Gyrational Modeling of Motion and Structure in the Common Governance Model

Citation: Korompilias, B. (2025). Common Governance Model: Mathematical Physics Framework. Zenodo. https://doi.org/10.5281/zenodo.17521384

Abstract

The Common Governance Model (CGM) adopts a gyrational modeling principle where motion and structure emerge from rotational and cyclic processes rather than linear trajectories. This framework posits angular momentum as the primitive structural invariant from which observable phenomena derive. CGM suggests that sustained motion requires underlying gyration, that spacetime configurations carry non-zero holonomy as structural memory, and that energy transforms across scales through reallocation among gyrational modes. The model reinterprets cosmic observables through rate-based rather than velocity-based descriptions and proposes connections between galactic morphology and the six degrees of freedom anticipated by CGM's recursive stages.

Definitions and Conventions

Angular momentum: L = r × p, treated as the primitive quantity generating rotational structure.

Aperture parameter: m_a = 1/(2√(2π)) ≈ 0.199471, representing the structural opening in CGM.

Completeness invariant: Q_G = 4π, the total solid angle for geometric completeness.

Gyration: Rotational or cyclic degrees of freedom underlying apparent linear motion.

Holonomy: Geometric memory accumulated through non-associative composition along closed paths.

CGM stages: CS (Common Source), UNA (Unity Non-Absolute), ONA (Opposition Non-Absolute), BU (Balance Universal).

1. Motivation: Gyrational Modeling Principle

CGM adopts a gyrational modeling principle: sustained motion at observable scales is supported by cyclic or rotational processes. Consider vehicles maintaining constant relative velocity. A car requires continuous wheel rotation against the road surface; an airplane depends on turbine rotation for propulsion. Removing these rotational components causes immediate deceleration through friction and drag.

This principle extends across physics. Quantum particles follow wave-like trajectories described by the de Broglie relation λ = h/p, suggesting oscillatory rather than purely linear motion. Atomic electrons occupy orbitals characterized by angular momentum quantum numbers. Planetary motion conserves angular momentum through orbital dynamics.

CGM posits that linear motion is an emergent approximation obtained by projecting underlying cyclic dynamics into co-moving reference frames. This modeling choice emphasizes rotational and phase degrees of freedom as fundamental, with linear trajectories as derived projections.

2. Angular Momentum as Primitive Structural Invariant

CGM identifies angular momentum as the most primitive structural invariant from which other observables emerge. This choice is motivated by angular momentum's role across scales and its generative relation to three-dimensional rotational structure.

The framework maps CGM stages to powers of angular momentum as a heuristic organizing principle:

CS (Common Source): Angular momentum persistence L corresponds to the principle that rotational states continue unless modified by external influences. The primordial chirality α = π/2 establishes handedness through non-identity left gyration.

UNA (Unity Non-Absolute): Quadratic expressions L² appear in rotational energy E_rot = L²/(2I). The orthogonal split β = π/4 activates three rotational degrees of freedom.

ONA (Opposition Non-Absolute): Coupling mechanisms generating translational degrees complete the SE(3) structure. Standard relations include torque τ = dL/dt and precession frequency Ω = τ/L.

BU (Balance Universal): Perpetual cycling through these stages via the 120° rotor maintains dynamic equilibrium while preserving structural memory.

This mapping represents a heuristic interpretation rather than fundamental physical derivation. The appearance of L² in energetic expressions and the role of angular momentum conservation in generating three-dimensional structure provide qualitative support for this organizational scheme.

3. Curvature, Holonomy, and Gyrational Memory

CGM suggests that physically realized configurations carry non-zero holonomy as gyrational memory. This holonomy records the order of operations in non-associative composition, creating structural memory that manifests as curvature.

In this interpretation, flatness represents an idealized local limit. Globally, non-trivial curvature and holonomy are expected because the model's alignment mechanism relies on non-associative composition that preserves order-dependent information.

Galactic structure provides qualitative support for this perspective. Disc-like galaxies exhibit approximately planar stellar distributions while frequently displaying collimated jets along axes close to the disc normal. CGM interprets this morphology as a signature of complementary degrees of freedom: planar rotation (UNA-associated rotational degrees) coexisting with axial features (ONA-associated translational degrees).

