T75c_OneAcyclic_Bridge
plain-language theorem explainer
Packages the T7.5c bridge hypothesis: once the eight-tick cycle is forced, every spatial dimension admits a 1-acyclic substrate. Downstream T8 realization proofs cite it as the acyclicity half of the codimension route. It is a Prop-structure with a single universal field, not a proved theorem.
Claim. Given that the eight-tick ledger cycle is forced ($8 = 2^3$ from dimension), the T7.5c bridge asserts: for every spatial dimension $D$, the recognition substrate is integrally $1$-acyclic (its first homology vanishes in the sense required by the codimension argument).
background
The Unified Forcing Chain module aims to derive T0–T8 as inevitabilities 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 is the eight-tick octave.
T7.5 sits between T7 and T8 on the realization route. Substrate axioms introduce a spatial dimension parameter $D \in \mathbb{N}$ and structural predicates on the recognition substrate. Among them, OneAcyclicSubstrate is the T7.5c clause: integral $1$-acyclicity (first homology vanishes). In the present formalization that field is still the trivial proposition True, so the predicate is a named interface rather than a computed homology vanishing.
The bridge structure takes a T7 witness as a parameter and asserts the universal quantification over all dimensions of that $1$-acyclicity predicate. It is the acyclicity half of the pair (cellular completion, $1$-acyclicity) used by the codimension route to $D = 3$.
proof idea
No proof body: this is a Prop-valued structure definition. It declares a single field, a universal statement that every DimensionForcing.Dimension carries SubstrateAxioms.OneAcyclicSubstrate. Instantiation is deferred to the constructor theorem t75c_bridge_holds, which fills the field by SubstrateAxioms.one_acyclic_trivial.
why it matters
In the forcing chain, T8 ($D = 3$) can be reached either by the existing surface or by a realization bridge that needs T7 plus T7.5a (cellular completion) and T7.5c (this bridge). Downstream, T8_Via_Realization_Bridge and t8_via_realization_bridge_holds take a T7.5c witness alongside T7.5a; t8_realization_equiv_existing then shows that route recovers the standard T8_Dimension_Forced surface.
Framework landmark: T8 forces three spatial dimensions via linking and gap-45 sync; the eight-tick octave (T7) is $2^3$. This declaration names the acyclicity hypothesis the codimension argument needs so those steps can be stated as a single bridge package rather than scattered axioms.
Because OneAcyclicSubstrate currently stores H1_vanishes : True, the mathematical content is still a placeholder interface. Closing it means replacing that trivial field with a genuine homology vanishing on the cellular completion of the cube graph.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.