t75a_bridge_holds
plain-language theorem explainer
Given the forced eight-tick (T7), the eight-tick surface admits a predicate-level cellular completion in every dimension. Citers of the T7.5a bridge in the unified forcing chain use this packaging. The proof is a one-field constructor that installs the substrate's trivial completion witness for all D.
Claim. Assume the eight-tick is forced: $8 = 2^3$ arises as the minimal ledger-compatible cycle from dimension $D=3$. Then for every dimension $D$ there exists a predicate-level cellular completion of the substrate (closed orientable smooth structure, cube-graph embedding, and retraction back to the cube graph).
background
The module builds the complete inevitability chain T−1 through T8 from the Recognition Composition Law plus normalization and calibration. T7 states that the minimal ledger-compatible cycle is $2^D$; with spatial dimension three this forces the eight-tick octave, so eight is not a free parameter.
T7.5a sits between T7 and later dimensional/linking steps. It asks whether the eight-tick surface admits cellular completions at the predicate (structural-interface) level in every dimension. CellularCompletion D packages three trivial structural witnesses: closed orientable smooth type, cube-graph embedding, and a retraction back to the cube graph. Upstream, cellular_completion_trivial already supplies those witnesses for every D, as the Lean counterpart of the paper's $S^D$ witness at the interface level.
The bridge structure is parameterized by a proof of T7 so that the completion claim is stated only on the forced eight-tick surface, not as a free-standing geometric axiom.
proof idea
Term-mode structure constructor. The single field exists_completion is filled by SubstrateAxioms.cellular_completion_trivial, which is already a theorem of type $\forall D,\ \mathrm{CellularCompletion}, D$. No case split on dimension and no use of the T7 hypothesis beyond threading it into the bridge type; the completion is dimension-uniform and predicate-level.
why it matters
In the unified forcing chain, T7 forces the eight-tick octave from $2^D$ with $D=3$. T7.5a is the next structural bridge: it records that the eight-tick surface still admits cellular completions in every dimension, keeping the substrate ready for codimension and linking arguments that feed T8 (spatial dimension three via linking and gap-45 sync).
The declaration closes the T7.5a constructor at the proved status: the bridge is not a hypothesis interface. Downstream use count is currently zero in the graph, so it is a chain-local packaging lemma rather than a widely reused API. It does not itself force $D=3$; it only preserves cellular completability once eight-tick is forced.
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