When Checking Takes Time

Evidence can arrive after the thing measured has changed

A diagnostic probe reads a hidden binary mechanism, but the mechanism can change before the reading arrives. Another probe supplies evidence and also exposes the mechanism to another transition. The controller can stop immediately or make at most two checks.

This technical addendum solves that small decision problem exactly. It uses classical Bayesian filtering and optimal stopping, not a new general theory of reflection.

A useful plan can start with an unhelpful check

In one exact example, immediate prediction has error 1/4. A mandatory single check raises gross error to 9/32. Yet a contingent plan, taking a second check only after one of the first readings, has total error-plus-cost 59/256, below 1/4.

This does not mean optional information is harmful. The controller can always choose to stop immediately. The harmful comparisons concern executing a check while its target changes, with policies evaluated at their own stopping times rather than at one common deadline.

What the result establishes

The branch-dominance proof reduces the optimum to immediate stopping or one contingent two-probe plan throughout the declared transition, disclosure and nonnegative-cost domain. Two exact implementations check 160 parameter/cost cases, 38 deterministic policies and 64 elementary outcomes per policy. The finite tests corroborate the analytic proof; they do not replace it.

The assumptions are essential: initial fault probability and observation error are both 1/4, transitions are known, independent and symmetric, and there is no extra delay after stopping. More probes, unknown hazards, repair actions and other observation contracts require fresh analysis.

Connection to B-Theory

Reflective Self-Models already separates using evidence from claiming an advantage for an explicitly named self-model. This addendum contributes a compact timing-and-cost example, not an architectural separation. Acquirable Predictive Models studies a different acquisition contract, and Diagnosing While Repairing treats broader diagnosis and intervention questions. Their assumptions and conclusions are not imported here.

The mechanism may equally be an external device. Nothing in this result establishes subjective awareness, personal identity or physical validation.

Read and reproduce

Read the complete addendum online, download the unchanged five-page PDF, or inspect the source and verification package.

The package includes the canonical text, offline HTML, PDF, standard-library Python checkers, proof and source ledgers, and reader-facing review summary and responses. Review scope and site provenance distinguish AI review, computation and publication. Historical priority remains unverified: Monahan’s full text was unavailable, and the 2013 Ahmad/Yu comparison was abstract-only. This release grants no additional reuse license.