C.29:1 - Problem frame

A mathematical representation can make a working question answerable by exposing the relations needed for an inference. It may combine source cases into a summary, change coordinates without losing distinctions, or embed the source in a larger mathematical domain. Each construction supports different uses. A queue may expose flow restrictions while omitting rework; an extension of the rational numbers to the reals makes limits available while retaining rational arithmetic.

C.29 addresses this representation choice and its use across a stated correspondence. It begins either with a working problem that needs a useful construction or with a proposed representation whose consequence needs to be derived, limited or rejected. Domain theory supplies the mathematical laws and application conditions.

A useful mathematical lens makes a needed inference possible through an explained correspondence.

Ask what the correspondence preserves, omits or introduces, which operations it supports, and how the resulting conclusion answers the working question.

C.29:1.1 - First-minute working situation

A production manager sees waiting work but cannot tell whether an extra station would help. The first task is to distinguish an arrival restriction, a service bottleneck and a batching effect. The candidate queue has to make at least one of those distinctions calculable or observable. The discovery cues in :4.2b help choose an object; they are not a required tour of mathematical families.

C.29:1.2 - Minimum scenario and anti-case set

Positive scenario. In :4.4’s two-station queue, the departure recurrence distinguishes a five-minute latency reduction from a change in sustained output. The next observation focuses on the slower station. Changed service time or finite-buffer blocking withdraws the affected calculation; actual capacity reliance requires checking the model against the line.

Anti-case. “The organization is a quantum system” is written without a candidate mathematical object, probe distinction or readout distinction, preserved structure, lost structure, LensUseBoundaryValue, or stop condition. The C.29 result is either a downgrade to local metaphor or a repaired use through C.29 and, where relevant, C.26.

Under-lensed anti-case. “The work stream has dynamics” or “this portfolio is a network” is used for a diagnosis that affects prediction, comparison, repair, or stop conditions, but no mathematical object changes what can be predicted, compared, diagnosed, repaired, or stopped. The repair is to choose a cheap candidate lens that exposes useful structure, or keep the sentence as ordinary prose.

False-positive scenario. A Markov kernel appears inside accepted local reliability modeling. When no separate representation or transfer question is open, complete the A.3.3 result without a C.29 output. If such a question arises, use the reliance rule in :4.4.

C.29:1.3 - Intended FPF use-value

The useful result is an answer, bound, obstruction, diagnostic split or next observation that the working reader could not recover from the prior account. Keep the correspondence and limitations inspectable when someone will reuse that result. A more elaborate record is useful only for the information its receiving use needs.