spatial_metric_at_identity
plain-language theorem explainer
The spatial metric coefficient at the identity evaluates to one half. This fixes the second derivative of the J-cost to one and confirms positive definite curvature along spatial axes in the emergent geometry. Researchers constructing Lorentzian spacetime from cost minimization cite this when building the diagonal metric from the T0-T8 chain. The proof reduces the constant arithmetic identity by direct numerical simplification.
Claim. At the identity the spatial metric coefficient equals $1/2$, which yields $J''(1)=1$.
background
Recognition Science derives 4D Lorentzian spacetime from the J-cost functional, where $J(x)=(x+x^{-1})/2-1$. The module establishes that metric signature, light cones, and the arrow of time are theorems of cost minimization along the forcing chain rather than background postulates. Spatial directions receive positive metric sign because displacement raises cost quadratically; the temporal direction receives negative sign because the eight-tick recognition operator lowers cost. Upstream constants supply the octave as eight ticks and the phi fixed point that sets the self-similar scale.
proof idea
The proof is a one-line wrapper that applies norm_num to simplify the constant expression $1/(2*(1+0))$ to $1/2$.
why it matters
This step supplies the spatial diagonal entry in the forced Minkowski metric diag(-1,+1,+1,+1) inside the central spacetime-emergence theorem. It closes the link from T5 J-uniqueness through J''(1)=1 to T8 three spatial dimensions. The result feeds the subsequent construction of causal structure and proper time with zero free parameters.
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