The Deployment-Sizing Law: How Much Budget a Frozen One-Byte Token Needs, and a Cheap Probe That Sizes a Deployment Before It Is Built

The Deployment-Sizing Law: How Much Budget a Frozen One-Byte Token Needs, and a Cheap Probe That Sizes a Deployment Before It Is Built Randolph James Ferlic, M.D. and Kimberly Kate Ferlic — Fieldstone Analytics, LLC, Austin, TX, USA Preprint · Zenodo DOI: 10.5281/zenodo.23323323 · 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 reduces 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 study converted the program's competence-zone law into a prospective separability screen that answers a deployer's first question — will my data pay an accuracy tax? — at a cross-domain area under the receiver-operating curve of 0.83. This paper answers the deployer's next question — how much budget do I need, and can I size it before I build the deployment? — by characterizing, pre-registered, how the token's accuracy scales with the three resources a deployer actually controls: the number of labeled commissioning windows N, the codebook size K (the token's bit-width), and the channel-partition ladder depth in bytes. We find a deployment-sizing law in four parts over a sixteen-dataset battery (bearings, robot force/torque, gesture, inertial activity, electromyography, electrocardiography, and eight held-out archive corpora), with the frozen encoder bit-identical to prior deposits and only the three budget knobs varied. One byte is the right budget, and often generous. Every one of the sixteen datasets reaches within 0.005 of its own best accuracy at a codebook of at most 256 cells, and on average 256 cells sit 0.032 below the per-dataset optimum — bigger tokens do not help, and on a twenty-four-channel gesture corpus a full byte is actively harmful (0.907 at 32 cells collapsing to 0.780 at 256). The deployer should therefore size the codebook down per deployment; this is a clean "information, not bits" confirmation. Bytes have sharply diminishing returns. On multichannel data the channel-partition ladder's first partition byte recovers a median 77 % of the recoverable accuracy tax (up to 86 % on robot force/torque), the curve is concave thereafter, and two-to-three bytes bring the residual tax under 0.05 on most corpora — so ladder bytes should be sized to the tax target, not maximized. The token is label-hungry, honestly. To reach 95 % of its own accuracy plateau the token needs a median 1.8× more labeled commissioning windows than a random forest needs to reach its plateau — a real limitation, reported as a pre-registered negative, that identifies the labeled commissioning budget as the cost that sizes the deployment. Achievable accuracy is predictable from a cheap probe — but not the way we pre-registered. Our pre-registered predictor of absolute accuracy from separability-plus-sample-size failed (leave-one-dataset-out Spearman −0.52), and the failure is instructive: separability predicts the tax, not the absolute accuracy, because absolute accuracy also carries an irreducible task difficulty a separability probe cannot see (an electrocardiography slice with 0.98 class purity yet a 0.63 ceiling inverts the naive predictor). Correctly specified — plateau ≈ a cheap random-forest ceiling on the probe, minus the separability-predicted tax — the sizing calculator predicts achievable accuracy at leave-one-dataset-out Spearman +0.85 (95 % interval 0.50–0.98), with a calibrated 80 % prediction interval that covers the truth on 75 % of held-out datasets at a median absolute error of 0.035, and the separability-to-tax relationship replicates the companion screen (Spearman −0.74 versus −0.76). We then harden the instrument: a class-balanced purity strengthens the tax screen from −0.71 to −0.90; the minimum labeled probe is about 100 windows for separable decisions (and more than 500 for label-hungry ones); and a codebook sized at commissioning transfers under multi-year drift to within 0.014 accuracy of a per-batch re-optimized codebook; and the token is bit-exact deterministic throughout, so the sizing curves and the calculator are reproducible to the bit. The synthesis is a deployment-sizing law and a pre-deployment sizing calculator: run a small labeled probe, read a cheap ceiling and a separability score, and the achievable accuracy, the codebook size, the ladder bytes, and the labeling budget all fall out — before the deployment is built, and with an honest prediction interval. The through-line is that the binding resource for this device is not compute or memory but the labeled commissioning budget, so the calculator answers a procurement question — will this frozen, free, auditable artifact hit my accuracy target on my data, and what one-time labeling will it cost — rather than a training-run optimization; it is the pre-deployment complement to the program's service-life