Honest caveats — where gate theory can fail

Parent: gate-theory

The theory is only as strong as three assumptions, and each can fail in practice. Naming them is part of the theory.

1. Finding the metric is the real work (proxy risk / Goodhart)

Banach-style guarantees are trivial to apply once you have a contraction — the entire engineering content is finding the metric in which steps actually contract. For an LLM harness the state is text/world-state with no natural metric. The harness's answer: don't seek a global metric, gate observable proxies (gate-deviation-is-observable-delta). The cost: every guarantee holds only up to the correlation between proxy and true progress — and optimizing against the proxy actively destroys that correlation. This is Goodhart's law, and it is the same thing as failure-modes's "gaming the gate", now upgraded from anecdote to a theory-level failure mode.

2. Tolerances compose badly

Softened guarantees degrade under composition: per-step errors $ε$ add linearly over $n$ steps if nothing eats them (total drift $n·ε$); confidence tolerances $α$ compose by union bound ($α_{total} ≤ \sum α_i$), so a thousand gates each 99% sound give no useful total guarantee. The only known repair is contraction: inexact-contraction caps total drift at $ε/(1−k)$ regardless of $n$. Conclusion: δ-gates alone are not a convergence argument; δ-gates plus a contraction are.

3. Correlated errors and self-induced distribution shift

The probabilistic axis of relaxation (generic-case typicality, probabilistic-undecidability) assumes errors are independent and inputs typical. Both fail for an agent loop: LLM errors are correlated (same model, same blind spots — union bounds are wildly optimistic), and the agent's own search steers inputs into atypical corners — the loop itself manufactures distribution shift, undermining exactly the typicality that generic-case guarantees need. Practical ranking for harness use: the certificates axis and the bounded-resources axis are worst-case-safe — lean on them; the typicality axis is the shakiest.

The theory's proof obligations are: find a contracting observable, budget the tolerance composition, and never lean on a typicality that the loop itself destroys.