compatibility_trivial
plain-language theorem explainer
Any spatial dimension satisfies the predicate-level compatibility condition with the realized T7 recognition cycle (the topological witness is a closed walk). Cited by the T7.5 substrate package and the T8 realization-bridge constructor. The structure field is literally True, so the proof is the trivial inhabitant.
Claim. For every spatial dimension $D \in \mathbb{N}$, the substrate is compatible with the realized recognition cycle: the topological witness is the closed walk produced by the T7 eight-tick cycle.
background
Module Foundation.SubstrateAxioms records the substrate-side structural inputs for the T7/T8 dimension route. Statements are deliberately predicate-level, matching the Alexander-duality style: Lean names the load-bearing commitments without claiming a full Mathlib smooth-topology chain (cellular completions, Thom isomorphism, Alexander/Lefschetz duality).
Spatial dimension here is just a natural number. Compatibility with the realized recognition cycle is a Prop-structure whose single field asserts that the topological witness is the closed walk coming from the T7 recognition cycle. In the present formalization that field is the unit proposition True, so the commitment is named rather than topologically discharged.
Upstream constants fix the forced value $D = 3$ elsewhere (T8 / linking); this lemma is parametric in $D$ and does not itself force the dimension.
proof idea
Term-mode structure constructor. The sole field witness_is_closed_walk has type True, so it is filled by trivial. No lemmas are applied; there is nothing to reduce.
why it matters
Closes the compatibility slot of the bundled T7.5 substrate package (substrate_package_trivial), which assembles cellular completion, one-acyclicity, loop entanglement, and this compatibility witness. That package feeds D3_compatible ("$D = 3$ is RS-compatible") in DimensionForcing and the constructor t8_via_realization_bridge_holds on the UnifiedForcingChain.
In the forcing chain, T7 forces the eight-tick octave (period $2^3$) and T8 forces $D = 3$ spatial dimensions. This declaration is the predicate-level bridge piece that says the realized T7 cycle is an admissible closed-walk witness on the substrate, so the T8 route can proceed without a full smooth duality development. It does not replace the geometric content; it marks where that content is assumed at Prop level.
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