IndisputableMonolith.Verification.DimensionLinking
Verification layer that records the Alexander-duality computation for an embedded circle in the D-sphere: the reduced first cohomology of the complement is isomorphic to the reduced cohomology of S^1 in degree D-2, hence nontrivial exactly when D=3. Cited by the T6–T8 spine audit when checking that spatial dimension is forced rather than postulated. The module mostly packages the Foundation.AlexanderDuality bridge and supplies explicit odd-dimension witnesses.
claimFor an embedded circle $K \subset S^D$, reduced Alexander duality gives $\widetilde{H}_1(S^D \setminus K) \cong \widetilde{H}^{D-2}(S^1)$. The right-hand side is nontrivial if and only if $D=3$, so nontrivial circle linking in the $D$-sphere forces $D=3$. The module also records odd-dimensional witness data for $D \in \{3,5,7,9\}$ used in sector and linking arguments.
background
Recognition Science forces spatial dimension at tier T8 of the UnifiedForcingChain. The topological engine is Alexander duality on spheres: non-trivial linking of an embedded circle exists in $S^D$ precisely when the complement carries a nontrivial reduced $H_1$.
Upstream, Foundation.AlexanderDuality replaces the old tautology "admits circle linking iff $D=3$" by a bridge predicate grounded in reduced cohomology degree (Hatcher, Algebraic Topology, Thm 3.44): $\widetilde{H}_1(S^D \setminus K) \cong \widetilde{H}^{D-2}(S^1)$, which is $\mathbb{Z}$ exactly at $D=3$.
This verification module sits one layer above that foundation. It re-exports the circle case, names the $H_1$ isomorphism, and packages loop-loop linking and same-sector parity constraints together with concrete witnesses at odd dimensions $D=3,5,7,9$.
proof idea
Not a single theorem: a thin verification façade over the cohomology argument already proved in Foundation.AlexanderDuality. The core claim is delegated to the predicate that $S^D$ admits nontrivial circle linking iff the dual degree $D-2$ hits the nontrivial reduced cohomology of $S^1$, i.e. iff $D=3$.
Sibling declarations then specialize: the $H_1$ complement isomorphism, the implication from loop-loop linking to $D=3$, a parity constraint that same-sector data force odd dimension, and explicit witness constructions (including reconstruction and an allowed-set characterization) at $D=3,5,7,9$. No new heavy algebraic topology is done here; the work is naming, packaging, and audit-facing witnesses.
why it matters in Recognition Science
T8 in the forcing chain asserts $D=3$ spatial dimensions. Without a genuine topological criterion, that step collapses to a definition. This module is the verification-side record that the criterion is Alexander duality for an embedded circle, nontrivial exactly at $D=3$.
It is imported by Verification.T6T8SpineAudit, the machine-checked honesty audit of tiers T6 (phi fixed point) through T8 (dimension). That audit distinguishes THEOREM content from FORCED-CONDITIONAL content; DimensionLinking supplies the topological spine item the auditor must see when scoring whether $D=3$ is forced.
Downstream readers checking the Recognition composition law chain or the eight-tick octave can treat this module as the place where "why three dimensions" is tied to a standard duality isomorphism rather than an axiom.
scope and limits
- Does not reprove Alexander duality; delegates to Foundation.AlexanderDuality.
- Does not treat higher-genus knots or links beyond an embedded circle.
- Does not force time dimension or spacetime signature; only spatial $D$ via circle linking.
- Does not by itself close T8; the spine audit still classifies tier status.
- Witnesses at $D=5,7,9$ do not claim physical linking in those dimensions.
used by (1)
depends on (1)
declarations in this module (12)
-
def
AlexanderDualityForCircle -
def
H1_complement_isZ -
theorem
loop_loop_linking_forces_D3 -
theorem
same_sector_forces_odd -
def
witness_p -
theorem
witness_p_ge_one -
theorem
witness_reconstruction -
theorem
allowed_set_A_characterization -
theorem
witness_D3 -
theorem
witness_D5 -
theorem
witness_D7 -
theorem
witness_D9