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Physical sensor thresholds
The live explorer and the archived talk calculation are separate numerical experiments. The archive is an independent check; it is not the live plot’s data source. Their target units, training-error conventions and interval summaries must be read accordingly.
The live model
Write γ = 1 + p/σ² for sensor quality. In eigencoordinates, features are x = (√γ s, z), with independent standard Gaussian signal s and nuisance coordinates z. The iid clean-score model predicts s. The physical sensor model predicts √(1 − 1/γ)s + e/√γ, where e is independent standard Gaussian noise. Its irreducible population error is 1/γ.
This corresponds to p = d sensors (the total feature dimension) and noise variance σ² = p/(γ − 1); γ = 1 is the no-signal limit. Matching the covariance with the clean-score model does not make their targets identical.
The “Training error / predicting zero” readouts and the archived training statistic both use ratios to predicting zero on the same dataset. Raw training mean squared error is used for the custom cap, the Train = test constraint, and the explicitly labeled numerical audits. Population test error uses predicting zero = 1. Best in span is an analysis benchmark that knows the population model, not a learner supplied with extra labels.
One dataset and summary curves
In One draw, Gaussian Schur-complement updates supply the sufficient statistics for prefixes of a growing dataset. Moving the sample count keeps that dataset; Fresh draw changes its seed. The clean and physical models share features and clean scores, while physical labels add an independent noise stream.
Best-in-span sensor medians and 10th–90th percentile bands are exact affine transforms of the clean-score beta law. Physical interpolation and training summaries use 2,048 deterministic Monte Carlo samples from the exact two-dimensional law. These bands contain the middle 80%; they are quantiles, not estimates of a potentially divergent mean.
For the current numerical method, crossing qualifications, sampler identity and selected values, open Model, formulas, and sampling calculation beside the live figure. The implementation is shared with Core in sample-threshold-models.js and sample-threshold-view.js.
The archived independent check
The talk archive fixed 50,000 sensors, noise variance 500, and independent Gaussian physical labels. Its training statistic divided each draw’s squared residual by that draw’s label energy. The generator used NumPy PCG64, seed 20261006, and 40,000 independent calibration sets at each of 193 saved sample counts. Archived bands contain the middle 50%, unlike the live explorer’s middle 80%.
The archived crossings of population error 0.1 are approximately 49.5225 examples for best in span and 1141.3399 for interpolation. They come from log-log interpolation between adjacent saved medians, not an exhaustive certificate of the first integer crossing. The corresponding exact stylized thresholds are 45 and 1,088.
The complete archived source and reproduction note (Markdown) records the archive checksum, generator commands, sampling law, and earlier shared-widget implementation notes. Those historical defaults and archived artifacts do not define the current interface; the live figure and its numerical-method drawer do.