temporal_sq
plain-language theorem explainer
The square of the temporal component of a spacetime displacement vector supports the decomposition of the interval in the Recognition Science derivation of 4D Lorentzian geometry from J-cost. Researchers on emergent spacetime would cite this when separating timelike and spacelike contributions to causal structure. It is realized as a direct projection onto the time coordinate of the 4-vector followed by squaring.
Claim. For a spacetime displacement vector $v = (Δt, Δx_1, Δx_2, Δx_3)$, the temporal square is defined as $(Δt)^2$.
background
The spacetime emergence module derives the complete structure of 4D Lorentzian spacetime from the J-cost functional and the forcing chain T0–T8. This includes J-uniqueness via the Recognition Composition Law, phi as the self-similar fixed point, the eight-tick octave for the temporal direction, and D = 3 for spatial dimensions, yielding the metric signature (−,+,+,+) with c = 1 as a tautology. Time carries the negative sign because the recognition operator decreases cost along that axis.
proof idea
This is a one-line definition that selects the zeroth entry of the displacement vector and squares the result.
why it matters
This definition is used by theorems that establish the spacetime interval as spatial norm squared minus temporal square, the light cone conditions, and proper time expressions. It advances the central claim that Lorentzian structure is forced by cost minimization without background postulates. It contributes to the framework step where the eight-tick octave and D = 3 combine to produce four-dimensional spacetime and the arrow of time.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.