Appendix B The Returns to Crystallization

This appendix gives the formal return structure behind the reusable capital argument of Section 4.3. Searching for a fresh, valid solution to a task of complexity $\kappa$ is expensive because a non-trivial objective is a product of constraints that must all hold at once; valid solutions occupy a sparse island whose measure decays geometrically in the number of binding constraints $m(\kappa )$. The expected search cost thus scales like ${\overline{q}}^{-m(\kappa )}\xi (\kappa )$ (with the per-constraint pass rate and $\xi \ge 1$ a backtracking overhead), while reusing a verified model costs only an instance-adaptation factor $\rho (\kappa )=O(1)$—for a GWM, the grounding inference of (13) that ties parameters to this instance’s evidence. Their ratio is the crystallization leverage

which grows exponentially in $\kappa$, since the numerator does and $\rho$ is bounded. Calibrated to current frontier per-step reliability, $\mathrm{\Lambda }\approx 160\times$ for a four-hour task and $>6,000\times$ for a full workday task.

Token leverage is only half the incentive. Writing $V(\kappa )$ for task value, $P_{\text{succ}}(\kappa )$ for the probability that an unverified search succeeds, and $\delta (\kappa )\ge 1$ for the damage multiplier when it fails and the bad output propagates downstream, the per-invocation gain from using a verified model decomposes into two additive terms:

The token-savings term shrinks as inference prices fall; the failure-avoidance term does not. For the high-stakes, many-constraint tasks that dominate the high-$\kappa$ regime it is the binding incentive—at $\kappa =10$ it already reaches $\approx 3.96V$ per invocation—and it persists even as token costs approach zero.