Recursion on a Frozen Single-Token Encoder: a geometry-gated drift-adaptation law, validated on silicon — within a pre-registered map of where recursion does not help

Recursion on a Frozen Single-Token Encoder: a geometry-gated drift-adaptation law, validated on silicon — within a pre-registered map of where recursion does not help Randolph James Ferlic, M.D. and Kimberly Kate Ferlic — Fieldstone Analytics, LLC, Austin, TX, USA Preprint · Zenodo DOI: 10.5281/zenodo.23423656 · CC-BY 4.0 · Community: spiral-domain-encoder-campaign · the geometry-gated recursive drift compensator described here is the subject of a filed U.S. provisional patent application; the remaining results characterize previously filed and published methods, and this characterization is published after that filing. Abstract A frozen, deterministic single-token encoder reduces a window of sensor inputs, by a fixed computation, to one 8-bit token and a bounded ≤256-cell lookup. A natural question for such a platform is whether adding recursive mathematical modeling — recursion in the codebook, in capacity, or in time — improves any of its metrics. We answer it with a pre-registration with frozen falsifiable bands, and the answer is sharply structured. On the representational axes recursion buys nothing: recursive codebook refinement does not raise the accuracy ceiling (+0.011 across sixteen domains, below a 0.02 band, unmoved by a stronger supervised codebook) — the single-token tax is a true information bottleneck; residual vector quantization is a capacity lever equal to the published token-ladder; offline refinement does not improve label-efficiency, even against active sample selection. A temporal monitoring-gate effect is real but fragile and, under a pre-registered test with a genuinely held-out open-set class arriving in real bursts, is not robust — dropped. The one durable positive is temporal drift adaptation, and it is governed by the *geometry* of the drift, not the domain. A label-free recursive running-mean compensator recovers drift that is a translation (mean-shift) of the feature distribution and fails on drift that changes its shape — established three ways (an injected additive-versus-shape contrast, a within-domain additive-fraction sweep, and six real-drift datasets whose recovery separates cleanly at the geometry boundary, with two independent real mechanisms — inter-patient and cross-device electrocardiography — recovering 0.4–1.1 of the label-recoverable gap and beating memoryless recentering). A steelman that tracks the distribution's shape does not rescue the rest, so the method carries an on-device mean-shift test that gates it. We then reduce the whole gated compensator to practice on a Cortex-M33: it recovers an injected translational drift (0.13→0.79) in a fixed-point runtime bit-identical to float, the gate opens on additive and declines on shape drift, the base token stays deterministic (0/256 flips), and the compensator adds +294 µs and a directly-measured +1.85 µJ (~20%) per inference — 2.6× inside a 5 ms budget. Highlights · Representational recursion buys nothing, and it survives a stronger learner. R1 does not raise the accuracy ceiling (+0.011, below the 0.02 band; a supervised class-conditional codebook moves it only +0.033); R2 is a capacity lever equal to the filed token-ladder; R4 gains no label-efficiency even with uncertainty/active selection (−0.80). The one-byte tax is a true information bottleneck. · The monitoring gate is dropped under realistic bursts. With a genuinely held-out open-set class arriving as real contiguous bursts, exactly one of twelve smoothing configurations crossed +0.05 AUROC and only on a bare majority — fragile across estimator and smoothing constant, and catastrophic at short bursts. · Drift adaptation is governed by geometry, not domain (the central result). A label-free recursive running-mean compensator recovers translational (mean-shift) drift and fails on shape-change drift — shown by an injected additive-vs-shape contrast (additive recovered on 13/13 domains, variance/rotation on none), a within-domain additive-fraction sweep (recovery rises with the translational fraction in 4/5 non-cardiac domains), and six real-drift datasets that separate cleanly at the boundary. · Two independent real mechanisms recover. Inter-patient (f_T ≈ 0.40) and cross-device (f_T ≈ 0.24) electrocardiographic drift are translational and are recovered (0.43–1.14 of the labeled-recalibration gap), beating memoryless recentering; sensor aging, electrode re-don, cross-subject activity, and four wearable channels are non-translational and are not recovered. · The method is gated and self-aware. Tracking the distribution's shape (online covariance/correlation-alignment, Kalman) does not rescue the non-translational cases, so the device computes a label-free mean-shift fraction and applies the recentering only when the drift is translational, otherwise withholding it and requesting recalibration. · Reduced