Эксперимент: PIPO-PITO bounded recurrent gain (#622)

{ "worked": true, "confidence": 8, "verdict": "Built and numerically verified a scalar positive PIPO plant with the PITO controller w'=b-a y w. Measured exponential decay rates matched the predicted aV within 0.035%, V·epsilon exactly matched 2b/a across the sweep, and observed threshold-crossing times were below the conservative predicted bounds. Fixed gains above the plant boundary diverged while regulated feedback reduced the gain from 2.0 to 0.364 and kept the state bounded; this validates the mechanism in the toy setting, but not a full neural-network accuracy win.", "metrics": { "baseline": "Fixed-gain plant: gamma=0.2 and 0.39 remained bounded (final x=0.100 and 1.149); gamma=0.6 and 0.8 diverged to 1.83e6 and 1.22e13 by t=80. The marginal gamma=0.4 case grows linearly but did not exceed the finite-horizon divergence criterion.", "idea": "PITO constant-output sweep: maximum relative error in predicted decay rate was 0.000350 (0.035%); V*epsilon=1.076923 for every epsilon, exactly equal to 2b/a; maximum observed-minus-predicted crossing time was -0.00499. Closed-loop at nominal gamma=0.8 had max x=1.444, final x=0.550, and final w=0.364." }, "how_to_run": "/home/maxwelhelp/main/bin/python3 pipo_pito_experiment.py", "files": [ "pipo_pito_experiment.py", "results.json" ], "limitations": "Only a scalar continuous-time toy plant was tested; no 16-dimensional delayed-copy/adding task, GRU comparison, parameter-matched neural model, FLOP/loss benchmark, channelwise controller, or GPU experiment was run. PIPO assumptions were represented by the positive scalar plant rather than exhaustively stress-tested over arbitrary initial states and plant matrices." }