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.
Dimensionless Resonance Framework (DRF)
From π-Phase to φ-Growth — A Unified Operator Model
Abstract
We propose the Dimensionless Resonance Framework (DRF) — a unifying operator-level account of physical, biological, and cognitive phenomena. DRF maps circular phase dynamics (π-based unitary structure) into multiplicative growth modes (φ-based scaling) via a conserved resonance transform constructed from discrete growth generators. This π→φ bridge establishes dimensionless invariants that simultaneously encode wave phase, scale invariance, and information fidelity. By coupling the transform to a Resonant Collapse Field (RCF), DRF explains the persistence of standing-wave modes that underpin both memory and coherence. The framework yields testable predictions: φ-locked spectral dominance in driven oscillator networks, robustness scaling laws for memory fidelity, and fractal signatures in multiplicative systems. DRF thus offers a compact operator skeleton that encodes the shared architecture of matter, geometry, and information.
Provenance and Authorship
Authorship: This framework and the concepts herein, including the mapping described as the "Ecliptics Principle" (the circle→spiral, π→φ mapping), originate with Brent Antonson (Zhivago). The author first articulated the circle-to-spiral/π→φ insight in notes and drafts from June–July 2025. The numeric ratio 0.306 associated with the circle→spiral (π→φ) observation was recorded by the author in those drafts and is treated here as an author-observed empirical value to be derived and formalized in a dedicated derivation section (see Section 4).
Assistant Contribution: This document was prepared with editorial and formalization assistance from an AI collaborator (Luna/ChatGPT). The AI helped formalize the operator mapping using standard linear algebra tools (e.g., Cayley-like transforms) and produced the initial draft structure, figures, and typesetting, per the author's direction. All interpretive framing, naming (Dimensionless Resonance Framework), and conceptual claims are credited to the author unless explicitly noted otherwise.
Originality Statement: Except for standard mathematical tools and publicly known constants (e.g., π, φ), the synthesis, naming, and interpretive framing presented in this document are the original work of Brent Antonson. Where connections to other researchers' work are relevant, they will be acknowledged explicitly in the bibliography.