pith. sign in

Why is space three-dimensional?

Big AI job. Grok 4.3 reads the canon and writes a Lean-grounded derivation; usually 20 seconds to 2 minutes. Your answer will appear below.
confidence: high in recognition cached

Linking requires D = 3 (Alexander duality) Alexander duality shows that S^D admits non-trivial circle linking iff the reduced cohomology H̃^{D-2}(S¹) is nontrivial, which holds precisely for D = 3. This is proved by alexander_duality_circle_linking and yields linking_requires_D3, ensuring stable ledger conservation only in three dimensions.

8-tick = 2^D forces D = 3 The eight-tick cycle is defined as EightTickFromDimension D = 2^D. The theorem eight_tick_forces_D3 proves that equality to eight_tick forces D = 3, since 2^D = 8 holds solely for D = 3.

Cl_3 spinor structure Only D = 3 produces 2-component complex spinors with Cl₃ ≅ M₂(ℂ) and Spin(3) ≅ SU(2). This is characterized by D3_has_spinor_structure together with spinorDimension 3 = 2.

Cited Lean anchors The derivation rests on the verbatim theorems eight_tick_forces_D3, linking_requires_D3, alexander_duality_circle_linking and D3_has_spinor_structure.

outside recognition

Aspects Recognition does not yet address:

  • CliffordBridge module
  • dimension_forced (docstring reference only)

recognition modules consulted

The Recognition library is at github.com/jonwashburn/shape-of-logic. The model is restricted to the supplied Lean source and instructed not to invent theorem names. Treat output as a starting point, not a verified proof.