The Emblem of the Triad: z⁴ = 1 — Sign Laws and Spectrum

This paper fixes a single equation as the emblem of the series: z⁴ = 1, with the legend "z is a readout (eigenvalue) of the triad." At the level of words in the three letters, the multiplicative closure is an eight-element group; the six orderings of the triple product take the two values ±1, coupled to the single orientation sign ε shared by the three mirror relations ΔΨ = εΛ, ΛΨ = εΔ, ΛΔ = εΨ; reversal of an ordering reverses its sign; even words are absolute — the square of any ordering is +1, the product of an ordering with its reversal is −1, and the three palindromes equal −1. A two-line parity principle organizes all of this: the value of a scalar word is a convention-free number times ε to the common parity of the letter counts. At the spectral level, the mirrors Λ and Ψ read ±1, the twist Δ reads ±i, the total spectrum of the represented group is exactly the fourth roots of unity μ₄ ≅ Z/4, and the ladder z ∈ {±1, ±i} ⟺ z² = ±1 ⟺ z⁴ = 1 collapses the sign at the fourth power; z⁴ = −1 has no solution in the spectrum. Every displayed identity holds for all letter-sign conventions and for every ratio λ; the external scale remains the sole blank, protected by axiom T2. All theorem-level claims are machine-verified (47/47). No new dynamics are introduced.