pure_temporal_is_timelike
plain-language theorem explainer
A four-vector with nonzero component only along the temporal axis satisfies the timelike condition under the interval quadratic form. Researchers deriving Lorentzian geometry from J-cost minimization cite this to fix the negative temporal sign. The argument rewrites via the timelike equivalence and reduces the inequality to positivity of the squared nonzero entry.
Claim. For nonzero real number $t$, the four-vector $v$ with components $(t,0,0,0)$ satisfies interval$(v)<0$.
background
The SpacetimeEmergence module derives the full 4D Lorentzian structure, including metric signature $(- , + , + , +)$, from the J-cost functional and the T0-T8 forcing chain. The interval function encodes the quadratic form that classifies displacements as timelike when negative. Upstream results supply A as the active edge count per tick together with consistency statements that keep the underlying combinatorial ledger and phi-ladder intact.
proof idea
The term proof rewrites the goal with timelike_iff_subluminal, producing the explicit inequality on summed spatial squares versus the temporal square. Norm_num simplifies the expression, after which sq_pos_of_ne_zero closes the proof from the hypothesis that $t$ is nonzero.
why it matters
The lemma confirms that the temporal direction alone produces a negative interval value, matching the requirement that the 8-tick recognition operator decreases cost along time. It supports the module's central claim that the Lorentzian signature is forced by cost minimization rather than postulated. The result aligns with the Recognition Composition Law and the derivation of light cones and proper time from the phi-ladder.
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