octave_matches_spatial
The eight-tick recognition period equals two raised to the spatial dimension count. Researchers deriving emergent Lorentzian geometry from J-cost minimization would cite this equality to confirm that the temporal octave aligns with three spatial axes. The proof is a one-line reflexivity that holds by the definitions of the eight-tick period and the spatial dimension.
claimThe eight-tick period equals $2^D$ where $D$ is the spatial dimension count.
background
The Spacetime Emergence module derives the full 4D Lorentzian structure from the J-cost functional and the T0-T8 forcing chain. J-cost near identity is quadratic, $J(1+ε) = ε² / (2(1+ε))$, producing positive curvature along spatial directions while the recognition operator reduces cost along the temporal direction. Upstream, eight_tick is the definitional natural number 8 from DimensionForcing, and spatial_dim is fixed at 3 by the T8 result in the same chain.
proof idea
The proof is a term-mode reflexivity. It equates the constant eight_tick (defined as 8) directly to 2 raised to spatial_dim (defined as 3) without further reduction steps.
why it matters in Recognition Science
This equality links the temporal octave (T7) to the spatial count (T8) and feeds the central derivation of the metric signature diag(−1,+1,+1,+1). It appears in the module's registry of forced spacetime properties and supports the claim that dimension 4 and the light-cone structure arise from cost minimization alone.
scope and limits
- Does not derive the spatial dimension value from axioms.
- Does not apply outside the T0-T8 forcing chain.
- Does not address the sign of the temporal metric coefficient.
formal statement (Lean)
75theorem octave_matches_spatial : eight_tick = 2 ^ spatial_dim := rfl
proof body
Term-mode proof.
76
77/-! ## §2 The Spatial Metric from J-Cost Curvature -/
78
79/-- **The exact J-cost quadratic form near identity.**
80 J(1+ε) = ε² / (2(1+ε)) for any ε with 1+ε > 0. -/