not_1_4_signature
Recognition Science forces the spatial dimension to equal 3. Researchers deriving Lorentzian spacetime from J-cost minimization cite this result to exclude a purely spatial 4D geometry. The proof reduces the negation directly to the definition of D_physical via simplification.
claimIn the Recognition Science framework the spatial dimension satisfies $D_s = 3$, hence $D_s$ is not equal to 4.
background
The module derives the full 4D Lorentzian structure from the J-cost functional and the T0-T8 forcing chain. Spatial directions carry positive metric sign because J''(1) = 1, while the 8-tick recognition operator supplies the single temporal direction with opposite sign. The upstream result D_physical from DimensionForcing fixes the spatial count at 3 and states that this value is RS-compatible.
proof idea
The proof is a one-line term wrapper that applies simp to the definitions of spatial_dim and D_physical, reducing the claim to the fact that D_physical equals 3.
why it matters in Recognition Science
Labeled SE-010, the theorem asserts that the (1,3) signature is the unique RS-compatible choice. It supplies the spatial half of the central module theorem that the metric diag(-1,+1,+1,+1), light cones, and proper time are forced by J-cost with zero free parameters. The result implements the T8 step of the chain and closes the spatial count before the full spacetime emergence is assembled.
scope and limits
- Does not derive the temporal dimension or the full Lorentzian metric.
- Does not supply empirical tests or numerical predictions for the dimension.
- Does not address forcing chains outside the T0-T8 sequence.
- Does not prove compatibility with quantum gravity models beyond the imported lemmas.
formal statement (Lean)
325theorem not_1_4_signature : ¬(spatial_dim = 4) := by simp [spatial_dim, D_physical]
proof body
Term-mode proof.
326
327/-- **SE-010: The signature (1, 3) is the UNIQUE RS-compatible signature.** -/