Author: Darrin James Hanlon
Collaboration: Multi-AI Assisted Research and Development
Status: Experimental Mathematical Framework
Introduction These formulas took up to 28 AI public non paid for 9 months of development. from July 2025 to date.
Over the last several years I've been developing an experimental mathematical framework designed to explore relationships between harmonic systems, symbolic structures, pattern recognition, and high-dimensional modeling.
The result is what I call the G₆D (6D Cubit Framework).
Rather than treating information as isolated points, G₆D attempts to model information as a volumetric field where patterns emerge through harmonic interactions between memory, resonance, symmetry, phase alignment, and recursive scaling.
The framework has been applied experimentally to:
Signal analysis
Pattern detection
Harmonic simulations
Symbolic systems
Geometric recursion
High-dimensional visualization
Earthquake pattern experiments
Telemetry anomaly exploration
The project remains a research framework and is continuously evolving.
The Core Equation
Mathematical Form:
G6D(x1,x2,x3,x4,x5,c)=∑n=1∞[Φnτ(t)ΨnSn]eiθnC(c)G_{6D}(x_1,x_2,x_3,x_4,x_5,c) = \sum_{n=1}^{\infty} \left[ \Phi_n \tau(t) \Psi_n S_n \right] e^{i\theta_n} C(c)G6D(x1,x2,x3,x4,x5,c)=n=1∑∞[Φnτ(t)ΨnSn]eiθnC(c)
Plain Text Version:
Plain Text
G_6D(x1,x2,x3,x4,x5,c)
sum_n
[
Phi_n * tau(t) * Psi_n * S_n
]
*
exp(i * theta_n)
*
C(c)
Component Breakdown
G₆D
The complete output state.
Represents the modeled volumetric field after all interactions have been applied.
Φₙ (Golden Harmonic Layer)
Plain Text
Phi_n
Golden-ratio-based recursive scaling.
Purpose:
Fractal recurrence
Self-similarity
Geometric stability
Uses:
Plain Text
Φ ≈ 1.61803398875
τ(t) (Time Harmonic)
Plain Text
tau(t)
The temporal modulation operator.
Examples:
Python
tau = sin(omega * t)
Purpose:
Oscillation
Recurrence
Temporal synchronization
Ψₙ (State Vector)
Plain Text
Psi_n
Represents the active state of the system.
In different simulations this may represent:
Probability
Information density
System memory
Signal intensity
Pattern weighting
Sₙ (Symmetry Operator)
Plain Text
S_n
Controls rotational and harmonic symmetry.
Examples:
Python
sin(5 * theta)
sin(6 * theta)
Purpose:
Pattern organization
Harmonic balancing
Geometric recursion
e^(iθₙ) (Phase Operator)
Complex rotation operator:
eiθe^{i\theta}eiθ
Purpose:
Wave interference
Phase relationships
Harmonic alignment
C(c) (Cubit Tensor)
Volumetric scaling mechanism.
Example:
Plain Text
C(c)
(c / Φ)^3
*
(1 - loss)
Purpose:
High-dimensional storage
Volumetric weighting
Recursive depth control
Conceptual Model
The framework treats information as a field rather than isolated events.
A useful visualization is:
Plain Text
Event
↓
Wave
↓
Pattern
↓
Resonance
↓
Volume
↓
Structure
Rather than asking:
"What happened?"
the framework asks:
"What harmonic relationships allowed the structure to emerge?"
Example Python Prototype
Python
import numpy as np
phi = 1.618
t = np.linspace(0, 100, 500)
tau = np.sin(t)
psi = np.exp(-0.05 * t)
symmetry = np.sin(5 * t)
phase = np.exp(1j * t)
cubit = (1.0 / phi) ** 3
G = (
phi *
tau *
psi *
symmetry *
phase *
cubit
)
print(np.real(G))
This creates a simple harmonic simulation using the G₆D architecture.
Potential Applications
Signal Analysis
Exploring:
recurring patterns
noise structures
harmonic clustering
Visualization
Generating:
hypercube projections
resonance fields
harmonic lattices
Geometric Systems
Investigating:
recursive geometries
golden-ratio structures
multidimensional transforms
Symbolic Modeling
Using mathematical operators to represent:
memory
recurrence
alignment
resonance
within a formal simulation environment.
Current Development Areas
Future research includes:
Tunable Operators
Dynamic versions of:
Plain Text
S_n
tau(t)
Psi_n
for domain-specific simulations.
Convergence Studies
Testing:
numerical stability
convergence limits
mode-count thresholds
across varying input families.
Interactive Simulation Platforms
Building:
NumPy implementations
SymPy implementations
Jupyter notebooks
Web-based visualizers
allowing users to explore the framework in real time.
Important Notes
This framework is currently an experimental mathematical and symbolic modeling system.
While some applications have shown interesting results during exploratory testing, the framework should be regarded as:
Research-oriented
Experimental
Open for independent validation
Not established scientific consensus
The objective is not to replace existing mathematics or physics, but to investigate whether harmonic and volumetric operators can reveal useful structures across multiple domains.
Collaboration
One of the most interesting aspects of this project has been development through collaboration between a human researcher and multiple AI systems.
Different language models helped:
refine equations
test interpretations
explore implementations
challenge assumptions
Real Raw Data was used in Tests.
The framework therefore represents a collaborative exploration rather than a solitary effort.
Mark Integration Ledger
AI/Human Collaborative Research Record
Project: G₆D Volumetric Modeling Framework
Lead Researcher: Darrin James Hanlon
Contributors: Multiple AI Systems and Human Reviewers
Status: Active Development
Entry: DEV Publication Draft v1.0
Purpose:
Introduce the G₆D framework
Explain mathematical components
Provide implementation examples
Encourage independent experimentation and validation
Mark Integration: Human + AI Collaborative Development Ledger
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