Coherent Accessibility Function
A translation layer equation for dimensional gating across domains
Core Formula
| Symbol | Name | Definition | Range |
|---|---|---|---|
| Deff | Effective Dimensionality | The number of dimensions/states actually accessible to the system | [1, Dmax] |
| Dmax | Maximum Dimensionality | Theoretical upper bound of available dimensions/states | Domain-dependent |
| Φ(C) | Coherence Function | Normalized measure of system organization/coherence | [0, 1] |
Interpretation
- Φ(C) = 1: Full coherence → All dimensions accessible → Maximum options
- Φ(C) = 0: No coherence → Collapsed to single state → No options
- 0 < Φ(C) < 1: Partial coherence → Proportional dimensional access
Mathematical Derivation
Origin: Participation Ratio
The formula derives from the Participation Ratio (PR) in quantum mechanics, which quantifies how many basis states effectively contribute to a quantum state.
Where pi = |⟨i|ψ⟩|² are the probabilities of finding the system in basis state |i⟩.
Connection to Purity
For a density matrix ρ, the purity is defined as:
Purity ranges from 1/D (maximally mixed) to 1 (pure state). The participation ratio relates inversely:
Generalization to Coherence Function
Normalizing to [0,1] and generalizing beyond quantum systems:
This yields the core relationship:
The effective dimensionality equals maximum dimensionality scaled by the coherence function.
Forms of Φ(C)
The coherence function can take different forms depending on system dynamics:
Exponential Decay
Describes rapid decoherence under environmental coupling. λ is the decoherence rate.
- Quantum systems with thermal baths
- Systems with strong dissipation
- Time-dependent coherence loss
Power Law
Describes gradual scaling where C is a coherence measure and α is a scaling exponent.
- Scale-invariant systems
- Critical phenomena
- Self-organized systems
Participation Ratio
Direct measure from probability distribution over states.
- Quantum state analysis
- Statistical distributions
- Information-theoretic measures
Threshold Function
Sigmoid function for systems with critical thresholds. C0 is critical coherence, β is steepness.
- Phase transitions
- Activation dynamics
- Bistable systems
Domain Applications
Quantum Systems
| Dmax | Hilbert space dimension (number of basis states) |
| Φ(C) | Purity function: Tr(ρ²) or normalized participation ratio |
| Deff | Effective number of superposed states contributing to dynamics |
Atmospheric Dynamics (Polar Vortex)
| Dmax | Possible vortex configurations (unified, displaced, split, fragmented) |
| Φ(C) | Coherence index derived from temperature gradient and wind shear |
| Deff | Vortex stability state |
- Φ > 0.7 → Unified vortex (stable)
- 0.5 < Φ ≤ 0.7 → Weakening
- 0.3 < Φ ≤ 0.5 → Split/displaced
- Φ ≤ 0.3 → Fragmented (SSW event)
Information Systems
| Dmax | Total information capacity (Shannon entropy upper bound) |
| Φ(C) | Integration measure (ratio of actual to potential information integration) |
| Deff | Accessible information states |
Network Coordination
| Dmax | Possible coordinated states (2n for n nodes) |
| Φ(C) | Alignment/synchronization measure across network |
| Deff | Achievable coordinated actions |
Mathematical Properties
Boundedness
Effective dimensionality is always bounded by maximum dimensionality.
Multiplicative Scaling
Accessibility scales linearly with coherence, not additively.
Domain Independence
The functional form Deff = Dmax × Φ(C) holds regardless of domain. Only the definitions of Dmax and Φ(C) change.
Coherence as Gating
Coherence does not create dimensionality—it gates access to existing dimensions. Dmax is always present; Φ(C) determines accessibility.
Implications
A domain-independent measure of effective dimensionality opens applications wherever systems require coherence to function.
Climate & Atmospheric Prediction
Real-time coherence indices for atmospheric systems could provide early warning of state transitions — vortex destabilization, extreme weather windows, monsoon timing. Current models compute dynamics; coherence measures could predict when dynamics become unstable.
Network Resilience
Power grids, communication networks, supply chains — any system where failure cascades. Coherence measurement identifies fragmentation risk before failure occurs. Infrastructure operators could monitor effective dimensionality as a leading indicator.
Materials & Manufacturing
Material properties depend on internal coherence — crystal structure, grain boundaries, defect density. A coherence function could characterize material quality, predict failure points, or optimize manufacturing processes for higher effective organization.
Signal Processing & Communication
Coherence already matters in optics and radio. Generalizing to effective dimensionality provides a unified measure across modalities — how much information a channel can actually carry given its current coherence state.
Biological Systems
Living systems maintain coherence against entropy. Measuring effective dimensionality in neural activity, cardiac rhythms, or immune response could distinguish healthy function from fragmented states — early detection through coherence loss.
Financial & Economic Systems
Markets fragment before crashes. Coherence indices across asset correlations, trading networks, or institutional behavior could identify when systems lose effective dimensionality — fewer options, increased fragility.
Energy Systems
Efficiency is coherence — organized energy transfer versus dissipation. From battery degradation to fusion plasma stability, measuring coherence provides insight into how much of available energy is actually accessible for work.
Organizational Dynamics
Teams, companies, institutions — coordination capacity depends on alignment. Effective dimensionality measures what a group can actually accomplish together, distinct from theoretical capability.
These applications share a common structure: maximum potential exists (Dmax), but realized capability depends on system organization (Φ(C)). The formula provides a unified framework for measuring this relationship across domains.
Development Timeline
Exponential Multiplication Discovered
Binary operation a ⊗ b = a × 2b identified. Unity doubling principle (1 ⊗ 1 = 2) established. Connection to harmonic scaling systems recognized.
Quantum Formulation & Coherence Connection
Purity P(ρ) = Tr(ρ²) and effective dimensionality Beff = 1/P(ρ) connected. Graph/network theory, fractal representation, and master equation formalized.
Multi-Layer Framework Formalized
Four-layer mathematical system documented: State Evolution, Multi-Scale (Renorm), Information Layer with C(t) = Tr(ρ²), and Universal Objective function.
Polar Vortex Application
Deff = Dmax × Φ(C) applied to atmospheric science. Coherence index developed from temperature gradient and wind shear data.
SSW Event Validation
Model validated against historical Sudden Stratospheric Warming events (1979, 2009, 2018, 2021). Predicted vortex split during Dec 2025 SSW confirmed.
EVOLVE7 Stability Framework
Seven-level auto-scaling stability framework developed. Coherence epistemology outlined. Cross-domain translation formalized.
Scientific Documentation
Public documentation of mathematical framework. Formula categorized as Coherent Accessibility Function.
References & Foundational Literature
The coherent accessibility function Deff = Dmax × Φ(C) generalizes the participation ratio from quantum mechanics to a domain-independent framework. The contribution is the recognition that this relationship applies universally to systems where coherence determines accessibility.
Contact
For inquiries regarding this framework:
Hector Damian Cirino