T75a_CellularCompletion_Bridge
plain-language theorem explainer
Once the eight-tick cycle is forced, every spatial dimension admits a predicate-level cellular completion of the cube graph. Researchers routing T8 through the realization bridge (rather than the classical linking/gap path) cite this interface. It is a Prop-structure with one universal field; the witness is the trivial substrate completion constructor, not a geometric construction.
Claim. Assume the eight-tick period is forced ($8 = 2^3$ from dimension three). Then for every spatial dimension $D \in \mathbb{N}$ there exists a cellular completion of the $D$-cube graph: a closed orientable smooth $D$-manifold substrate, a tame cube-graph embedding, and a retraction back onto the cube graph.
background
The Unified Forcing Chain module aims to show that every step from the absolute floor through T0–T8 is forced by the Recognition Composition Law plus normalization and calibration, not merely compatible with them. In that ladder, T7 asserts that the minimal ledger-compatible cycle is $2^D$, and with $D = 3$ this is the eight-tick octave.
Spatial dimension here is just a natural number. A cellular completion (T7.5a in the substrate axioms) packages three structural clauses used in the dimension paper: a closed orientable smooth $D$-manifold substrate, a tame embedding of the cube graph, and a retraction back to that graph. In the present formalization those three fields are proof placeholders equal to True, so the predicate is available in every dimension at the interface level.
This bridge takes a T7 hypothesis and asserts that such a completion exists for every $D$. It sits between the eight-tick surface and the realization route to T8.
proof idea
No proof body: this is a Prop-structure definition, not a theorem. It packages a single field exists_completion, a universal quantification over spatial dimensions demanding a cellular completion. The inhabited instance is built downstream by the one-line constructor that applies the trivial substrate completion witness to every $D$.
why it matters
T8 (spatial dimension forced to three) is the terminal landmark of the forcing chain, classically via linking and gap-45 sync. This bridge opens an alternate realization route: T7 plus T7.5a (cellular completion) and T7.5c (one-acyclic substrate) feed the structure that recovers the same $D = 3$ conclusion, and the equivalence theorem shows that route agrees with the existing T8 surface.
Downstream, the bridge is required by the realization-bridge structure, its constructor, and the equivalence to classical T8. In the paper numbering it is the T7.5a step between the eight-tick octave and dimension forcing. Because the cellular-completion fields are still trivial placeholders, the geometric content of the dimension paper remains an open formalization target; this declaration only locks the predicate-level interface into the chain.
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