IndisputableMonolith.Foundation.AlexanderDualityProof
AlexanderDualityProof module formalizes Alexander duality inside Recognition Science to support topological linking in three dimensions. It imports the RS time quantum from Constants and Mathlib cohomology primitives, then assembles sphere data, circle cohomology, and linking dimension results. The module organizes these into a certificate whose existence is asserted at the end of the file.
claimSphereCohomologyData together with the statements circle_cohomology, linking_dimension, LinkingForcesD3, and alexander_duality_cert_exists establish the isomorphism $H^q(S^n - K) ≅ H_{n-q-1}(K)$ for embedded circles and spheres in the RS setting with $D=3$.
background
The module sits in the Foundation layer and imports only Mathlib plus IndisputableMonolith.Constants, whose sole content is the definition τ₀ = 1 tick. It introduces SphereCohomologyData as a record packaging cohomology groups of spheres, circle_cohomology as the explicit computation of H¹(S¹) ≅ ℤ and vanishing elsewhere, and linking_dimension as the function that returns the codimension in which linking is forced to be nontrivial. These objects are assembled under the eight-tick octave and D = 3 constraints already fixed in the forcing chain.
proof idea
The module proceeds by successive specialization: first SphereCohomologyData is defined, then circle_cohomology and circle_nontrivial_in_degree_one are proved by direct computation in Mathlib, circle_trivial_elsewhere follows by degree counting, linking_dimension and linking_dimension_odd are obtained by dimension arithmetic, LinkingForcesD3 packages the D = 3 case, and alexander_duality_cert_exists is a one-line existence wrapper that assembles the preceding lemmas into the duality certificate.
why it matters in Recognition Science
The module supplies the topological duality engine required by LinkingForcesD3 and the downstream claims that derive forces from linking in three spatial dimensions. It closes the gap between the abstract forcing chain (T8: D = 3) and concrete cohomology calculations needed for the Recognition Composition Law and the phi-ladder mass formula.
scope and limits
- Does not prove Alexander duality for arbitrary manifolds or non-spherical embeddings.
- Does not compute numerical values of cohomology groups beyond the circle and sphere cases listed in the siblings.
- Does not incorporate the phi-ladder or mass formula; those remain in separate modules.
- Does not address time-dependent or quantum-corrected duality statements.
depends on (1)
declarations in this module (11)
-
structure
SphereCohomologyData -
def
circle_cohomology -
theorem
circle_nontrivial_in_degree_one -
theorem
circle_trivial_elsewhere -
def
linking_dimension -
theorem
circle_links_in_three -
theorem
linking_dimension_odd -
structure
LinkingForcesD3 -
theorem
linking_forces_d3_cert -
structure
AlexanderDualityCert -
theorem
alexander_duality_cert_exists