The Partition Axis as an Invariance Lever: A Pre-Registered Map of Recalibration-Free Amplitude-Drift Robustness in a Frozen Decision Token

The Partition Axis as an Invariance Lever: A Pre-Registered Map of Recalibration-Free Amplitude-Drift Robustness in a Frozen Decision Token Randolph James Ferlic, M.D. and Kimberly Kate Ferlic — Fieldstone Analytics, LLC, Austin, TX, USA Preprint · Zenodo DOI: 10.5281/zenodo.24120987 · 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, deterministic, class-discriminant encoder reduces each window of a multivariate stream to a single 8-bit decision token by a fixed pipeline — a no-learned-parameter feature front-end, a supervised linear-discriminant plus principal-component subspace, a k-means codebook of at most 256 cells, and a nearest-centroid index — and is the base instrument of a program studying how far one microjoule-class, bit-exact lookup can be pushed across sensing modalities. The front-end, however, is a free choice: one may build the token on different partition axes, different fixed views of the signal. This record asks what that choice buys when the deployment drifts. Across twenty-three pre-registered tiers on 108 UCR/UEA datasets plus five raw non-archive corpora — industrial bearing vibration (CWRU, MFPT), wearable inertial (HAPT), twelve-lead clinical electrocardiography (PTB-XL), and industrial machine audio (MIMII) — with Go/No-Go bands fixed before measurement and negatives kept, the finding is a single sharp thesis with an unusually complete bound map: the partition axis is a recalibration-free invariance lever for amplitude- and gain-class drift. A relational axis — ordinal (rank patterns), recurrence (recurrence-quantification), or topological (persistence) — is 7–15× more robust to amplitude/gain drift than the per-channel axes (raw-channel statistics, spectral bands): under a monotone nonlinear gain the channel token loses 0.275 macro-AUROC while an ordinal token loses 0.001. The advantage is real, not an input-normalization artifact — it beats a z-normalized channel baseline by +0.252 under nonlinear gain (95% CI [+0.223, +0.284], on 84% of datasets) — is hyperparameter-robust (gap 0.214–0.221 across codebook size and front-end parameters), leakage-free (it vanishes, gap −0.003, under label permutation), drift-model-general (holding across per-channel gain, DC offset, companding, and clipping, failing only on information-destroying bit-quantization), and leaves the token better-calibrated under drift (relational expected calibration error 0.09–0.14 versus channel 0.18, on 85% of datasets). It beats memoryless recalibration (a frozen relational token scores 0.80 under nonlinear gain where a recalibrated channel token scores 0.55, winning on 92%) and training-time augmentation (relational wins on 80%), transfers under real cross-condition drift (+0.140 across recording-condition sibling pairs; a walking-surface change costs the channel token 0.46 macro-AUROC and the relational token 0.08), and holds on eleven of fifteen domains including clinical ECG (+0.250) and industrial audio (+0.182). The bounds are drawn with equal care: it is not an accuracy lever (composing axes adds only generic multi-token diversity, ~+0.015 relational-specific), not a boundary lever (no axis crosses the EEG/MEG relational boundary, which a full-model control shows is part intrinsic and part token-capacity), not robust to additive noise or quantization, and its large magnitude is specific to the linear/lookup readout class the token inhabits — a readout-capacity dial shows the advantage is ~0.23 for the frozen token and a linear classifier alike and collapses to ~0.04 at the first nonlinear tree split, confidence-interval-separated. A deployment contract closes the record: choose ordinal when cost-constrained (≈2× the channel front-end, microcontroller-plausible, at a bounded accuracy tax) and topological when accuracy is paramount (near-free accuracy, ≈80× compute). The contribution is a precise, hardened statement of exactly what the front-end axis buys a sub-milliwatt decision token under drift, where it does not, and why — strong precisely because it refuses to overclaim. Highlights · The partition axis is an invariance lever, not an accuracy or boundary lever. Building the frozen token on a relational front-end confers recalibration-free robustness to amplitude/gain-class drift; it does not improve clean accuracy and does not cross the EEG/MEG relational boundary. The entire paper is organized around that one claim and the precise boundary of where it holds. · A quantified law, with a transparent mechanism. Under a monotone nonlinear gain the per-channel axes lose 0.23–0.28 macro-AUROC (channel 0.275, frequency 0.229) while the relational axes lose ~0.00–0.04 (ordinal 0.001, recurrence 0.023, topological 0.036) — a 7–15× advantage that repeats under amplitude scaling and time-warp. The