Sensor Decisions and Paid Calibration

Learning a measurement law before making decisions

A controller may have several imperfect readings of a target and still lack the information needed to trust them. If two reference sensors make correlated errors, agreement can be less reassuring than it appears. Paying for repeated calibration observations can narrow that uncertainty, but the observations themselves consume resources.

This technical addendum extends the supplied sensor-reliability research report. It separates the current target state from the measurement law shared across independent episodes. The calibration records contain no truth labels or mode labels. The controller learns from the joint pattern of two primary readings and two reference readings, then freezes a policy for fresh deployment episodes.

The setting is deliberately finite and explicit. Sensor marginal accuracies, polarity and primary dynamics are supplied assumptions. The unknown parameter describes reference-error dependence in the interval from zero to 1/25. Calibration uses a fixed sample count chosen in advance. All charges are stipulated model units.

What the addendum proves

• The complete four-reading record reduces to three minimal-sufficient categories. Keeping the primary readings is strictly more informative than keeping only reference disagreement, and increases Fisher information. This does not prove a strict sample-size saving for the particular certification task.
• For any closed parameter interval in the stipulated range, the paper gives the exact achievable coverage frontier and the minimum worst-law, worst-current-mode expected reference count. The converse covers every allowed randomized history policy, and an explicit policy attains the bound.
• Positive coverage and cost margins give matching upper and lower sample-size orders outside the no-training region, for failure probabilities at most 0.1. The constants do not match, and the exact finite-sample optimum is unresolved.
• No finite fixed calibration sample can uniformly certify the exact worst-law oracle budget or maximum coverage. These endpoint results do not construct an infinite-sample procedure.

At the full uncertainty interval, the certain no-training policy can guarantee coverage 11/16, compared with 3/4 when the law is known. Calibration can certify targets between these limits when the required cost and coverage margins are positive. The inherited robust threshold 3/40 belongs to the earlier report; the calibration constants here remain restricted to 1/25.

Evidence and limits

The general frontier and all-learner statements are analytic proofs. A fresh run of the unchanged public checker passes 25 exact likelihood and symbolic-algebra checks, reproducing the supplied mathematical results. These finite checks support the algebra rather than replace the proofs or establish physical performance.

Separate internal AI mathematical and hostile reviews closed their requested corrections. After materialization, both original reviewers read the complete final UTF-8 source and independently calculated its SHA-256, returning PASS with no required changes. This closes the binding to the released file despite presentation and citation differences from the earlier conversation rendering. Their review does not certify executable reproduction, PDF rendering or historical priority, and is distinct from external human peer review and formal certification.

The statistical machinery is classical: experiment comparison, concentration and change-of-measure methods already have substantial antecedents. The contribution is their stated specialization to this unlabeled sensor record and its interval decision geometry. Exact finite sample requirements, optimal leading constants and demonstrated strict full-record sample savings remain open.

The cost comparison gives a sufficient condition for a certified expected saving on the good-training event. It is not a necessary break-even threshold for an actual policy, which may spend less than its certified budget. Conditional and unconditional expected costs are treated separately. No monetary quote or deterministic realized bill is implied.

Place in B-Theory

The distinction is between evidence about a present state and evidence about a law shared across episodes. That distinction fits the Being/Becoming programme’s interest in how a system obtains the information its decisions require. It also complements Reflective Self-Models, while preserving the same architectural caution: an explicit self-model and a compiled controller using the same evidence can implement the same policies.

B-Theory here does not invoke the metaphysics of time. The addendum identifies no natural self/world boundary, inherent self-model advantage or criterion of phenomenal awareness. Unknown sensor marginals, changing dependence, persistent targets, physical hazards, adaptive stopping of calibration and deployment-time learning are outside its contract.

Read and reproduce

Read the complete addendum online with mathematics, download the unchanged ten-page PDF, inspect the exact LaTeX or Markdown, or download the complete public source and verification release.

The release includes the inherited sensor report, source/proof ledger, bibliography, public AI-review summary, exact checker and recorded results. Public provenance records source/download hashes, verification scope and presentation changes. The source byline is B-Theory research programme; no individual author is inferred. Native MathML supports reflowing reading, but specialist screen-reader testing remains outstanding and the supplied PDF is untagged.