Hybrid Adaptive Calibration under Bounded Oscillator Coupling Budgets

The algebraic core of the descent argument has been machine-verified in Lean 4 / Mathlib (lean/RequestProject/LyapunovDescent.lean). The formalization establishes convexity and closedness of the constraint set, the key symmetry-based phase contribution identity, and the descent conclusion under explicit hypotheses on the dL/dt decomposition and the projection's variational inequality. Closing the remaining gaps (weight dynamics, chain-rule derivation, projection existence, full KKT stationarity) is the next formalisation step — see open issues.

The formal foundations have been extended and published as a separate sorry-free library: kuramoto-lean (Ben Cassie, 2026), available at https://github.com/velvetmonkey/kuramoto-lean, with a companion paper at https://doi.org/10.5281/zenodo.20468619. That library proves the unprojected algebraic joint descent core in hebbian_joint_lyapunov_descent with zero sorry, zero admit, and no new axioms.

1. What this is

This repository studies budgeted Hebbian Kuramoto dynamics with a fixed sparsity support and symmetric-Frobenius projection. The model maintains a symmetric coupling matrix on a fixed edge mask, with per-node row-sum budgets enforced by exact projection at every step of an alternating phase / weight update. The contribution is a constrained descent flow on the joint state (θ, W) with an identified Lyapunov function, a hybrid freeze schedule that splits adaptive weight learning from fixed-weight phase settling, and a hardware-constraint framing in which the per-node budget represents the physical coupling-resource limit of an oscillator-based Ising machine. This is not a state-of-the-art Max-Cut solver, not a device-native learning law, and not a validated hardware primitive.

Paper

Budgeted Hebbian Kuramoto Dynamics for Max-Cut under Amplitude Heterogeneity: Robustness, Not Cut Quality, Is the Signal Ben Cassie (2026). Zenodo. https://zenodo.org/records/20303914

2. The theorem (informal)

Under zero detuning (ω = 0), a fixed support mask, symmetric weights, and exact symmetric-Frobenius projection onto the budget polytope at each step, the joint dynamics (θ̇, Ẇ) descend a constrained surrogate energy. Limit points satisfy KKT stationarity for that energy under the row-budget, non-negativity, and support constraints.

3. What is tested

• Graph families. Sparse Erdős–Rényi (n=200, p=0.05), dense Erdős–Rényi (n=200, p=0.15), random 10-regular (n=200), small instances with proven optima via Gray-code enumeration (n ∈ [20, 30]), and GSet G1 (n=800) as a literature calibration anchor.
• Methods compared. Static Kuramoto on the original adjacency; static plus randomised-hyperplane rounding; greedy single-flip local search; static-projected, static-budgeted (top-k symmetric support), random-budgeted (random non-negative weights then projected), learned-support-random-weights (support inherited from a converged Hebbian run, random weights on top); topology-scrambled control (double-edge-swap mask); Hebbian and hybrid variants under both Sinkhorn-style and symmetric-Frobenius projection; SDP relaxation with Goemans–Williamson rounding.
• Budget levels. Row-sum budget set to mean_degree and to 0.5 · mean_degree.
• Amplitude heterogeneity sweep. Lognormal node gains a_i ∼ LogNormal(−σ²/2, σ) with σ ∈ {0.0, 0.25, 0.5, 1.0}, effective coupling K_eff[i,j] = a_i a_j W_ij. This targets the amplitude-heterogeneity regime identified in Khan et al. (arXiv:2510.24416), where quasi-steady amplitude variations in parametric-oscillator Ising machines rescale the effective spin couplings and degrade solution quality. Methods compared: static_budgeted, hebbian_frobenius, hybrid_frobenius, and an oracle-compensation reference.

All experiments use 10 method seeds per cell. The benchmark tests whether the stationary coupling configuration produced by budgeted Hebbian adaptation is useful for graph optimisation under hardware-imposed coupling-resource constraints — specifically, whether the adaptive weight allocation provides signal beyond what the support sparsity and budget alone already provide.

4. What the results show

Results from the full benchmark suite will be added here.

5. What failed or is not claimed

• The Frobenius projection is implemented as a digital simulation proxy, not a hardware primitive. The dual root-finder satisfies KKT to machine epsilon on validation, but no physical realisation is claimed.
• The theorem requires zero detuning and a fixed mask. Detuning and adaptive topology are studied as ablations but lie outside the theorem.
• Classical Max-Cut solvers are not the target of this work. They appear as calibration baselines to anchor the cut-quality axis, not as competitive comparators.
• Biological plausibility of the quadratic weight decay term in the Lyapunov function is not claimed. The λ‖W‖²_F / 4 regulariser is a control-theoretic ingredient, not a model of synaptic plasticity.
• This is a control / calibration algorithm. It is not yet a device-native learning law.

6. How to reproduce

git clone https://github.com/velvetmonkey/flywheel-universe
cd flywheel-universe
pip install -r requirements.txt
python validate_projection.py
python phase2_benchmark.py pilot

7. Citation / preprint

Theory note and benchmark paper in preparation. Zenodo DOI will be added here.

Interactive Demo

Live: https://velvetmonkey.github.io/flywheel-universe/ — no install, no download.

A network of Kuramoto oscillators synchronises via Hebbian learning. Five topologies (fully connected, ring, sparse random, hub-spoke, island chain), speed and node-count controls, senescence and learning toggles, and a perturbation slider to find the basin boundary.

Source: demos/hebbian-kuramoto.html (also runs standalone in any browser).

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Earlier framing of this repository — the universe / boundary-rider / cosmology-web analogies, the "six primitives" rhetoric, and the pre-Phase-2 Max-Cut numbers — is preserved in EXPLORATORY_NOTES.md and is not part of the main technical claim.

Related repositories

Part of the Flywheel suite — local-first knowledge infrastructure over a plain-markdown Obsidian vault:

• vault-core — Shared infrastructure for the Flywheel ecosystem.
• flywheel-memory — Persistent knowledge-graph memory MCP server: semantic search, read, and write over your vault.
• flywheel-crank — Desktop window into your vault's Flywheel MCP server.
• flywheel-gravity — A compressed, reality-filtered context field over a vault.
• flywheel-ideas — Local-first decision ledger: falsifiable bets, accepted outcomes, reusable lessons.
• mega-monkey — Telegram-native AI research cockpit over an Obsidian vault.
• roundtable — Local MCP server for delegating tasks to multiple AI models.

Research and experiments:

• flywheel-concept — A falsifiable study of cross-model concept geometry.
• flywheel-geometry — A pre-registered study of cross-domain knowledge retrieval.
• flywheel-universe (this repo) — Lean 4 / Mathlib-verified core of the descent argument.
• flywheel-velvetgram — Local widescreen Telegram reader for long-form reading.

Verified-cognition demo: mcp-seal (verified MCP approval gate) and canary (the seal demo host).