coefficient_is_one_quarter_derived
plain-language theorem explainer
Under the rank reading of the holographic selector, the pixel-to-sector ratio equals the Bekenstein–Hawking coefficient 1/4 as a rational identity. Holography and black-hole entropy arguments that adopt one generator per face cite this identity. The proof is a one-line application of the coefficient bridge to the already-derived selector instance.
Claim. In $\mathbb{Q}$, one divided by the number of admissible sectors of a local pixel equals $1/4$. The equality is conditional on the rank reading of the selector (one posted distinction per face).
background
This module sits in the holography layer and treats recognition multiplicity as a T-1 ledger consistency check, not a bare forcing from the foundation chain. For a cell of $k$ unit faces (each a minimal closed recognition loop in $D=3$), three independent counts are compared: ledger multiplicity (unit-weight ledger cost with one primitive double-entry distinction per face), closure rank of the local map, and nullity. Multiplicity equals $k$ by construction of the cell ledger.
The module header retracts an earlier claim that the Bekenstein $1/4$ was forced from the ledger floor under T-1 alone. The cell ledger types one generator per face by modeling choice; a nullity-shaped mirror ledger is equally T-1-clean and yields the opposite branch. Bare T-1 underdetermines the shape, so the shape encodes the selector rather than deriving it.
The coefficient identity here is the rational payoff of that rank reading: admissible sector cardinality enters the classical area-law prefactor. Live candidate forcing of the reading itself is deferred to gluing extensivity in the quad-plaquette development.
proof idea
Term-mode one-liner. Apply the coefficient-bridge lemma that turns a selector instance at weight $1$ into the rational identity $1/|\mathrm{admissible\ sectors}|=1/4$, feeding the already-proved derived selector certificate. No local arithmetic is unfolded; the bridge owns the card-to-quarter reduction.
why it matters
Closes the conditional coefficient leg of the holography consistency bundle. The sole downstream consumer packages it with multiplicity–rank equalities at one and two faces, the multiplicity–nullity divergence on the domino, the image-times-kernel factorization, and the derived selector, as the verify-target certificate for the holography loop.
In framework terms this is the classical Bekenstein–Hawking prefactor $1/4$ under RS modeling, not a T5–T8 forcing step. The doc-comment and module audit (holo_mult_fable_20260702) mark it explicitly conditional on the one-generator-per-face ledger: true and stable for downstream names, but not selected by T-1 alone. The open path to non-circular selection is gluing-invariance of rank (versus non-extensivity of nullity) in the quad module.
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