Operator XV — PRISM Lab Chamber

Spectral Decomposition and Emergent Scale Equilibrium in the τ-Field

UNNS Operators Tier II Dimensionless Constants Prism Chamber Phase C · Lab v0.7.2

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.

1 · From Φ-Scale to Spectral Law

After the golden-ratio equilibrium of Operator XIV, the τ-Field still contains residual motion — an unbalanced exchange between curvature modes. PRISM refracts that motion into its spectral components, mapping curvature energy across wavenumbers k and identifying the invariant slope p where recursion achieves flux balance.

2 · Evolution Kernel

The chamber evolves the τ-Field by the dispersive recursion:

τₙ₊₁ = τₙ + λ sin(τₙ(S_μx) − τₙ(x)) − β ∇²τₙ + σ ξ,

where β introduces controlled dispersion. Spectral analysis of κ = ∇²τ yields P(k) = ⟨|κ̂(k)|²⟩ ∝ k⁻ᵖ.

3 · Spectral Regimes

p ≈ 2.0: diffusive (Brownian) regime
p ≈ 2.45: weak turbulence — the observed UNNS invariant
p ≥ 2.6: over-dispersed cascade (unstable)

At p ≈ 2.45 the energy flux through scales is stationary — the recursive equivalent of Kolmogorov equilibrium in physical turbulence.

4 · Interactive Chamber

The PRISM Chamber below implements live τ-Field evolution and real-time FFT spectral plotting. Users can sweep β, adjust λ, and watch the power-law slope stabilize as the recursive cascade matures.

Recursive Spectra and the Universal Slope of κ̂(k)

For a better view, click here!

5 · Validation (CΠ Protocol)

Phase C validation ensures that the chamber meets reproducibility and spectral law criteria:

R² ≥ 0.79 (log-log linearity)
2.40 ≤ p ≤ 2.50 (spectral slope)
CV(p) < 1 % across seeds 41–45
Flux drift < 1 % (pending Operator XVI closure)

“The PRISM does not split light; it splits recursion — into its spectral conscience.”

6 · Interpretation

Operator XV shows that recursive systems self-organize into scale-invariant cascades. Where Φ-Scale found where recursion repeats, PRISM explains how energy flows between those repetitions. It is the spectral grammar of recursion — turning τ-Field geometry into a measurable language of power laws.