Recommend the Cantor-preserving operating point

[Al-does]

Stacked on #16, which traded the fractal away for global-R² headroom. That was the wrong trade, and this PR supersedes its recommendation.

Why less visible structure looked like more sensitivity

The two things are the same number. The mixed-state set is the attractor of an iterated function system with one contractive map per token. Disjoint first-level images need a contraction ratio below 1/sqrt(3), which puts at least two thirds of belief variance between the branches — and between-branch variance is exactly what a rank-2 probe on the last token reads.

alpha at stay=0.95	0.85	0.80	0.75	0.70	0.55
box dimension	0.85	1.01	1.22	1.38	1.82
last-token probe floor	0.730	0.633	0.543	0.461	0.264

So a low global-R² floor requires closing the gaps. stay carries no such cost: it moves the accuracy axis while leaving the geometry alone.

Mixed-state geometry at four operating points

Recommendation: stay = 0.95, alpha = 0.75

Four points, matched PPO and IQN arms, three seeds, 2.5M steps, identical recipe and probe.

All four operating points on three axes

point	stay/alpha	band	arm	belief R²	within-branch R²	accuracy range share
shipped	0.90 / 0.85	0.036	untrained	0.883	−0.717	—
			ppo	0.8461 ± 0.0065	−1.252 ± 0.091	−0% ± 0%
			iqn	0.9756 ± 0.0032	+0.642 ± 0.048	47% ± 2%
cantor_sharp	0.95 / 0.85	0.056	untrained	0.866	−0.089	—
			ppo	0.9138 ± 0.0356	+0.300 ± 0.290	41% ± 31%
			iqn	0.9793 ± 0.0015	+0.832 ± 0.011	90% ± 1%
cantor	0.95 / 0.75	0.085	untrained	0.823	+0.300	—
			ppo	0.8990 ± 0.0188	+0.601 ± 0.076	63% ± 13%
			iqn	0.9737 ± 0.0012	+0.896 ± 0.005	92% ± 1%
proposed	0.96 / 0.55	0.168	untrained	0.833	+0.715	—
			ppo	0.8281 ± 0.0548	+0.708 ± 0.093	52% ± 22%
			iqn	0.9646 ± 0.0056	+0.940 ± 0.010	91% ± 1%

cantor is the only point where the three conditions order cleanly with non-overlapping error bars on the metric that has a floor of zero by construction: untrained 0.300 < PPO 0.601 ± 0.076 < IQN 0.896 ± 0.005. It also has the lowest IQN seed spread of the four, keeps visible level-one gaps at box dimension 1.22, and takes 2.4× the global-R² band and 4.1× the accuracy resolution of the shipped point.

cantor_sharp is the zero-risk option: alpha is untouched so the picture is identical in character, and PPO still moves from 0% to 41% of the accuracy range.

The deeper fix is the metric

Nearly all of a Cantor-structured belief's variance is which cluster you are in, which the last token already gives away. conditional_residual_r2 at depth two discards exactly that term, so any depth-two token model scores zero there by construction at every operating point. That is why the sparse points still discriminate well on it while their global-R² bands look hopeless — and why proposed, with the widest global band, is the most compressed on it: a smooth belief is something a random causal filter already tracks (untrained scores 0.715 there).

Reward-only PPO fails to clear the global-R² floor at all four points, so that finding is not an artifact of a badly chosen process.

Cost

One RTX 4090 for ~1.2 h, destroyed on completion; no instances remain.

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