πΊοΈ 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 roadmap records the current state of the implementation and outlines the next milestones.
2. Current Core (Implemented)
| Layer | Status | Description |
|---|---|---|
| Field Space | β Stable | Defines multidimensional field container (state manifold) |
| Operator IR | β Stable | Internal representation for analytic, stochastic, and differential kernels |
| Operator Base & Split | β Stable | Base class for composable operators; implements Strang/Lie splitting |
| Integrators | β Stable | Deterministic and stochastic time-stepping functors (RKF45, EM, etc.) |
| Scheduler | β Stable | Multithreaded runtime orchestrator for operator graphs |
| Profiler | β Stable | Records execution time and performance metrics per operator |
| Logger / Config | β Stable | Asynchronous logging and runtime configuration loader |
| Stimulus & Feature Operators | β Stable | Base signal sources and phase coordinate extractors |
| Analytic Warp | β Stable | Core analytic transformations (Digamma, Bessel, Gaussian, etc.) |
| Measurement Operators | β Stable | Measurement loop with C + Lua regression suites |
These form the core operator backbone of the Oakfield runtime.
3. Operator Inventory & Status
β Fully Implemented Operators
| Operator | Category | Notes |
|---|---|---|
| Linear Dissipative | Diffusion | Fractional Laplacian with FFT backend; supports Ξ±-parameterized viscosity |
| Analytic Warp | Advection | q-Hyperexponential, Trigamma, Tetragamma, Power, Tanh and more |
| Remainder (Nonlinear) | Reaction | Constructs R = f(u) - L[u]; supports cubic, quadratic, identity modes |
| Mixer (Coupling) | Coupling | Linear/bilinear field coupling; can construct ΞΈ-methods |
| Sieve (Measurement) | Measurement | Low-pass, high-pass, band-pass spectral filtering with stochastic taps |
| Phase Feature | Measurement | Amplitude/phase extraction from complex fields |
| Fractional Memory | Diffusion | History-based fractional derivatives DΞ±t |
| Stochastic Noise | Diffusion | ItΓ΄/Stratonovich noise; fractional OU process with Ξ±-parameterized decay |
| Thermostat | Thermostat | Energy regulation (soft-lambda, additive, multiplicative modes) |
| Stimulus Sinusoidal (Advanced) | Potential | Sine, Traveling, standing, chirp, Gaussian-envelope waves |
| Stimulus Gaussian | Potential | Gaussian spatial envelopes with temporal modulation |
| Stimulus Spectral Lines | Potential | Discrete line spectrum excitation |
| Stimulus Random Fourier | Potential | Randomized Fourier mode stimulus |
| Stimulus Checkerboard | Potential | Structured checkerboard forcing |
| Stimulus fBm | Potential | Fractal Brownian motion drive for multiscale patterns |
| Stimulus Gabor | Potential | Gabor/edge-like localized stimulus |
| Stimulus White Noise | Potential | Spatial white-noise source |
| Spatial Derivative | Differential | Gradient/Laplacian-style derivative stencil operators |
| Minimal Convolution | Convolution | Lightweight convolution/min-filter operator |
| Warp Safety | Safety | Guards analytic warp domains and branch cuts |
π High-Priority Additions
| Operator | Priority | Notes |
|---|---|---|
| Operator Metadata Registry | β Done | Category, precision, stochastic flags, preferred_dt |
| SDE Milstein Integrator | π₯ High | Second-order stochastic integrator (extends Euler-Maruyama) |
| Spectral Band Stimulus | π₯ High | Bandlimited noise excitation for specific frequency ranges |
| Gradient Noise Stimulus | π Medium | Perlin/Simplex-style coherent noise for spatial patterns |
| Turbulence Stimulus | π Medium | Multi-octave fractal noise (fBm composition) |
| Measurement Metrics Suite | π§ Mostly implemented | Core remainder/complexity metrics landed; extending summaries/visuals |
π‘ Future / Research Track
| Operator / System | Category | Status | Notes |
|---|---|---|---|
| BSDE Solver | PDE/Learning | π§ͺ Research | Backward SDE via deep learning (requires neural operator infrastructure) |
| Autograd / Differentiable | Learning | π§ͺ Research | Automatic differentiation for sensitivity analysis |
| Neural Operator (Learned) | Learning | π§ͺ Research | Parametric operator OΞΈ learnable from data (PyTorch/ONNX integration) |
| Worley/Voronoi Noise | Stimulus/Geometry | π Backlog | Cellular noise patterns (lower priority vs core dynamics) |
