Dirac Equation from the SL(2,ℂ) Pulsation of K₄ in the PDL Framework

The Projective Dynamic Logo (PDL) framework derives the Schrödinger equation from the (A) ∧ (B) stability criterion (D32). This paper extends the derivation to the relativistic regime and proves that the Dirac equation follows from the same criterion via a closed algebraic chain requiring no additional postulate. Four results are established. Lemma 1 derives the half-cycle operator conditions (T-a): T|c₊⟩ = |c₋⟩ and (T-b): T|c₋⟩ = −|c₊⟩ from the PDL pulsation axiom and condition (B), with the sign −1 in (T-b) shown to be structurally forced, not conventional. Theorem 1 proves T² = −I₂ in three lines from (T-a) and (T-b) alone, without invoking any Clifford algebra, spinor formalism, or relativistic postulate; the spin-½ double cover structure follows as a corollary, not a postulate. Theorem 2 establishes uniqueness of T = −iτ₂ by exhaustive check over all eight canonical candidates I₂, τ₁, τ₂, τ₃×1, i, with τ₁ excluded algebraically by τ₁² = +I₂ ≠ −I₂. Theorem 3 derives the Clifford algebra γμ, γν = 2ημνI₄ with γ⁰ = τ₃⊗I₂ and γⁱ = iτ₂⊗σᵢ as a theorem, with numerical residual 0. 00×10⁰. The non-relativistic limit reproduces D32 exactly, using the same αPDL and mₑ with no adjustment, confirming PDL self-consistency at two relativistic levels. This result closes OP4 of D32 and establishes that the period-4 pulsation of K₄ is the combinatorial origin of the relativistic spinor structure of matter.