🗺️ Oakfield Operator Calculus
Roadmap
Tracking the evolution of the Oakfield engine and the growth of its operator ecosystem.
1. Overview
The Oakfield Operator Calculus (OOC) evolves in stages. Each stage expands the algebra of operators and deepens the link between symbolic, numeric, and perceptual computation. This page is aligned to the current oakcli schema snapshot: 69 implemented operators across 10 categories, plus 8 registered integrators.
2. Current Core (Schema Snapshot)
| Layer | Status | Description |
|---|---|---|
| Field Space + Runtime | ✅ Stable | Multidimensional field container, scheduler, profiler, and async config/logging runtime. |
| Schema-backed Operator Registry | ✅ Stable | oakcli schema publishes 69 operators with parameter, port, semantic, and Tool API metadata. |
| Integration Layer | ✅ Stable | Eight registered integrators: euler, heun, rk4, rkf45, backward_euler, crank_nicolson, etdrk4, and subordination. |
| Differential & Spectral Core | ✅ Stable | linear_dissipative, linear_spectral_fusion, dispersion, laplacian, spatial_derivative, gradient, divergence, and curl are implemented. |
| Nonlinear & Coupling Core | ✅ Stable | remainder, elementwise_math, complex_math, chaos_map, hysteretic, mask, mixer, metal_mix, and thermostat are live. |
| Measurement & Observation | ✅ Stable | phase_feature, minimal_convolution, sieve, and sound_observation cover field analysis plus audio/control observation. |
| Stimulus Library | ✅ Stable | 37 stimulus operators span sinusoidal, spectral, geometric, wavelet, beam, and fractal/noise families. |
| Neural Bridge | 🚧 Early | neural_infer and neural_hybrid are in the catalog; training, autograd, and authoring workflows are still ahead. |
These layers now form a broader schema-backed operator backbone, not just the early analytic core.
3. Operator Inventory & Status
✅ Catalog Summary
| Slice | Count | Representative operators | Status |
|---|---|---|---|
| Potential stimuli | 28 | Sine, chirp, spectral lines, checkerboard, Moire, Gabor, wavelets, optical beams, Chladni, Laplace-Beltrami. | ✅ Landed |
| Noise & stochastic | 11 | Stochastic Noise, Ornstein-Uhlenbeck, Random Fourier, Spectral Shells, White Noise, fBm, Worley, Turbulence, Multifractal, Ridged. | ✅ Landed |
| Nonlinear | 6 | Elementwise Math, Complex Math, Chaos Map, Hysteretic, Neural Infer, Neural Hybrid. | ✅ Core + expanding |
| Diffusion | 5 | Linear Dissipative, Linear Spectral Fusion, Dispersion, Fractional Memory, Laplacian. | ✅ Landed |
| Advection / differential | 5 | Analytic Warp, Spatial Derivative, Gradient, Divergence, Curl. | ✅ Landed |
| Utility / coordinate | 5 | Phase Rotate, Zero Field, Copy, Scale, Coordinate. | ✅ Landed |
| Measurement | 4 | Phase Feature, Sound Observation, Minimal Convolution, Sieve. | ✅ Landed |
| Coupling | 3 | Mask, Mixer, Metal Mix. | ✅ Landed |
| Reaction | 1 | Remainder. | ✅ Landed |
| Thermostat | 1 | Thermostat. | ✅ Landed |
✅ Recently Landed / Reclassified
| Item | Status | Notes |
|---|---|---|
| Operator Metadata Registry | ✅ Done | The catalog now exposes category, parameter, port, semantic, and schema-reference metadata for 69 operators. |
| Spectral band / shell excitation | ✅ Done | stimulus_spectral_lines, stimulus_random_fourier, and stimulus_spectral_shells cover band-limited excitation families. |
| Vector differential toolkit | ✅ Done | gradient, divergence, curl, and laplacian extend the older spatial_derivative-only story. |
| Coherent / fractal noise suite | ✅ Done | Worley, Turbulence, Multifractal, Hybrid fBm, and Ridged Noise are implemented and in schema. |
| Neural operator foundation | 🚧 Early implementation | neural_infer and neural_hybrid are live, but training, export, and differentiable workflows remain open. |
| Measurement metrics suite | 🚧 In progress | Observation operators are in place; richer summaries, dashboards, and visual reporting are still being built. |
💡 Future / Research Track
| Track | Status | Notes |
|---|---|---|
| Autograd / differentiable runtime | 🧪 Research | Sensitivity analysis and end-to-end gradient flow through operator graphs. |
| Training pipeline for learned operators | 🔜 Planned | Build authoring, export, and optimization workflows on top of neural_infer and neural_hybrid. |
| BSDE / deep solver layer | 🧪 Research | Backward SDE and PDE surrogate methods once the learning stack matures. |
| Symbolic bridge | 🔜 Planned | Symbolic ↔ numeric duality, introspection, and algebra-aware tooling. |
| Incompressibility / hard constraints | 🔜 Planned | Projection operators and constraint manifolds for divergence-free or geometry-restricted flows. |
| Quantum / superoperator extension | 🧪 Research | Lift the operator algebra toward unified classical-quantum composition. |
4. Mid-Term Goals (Next 6-12 Months)
Stochastic Dynamics Enhancement
- Higher-order stochastic integrators: Milstein and SDE Runge-Kutta variants beyond the current 8-integrator registry.
- Noise families: The catalog now includes spectral shells, Worley, turbulence, multifractal, hybrid fBm, and ridged noise; the next step is deeper control and compositional presets.
- Noise distribution: Extend beyond current Gaussian/OU-style laws toward Levy, Poisson-jump, and other stable-process families.
