Эксперимент: Puiseux Arclength Continuation for Implicit Layers (#1350)

{ "worked": false, "confidence": 8, "verdict": "Built a scalar Puiseux/arclength continuation MVP for the singular implicit branch h(z,t)=z^2-t, including coefficient fitting, exponent estimation, pseudo-arclength Newton correction, and a linear-predictor baseline. The coefficient-system check passed with 6.7e-16 equation error and 7.2e-16 scaled-coefficient error. On 80 matched geometric steps from t=1 to 1e-8, Puiseux correction averaged 2.71 Newton iterations versus 3.26 for linear prediction, but both had zero failures; the idea was ~31x slower in this tiny implementation, prediction error was slightly worse, and exponent fitting degraded near machine precision, so there is no clear robust promised effect.", "metrics": { "baseline": "80 steps; 0 failed; mean Newton iterations 3.2625; mean prediction error 0.0038562; max residual 9.50e-12; runtime 0.00111 s", "idea": "80 steps; 0 failed; mean Newton iterations 2.7125; mean prediction error 0.0048366; max residual 6.44e-12; fitted alpha mean 1.482 (last clamped to 0.1); runtime 0.0354 s" }, "how_to_run": "/home/maxwelhelp/main/bin/python3 puiseux_experiment.py", "files": [ "puiseux_experiment.py", "results.json" ], "limitations": "Only a scalar analytic branch was tested, not a neural equilibrium layer, multidimensional rank-deficient Jacobian, adaptive step rejection, GPU execution, training accuracy, or wall-clock performance at realistic scale. The endpoint exponent estimate is contaminated by floating-point/finite-difference effects, and the benchmark uses a fixed schedule rather than a full adaptive continuation policy." }