Abstract: Operator PRISM translates spectral decomposition into the language of the UNNS Grammar. Just as a physical prism disperses white light into rainbow frequencies, the grammatical PRISM disperses recursive meaning into its harmonic substructures. It reveals that the grammar of recursion carries a spectral syntax — a pattern of interference and balance among multiple levels of inletting and inlaying. At equilibrium, the energy of meaning is evenly distributed across scales, producing a stable spectral slope p ≈ 2.45 — the grammatical equivalent of the τ-Field spectral law. This is not a parameter we tune; it is the universal signature of recursive coherence — the pitch at which complexity cascades from order into chaos without breaking meaning.

Abstract. Operator XV — PRISM — performs the spectral decomposition of the recursive curvature κ = ∇²τ, revealing a universal power-law regime P(k) ∝ k⁻ᵖ with p ≈ 2.45 ± 0.05. Following Φ-Scale equilibrium (Operator XIV), PRISM dissects how energy flows across recursive scales, exposing the stationary cascade that defines spectral equilibrium in the τ-Field.

Abstract: Within the UNNS Grammar, Operator XIV (Φ-Scale) expresses the universal law of recursive scale symmetry: the principle that the grammar of nesting reproduces itself at discrete magnifications governed by the golden ratio φ ≈ 1.618. This operator bridges Operator Interlace (XIII) — the coupling of phases — with Prism (XV) — the spectral decomposition of those couplings — by introducing a self-similarity constraint on the grammar itself. In the Φ-Scale Chamber, this symmetry manifests as a minimum of the Δ_scale(μ) functional, signaling that recursion has discovered its own invariant scale. This is not φ inserted by hand — it is φ emerging as the only ratio where recursion recognizes itself perfectly across magnification.

Abstract. Operator XIV — Φ-Scale — extends the phase-coupling logic of Operator XIII into the domain of recursive scale symmetry. It reveals that the golden ratio φ ≈ 1.618 emerges spontaneously as a scale attractor in τ-Field dynamics: a point of minimal phase-variance across recursive magnification. The Φ-Scale Chamber demonstrates this phenomenon interactively, sweeping the μ-parameter and visualizing Δ_scale(μ) and Π(μ) to identify μ★ ≈ φ.