characterization, which shows that same labeling budget to be the token's binding cost over a multi-year deployment. No new algorithmic subject matter is disclosed; the budget knobs are parameters of previously filed methods, the sizing curves are properties of those methods, and the calculator composes a previously published separability screen with standard off-the-shelf tools, all measured on public data. Highlights · One byte is the right budget, and often generous — on all sixteen datasets the token is within 0.005 of its best accuracy at a codebook of at most 256 cells, and 256 cells sit 0.032 below the per-dataset optimum on average (0.127 below on a hard 24-channel gesture corpus, where a full byte over-partitions a low-separability manifold and loses accuracy). The optimum is typically 32–128 cells (five to seven bits). Size the codebook down per deployment — a clean "information, not bits" confirmation. · Bytes have sharply diminishing returns — the channel-partition ladder's first partition byte recovers a median 77 % of the recoverable multichannel accuracy tax (up to 86 % on robot force/torque), the curve is concave thereafter, and two-to-three bytes bring the residual tax under 0.05 on most corpora. Size ladder bytes to the tax target, not to the channel count. · The token is label-hungry (an honest pre-registered negative) — to reach 95 % of its own plateau the token needs a median 1.8× more labeled commissioning windows than a random forest needs for its plateau, driven by the low-separability decisions. The labeled commissioning budget — not compute or memory — is the cost that sizes the deployment. · A pre-deployment sizing calculator — a pre-registered predictor of absolute accuracy from separability-plus-sample-size failed (leave-one-dataset-out Spearman −0.52), because separability predicts the accuracy tax, not the absolute level (an electrocardiography slice is cleanly separable yet intrinsically hard). Re-specified as cheap random-forest ceiling minus separability-predicted tax, the calculator predicts achievable accuracy at leave-one-dataset-out Spearman +0.85 (95 % interval 0.50–0.98), with an 80 % prediction interval (coverage 0.75, median error 0.035). · A hardened instrument — a class-balanced purity strengthens the tax screen from −0.71 to −0.90 by removing base-rate inflation; the minimum labeled probe is about 100 windows for separable decisions and more than 500 for label-hungry ones (the concrete "label this many first"); and a codebook sized at commissioning transfers under multi-year sensor drift to within 0.014 accuracy of a per-batch re-optimized one, so the sizing is a one-time cost. · Honest integrity checks — the one pre-registered predictor that failed is reported as a failure and re-specified in the open; the battery was doubled from eight to sixteen datasets as an adversarial robustness check, which strengthened the calculator (leave-one-out 0.74 → 0.85) and the byte-value law (60 % → 77 %) while leaving the label-hungry negative intact; and the token is bit-exact deterministic (float32 = float64) throughout. What this record contains · Manuscript_Paper59.pdf — the deployment-sizing-law characterization (6 figures, 43 references); and Manuscript_Paper59.docx, the editable source. · PAPER_59_ZENODO_ARCHIVE.zip — the reproducibility archive (md5 in ARCHIVE_MD5.txt): the frozen pre-registrations (the sizing-law pre-registration for S1–S4, and the deployment-grade stress-test pre-registration for the hardening probes SZ-1 through SZ-4); the shared dataset loaders and the frozen token encoder module; the core sizing runner (learning curves, codebook sweep, byte sweep, and the pre-registered and corrected calculator); the sixteen-dataset expansion runner; the hardening-probe runner (the calibrated prediction interval, the minimum-probe sweep, the budget-transfer-under-drift test, and the class-balanced screen); the result records (JSON); the six figures and their builder; the manuscript source; and a README. Public data is not redistributed (two rolling-element bearing corpora; two robot force/torque profiles; a 24-channel aircraft-handling gesture corpus; a smartphone inertial human-activity corpus; a surface-electromyography corpus; a two-class electrocardiography slice; eight held-out multivariate archive corpora; and, for the drift-transfer probe, a 36-month chemical gas-sensor-array drift corpus; each fetched or read from a local cache at run time); all paths and handles are scrubbed and leak-scanned, and the productization know-how — the unlabeled few-window onboarding variant of the probe and the per-deployment configuration-selection and probe-execution procedure — is retained privately and excluded (the study is characterized with labeled probes only). Cite as R. J. Ferlic and K. K. Ferlic, "The deployment-sizing law: how much budget a frozen one-by