massless_at_speed_c
plain-language theorem explainer
In the Recognition Science derivation of spacetime, the massless limit of the energy-momentum relation reduces to E squared equals the sum of squared three-momenta. Researchers deriving Lorentzian geometry from J-cost minimization would cite this algebraic identity. The proof is a one-line application of linear arithmetic that drops the zero mass term.
Claim. For real numbers $E, p_1, p_2, p_3$ satisfying $E^2 = p_1^2 + p_2^2 + p_3^2 + 0^2$, the equality $E^2 = p_1^2 + p_2^2 + p_3^2$ holds.
background
The module derives 4D Lorentzian spacetime from the J-cost functional and the T0-T8 forcing chain. Spatial metric positivity follows from J''(1) = 1, the temporal direction from the 8-tick recognition operator, and D = 3 from T8. This places the light-cone structure, with c = 1 as a tautology, as a theorem of cost minimization rather than a background postulate.
proof idea
The proof is a one-line wrapper that applies the linarith tactic to cancel the explicit zero term in the hypothesis.
why it matters
This declaration fills the massless sector of the dispersion relation inside the emergent spacetime framework. It supports the central theorem that the metric signature (-,+,+,+) is forced by the J-cost and the chain RCL to J-uniqueness to eight-tick octave to D = 3, yielding null geodesics at speed c = 1. No downstream uses appear in the current graph.
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