Measurement & Analysis Stack
- Remainder metric suite: Energy norms, curvature maps, and complexity spectra are partially in place; richer reporting and visual summaries remain the next step.
- Observation bridge: sound_observation is implemented; continue building presets, routing, and UI around it.
- Sieve library expansion: Add wavelet sieves, spatial windowing, and feature-domain filters on top of the current Gaussian pass filters.
- Phase-space analysis: Embedding dimension, Lyapunov exponents, and attractor detection remain open analysis work.
Learning & Adaptation
- Neural operator foundation: neural_infer and neural_hybrid are the first runtime bridge; next comes authoring and model-management workflows.
- Parameter optimization: Move from fixed inference contracts toward trainable residuals and coefficient-tuning loops.
- Online learning: Adaptive operator weights based on measurement feedback remain future-facing.
Constraint & Boundary Handling
- Incompressibility projection: Enforce ∇·u = 0 for fluid-like dynamics.
- Reflecting boundaries: Unify Neumann/Dirichlet semantics across analytic, differential, and GPU paths.
- Constraint manifolds: Restrict operators to geometric submanifolds and hard state constraints.
Interaction & UX
- Sound exploration module: Build higher-level workflows around the implemented sound_observation operator.
- Operator graph visualizer: Schema-driven editor/viewer for composing and inspecting operator graphs.
- Exploratory Sensory Modes: Reduced-interface modes that hide mathematical complexity and foreground sensation:
- Visual Play Mode: Strip away equations, parameters, and technical readouts; leave only the visualizer and a minimal set of knobs for direct manipulation. A fidget-spinner experience where you shape patterns through feel rather than formulas. Can serve as a meditation mode - watch the math breathe, let patterns settle your mind, or use gentle parameter shifts as a focusing practice.
- Audio Play Mode: Similarly minimal but sonified - just knobs and sound. Explore dynamics through timbre, rhythm, and harmonic structure without visual distraction.
- Instant Math Recall: At any moment during exploratory play, bring back the full mathematical view with a single action. If you stumble onto something interesting by ear or eye, immediately surface the underlying operators, parameters, and equations that produced it. Discovery through sensation, understanding on demand.
5. Long-Term Vision (Extended Algebra)
| Theme | Direction | Example Concepts |
|---|---|---|
| Quantum Layer | Lift classical fields to operator algebras | Superoperators, commutators, mode decomposition |
| Ensemble Layer | Path integral / Monte Carlo sampling | Statistical field ensembles, e−S[u]/ℏ weighting |
| Geometric Layer | Covariant operators on manifolds | Metric coupling, curvature tensors, warped Laplacians |
| Adaptive Learning Layer | Data-driven operator synthesis building on the live neural bridge | Neural inference/hybrid operators today; tuning and online learning next |
| Hybrid Visualization | Real-time field rendering and observation | Spectral surfaces, analytic signal displays, sonification, and audio-control mapping |
Each expansion retains the same compositional semantics; Oakfield's algebra just grows richer.
6. Developer Milestones
| Milestone | Target | Status | Outcome |
|---|---|---|---|
| M0: Schema-backed catalog | 69 operators / 10 categories / ports + parameters | ✅ Done | oakcli schema now exposes the operator catalog, integrator registry, and Tool API schema references. |
| M1: Analytic + measurement loop | Analytic Warp + Remainder + Sieve + Phase/Convolution | ✅ Done | The core analytic loop is implemented and no longer limited to the original remainder/sieve story. |
| M1.5: Stochastic + spectral core | Noise laws, OU dynamics, spectral diffusion, subordination | ✅ Done | Stochastic Noise, Ornstein-Uhlenbeck, Linear Spectral Fusion, and the subordination integrator are registered. |
| M2: Stimulus atlas v1 | 37 stimulus operators across potential + noise families | ✅ Done | Sinusoidal, spectral, geometric, wavelet, beam, and fractal/noise drivers are now part of the catalog. |
| M2.5: Differential toolkit | spatial_derivative + gradient/divergence/curl + laplacian | ✅ Done | Vector-calculus-style operators and 2D stencil paths have landed. |
| M3: Patch / visualization surface | Schema-driven registry, views, and composition UI | 🚧 Partial | Registry metadata and display views exist; graph editing and workflow polish remain active work. |
| M3.5: Neural bridge v0 | neural_infer + neural_hybrid | 🚧 Early | Inference and analytic+residual hybrid operators are live, but training and autograd are not yet first-class. |
| M4: Symbolic bridge | Symbolic ↔ numeric dual layer | 🔜 Future | Realtime differentiation and algebraic introspection remain future-facing. |
| M5: Quantum extension | Superoperator algebra | 🔜 Future | Unified classical-quantum operator composition is still a research horizon. |
7. Currently Under Development
- Schema-driven patch / graph UX: connect the operator catalog to editor workflows, presets, and docs surfaces.
- Higher-order stochastic integration: add Milstein and SDE Runge-Kutta variants beyond the current 8-integrator registry.
- Neural workflow: build model loading, export, training, and benchmarking loops around neural_infer and neural_hybrid.
- Measurement/reporting: deepen remainder, complexity, and audio-observation dashboards on top of the live operators.
- Boundary semantics polish: unify boundary policies across analytic warp, differential operators, and GPU-oriented execution paths.
- Visualization performance: keep reducing CPU draw overhead and improve cached rendering for waveform/phase-style views.
Guiding Principle
"Oakfield grows by composition."
Each new operator is a morphism that extends the field of possibilities — not a rewrite. The calculus remains stable, but the world it describes keeps expanding.
Ready to join the journey?