Spectral-Ordering Block Optimizer MVP

Implementation

spectral_ordering_experiment.py implements block error maps T_b, sequential composition J_pi = T_last ... T_first, exhaustive permutation selection by spectral radius, a finite-trajectory verification, and a two-block nonlinear regression comparison.

Run with:

/home/maxwelhelp/main/bin/python3 spectral_ordering_experiment.py

Results are written to results.json.

Toy math verification

The fixed step size in the coupling sweep is gamma=0.42.

1. Decoupling prediction: at lambda=0, all 24 permutations have exactly the same radius: best=worst=0.580000 and gap=0. This is the predicted disappearance of an ordering effect when cross-block sensitivity is zero.
2. Coupling prediction: the order gap increases monotonically with lambda: 0.0000 at lambda=0, 0.01549 at 0.2, 0.04075 at 0.5, and 0.06899 at 0.8. Thus directed cross-block coupling creates a measurable ordering effect.
3. Stability prediction: the predicted rho=1 boundary versus the first observed growth point was: lambda=0.2, 1.410 vs 1.45; lambda=0.4, 1.089 vs 1.10; lambda=0.6, 0.887 vs 0.90; lambda=0.8, 0.748 vs 0.75. The coarse empirical grid agrees within 0.04 step size.
4. Asymptotic ratio: at lambda=0.5 and gamma=0.42, predicted rho=0.58098 and measured tail norm ratio=0.57788 (0.53% relative difference).

These checks support the claimed local spectral-radius mechanism in the explicitly constructed non-symmetric block model.

Neural mini-experiment

A deterministic 2-layer tanh regression model was trained for 180 full-batch steps on 256 points. The simultaneous update, fixed sequential order, and adaptive sequential order used the same initialization and learning rate. Final MSE was:

• simultaneous: 0.383931
• fixed sequential: 0.383590
• adaptive sequential: 0.382999

At step 30 the values were 0.456103, 0.456093, and 0.456006 respectively. The adaptive policy selected order (1,0) at all 180 recorded steps and had a small improvement over simultaneous updates (about 0.24% final MSE).

Honest assessment

The mechanism worked in the toy verification and produced a small favorable signal in the neural MVP. However, the neural adaptive selector is only a cheap one-step candidate-loss proxy; it does not yet estimate block Jacobians with perturbations, power iterations, or hysteresis, and the comparison is a tiny regression rather than a transformer/CIFAR/language-model task. Therefore this is evidence for the mechanism, not evidence of a robust practical speedup.