This represents an alignment of expected degrees of freedom with observed large-scale structure rather than a definitive proof. The perpendicular relationship between disc and jet orientations qualitatively matches the orthogonal splits β = π/4 and γ = π/4 anticipated by the UNA and ONA stages.

4. Scale Transitions: Energy as Gyrational Mode Reallocation

CGM suggests that energy transforms across scales through reallocation among gyrational modes rather than propagation through space. Quantum well energy levels En = n²π²ℏ²/(2mL²) display n² scaling, which CGM interprets as an L² signature reflecting the quadratic appearance of angular momentum in energetic expressions.

This perspective reframes various phenomena:

Atomic transitions: Angular momentum exchange between electron and photon states rather than particle emission and absorption.

Molecular vibrations: Coupled rotational modes between atomic centers rather than oscillating point masses.

Stellar processes: Angular momentum cascading through nuclear states rather than linear energy flow.

Galactic dynamics: Angular momentum distribution across orbital configurations rather than gravitational particle interactions.

This interpretation emphasizes transformation between discrete gyrational states rather than continuous propagation through intervening space. Energy appears conserved locally while transforming scales through the discrete ladder of angular momentum modes.

5. Cosmological Reframing: Rates Rather Than Velocities

CGM reinterprets cosmological observables through rate-based rather than velocity-based descriptions. The Hubble parameter H₀ has dimensions of inverse time and represents a fundamental rate rather than recession velocity. In this view, cosmic expansion is a rate of structural unfolding rather than linear motion through space.

High cosmic velocities, such as the Milky Way's motion relative to the cosmic microwave background, are reinterpreted as phase relationships within larger-scale gyrational frames. This perspective explains why such velocities are not directly felt: observers participate in co-moving frames defined by the local gyrational structure rather than translating through external space.

Variations in Hubble measurements (67.4 to 73.4 km/s/Mpc) are interpreted as different aspects of the rate structure rather than measurement uncertainties. This spread may reflect the interplay between different scales of gyrational dynamics rather than systematic errors in distance calibration.

6. Relations to Established Formalisms

CGM aligns with and departs from standard physics in specific ways:

Angular momentum primacy: While standard mechanics treats linear momentum p = mv as fundamental and angular momentum L = r × p as derived, CGM reverses this hierarchy, treating rotational structure as primitive.

Holonomy as memory: General relativity incorporates holonomy through parallel transport, but CGM emphasizes holonomy as accumulated structural memory rather than purely geometric effect.

Non-associativity as curvature: Standard differential geometry relates curvature to commutator brackets [∇μ, ∇ν]. CGM extends this to non-associative composition, where the associator (AB)C - A(BC) measures accumulated gyrational memory.

Operator framework: The relations [X^i, P_j] = iK_QG δ^i_j with K_QG ≈ 3.937 provide a modified uncertainty principle incorporating gyrational effects. In physical units, this reduces to standard quantum mechanics through the bridge relation S_min × κ = ℏ.

7. Open Derivations and Future Directions

Several aspects of the gyrational framework require further development:

Field-level formulation: A complete field theory would define angular momentum density fields ℓ(x) with constitutive relations analogous to Einstein's field equations. The precise form of such equations remains an open derivation.

Holonomy quantization: The measured holonomy values (ω(ONA↔BU) = 0.097671 rad, δ_BU = 0.195342 rad, etc.) suggest a discrete ladder structure, but the underlying quantization principle requires clarification.

Scale coupling: The mechanism by which gyrational modes at different scales influence each other needs precise mathematical formulation beyond the heuristic mappings presented here.

Experimental signatures: Specific predictions distinguishing gyrational dynamics from standard linear descriptions would strengthen the framework's empirical foundation.

Conclusion

The gyrational modeling principle offers a systematic reinterpretation of motion and structure emphasizing rotational and cyclic processes. CGM's identification of angular momentum as the primitive structural invariant provides a organizing framework for understanding phenomena across scales. While many aspects require further development, the approach demonstrates internal consistency and qualitative alignment with observed structures such as galactic morphology and quantum energy level scaling. The framework suggests that what appears as linear motion and energy propagation may be emergent projections of underlying gyrational dynamics maintaining structural coherence through holonomic memory.