The Sub-Bit Decision Token: An Information Ceiling on a One-Byte Codebook, a Near-Universal Escape via a Frequency Ladder, a Cheap Commissioning Meter, and a Bit-Exact Reduction to Practice on Commodity Silicon

The Sub-Bit Decision Token: An Information Ceiling on a One-Byte Codebook, a Near-Universal Escape via a Frequency Ladder, a Cheap Commissioning Meter, and a Bit-Exact Reduction to Practice on Commodity Silicon Randolph James Ferlic, M.D. and Kimberly Kate Ferlic — Fieldstone Analytics, LLC, Austin, TX, USA Preprint · Zenodo DOI: 10.5281/zenodo.23962249 · CC-BY 4.0 · Community: spiral-domain-encoder-campaign · This work is a scientific characterization of previously filed and published methods; it discloses no new mechanism and is accompanied by no new patent application — see the notice below. Abstract A frozen single-byte "decision token" — features mapped by a closed-form affine embedding to a ≤256-cell k-means codebook, read out by a per-cell class posterior — is the base instrument of a program that studies how far one deterministic, microjoule-class lookup can be pushed across sensing modalities. This record asks the sharpest quantitative question about it: how much decision-relevant information does one token actually carry, can any cleverer way of placing its cells raise that number, and when it cannot, what does? Across an eleven-tier pre-registered battery on the full UCR and UEA time-series archives, with decision bands fixed before measurement and negatives kept, four things are established. (1) A bound. The token carries, bias-corrected and shuffle-null-subtracted, only ≈0.42 bits of net label information on average (bootstrap 95% confidence interval 0.19–0.68; naive plug-in mutual information reads 1.17 bits but the finite-sample bias floor alone is 0.75 bits), and the ceiling is front-end-independent — it moves by at most 0.16 bits across five embeddings including a nonlinear one. It is a sub-bit decision stamp, not the multi-bit code a naive estimate suggests. (2) A hardened negative. No re-placement of the cells beats plain reconstruction-driven k-means: an information-optimal (agglomerative information-bottleneck) quantizer overfits (held-out −0.014), curved-space quantization is a wash (+0.001), and a flow-aware "generating" symbolic partition is refuted (−0.021); a test-tuned oracle finds apparent headroom that a realizable train-selected rule cannot capture (information-bottleneck realizable −0.007, curvature −0.001). (3) A near-universal escape, and a map. The only lever that moves held-out accuracy is admitting more independent information — a partition-and-aggregate ladder — validated on a new axis, frequency: a dyadic-band ladder lifts accuracy by +0.044 on average, stacks on the channel ladder, and on a 105-dataset census escapes the single-token bound on 88% of the archive. The picture is a four-regime law — bound (one token saturates), ceiling (easy tasks already saturated; canonical bearing-fault identification sits at 0.998, no room), escape (multi-scale morphology: handwriting +0.221, valve-acoustic anomaly +0.204), and boundary (electroencephalography, where the ladder cannot help even with a healthy token). (4) A cheap meter, and silicon. Escape is not predictable from a label-free energy statistic, but a cheap, codebook-free joint-separability meter predicts it (Spearman ρ 0.75 archive-wide; leave-one-out precision 0.99), validated out-of-source on real bearing and acoustic data (3/3). Finally, the laddered token runs bit-exact (0/54 decision flips, float32) and deterministic at 1.66 ms per inference on a commodity Arm Cortex-M4F, with a calibrated int8 path agreeing 0.98 and an exact host port; and measured head-to-head against a modern random-kernel (ROCKET-family) classifier on identical folds, the laddered token is a cost–accuracy dial whose mean accuracy climbs from 91% to 96% of ROCKET across one to four deterministic bytes — most of it in the first two — reaching 95.9% at ~500× fewer features, a tunable, auditable operating point on the cost–accuracy plane rather than a higher accuracy. A deployment contract closes the record. The contribution is a precise, hardened, hardware-validated statement of exactly how much one byte decides, when more bytes help, and how to tell in advance — strong precisely because it refuses to overclaim. Highlights · One byte is a sub-bit decision stamp. Bias-corrected and shuffle-null-subtracted, a frozen single-token codebook carries only ≈0.42 net bits of label information on average (95% CI 0.19–0.68) — the naive plug-in estimate (1.17 bits) is mostly a finite-sample bias floor (0.75 bits). The ceiling is front-end-independent (0.16-bit spread across five embeddings, including a nonlinear one) and sits far below the code's 6-bit capacity. A methodological corollary: naive mutual-information estimates of quantized decisions overstate the usable information and must be bias-corrected. · No cleverer placement of the cells escapes the bound — even against its own oracle. An information-optimal (agglomerative