Icosahedral Congruence-Robust Strain Sensor

Implementation

experiment.py implements:

• six normalized icosahedral axis representatives;
• Frobenius-isometric symmetric vectorization (S11,S22,S33,sqrt(2)S12,sqrt(2)S13,sqrt(2)S23);
• transformed projectors P_i(F)=(Fv_i)(Fv_i)^T;
• ridge least-squares reconstruction and optional trace-free projection;
• comparisons against coordinate directions and a fixed random six-direction frame.

Run with:

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

The run writes results.json and prints the same results to stdout.

Mechanism predictions and observations

All results use fixed seed 1468, random trace-free tensors, and determinant-one deformations generated as F=expm(H) for trace-free symmetric H.

1. Identity spanning prediction: the six icosahedral projectors should have rank 6, while coordinate projectors should have rank 3. Observed ranks were 6 and 3.
2. Congruence rank prediction: because congruence by invertible F is an invertible map on Sym(3), the icosahedral frame should remain rank 6 and the coordinate frame should remain rank 3 for every deformation. Over anisotropy strengths 0, 0.5, 1, 1.5, 2, 2.5, 3, observed icosahedral ranks were [6,6,6,6,6,6,6] and coordinate ranks were [3,3,3,3,3,3,3].
3. Conditioning/noise prediction: reconstruction error should increase as the smallest singular value decreases. With measurement noise standard deviation 1e-3, 100 trials per strength, the icosahedral relative error increased from 0.00127 at strength 0 to 0.01612 at strength 3 (12.68x), while inverse smallest singular value increased from 1.118 to 22.456 (20.09x).

Noiseless maximum relative reconstruction error was 1.78e-9 for the icosahedral frame and 8.47e-7 for the random six-direction frame (ridge disabled for this check).

Baseline comparison

The random six-direction frame was a full-rank six-channel comparator. At anisotropy 0, mean noisy relative errors were 0.00127 (icosahedral) versus 0.01004 (random); at anisotropy 3 they were 0.01612 versus 0.16886. The coordinate-direction baseline is structurally non-identifiable for general symmetric tensors and had errors around 0.60--0.75 because its off-diagonal components cannot be reconstructed.

Thus the proposed frame shows a clear numerical signal: full identifiability, rank preservation, and substantially better noise robustness than this fixed random-frame baseline in this toy sweep. This validates the mathematical mechanism, not the claimed end-to-end graph-fluid improvement.

Limitations

The planned 3D graph neural fluid predictor, equal-parameter rollout comparison, learned directions, and training/rollout stability were not implemented. Conditioning under arbitrary nonsymmetric SL(3) matrices was not swept separately (the tested deformations were symmetric volume-preserving exponentials), although the rank argument applies to every invertible F. The coordinate baseline has only three unique channels, intentionally exposing its identifiability failure rather than matching six channel count with duplicated axes.