When Does a Frozen One-Byte Token Suffice? A Cross-Domain Separability Law and a Foundation-Model Ceiling

When Does a Frozen One-Byte Token Suffice? A Cross-Domain Separability Law and a Foundation-Model Ceiling Randolph James Ferlic, M.D. and Kimberly Kate Ferlic — Fieldstone Analytics, LLC, Austin, TX, USA Preprint · Zenodo DOI: 10.5281/zenodo.23288053 · CC-BY 4.0 · Community: spiral-domain-encoder-campaign · a characterization of previously filed and published methods; no new algorithmic subject matter is disclosed. Abstract A frozen, deterministic, class-discriminant single-token encoder maps a window of sensor features, through a fixed supervised projection and a small quantized codebook, to one 8-bit symbol and a bounded lookup-table decision. A companion program established where this device wins and pays as four falsifiable laws; the first of those — a competence zone in which the token is at near-parity with a strong model when the decision is separable in cheap features, and pays a real tax otherwise — is the one a sophisticated evaluator will press hardest, with two questions. First: if the token merely trails a bigger model, why not deploy the bigger model — in particular a pretrained time-series foundation model? Second: can you tell, before a pilot, whether the token will land at parity or pay a tax? We answer both, pre-registered, across a twenty-dataset battery spanning bearing vibration, robot force/torque, inertial human activity, gesture, cardiac electrocardiogram, wearable electromyography, and eight further multivariate archives. The foundation-model ceiling. Across twelve datasets, two pretrained time-series foundation models (MOMENT-1-large, 341M parameters; Chronos-T5-large, ~710M) — evaluated by frozen linear probe, by a nonlinear multilayer-perceptron probe, and by full end-to-end fine-tuning — never beat a classical RandomForest ceiling (mean foundation-minus-classical −0.08 and −0.07; dataset-bootstrap 95% confidence intervals entirely below zero; no dataset exceeded the classical ceiling by the pre-registered 0.03 margin). The result is robust to stronger ceilings: a MiniROCKET transform and a from-scratch one-dimensional convolutional network often beat the RandomForest, yet no foundation model beats the best of them either. On one modality (electromyography) both foundation models collapse to chance (~0.51) while the one-byte token holds 0.958. The separability law. A single cross-validated measure of class separability in the feature space — nearest-neighbour label purity, or a cross-validated linear-discriminant silhouette — predicts the token's accuracy tax across datasets (Spearman −0.76 and −0.92; permutation p < 0.01), generalizes out-of-sample to eight held-out archives (−0.76), and functions as an actionable pre-deployment screen (receiver-operating-characteristic area 0.83 for flagging a dataset that will pay a tax). Recovering the tax, and the real cost. A filed channel-partition token-ladder recovers ~79% of the single-token tax on multichannel data at a few bytes, with the residual still predicted by separability; the token is bit-exact deterministic (32- vs 64-bit agreement 1.000), abstains usefully (selective accuracy rises +0.027 at 70% coverage), is cheaply recalibratable (temperature scaling cuts its calibration error 45%, though it is less calibrated than a full model out of the box), is imbalance-robust in ranking (macro-AUC), and is 350–1,040× faster than a foundation model on a CPU (9–18× faster than the RandomForest). We also report two single-fold false positives — an electrocardiogram split on which a fine-tuned and a nonlinear-probed foundation model appeared to beat the classical ceiling — that both dissolved under cross-validation, a worked example of re-testing a surprising result on a harder protocol before accepting it. The honest synthesis is that accuracy is not the axis on which this device competes: no model, foundation or classical, buys an accuracy ceiling over a RandomForest on structured sensor data, so the decision is footprint, determinism, auditability, and measured speed — at a tax that is predictable before deployment. No new algorithmic subject matter is disclosed; every result is a property of a previously filed and published method. Highlights · A foundation-model ceiling — across twelve datasets and two large pretrained time-series foundation models (MOMENT-1-large, 341M; Chronos-T5-large, ~710M), evaluated by frozen linear probe, nonlinear multilayer-perceptron probe, and full end-to-end fine-tuning, no foundation model beats a classical RandomForest (mean margin −0.08 / −0.07; dataset-bootstrap 95% confidence intervals entirely below zero; no dataset beaten by more than 0.03). On electromyography both foundation models collapse to chance (~0.51) while the one-byte token holds 0.958. · Not a weak-ceiling artifact — stronger comparators (a MiniROCKET transform, a from-scratch one-dimensional convolutional network, gradient boosting) often beat