C.29.3:5.3 - Preserve a bound during material admission operations

A demonstration room has a capacity of three visitors. It starts empty with three distinct cards in a free stock. One card is issued to a visitor before entry; the visitor retains it inside and returns it after exit. Every entry follows this rule, each visitor inside holds one card, each card is exclusively assigned, and cards are not lost or duplicated.

The computation is the continuing admission decision and preservation of the capacity bound. People and material transfers execute its finite-state procedure.

A card can be Free, Reserved outside, Inside with its visitor, or Awaiting return after exit. Let F, R, I and E count those states. The fixed stock gives:

F + R + I + E = 3
occupancy = I
occupancy = 3 − F − R − E ≤ 3 − F ≤ 3

Issuing a card changes Free to Reserved. Entry changes Reserved to Inside. Exit changes Inside to Awaiting return. Return changes Awaiting return to Free. Cancellation can return a Reserved card to Free while its visitor remains outside.

Every operation moves one existing card between states. Entry requires its exclusive reservation. Return requires that its visitor is outside. These conditions preserve both the stock and the association between visitors inside and Inside cards. They establish the capacity bound throughout permitted histories, including overlapping admissions.

After two cards have been issued, one visitor may be inside and the other waiting outside. Then F = 1, R = 1, I = 1 and E = 0. The count of unavailable cards is 2, while occupancy is 1. Reading occupancy as 3 − F would be wrong. That count is an upper bound until R = E = 0.

A second entrance can share the same stock or receive a partition of it, such as two cards at one entrance and one at the other. Both preserve the total of three. Giving the second entrance three copied cards instead permits six simultaneous admissions, despite each attendant following their local procedure.

Partitioning can cause waiting at one entrance while a card is free at the other. Moving an existing free card preserves the total; creating another accepted card changes it. This separates the capacity property from the additional question of useful service across entrances.

The result is a conditional admission arrangement, its preserved bound and a qualified reading of the free stock. The administrative method supplies controlled entry, exclusive possession and return. The mathematical construction makes explicit which physical and procedural properties the computation consumes.