Budgeted Hebbian Kuramoto Dynamics for Max-Cut under Amplitude Heterogeneity: Robustness, Not Cut Quality, Is the Signal

We study budgeted Hebbian-coupled Kuramoto dynamics as a coupling-resource allocation mechanism for oscillator-based Ising machines applied to graph Max-Cut. The Hebbian rule learns a symmetric, non-negative coupling matrix on a fixed edge support, with per-node row-sum budgets enforced by exact symmetric-Frobenius projection.

The predeclared CORE test fails: random-budgeted coupling recovers more cut than Hebbian-Frobenius on every tested graph family at zero amplitude heterogeneity. The predeclared HARDWARE-ROBUSTNESS test holds: under lognormal amplitude heterogeneity (σ ∈ {0, 0.25, 0.5, 1.0}), hybrid-Frobenius degrades less than the registered random-budgeted comparator, with paired slope difference +12.3 cut/σ and bootstrap 95% CI [+8.5, +14.7], direction consistent across all three graph seeds. The oracle reference, which compensates for known amplitude variation, collapses fastest (−17.8 cut over the range), indicating that knowing the noise model without adapting is brittle.

The theoretical contribution is a Lyapunov-descent and KKT-stationarity theorem for the continuous-time joint phase/weight flow, with the algebraic core formalised in Lean 4 / Mathlib. Open proof issues are explicitly catalogued.

Code, raw CSVs, and Lean source: https://github.com/velvetmonkey/flywheel-universe