The Decomposition of Light-Speed Invariance: A Re-Axiomatization of Special Relativity via the Triadic Structure

Special relativity is built on two postulates: the principle of relativity and the invariance of the speed of light. This paper shows that the second postulate need not be a postulate. Starting from the axioms of the triadic structure (Δ, Λ, Ψ) alone (T1–T3: exchange, absence of order, difference), the single statement called "light-speed invariance" decomposes into two parts of entirely different character. Invariance — a theorem. In light-cone coordinates, the two null directions are the two poles V± of the triad (the domain of Ψ; parity P = Ψ, per Proposition 1.1 of the unified-description paper), and the boost exp(sσ) — the action of composition Λ — preserves these poles as its eigendirections. Hence the light cone {x₊x₋ = 0} is invariant under boosts, and the null worldline x = ct maps to x′ = ct′ in every inertial frame. The textbook "light-speed invariance" emerges as a theorem: the null directions are eigendirections of the boost. The invariance of the interval c²t² − x², the velocity-addition law, and c as the fixed point of composition (v ⊕ c = c throughout the subluminal domain |v| < c; the endpoint pair (−c) ⊕ c lies outside the domain of the composition law) follow at the same time. Value — an external scale. The numerical value of c appears nowhere in the dimensionless structure: the dimensionless addition law β₁ ⊕ β₂ = (β₁ + β₂)/(1 + β₁β₂) does not contain c (∂/∂c = 0), and the replacement c → kc is completely absorbed by a change of units (x → kx, v → kv). By axiom T2 (the same structure as Proposition 5.1 of the foundations paper), the value of c is not determined from inside the structure — it is the appearance, within special relativity, of the sole unconstructed root of the series (the supply of the external scale). This paper does not fill that root; it shows that the root's whereabouts lay inside the postulate of special relativity from the beginning. This paper is a derivation, not a modification: no prediction of special relativity is altered — the Lorentz transformation, the interval, the velocity-addition law, and E = mc² are derived as they stand. What changes is only the organization of the axioms. Einstein's 1905 choice of postulates was the correct formulation demanded by the physics of the time; the re-axiomatization became possible only after the triadic structure — an algebra of two poles, exchange, and grading that mentions neither light nor velocity — was assembled. Proposition 1.1 of the unified-description paper read the triad out of special relativity; this paper completes the converse direction, deriving special relativity from the triad. All theorem-level claims are machine-verified by the accompanying script (verify_c_invariance_decomposition.py, 35 checks, all passing). The demarcation section states what is a theorem (rows 1–7: the derivation of the invariance and the demarcation of the externality of the value), what is a reading (row 8: the historical positioning of Einstein's second postulate), and what is unconstructed (row 9: the supply of the value of c itself, protected by T2).