information-bottleneck) quantizer overfits the commissioning labels (held-out −0.014); curved-space (spherical / hyperbolic) quantization is a wash (+0.001); a flow-aware "generating" symbolic partition is refuted (−0.021). A test-tuned oracle finds apparent headroom (information-bottleneck +0.030, curvature +0.018) that a realizable inner-validation rule cannot capture (−0.007, −0.001) — the decisive control separating a real lever from selection-bias headroom. · The one lever that works is a ladder, and it escapes the bound on 88% of the archive. Distributing capacity across independent tokens and aggregating — here on a new axis, frequency (dyadic bands) — lifts held-out accuracy by +0.044 on average, stacks on the channel ladder (+0.032), and is ~⅔ structural (beats a byte-matched single token by +0.028, and a capacity-matched 256-cell single codebook by +0.067). A 105-dataset census of the full UCR/UEA archive finds the ladder escapes the single-token bound on 92 of 105 datasets. · A four-regime law: bound · ceiling · escape · boundary. One token ≈ 0.42 net bits (bound); on easy tasks the single token is already saturated and the ladder has no room (ceiling — canonical bearing-fault identification at AUROC 0.998); on multi-scale morphology the ladder breaks the bound, sometimes dramatically (escape — handwriting +0.221, the four-byte ladder lifting the single token from 0.669 to 0.891, near a modern classifier; valve-acoustic anomaly +0.204); and on neural signals the ladder cannot help even when the token is healthy (boundary — a self-regulation EEG task where a 0.835 token still loses −0.022). · A cheap commissioning meter predicts escape — out of sample and out of source. A label-free energy statistic does not predict escape (ρ 0.13); a cheap, codebook-free joint-separability meter (a three-fold shrinkage-discriminant on stacked multi-band features) does, at Spearman ρ 0.75 archive-wide and leave-one-out precision 0.99. The development-set threshold correctly classifies three genuinely out-of-source real industrial datasets (bearing vibration as ceiling; machine-acoustic anomaly as escape). · The escape lever is deterministic silicon. The frequency-ladder token — four band-tokens plus a joint readout — runs on a commodity 32-bit Arm Cortex-M4F with its float32 decision bit-exact to the reference (0/54 flips) at 1.66 ms per inference, a calibrated int8 path agreeing 0.98 and porting exactly. A reconciliation worth stating: naive post-hoc int8 is lossy (a quantizer-choice artifact), while the per-dimension min-max–calibrated scheme is bit-exact; and the ladder prunes two-fold free but not four, because its tokens use their capacity. · Placed on the accuracy frontier as a tunable dial, not hidden from it. Measured head-to-head on identical folds against a modern random-convolutional-kernel classifier in the ROCKET family, the single token trails on raw accuracy on all ten test datasets — as a one-byte lookup against a two-thousand-feature classifier should, and as it never claimed otherwise. The point is the shape of that gap: read as a progression, the frequency ladder is a cost–accuracy dial whose mean accuracy climbs from 91% to 96% of ROCKET across one to four deterministic bytes, with about four-fifths of the gain already at two bytes and a clean plateau after — reaching 95.9% at roughly 500× fewer features, as a deterministic, bit-exact table lookup, with the three methods converging on already-saturated tasks. A deployer picks an operating point on a short, monotone, diminishing-returns curve rather than choosing between "one byte" and "a full classifier." One honest nuance sharpens the boundary: on the EEG task the ladder cannot help (the dial runs downhill) but the far costlier ROCKET can (0.957) — so the neural boundary is a limit of the cheap representation, not of the task, which is exactly the cost–accuracy trade the token exists to let a deployer price. · Three reusable methods, not only one result. Beyond the token, the record contributes a bias-corrected information measurement (a shuffle-null subtraction that isolates the finite-sample floor — a guard any study estimating the information a discrete code carries about a label should adopt before making a multi-bit claim); the oracle-versus-realizable construction (grant a candidate mechanism a test-tuned upper bound, then show an honest, validation-selected rule cannot reach it — which forecloses the "you under-tuned the baseline" objection before it is raised); and a predict-before-building meter (answer "will a bigger model help?" cheaply, from a modest labeled sample, with a stated error rate, ahead of building the bigger model). Each is small, general, and separable from the specific encoder. · A deployment contract, not only findings. The map distils into testable rules: size the codebook for the information it carries (≈0.4 bits), not the 256 cells it has; do not try to out-place plain k-means; add accuracy by laddering an independent axis — channels, time, or now frequency — and expect roughly