Operator XV — PRISM Lab Chamber

Details: Category: Labs; Published: 01 November 2025

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.

Operator XIV — Φ-Scale: When Recursion Rhymes With Itself | UNNS.tech

Details: Category: UNNS Grammar; Published: 01 November 2025

When Recursion Rhymes With Itself — The Golden Ratio as Emergent Symmetry

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UNNS Operators Tier II Dimensionless Constants Phase V Research Φ-Scale Chamber Ready

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.

Operator XIV — Φ-Scale Lab Chamber | UNNS.tech

Details: Category: Labs; Published: 01 November 2025

Emergent Scale Symmetry and the Golden-Ratio Attractor in the τ-Field

UNNS Operators Tier II Dimensionless Constants Φ-Scale Chamber Lab v0.7.2

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.

UNNS Operator XIII: Interlace as Cross-Recursive Grammar | UNNS.tech

Details: Category: UNNS Grammar; Published: 01 November 2025

Operator XIII — Interlace

When Two Recursions Dance — The Birth of Mixing Angles

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UNNS Lab · Operator XIII – Interlace

Details: Category: Labs; Published: 01 November 2025

UNNS Lab confirms a major milestone: Operator XIII — Interlace has been calibrated and theoretically closed, reproducing the Weinberg angle from pure τ-field recursion.

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