Critical Spectral Mode Compression MVP

What was implemented

experiment.py constructs a D=256 diagonal complex spectrum with stable eigenvalues lambda_j=rho exp(i omega_j) and evaluates

R(k) = |sum_j q_j lambda_j^k|^2.

It compares frequency-quantile selection, random selection, and equal-magnitude tie-break selection at M in {8,16,32,64}. It also sweeps the spectral radius through and beyond the stability boundary.

Quantitative predictions and observations

Stability-rate prediction. For a single retained mode, R(k)=rho^(2k), so the log slope must be 2 log(rho). The boundary sweep observed:

| rho | predicted slope | observed slope | |---:|---:|---:| | 0.999 | -0.0020010007 | -0.0020010007 | | 1.000 | 0 | approximately 0 | | 1.001 | 0.0019990007 | 0.0019990007 | | 1.010 | 0.0199006617 | 0.0199006617 |

This confirms the stability boundary: rho<1 decays, rho=1 is non-decaying, and rho>1 grows.

Mode-count prediction. Increasing a frequency-quantile quadrature should reduce return-proxy discrepancy. Log-RMSE against the D=256 teacher was 5.37, 4.67, 4.04, 3.45, 2.74, and 1.80 for M={4,8,16,32,64,128}, respectively. Thus the error decreases with mode count, but does not become small at M/D <= 0.25.
Power-law preservation prediction. The teacher's fitted slope over k=20..180 was 1.236 (the fit is not a clean power law because the finite oscillatory quadrature has interference zeros). At M=64 (25% of the modes), frequency selection produced slope 0.765, a 38% relative exponent error; at M=32 it produced 0.196, an 84% error. The <=10% exponent-preservation criterion was not met. Frequency selection was not better than the controls in log-RMSE at these settings.

Verdict

The core scalar stability mechanism is numerically correct, including the exact transition at |lambda|=1. However, this toy experiment does not validate the central compression claim: sparse frequency selection did not preserve the teacher's fitted return exponent or accurately reproduce its return curve at M/D <= 0.25. Therefore the proposed mechanism is marked worked=false for this MVP.

Reproduction

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

This writes results.json and prints the same JSON to stdout.

Limitations

No learned SSM, Copy task, Adding task, fine-tuning, complex-conjugate real implementation, greedy pair-difference optimization, or GPU benchmark was tested. The chosen finite spectrum also does not produce a robust decade-long power law, so the failed exponent result is evidence against this setup, not a universal impossibility result.