Dimensionless Resonance Framework (DRF): From π-Phase to φ-Growth — A Unified Operator Model
1. Introduction
Physics has long balanced between two archetypes: the circle and the spiral. Waves, oscillations, and orbits embody the circular story of phase (π), while branching growth, self-similarity, and biological scaling tell the spiral story of φ. Traditionally, these domains have been treated separately: unitary operators for phase dynamics, and multiplicative matrices for growth.
The Dimensionless Resonance Framework (DRF) proposes that these stories are two views of a single operator structure. By mapping discrete growth generators into the unitary domain, DRF reveals a universal π→φ bridge. In this formulation, the golden ratio (φ) emerges not just as a number in nature but as a stable eigenphase of a resonance operator.
This bridge provides more than symbolism. It defines dimensionless invariants that survive changes of scale, allowing one operator framework to describe matter, geometry, and information simultaneously. When coupled to a Resonant Collapse Field (RCF) — a trace-preserving, resonance-biased projection — DRF explains how standing-wave modes persist across domains, embedding memory and coherence within the same algebra.
DRF therefore unites three themes:
- Matter: standing-wave persistence and resonance in physical systems.
- Geometry: φ-locked growth and self-similarity.
- Information: fidelity of memory lattices under recursive collapse.
2. Formalism: The π→φ Operator Bridge
2.1 Growth Generator
Begin with the Fibonacci generator:
A = [[1, 1], [1, 0]]
Its eigenvalues are:
λ₁,₂ = (1 ± √5)/2 = φ, 1−φ,
where φ ≈ 1.618... is the golden ratio. Thus, growth dynamics (discrete recurrence → exponential scaling) are encoded directly in A.
2.2 Unitary Dressing
We define the resonance operator:
U = (A − iI)(A + iI)⁻¹.
This Cayley-like transform maps real growth into the unitary domain. The eigenvalues of U lie on the unit circle, expressible as:
μⱼ = e^(iθⱼ), j=1,2.
Thus, growth eigenvalues λⱼ become eigenphases θⱼ.
2.3 The π→φ Mapping
Explicit calculation yields:
θ(φ) ≈ 116.565°, θ(1−φ) ≈ −63.435°.
The golden ratio φ, a growth constant, is thereby encoded as a phase angle on the unit circle. This establishes the π→φ operator bridge: multiplicative scaling eigenmodes become locked phase rotations.
2.4 Dimensionless Resonance
Because the mapping is via U, the result is dimensionless: only ratios and phases survive. This is the heart of DRF — a resonance framework that is invariant under rescaling, bridging growth and oscillation without dependence on physical units.
2.5 Generalization
For any Perron–Frobenius growth matrix Aₙ, DRF constructs:
Uₙ = (Aₙ − iI)(Aₙ + iI)⁻¹,
and defines its resonance spectrum {θⱼ}. The conjecture: across families of growth matrices, φ-like eigenmodes dominate as robust resonances, explaining the ubiquity of golden-ratio scaling in physics, biology, and cognition.