# Эксперимент: Symmetry-Block Neural PDE Solver (#541)

{
  "worked": true,
  "confidence": 8,
  "verdict": "Built a symmetry-adapted implicit diffusion solver for a cyclic mesh using character projectors and the Fourier representation basis. All three quantitative mechanism predictions were confirmed across n={16,32,64,128,256}, theta={0,0.1,1,10}, and tau={0.01,0.1,1}: commutator error was 0, transformed off-block error was 4.03e-14, and relative solution disagreement was 3.67e-14, with residual below 4.39e-14. The algebraic mechanism is real, but this toy does not establish a consistent runtime win: median transformed-solve time was 0.223 ms versus 0.697 ms for sparse solving, while timings varied by size because dense transforms and small-problem overhead dominate.",
  "metrics": {
    "baseline": "SciPy sparse direct solve: median 0.697 ms; maximum relative residual 4.39e-14.",
    "idea": "Fourier symmetry-basis scalar block solve: median 0.223 ms; maximum relative solution error 3.67e-14; maximum transformed off-block ratio 4.03e-14; maximum commutator ratio 0.0; projector resolution, idempotence, and orthogonality errors 1.08e-14, 6.80e-15, and 1.90e-15."
  },
  "how_to_run": "/home/maxwelhelp/main/bin/python3 symmetry_block_solver.py",
  "files": [
    "symmetry_block_solver.py",
    "results.json"
  ],
  "limitations": "This is a 1D cyclic graph toy, not the proposed 2D/3D symmetric simplicial torus or a neural PDE model. It uses complex one-dimensional irreducible blocks and dense Fourier transforms, and does not measure GPU execution, peak memory, batched block parallelism, training loss, FLOPs, or scaling to 10k vertices. Timing is a small single-right-hand-side comparison and is not a robust production speedup estimate."
}