to the single-coalition path.
Also corrects §8.4's closing paragraph, which still quoted a pre-countable
D-hat/D of 1.001 and fork rates that contradicted its own table.
Co-Authored-By: Claude Opus 5 <noreply@anthropic.com>