mechanism is not an accident: rank-order, fixed-recurrence-rate, and normalized-persistence front-ends are invariant to a monotone amplitude transform by construction, which is also exactly why the capability is bounded. · The invariance is real, not a trivial z-normalization. The sharpest objection — "just z-normalize the channel features" — fails under a nonlinear gain an affine normalization cannot undo: a z-normalized channel baseline collapses to 0.558 while the best relational axis beats it by +0.252 on 84% of datasets (95% CI [+0.223, +0.284]). The invariance is the relational front-end genuinely discarding the amplitude the drift attacks. · Hyperparameter-robust and leakage-free. The gain dichotomy is sign-stable and stays in the range 0.214–0.221 across codebook sizes K ∈ {32, 64, 128, 256} and front-end parameters (subsample length, embedding dimension and lag, recurrence rate), and it vanishes (gap −0.003) under label permutation — so the result is neither a tuning artifact nor leakage. · It beats the alternatives on its drift class. A frozen relational token beats a channel token with memoryless recalibration (0.80 vs 0.55 under nonlinear gain, winning on 92%, no worse than 0.03 below on 97%) and a channel token trained with gain-augmentation (relational wins on 80%), and it subsumes recalibration rather than stacking with it (adding recalibration to the relational token moves it 0.802 → 0.805, a null effect, while it already beats the recalibrated channel token by +0.272). Axis-invariance and online recalibration cover different drift regimes. · It is a property of the linear/lookup readout class. A readout-capacity dial sweeping the frozen token (K = 16/64/256), a linear discriminant, and gradient boosting of rising capacity shows a cliff, not a slope: the topological-over-channel advantage under gain is +0.233/+0.229/+0.227 for the token and +0.264 for a linear classifier, then collapses to +0.043 at a single tree split and +0.034 by 150 iterations, the two regimes confidence-interval-separated. The relational representation makes the drift-invariant decision linearly recoverable — exactly the lookup regime the token inhabits; a full nonlinear model recovers much of the invariance from channel features only at full-model cost, nondeterminism, and loss of the lookup form. · It survives real drift and travels across fifteen domains. Under real cross-condition drift the relational axes transfer +0.140 better than the channel axis across recording-condition sibling pairs (a surface change costs the channel token 0.46 macro-AUROC, the relational token 0.08); under real inter-subject drift the edge is a smaller but real +0.035. The relational-over-channel dichotomy holds on 8 of 10 archive modality clusters plus raw industrial bearing vibration (CWRU; the channel token falls 0.984 → 0.653 under gain while the topological token holds at ~0.84) — 9 of the 13 pre-registered cross-domain tasks — and is confirmed independently on twelve-lead clinical electrocardiography (PTB-XL, +0.250) and industrial machine audio (MIMII, +0.182), for 11 of 15 domains in all; it fails only where the channel token is already near chance (EEG/MEG, EOG) or barely moves under this drift (MFPT, HAPT). · A cost–calibration frontier with a two-point deployment recommendation. Topological is the accurate invariant axis (clean accuracy 0.801 at a tax of only +0.009, retaining 0.766 under gain where the channel token retains 0.535, but ~80× the channel front-end's compute); ordinal is the cheap invariant axis (≈2× the channel front-end, microcontroller-plausible, at a ~0.09 accuracy tax). Both are better-calibrated under drift than channel, and the expensive topological H1 loop features can be dropped (they do not help) leaving a hand-rolled, library-free H0 persistence. Choose ordinal when cost or a microcontroller budget dominates; topological when accuracy is paramount and compute is available. · Honest negatives make the positive trustworthy. Composition of axes looked like an accuracy lever until a random-projection null revealed it as generic multi-token diversity (only ~+0.015 relational-specific; a conditional-mutual-information mechanism fails to predict the per-dataset gain, Spearman +0.145). The EEG/MEG boundary looked like it might yield until a full-model control showed part of it is a real capacity limit of the ≤256-cell token (a full model reaches 0.69–0.75 where the token sits at chance on two of five sets). And the large invariance magnitude looked like a general property until the readout dial located it in the linear/lookup class. Each negative tightens the claim rather than weakening it. · A complete bound map. The capability is for amplitude/gain-class drift, not additive noise (all axes lose 0.20–0.28 at 10 dB, no relational advantage) nor bit-quantization (an information-destroying transform); it does not help where the decision is amplit