T8_TopologyDependencyAudit_Bridge
plain-language theorem explainer
Audit certificate packaging six closed topological obligations behind T8 (spatial dimension forced to three). Given a T8 witness, it records that circle reduced cohomology is nontrivial exactly in degree 1, sphere–circle linking holds iff D=3, the dimension module’s linking predicate is the Alexander-duality one, and uniqueness plus low/high-D exclusions follow. Downstream CompleteForcingChain and the holds theorem cite it. Pure Prop-structure definition, not a derived proof.
Claim. Assuming spatial dimension is already forced to three, the following hold: reduced cohomology $\tilde H^k(S^1;\mathbb{Z})$ is nontrivial iff $k=1$; the $D$-sphere admits nontrivial linking of disjoint embedded circles iff $D=3$; the dimension-forcing linking predicate is equivalent to that Alexander-duality sphere-linking predicate; any dimension supporting nontrivial linking equals $3$; there is a unique RS-compatible dimension; and dimensions $1$, $2$, and all $D\ge 4$ fail nontrivial linking.
background
The Unified Forcing Chain module claims T0–T8 are inevitabilities from the Recognition Composition Law plus normalization and calibration. T8 is the spatial-dimension step: $D=3$ is the unique value satisfying nontrivial linking (ledger conservation), eight-tick synchronization $2^D=8$, and gap-45 sync.
Alexander duality supplies the topological content. CircleReducedCohomologyNontrivial k is the concrete predicate that $\tilde H^k(S^1;\mathbb{Z})$ is nontrivial; by Hatcher §2.2 it holds exactly at $k=1$ (status closed, no axiom). SphereAdmitsCircleLinking D says $S^D$ admits nontrivial circle linking, defined via Alexander duality (Hatcher Thm 3.44) as nontriviality of $\tilde H^{D-2}(S^1)$, i.e. the circle-cohomology predicate at degree $D-2$.
The parent T8 structure already asserts linking forces $D=3$, eight-tick forces $D=3$, and unique RS-compatible dimension. This audit structure sits on top of a T8 witness and checks that those claims rest on the closed Alexander-duality bridge rather than an eight-tick tautology or a hidden axiom.
proof idea
No proof body: this is a Prop-valued structure definition whose fields are the audit obligations. Instantiation is separate. The companion theorem t8_topology_dependency_audit_bridge_holds builds a value by discharging each field from Alexander-duality lemmas (e.g. circle_reduced_cohomology_iff for the cohomology biconditional, and the sphere-linking characterization of $D=3$) together with the assumed T8 witness for uniqueness and the topological forcing route. A Subsingleton instance records that any two such certificates are definitionally equal.
why it matters
T8 is the chain step that forces spatial dimension three (primer landmark T8; module chain: linking + gap-45 sync, with T7’s eight-tick as $2^D$). Earlier surfaces risked treating linking as $D=3$ by definition or leaving circle cohomology axiomatic. This certificate documents that the topological route is theorem-backed: cohomology closed, sphere linking iff $D=3$, dimension-module linking unfolds to Alexander duality, and non-linking outside three is a proved consequence of the same predicate.
CompleteForcingChain consumes the T8 layer as part of the full T−1 through T8 inevitability bundle. The holds theorem is the constructive witness that the current T8 surface passes the audit. Together they keep the dimension step from reopening as scaffolding when the chain is assembled.
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