| Schwarzschild Geodesic | Geometry/Geodesic | π Backlog | GR-inspired geometric operators (niche, far future) |
| Incompressibility Enforcer | Constraint | π Planned | Projection operator for divergence-free fields |
4. Mid-Term Goals (Next 6-12 Months)
Stochastic Dynamics Enhancement
- SDE Integrators: Milstein (2nd-order stochastic), Runge-Kutta SDE variants
- Noise Operators: Spectral band noise, gradient noise (Perlin/Simplex), turbulence composition
- Noise Distribution: Extend beyond Gaussian (LΓ©vy, Poisson jumps, stable distributions)
Measurement & Analysis Stack
- Remainder Metric Suite: Energy norms, curvature maps, complexity spectra atop remainder output (core features implemented; expanding visual/reporting hooks)
- Sieve Library Expansion: Wavelet sieves, spatial windowing, feature-domain filters
- Phase-Space Analysis: Embedding dimension, Lyapunov exponents, strange attractor detection
- q-Hyperexponential: Implemented in analytic warp; further optimizations/parameter sweeps optional
Learning & Adaptation
- Neural Operator Foundation: PyTorch/ONNX integration for learned operators OΞΈ
- Parameter Optimization: Gradient-based tuning of operator coefficients
- Online Learning: Adaptive operator weights based on measurement feedback
Constraint & Boundary Handling
- Incompressibility Projection: Enforce βΒ·u = 0 for fluid-like dynamics
- Reflecting Boundaries: Proper handling of Neumann/Dirichlet conditions in operators
- Constraint Manifolds: Operators restricted to geometric submanifolds
Interaction & UX
- Sound Exploration Module: Fine-tune models by ear; map operator parameters to audio feedback
- Operator Graph Visualizer: Node-based 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, eβS[u]/β weighting |
| Geometric Layer | Covariant operators on manifolds | Metric coupling, curvature tensors, warped Laplacians |
| Adaptive Learning Layer | Data-driven operator synthesis | Neural parameter tuning, online learning of dynamics |
| Hybrid Visualization | Real-time field rendering | Spectral surfaces, analytic signal displays, sonification |
Each expansion retains the same compositional semantics; Oakfield's algebra just grows richer.
6. Developer Milestones
| Milestone | Target | Status | Outcome |
|---|---|---|---|
| M0: Operator Metadata | Category/Precision/Stochastic | β Done | All 12 operators tagged with metadata for scheduling hints |
| M1: Analytic Loop | Remainder + Sieve + Mixer | β Done | C and Lua regression tests guard the measurement loop |
| M1.5: Stochastic Calculus | ItΓ΄/Stratonovich noise laws | β Done | Stochastic operator supports both interpretations |
| M2: Interactive Patch System | Operator registry + visual UI | π§ Partial | Registry + operator metadata wired; UI patching in progress |
| M2.5: Display Views v1 | Waveform/Phase/Polar (CPU-first) | β Done | Waveform/phase/polar views landed; baseline image tests in place |
| M3: Symbolic Bridge | Symbolic β numeric dual layer | π Future | Realtime differentiation and algebraic introspection |
| M4: Learning Integration | Neural operators | π Future | Adaptive / learned operator layers |
| M5: Quantum Extension | Superoperator algebra | π Future | Unified classicalβquantum operator framework |
7. Currently Under Development
- 2D field upgrade (Phase 2): Generalizing stimuli (sinusoidal variants, spectral lines, Gabor, random Fourier), spatial_derivative, and minimal_convolution to stride/rank-aware paths with per-axis spacing and boundary options; adding 2D Laplacian/convolution variants plus schema/Lua params.
- Boundary semantics polish: Threading unified boundary policies through analytic warp sampling and GPU codegen; keeping CPU fallback until Metal/CUDA handle N-D boundary-aware indexing.
- Remainder/complexity metrics: Core energy/curvature/complexity metrics in place; expanding visual/reporting hooks.
- Visualization performance: Waveform/phase-portrait CPU paths reducing heap churn and batching draws; refining cached text/envelope handling and buffer caps for smoother UI.
Guiding Principle
"Oakfield grows by composition."
Each new operator is a morphism that extends the field of possibilities β not a rewrite. The calculus remains stable, but the world it describes keeps expanding.
Ready to join the journey?