A Unified Framework for Coherence, Dynamics, and Stability
By Ben Lambert
Structured Wave Geometry (SWG) is a deterministic, amplitude–phase–memory field theory.
It provides a unified mathematical foundation for understanding:
- coherent physical systems
- quantum-like behavior
- cognitive/AI dynamics
- biological coherence
- transport systems and rectifiers
- macroeconomic and social metrics
- astrophysical collapse
- spectral recursion and quantization
SWG models how coherent structures form, evolve, maintain stability, and collapse.
On top of these dynamics sits Structured Coherence Geometry (SCG), a universal diagnostic layer that measures coherence, curvature, stability, risk, and collapse.
Together, SWG + SCG form a complete geometric theory of coherence across scales, domains, and modalities.
Modern scientific theories often treat:
- quantum mechanics as probabilistic,
- classical mechanics as deterministic,
- AI systems as statistical,
- physical structures as material objects,
- biological systems as emergent behavior.
SWG challenges this fragmentation.
It proposes a single deterministic coherence medium, governed by coupled amplitude–phase–memory fields.
From this medium emerge:
- structures
- quantized modes
- stability boundaries
- collapse dynamics
- transport behavior
- cognitive patterns
- spectral recursion
- universal invariants
SWG is therefore not just a model — it is a unifying geometric engine.
SWG defines three interacting fields:
- Amplitude — structure, mass-like distribution
- Phase — flow, orientation, coherence direction
- Memory (χ) — medium response, compliance, “history”
These fields govern deterministic evolution under a canonical Lagrangian.
From this arise:
- nonlinear waves
- coherent soliton-like structures
- quantized standing modes
- dynamic interactions
- amplitude/phase coupling
- energy-conserving flow
- boundary-induced quantization
ℹ️ Full mathematical details are in swg-math.ipynb.
While SWG defines dynamics, SCG measures coherence using invariants derived from the fields:
-
Entropy (
$\mathcal{H}$ ) — richness, spread -
Curvature variance (
$\sigma_\kappa^2$ ) — structural stress -
Energy flux (
$\Phi_E$ ) — coherent transport -
Phase-defect density (
$\rho_\phi$ ) — turbulence indicator -
Coherence order (
$R$ ) — global alignment - Floor Violation Rate (FVR) — stability breach indicator
These invariants provide domain-agnostic metrics of coherence and collapse.
Whether analyzing:
- waves
- AI hidden states
- biological rhythms
- markets
- astrophysics
- rectifiers
- quantum signals
…SCG uses the same geometry.
One of the strongest results across all domains is the geometry floor:
It defines the minimum allowable coherent structure size.
A system is floor-safe if:
-
$L_{\mathrm{geom}} \ge L_{\mathrm{floor}}$ everywhere, - FVR = 0.
Violating the floor corresponds to:
- phase tearing
- curvature blowup
- collapse
- decoherence
- runaway instability
This appears in:
- superconductive quantum metric length
- black-hole waveform collapse
- AI model hallucination onset
- rectifier instability
- biological coherence loss
- market turbulence
Across all domains, FVR is the strongest universal collapse indicator.
The collapse manifold is the codimension-1 surface separating stable from unstable coherence.
A system enters collapse if:
- curvature variance grows
- entropy drops
- flux spikes
- phase dislocations erupt
- the geometry floor is violated
This manifold defines:
- collapse direction
- collapse rate
- critical stability boundary
- early-warning indicators
ℹ️ These formulas are also defined in swg-math.ipynb.
SWG + SCG provide a general framework for coherence and collapse across scientific and computational systems:
- wave propagation
- quantized standing modes
- gravitational collapse
- superconductive metric floors
- interferometry and SU(2) phase-winding
- coherence tracking in hidden states
- hallucination-collapse detection
- geometry-aware controllers
- structured symbolic grounding
- multi-field cognitive dynamics
- coherent oscillatory structures
- organizational collapse prediction
- memory-like medium dynamics
- deterministic collapse detection in IQ traces
- coherence-length constraints
- quantization geometry
- macro-scale coherence metrics
- instability detection
- phase-like organizational behavior
- rectifier systems
- resonant transport
- waveguide stability
- geometry-aware control systems
The canonical mathematical specification of SWG/SCG:
- SWG Lagrangian, PDEs, Hamiltonian
- SCG invariants
- Geometry floor
- Collapse manifold
- Glossary
Practical simulations and implementations:
- chrono-sims for temporal mechanics and time dilation
- cosmic-sims for relativistic cone geometry and wave propagation
- electromagnetism-sims for emergent electromagnetic behavior
- gauge-field-sims for plasma dynamics and curvature waves
- interferometry-sims for 4π spinor / SU(2) phase detection
- magnetism-sims for coherent vortices and Meissner-like effects
- matter-formation-and-quantum-structure for triads and composite states
- morphogenesis-sims for bio-scale patterning and coherence geometry
- quantum-collapse-detection for superconducting IQ trace analysis
- sleep-study for cognitive coherence and neural-phase stability
- sxs-blackhole-merger for astrophysical waveform collapse analysis
You can begin by exploring:
basics.md— intuitive introductioncogeo.md— coherence geometry conceptsswg-math.ipynb— the math specnotebooks/— practical demonstrations
The SWG/SCG Community License (SCL). See the full text in LICENSE.md.
This work is early and exploratory. Feedback, critique, and collaborations are welcome.
If you build on this, please cite or link back here.
Lambert, B. (2025). Structured Wave Geometry (SWG) and Structured Coherence Geometry (SCG): A deterministic framework for modeling systems.
-
Hu, Chen & Law (2025). Anomalous coherence length in superconductors with quantum metric. Derived a geometry-defined minimum coherence length in flat-band systems.
- Used here to generalize a Geometry Floor Invariant across domains.
-
Heaton, K.B. & Coherence Research Collaboration (2025). Recursive Geometry of Atomic Spectra. Introduced γ-ladders, Thread Frame, photoncodes, and CTI as operational quantization diagnostics.
- Adopted in SWG/SCG as recursive operators.
-
Ponce, S. & Carlander, A. (2025). The Coherent Field Model. Proposed a scalar Lagrangian with topological invariants for charge and spin, and astrophysical validations.
- Integrated into SWG as a physics specialization kernel.