the RandomForest, and still no foundation model beats the best of the four on any dataset; the token stays within 0.05 of that best ceiling on every genuinely separable dataset. · A separability law, usable before a pilot — a single cross-validated separability measure (k-nearest-neighbour label purity; a cross-validated linear-discriminant silhouette) predicts the token's accuracy tax across datasets (Spearman −0.76 / −0.92; permutation p < 0.01), generalizes out-of-sample to eight held-out archives (−0.76), and operates as a go/no-go screen (receiver-operating-characteristic area 0.83) a deployer can run on a labeled sample before committing to a pilot. · The tax is recoverable — the filed channel-partition token-ladder recovers ~79% of the single-token tax on multichannel data at a few bytes, and separability still predicts the residual (Spearman −0.68). · Deployment and trust — the token is bit-exact deterministic (32- vs 64-bit agreement 1.000), abstains usefully (selective accuracy +0.027 at 70% coverage), is cheaply recalibratable (temperature scaling cuts expected calibration error 45%, though it is less calibrated than a full model out of the box), is imbalance-robust in ranking under 50:1 injected imbalance, and is 350–1,040× faster than a foundation model on a CPU (9–18× faster than the RandomForest). · Honest integrity checks — two single-fold false positives, on an electrocardiogram split where a fine-tuned and a nonlinear-probed foundation model appeared to beat the classical ceiling, both dissolved under cross-validation and are reported as a worked example of re-testing a surprising result before accepting it. · The synthesis — accuracy is not the axis on which this device competes; no model a customer would reach for buys an accuracy ceiling over classical machine learning on structured sensor data, so the decision is footprint, determinism, auditability, and measured speed, at a tax that is predictable before deployment and recoverable at a few bytes. What this record contains · Manuscript_Paper57.pdf — the foundation-model-ceiling and separability-law characterization (6 figures, 43 references); and Manuscript_Paper57.docx, the editable source. · PAPER_57_ZENODO_ARCHIVE.zip — the reproducibility archive (md5 in ARCHIVE_MD5.txt): the four frozen pre-registrations (PAPER57_PREREG.md, PAPER57_TIER3_PREREG.md, PAPER57_TIER3B_PREREG.md, DERISK_PREREG.md); the foundation-model runners (MOMENT and Chronos; frozen linear probe, nonlinear multilayer-perceptron probe, and full fine-tuning); the stronger-ceilings runner (MiniROCKET, gradient boosting, one-dimensional convolutional network); the separability-predictor and external-validity runners; the composed channel-partition-ladder runner; the determinism, corruption, calibration/abstention, class-imbalance, and latency runners; the statistics and merge scripts; the shared dataset loaders and the frozen token encoder module; the result records (JSON); the six figures; the figure builder; the manuscript source; and a README. Public data is not redistributed (CWRU, MFPT, and UCI robot/activity/EMG benchmarks; MIT-BIH electrocardiogram; and the UEA multivariate archive, fetched or read from a local cache at run time); all paths and handles are scrubbed (PATH_TO_DATA / PATH_TO_SCRATCH, any cloud handle → MODAL_USER) and leak-scanned, and the per-deployment configuration-selection procedure and the unlabeled onboarding fit-diagnostic are retained privately and excluded. Cite as R. J. Ferlic and K. K. Ferlic, "When does a frozen one-byte token suffice? A cross-domain separability law and a foundation-model ceiling," Zenodo, 2026, doi: 10.5281/zenodo.23288053. License and patent notice Released under CC-BY 4.0. Consistent with that license, no patent or other intellectual-property right of the authors is licensed, waived, or conveyed by this deposit. This work characterizes previously described, filed, or published methods and discloses no new algorithmic subject matter; random forests, gradient boosting, k-means / vector quantization, Fisher discriminant analysis, the information bottleneck, nearest-centroid classification, temperature-scaling calibration, selective prediction, the MiniROCKET / convolutional-kernel transform, and pretrained time-series foundation models (MOMENT, Chronos) are established prior art or third-party models, used only as tools or comparators. The methods characterized — the class-discriminant single-token codebook encoder, the multi-token / channel-partition token-ladder, and the soft-readout mechanism — are the subject of filed and pending U.S. patent applications held by the authors, spanning the encoder, the multi-token / token-ladder application, and the personalization / on-device-adaptation applications. The foundation-model-ceiling, separability-to-tax law, go/no-go-screen, ladder-recovery, determinism, abstention, calibration, imbalance, and latency results reported here are properties of the frozen filed method, or comparisons against