Realised over Provisioned

Proposition 4 — The Form.

Statement. The realised coupling is the structural coupling divided by (1 + ε): α_G∗ = α_G / (1 + ε).

Proof. The axiom names both totals. What The Snap provisioned is its full right-hand side: 1:1 + 1×ε — the balance, plus the one element whose holding makes the balance persist (Propositions 2 and 3). What persists as the realised arena is the 1:1. The realised fraction of the provisioned is therefore 1 / (1 + ε): the balance over the balance-plus-commitment. The correction is a quotient because the axiom is a sum. ∎

The first-order shadow (1 − ε) differs from the quotient at order ε² — about 53 parts per million in G — below anything current metrology can resolve. The structural form is the quotient, because the quotient is what the axiom writes, and every number in this paper is computed in the quotient convention.

The value of ε in the quotient — a selection, argued and booked. The form above is derived; the value is not, and the paper says so. AP06 §0.7 holds open a family of domain readings of ε — the leakage ratio at the gravitational boundary near 10⁻¹⁷, at the retinal boundary between 10⁻⁶ and 10⁻⁵, at the cosmological boundary near 10⁻⁵ — beside the corpus-wide ε = α_em at the coupling boundary. Choosing among them is a selection, and it is graded here as AP28 grades its 1/π: SELECTED AMONG STRUCTURAL CANDIDATES, with the argument stated and the closure owed (D-CCC.4).

The argument for the coupling denomination. The commitment is a fraction of the provisioned coupling — a discount is denominated in the currency of what it discounts, and the currency of α_G is fixed by AP28’s own Proposition 5: α_em is, by definition there, the unpaired element’s one-channel coupling worth. One committed element, read in coupling currency, is one α_em. The domain values AP06 catalogues are of a different kind — they are leakage ratios of specific absorbing systems, flow quantities measured at boundaries, not the element’s denomination in the coupling it discounts. The selection is therefore argued, not arbitrary — and not derived. If the constitution is shown to read at a different boundary, the correction rescales and §9 fails. That sentence is the honest price of this section, and D-CCC.4 keeps it visible.