single_flip_posts
plain-language theorem explainer
Flipping any single vertex bit on the forced D=3 eight-vertex cell changes the six-face boundary record, for every one of the 256 configurations. Holography and complementarity arguments cite this as the local half of the cell-injection dichotomy: no single bulk distinction is unrecorded. The proof is pure finite decision over the product of configurations and vertices.
Claim. For every cell configuration $c$ (one recognition bit on each of the eight vertices of the $D=3$ cube, packed as an element of a $256$-set) and every vertex index $i\in\{0,\ldots,7\}$, the six-face boundary record after flipping vertex $i$ differs from the boundary record of $c$.
background
The module runs the cell-injection test from the entropy-fork program: flip one interior bit of the forced D=3 cell and ask whether the boundary record must change. The substrate is forced, not free modeling. The forcing chain gives D=3 and the eight-tick octave as the cube $2^3$: eight vertices, six faces.
A cell configuration places one recognition bit on each vertex (256 total states). The boundary record is the six face-closure parities, one per face, read by the per-face closed functional on that single cell. The local bulk move under test is flipping one vertex bit.
The module's verdict is a dichotomy, not a bare yes or no. This result is the local half: every single-vertex flip posts. The global half is that the record map is still non-injective, with a rank-4 kernel of whole-face and related moves invisible from the boundary.
proof idea
The domain is finite: 256 configurations times 8 vertex indices. The claim is a decidable boolean predicate on that product (equality of two six-bit face records after a bit flip). The proof is the single tactic decide, which exhaustively evaluates every pair and closes the goal. No intermediate algebraic lemmas are required; the check is pure enumeration of the cell configuration space against the face-record map.
why it matters
This is the positive local answer to the panel's cell-injection question: every bulk single-bit distinction posts to the boundary. It is packaged into the target cell-injection certificate (together with the sharper three-face incidence claim, non-injectivity, kernel size, rank-nullity, and global-only blindness). It also supports the first-isomorphism check that $|\mathrm{image}|\cdot|\mathrm{kernel}|=256$.
In Recognition Science this sits on the forced D=3 eight-tick cell (T7/T8). The entropy-fork program uses it to separate two futures for complementarity: unrecorded bulk degeneracy would leave complementarity axiomatic; universal local posting pushes it toward a theorem. The local half succeeds, while global face-flip blindness remains, so complementarity is not yet discharged. The companion three-face result sharpens locality: exactly the three faces incident to the flipped vertex change.
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