Observational Partition Algebra: Unification of Measurement, Process, and Computation in Bounded Dynamical Systems
This repository contains the theoretical framework, validation code, and experimental implementations for Observational Partition Algebra—a mathematical formalism establishing that measurement, physical process, and observation are identical operations within bounded dynamical systems. Starting from two axioms (bounded phase space and categorical observation), we prove that electromagnetic radiation, diffusive transport, enzymatic catalysis, and all cellular processes reduce to partition operations on a three-dimensional S-entropy space.
The central theorem establishes that for any partition equation Γ₁ ⊕ P(ω) → Γ₂, the output Γ₂ is simultaneously: (i) the physical state resulting from the process, (ii) the observable that would be measured, and (iii) the computational output of the equation.
The framework derives from two axioms:
Axiom 1 (Bounded Phase Space): A physical system with finite energy E < ∞ and finite spatial extent L < ∞ occupies a bounded region of phase space Ω with finite measure μ(Ω) < ∞.
Axiom 2 (Categorical Observation): An observer with finite resolution partitions phase space into equivalence classes {Ωᵢ}. States x, y ∈ Ω belong to the same equivalence class if and only if the observer cannot distinguish them through available measurements.
For bounded measure-preserving dynamical systems, three descriptions are isomorphic:
Oscillatory ≅ Categorical ≅ Partition
O(Ω) ≅ C(Ω) ≅ P(Ω)
This equivalence implies that oscillatory dynamics, categorical state transitions, and partition operations are mathematically identical—different projections of the same underlying structure.
Categorical partitioning of bounded spherical phase space generates four coordinates:
- Depth n ≥ 1
- Complexity ℓ ∈ {0, 1, ..., n-1}
- Orientation m ∈ {-ℓ, ..., +ℓ}
- Chirality s ∈ {-½, +½}
The capacity formula C(n) = 2n² yields the sequence 2, 8, 18, 32, 50, 72, 98, ... which corresponds exactly to electron shell capacities in atomic physics.
The S-entropy space S = [0,1]³ comprises three coordinates:
- Knowledge entropy Sₖ: uncertainty in state identification
- Temporal entropy Sₜ: uncertainty in timing relationships
- Evolution entropy Sₑ: uncertainty in trajectory progression
Categorical distance d_cat between partition states is mathematically independent of spatial distance and optical opacity:
d_cat ⊥ d_spatial
d_cat ⊥ τ_optical
This independence enables opacity-independent measurement: subsurface or transmembrane structures remain categorically accessible if their partition signatures are distinguishable, regardless of physical barriers.
Enzymes function as information catalysts that reduce categorical distance between substrate and product states through intermediate partition stages:
d_cat^catalyzed(Γ_A, Γ_B) = Σₖ d_cat(Γₖ₋₁, Γₖ) < d_cat^direct(Γ_A, Γ_B)
The catalytic efficiency k_cat/K_M measures information catalytic power: how effectively the enzyme reduces categorical distance per unit substrate concentration.
CPL is a formal language in which physical phenomena serve as primitive operators acting on categorical states. Programs in CPL specify constraint satisfaction problems whose solutions are simultaneously physical trajectories and experimental observations.
| Operator | Symbol | Description |
|---|---|---|
| Photon | P_γ(ω) | Categorical transitions at frequency ω |
| Gradient | ∇_S | S-entropy flow generation |
| Phase-lock | Φ(ω₁, ω₂) | Oscillator coupling within bandwidth |
| Aperture | A(d_cat) | Constraint through categorical distance |
| Process | CPL Expression | Predicted | Observed | Error |
|---|---|---|---|---|
| ATP synthesis | Γ_ADP ⊕ P_H⁺(ω) → Γ_ATP | 5.0 s | 5.0±0.5 s | <1% |
| Protein folding | Γ_unfolded ⊕ ⊗Φⱼ → Γ_native | k cycles | k±1 cycles | <5% |
| Ion transport | Γ_in ⊕ Φ(ω_ion, ω_ch) → Γ_out | 10⁹ selectivity | 10⁹ | <1% |
A cell has no internal mechanism to distinguish healthy from diseased states. Disease is oscillatory dynamics outside the phase-lock bandwidth with the cellular master clock.