to practice on a Cortex-M33. On-device recovery 0.13→0.79, fixed-point == float (359/360 tokens identical), gate opens on additive / declines on shape, base token 0/256-flip deterministic; the compensation adds +4 µs decision-only (its single search replaces the base search) and +294 µs in the dual-output fingerprint configuration (2.6× inside a 5 ms budget), with a directly power-meter-measured +1.85 µJ (~20%) per inference. The transferable lesson A label-free drift compensator's reach is set by the geometry of the drift, which is measurable on-device, and it must be characterized against that geometry. An injected mean-shift neither proves the method (too clean) nor, by its failure on other injected forms, refutes it; and a single real modality can mislead. A pre-registered translational / non-translational decomposition across datasets that span the geometry is what reveals the true law — here, that a recursive mean-tracker recovers translational drift wherever it occurs (including real inter-patient and cross-device physiological drift) and fails on shape-change drift wherever it occurs. What this record contains · Manuscript_Paper63.pdf / .docx — the manuscript (16 pages, 5 figures, 38 references). · PAPER_63_ZENODO_ARCHIVE.zip — the reproducibility archive: the frozen pre-registrations (the recursion-master battery, the gate operating-regime probe, the real-drift transfer test, and the Tier-3/Tier-4 hardening set, with their a-priori bands and decision rules), the runners for all four recursive derivatives and the real-drift and geometry-sweep validations, the shared dataset loaders and the frozen token encoder, the per-cell result records (JSON) and verdict summaries, the on-device drift bench log, the five figures and their builder, and a README. Public corpora are not redistributed; absolute paths, handles, and productization know-how are scrubbed. Cite as Ferlic, R. J., & Ferlic, K. K. (2026). Recursion on a Frozen Single-Token Encoder: a geometry-gated drift-adaptation law, validated on silicon — within a pre-registered map of where recursion does not help. Zenodo. https://doi.org/10.5281/zenodo.23423656 Corresponding author: Randolph J. Ferlic — [email protected]. ORCID: 0009-0006-2367-5205 (R.J.F.), 0009-0007-3012-3430 (K.K.F.). License and patent notice Released under CC-BY 4.0. Consistent with that license, the deposit licenses the copyright in the text and figures only, and no patent or other intellectual-property right of the authors is licensed, waived, or conveyed by this deposit. The geometry-gated recursive drift compensator — the label-free recursive running-mean recentering, the on-device translational-drift (mean-shift-fraction) gate, the recalibration fallback, the retained-fingerprint dual output, and the fixed-point embodiment — is the subject of a U.S. provisional patent application filed by the authors, and this characterization is published after that filing. The remaining results characterize previously filed and published methods and disclose no new subject matter: the base single-token encoder, the residual/channel-partition token-ladder (R2), and the offline codebook refinement (R4) are the program's filed and published methods; R1 confirms a null; the monitoring gate is dropped. k-means and vector quantization, Fisher discriminant analysis, nearest-centroid methods, exponential/recursive-mean and Kalman filters, CUSUM detectors, correlation-alignment (CORAL) and batch-norm/test-time adaptation, and principal-component analysis are established prior art or standard tools, used only as tools or comparators. The per-deployment configuration-selection and input-conditioning procedure and the silicon productization and commissioning toolchain are retained as trade secrets and deliberately not disclosed. No third-party partner, customer, or commercial collaborator is named; the public datasets named are cited for reproducibility. © 2026 Fieldstone Analytics, LLC and the authors. Inquiries: [email protected]. Companion deposits (spiral-domain-encoder-campaign) · "The Frozen Token" capstone — the four-laws / competence-zone theory this study belongs to (doi:10.5281/zenodo.23002239). Is part of. · The Composed Platform — the multi-capability frozen-token stack and on-silicon track this study extends with drift adaptation (doi:10.5281/zenodo.23401467). · When Does a Frozen One-Byte Token Suffice? — the separability law the geometry result relates to (doi:10.5281/zenodo.23288053). · Watch the Input, Not the Representation — the self-twin monitor whose gate this study tightens and drops (doi:10.5281/zenodo.23351878). · The channel-partition token ladder — the capacity lever R2 equals (doi:10.5281/zenodo.22003179). · The Cheapest Digital Twin — the label-free self-twin / drift context (doi:10.5281/zenodo.22922678). · The Service-Life Trajectory — drift / re-commission (doi:10.5281/zenodo.23321165). · Class-discriminant single-token codebook — base encoder (doi:10.5281/zenodo.20