Continuous Acquisition Timing

When waiting changes what a sensor can tell you

A threshold sensor reports whether a moving state lies above a chosen level. Two possible states can give the same answer now and different answers later. That possibility alone does not show that waiting helps when every prediction must arrive before the next transition.

This scoped addendum to Acquirable Predictive Models resolves one exact scalar model. The inputs are announced in advance and cannot adapt to observations. The disturbance is adversarial within a fixed bound. Every observation is paid, and forecasts must be correct for every admissible execution.

The original alphabet leaves no timing advantage

For inputs drawn from {-1, 0, 1}, delayed and initial-time acquisition have the same optimum for every positive finite horizon, every nonempty initial set in [-2, 2], and every finite threshold library, including an empty library.

Both acquisition methods are infeasible if disturbance leaves a required sign ambiguous or if the initial threshold answers fail to separate the robust output words. Otherwise both need the ceiling of log base two of K probes in the worst case, where K is the number of distinct disturbance-independent output words.

Here initial-time “batch” acquisition can choose its queries adaptively at one physical instant. It is not a nonadaptive list of tests. The equality is about a hard worst-case budget, not expected cost or unequal query prices.

One added input changes feasibility

Adding the input value 1/2 permits a two-step example with positive-width initial uncertainty. The only available threshold is useless initially but separates the two possible state sets after the first transition. One delayed probe is necessary and sufficient, while initial-time acquisition is impossible.

This is a change from infeasible to feasible, not a reduction between two finite optimum budgets. Minimality means one additional input value; it does not mean the smallest numerical perturbation, energy or distance.

What connects this result to B-Theory

The parent report separates a sufficient predictive representation from the information a permitted experiment can obtain. This note studies that distinction on exact real intervals under a different, explicit timing contract. It does not import the parent’s topological query-admissibility rule as a physical safety condition.

Diagnosing While Repairing studies diagnosis alongside interventions and repair order. Here the input word remains fixed in both comparator classes; the issue is sensing time. Binary decision trees, interval reachability, active testing and indistinguishable-transcript arguments are classical antecedents. The candidate contribution is the specialized obstruction and its boundary example, with historical priority still unresolved.

The model establishes no intersample safety guarantee, noisy-sensor theorem, stochastic prediction result, physical validation or cognitive property.

Read and reproduce

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

The complete reader preserves six numbered sections, five results, five proofs, nine references and 142 source-based mathematical expressions. It includes exact TeX annotations and a recorded repair to the final row of the three-case optimum. The original PDF, LaTeX source, review summaries and archive are unchanged.

Fresh exact-arithmetic checks reproduce 1,092 original-alphabet input words and 7,182 robust pairs. The separately implemented checker also passes 21,845 ordered-tree instances, 13,284 endpoint cases and all 10 small-center representations. These finite checks corroborate the analytical universal proofs; they do not replace them.

Review scope, source-access limits, fresh reproduction results, and site provenance distinguish the manuscript, internal AI review, computation and site presentation. Hu and Tucker’s full text was not inspected in the priority comparison. Specialist screen-reader testing remains outstanding and the PDF is untagged.

The frozen source retains its earlier local-unpublished wording and missing companion-location note. The reader supplies the now-available companion link outside that unchanged text. No additional reuse license is granted by this release.