The Double Entry

Proposition 6 — Two Readings, One Element: an Extension, Stated as an Extension.

Statement. The puncture factor (1 + 1/π) and the commitment quotient 1/(1 + ε) are two independent readings of the one unpaired element — what it contributes and what it committed — and their joint appearance in the realised formula is not double-counting.

Argument. Be precise about the licence. AP24 licenses reading one ε through different instruments — as G, as c, as α_em: across instruments. This section needs more: two readings of the element inside one instrument, multiplicatively, in α_G — and α_em itself already plays the per-channel role twenty-one times in the same expression. That is an extension of the licence, and the paper claims it as one, not as a straight application.

What the extension requires, and what supports it. It requires that “what the element contributes” and “what the element committed” be independent questions even at a single boundary — that contribution and cost not be one projection wearing two names. The support is structural, from §8’s graph: the puncture factor is an edge-stratum statement — the capacity the element’s round hole adds to a coupling channel (AP28, Proposition 4) — while the commitment is a vertex-stratum statement — the provision the element’s constitution required (§3). The two within-formula readings are readings of different strata of one proven structure, which is as close as the corpus currently comes to within-instrument independence. Close is not closed: the extension is claimed, supported, and guarded — not proved. And the dependency is declared rather than hidden: §8 is load-bearing for this section’s support — for the independence of the two readings — and for nothing arithmetic; the MOTIVATED grade of §8’s assignment is inherited here as the grade of the support. ∎

The guard is KS-CCC.6, and it is coupled to a debt the corpus already carries. AP28’s KS-R.9 sub-debt owes the formal normalisation of the boundary coupling. If that calculation, when completed, is shown to already contain the persistence provision — one reading, not two — then the commitment double-counts, the correction is removed, and the puncture factor absorbs it under a reinterpreted normalisation. The switch is stated so that the hostile reading of this section — “first a factor was added to fit, now a factor is divided to fit better” — meets its answer in the register where it belongs: a named test, not a rhetorical defence.