# Эксперимент: Resistance-certified tree attention (#1010)

{
  "worked": true,
  "confidence": 8,
  "verdict": "Built an exact small-graph MVP for resistance marginals, Theta projection, weighted spanning-tree sampling, and sparse aggregation in verify_and_experiment.py. Complete-graph and cycle resistance predictions matched numerical results to near machine precision, and the star feasibility boundary appeared exactly at degree 3. Projection enforced the degree cap and tree-mass constraint; sampled trees reduced edges 3x but incurred nonzero aggregation error, so this is a sparsity tradeoff rather than a demonstrated accuracy win.",
  "metrics": {
    "baseline": "Dense resistance-weighted aggregation on K6: 15 undirected edges.",
    "idea": "Sampled spanning-tree aggregation on K6: 5 undirected edges, a 3.0x reduction; tree-vs-dense MSE 0.3837 +/- 0.0050 over 500 samples. Empirical weighted-tree marginal RMSE was 0.00734 over 3000 samples.",
    "predictions": "For K_n, observed mean degrees exactly matched 2-2/n for n=4,...,10, with edge marginal error below 2.3e-16. On cycles, the resistance formula error was at most 3.2e-15 over conductance ratios 0.01-100, and maximum degree stayed below 1.982. Stars were feasible at n=3 and infeasible for n=4,...,8. Projection changed maximum degree from 2.8656 to 2.0000 and produced total mass 5.0000."
  },
  "how_to_run": "/home/maxwelhelp/main/bin/python3 verify_and_experiment.py",
  "files": [
    "verify_and_experiment.py",
    "results.json"
  ],
  "limitations": "No Cora, PubMed, or learned GAT/node-classification experiment was run. The sampler enumerates Prüfer trees and is practical only for tiny complete graphs; scalable sparse Laplacian solves, Hutchinson estimates, dependent rounding, GPU execution, timing benchmarks, and seed-to-seed accuracy comparisons were not tested. The MVP projection includes all spanning-tree cuts and vertex degree caps but does not encode the full odd-set inequalities of the matching polytope."
}