One Arena, One Snap, One ε

The question this section answers is the one a careful reader asks first: every persisting configuration holds a commitment, and there are on the order of 10⁸⁰ configurations — so why does the coupling carry one factor of (1 + ε) rather than (1 + ε) to some enormous power? The answer is that the exponent counts arenas, not configurations, and there is one arena.

Proposition 3 — One Snap, One ε.

Statement. The arena’s commitment is exactly one ε.

Proof. Two steps, from two sources, kept separate. First, the arena constitutes entire: AP28 §4’s clause is arena-scoped in its own words — “Axiom R does not permit selective persistence. The crack is open everywhere or nowhere.” The arena does not come into being region by region or channel by channel; its constitution is a single event, and Proposition 1 gives that event a unique record: the break. Second, one event carries one ε: an Actualization State instant produces exactly one unpaired element (AP43 D5). One arena, one snap, one instant, one ε. By Proposition 2, held. ∎

Corollary — Already Booked. Matter configurations hold constitution commitments of their own, by the same Proposition 2 — and every one of them is already on the books. AP29 establishes E ≡ κ(B) · AS: mass is held coupling capacity, actualised. A configuration’s held ε is held coupling capacity; therefore it is a constituent of the configuration’s mass — and the masses the corpus uses are measured masses, which contain everything the configuration holds. A configuration’s commitment is not a correction waiting to be applied to its mass. It is part of what its mass has been all along. Nothing dresses any mass; no mass ratio can carry a commitment signature; the proton-to-electron ratio of AP30 is invariant trivially — not by cancellation, but because there was never anything to cancel — and AP30 §7’s reading of its own residual, “It is not a missing term,” stands exactly as locked. The one commitment in the universe that no measured quantity contains is the arena’s, because the arena is the one configuration that is not a mass inside itself. Its account alone was open. ∎

This corollary is guarded by KS-CCC.2, kept deliberately as a fence against any future mass-side drift, and the guard is unforgiving for a reason worth restating: a commitment that scaled with a configuration’s channel count would shift the proton-to-electron ratio at eight orders of magnitude above what measurement allows. The arena reading does not merely survive that constraint. It explains why the constraint was never in danger: matter’s commitments were priced into matter from the start.