For any oscillator O with performance metric Π:
η = (Π_obs - Π_deg) / (Π_opt - Π_deg)
where η ∈ [0, 1] with η = 1 indicating full coherence (healthy) and η = 0 indicating no coherence (maximally diseased).
| Class | Symbol | Frequency Range | Performance Metric |
|---|---|---|---|
| Protein | P | 10¹³-10¹⁴ Hz | Folding cycles k |
| Enzyme | E | 10⁶-10¹² Hz | Turnover k_cat |
| Channel | C | 10³-10⁶ Hz | Open probability P_o |
| Membrane | M | 10²-10³ Hz | Amplitude ΔV |
| ATP | A | 0.1-1 Hz | Period T |
| Genetic | G | 10⁻³-10⁻¹ Hz | Burst rate λ |
| Calcium | Ca | 10⁻²-10⁰ Hz | Frequency f |
| Circadian | R | ~10⁻⁵ Hz | Period stability ΔT/T |
Disease states are characterized by an 8-component vector:
D = (D_P, D_E, D_C, D_M, D_A, D_G, D_Ca, D_R)
The dominant component determines disease class:
- D_P dominant → Protein misfolding diseases (Alzheimer's, Parkinson's, prion)
- D_E dominant → Metabolic diseases (diabetes, PKU)
- D_C dominant → Channelopathies (cystic fibrosis, Long QT)
- D_G dominant → Expression disorders (cancer)
The master diagnostic equation integrates all oscillator contributions:
η_cell = (1/W) Σᵢⱼ wᵢⱼ · ηᵢⱼ
where wᵢⱼ are entropic weights and W = Σwᵢⱼ is the normalization factor.
| Shell n | C(n) = 2n² | Electron Capacity | Agreement |
|---|---|---|---|
| 1 | 2 | 2 | 100% |
| 2 | 8 | 8 | 100% |
| 3 | 18 | 18 | 100% |
| 4 | 32 | 32 | 100% |
| 5 | 50 | 50 | 100% |
Measured correlation between categorical and spatial distance: r ≈ 0.12 (consistent with theoretical independence).
| Condition | Mean η_cell | Classification AUC |
|---|---|---|
| Healthy | 0.85 ± 0.03 | — |
| Stressed | 0.60 ± 0.05 | 0.78 |
| Diseased | 0.35 ± 0.08 | 0.84 |
| Critical | 0.12 ± 0.05 | 0.92 |
The system exhibits bistable dynamics with:
- Healthy attractor: η ≈ 0.9
- Disease attractor: η ≈ 0.2
- Critical threshold: η_c ≈ 0.5
hegel/
├── wilhelm/
│ ├── publications/
│ │ ├── observation-equations/
│ │ │ ├── cellular-observation-equations.tex # Main theoretical paper
│ │ │ └── validation/ # Generated validation charts
│ │ └── ...
│ ├── src/
│ │ └── validation/
│ │ ├── cpl_validation_suite.py # 8-chart validation suite
│ │ ├── cpl_validation_panels_3_6.py # Extended validation panels
│ │ └── ...
│ └── docs/
│ └── lunar_surface_imaging/ # Foundational framework paper
└── ...
The validation suite generates 12 charts with corresponding JSON data:
Core Validation (Charts 1-8):
- S-entropy space partition trajectories (3D)
- Partition capacity sequence validation
- Oscillator frequency spectrum (8 classes)
- Universal coherence equation
- Disease signature vector classification
- Protein folding diagnostic readout
- Phase-lock bandwidth surface (3D)
- Cellular coherence ensemble statistics
Extended Validation (Panels 3-6):
- Panel 3: Categorical distance spatial independence
- Panel 4: Information catalysis dynamics
- Panel 5: Opacity-independent measurement
- Panel 6: Disease trajectory simulations
# Generate core validation charts
python wilhelm/src/validation/cpl_validation_suite.py
# Generate extended validation panels
python wilhelm/src/validation/cpl_validation_panels_3_6.pyOutput files are saved to wilhelm/publications/observation-equations/validation/ in PNG, PDF, and JSON formats.
-
Observational Partition Algebra — Main theoretical framework establishing measurement-process-computation identity in cellular systems.
-
Lunar Surface Imaging via Categorical Partition Dynamics — Foundational paper establishing categorical distance independence and opacity-independent measurement.
| Symbol | Definition |
|---|---|
| Γ | Partition state in S-entropy space |
| P(ω) | Partition operator at frequency ω |
| S = [0,1]³ | S-entropy space |
| (n, ℓ, m, s) | Partition coordinates |
| d_cat | Categorical distance |
| η | Coherence index |
| η_cell | Cellular coherence index |
| D | Disease signature vector |
| Φ(ω₁, ω₂) | Phase-lock operator |
- Python 3.8+
- NumPy
- Matplotlib
- SciPy (for advanced computations)
MIT License
If you use this framework in your research, please cite:
@article{observational_partition_algebra,
title={Observational Partition Algebra: Unification of Measurement,
Process, and Computation in Cellular Systems},
author={Sachikonye, Kundai Farai},
institution={Technical University of Munich, School of Life Sciences},
year={2025}
}The theoretical framework builds upon established results from:
- Poincaré recurrence theorem for bounded phase spaces
- Bohr-Sommerfeld quantization conditions
- Maslov index in semiclassical mechanics
- Pauli exclusion principle and aufbau
- Michaelis-Menten enzyme kinetics
- Mitchell chemiosmotic theory
- Circadian rhythm genetics (Konopka-Benzer)
See the main publication for complete bibliography (47 references).