What a Single Decision Token Can and Cannot Reconstruct: The Shape- versus-Phase Boundary of Extreme Signal Compression

Randolph James Ferlic, Kimberly Kate Ferlic

What a Single Decision Token Can and Cannot Reconstruct: The Shape-versus-Phase Boundary of Extreme Signal Compression Randolph James Ferlic, M.D., and Kimberly Kate Ferlic — Fieldstone Analytics, LLC. Summary A class-discriminant codebook compresses each window of a sensor stream to a single ~8-bit token chosen to preserve a decision. This work asks — definitively, on real public benchmarks (bearing vibration, articulated-motion gesture, robotic force/torque) — how much of the raw signal can be recovered from that token (“token inversion”), and what governs the answer. The result is a clean, general boundary: the token recaptures a signal’s decision-relevant structure and never its high-entropy detail, and whether that detail is recoverable depends on whether the signal’s information lives in its shape or its phase. Every experiment is pre-registered with a frozen honest prior; all datasets are public. Key results • Single-token reconstruction recovers the spectral signature, not the waveform: on bearing vibration the raw-waveform R² is negative (worse than the global mean, because the phase averages out), yet the token holds 46–65% of the magnitude spectrum from one token and the reconstruction re-classifies at 0.60–0.89 AUC. Trajectory-shaped gesture recovers positively (R² ≈ 0.57 by two tokens). • The waveform is unrecoverable at any bit budget: even sixteen residual tokens (128 bits) cannot beat the global mean on vibration. Reconstructing from the full feature vector before any quantization also yields R² ≈ 0, which places the loss at featurization, not quantization. A learned decoder collapses to the global-mean baseline rather than recovering the waveform. • A generative phase-fabricating decoder is decision-faithful and spectrally correct (magnitude R² up to 0.81; re-classifies at 0.68–0.83) but not statistically realistic — a classifier separates fabricated from real windows at AUC 0.98–1.00. It is a class-visualization capability, not realism or recovery. • Operationally, the regime is predictable in advance: signals whose energy concentrates in a few low-frequency, cross-window-stable components (trajectory shape) are recoverable; signals whose energy spreads across broadband, per-window-random phase (vibration) are not. Honest boundary Reconstruct-ability tracks the shape-versus-phase character of the source. What comes back is the decision-relevant structure (spectral fault signature; low-frequency trajectory shape); what never comes back is the high-entropy phase, discarded at the moment of summarization. This is the irreducible cost of decision-oriented compression, and the same property that makes the token so small. A richer generative decoder can make a fabrication more convincing but, by the data-processing inequality, cannot move the boundary. Contents of this deposit Manuscript (PDF/DOCX), Figure 1, three frozen pre-registrations (PREREGISTRATION.md), and a reproducibility folder with deterministic runners and per-run result summaries for all three studies. Means over five seeds; shared stratified 25% test splits; PYTHONHASHSEED=0. Datasets: CWRU Bearing Data Center, MFPT, UEA/UCR NATOPS, UCI Robot Execution Failures (all public). Keywords extreme compression; token inversion; vector quantization; reconstruction; information bottleneck; rate–distortion; condition monitoring; task-oriented communication; pre-registration; honest negatives. References [1] R. J. Ferlic and K. K. 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Companion deposits (single-token codebook family) Paper 19 — 10.5281/zenodo.20788187 · Paper 20 — 10.5281/zenodo.20802759 · Paper 21 (residual VQ) — 10.5281/zenodo.20802826 · Paper 22 — 10.5281/zenodo.20805321 · Paper 23 — 10.5281/zenodo.20821668 · Paper 24 — 10.5281/zenodo.20821779 · Paper 25 — 10.5281/zenodo.20821903. License Creative Commons Attribution 4.0 International (CC-BY 4.0). Consistent with that license, no patent, patent-application, or other intellectual-property right of the authors is licensed, waived, or granted by this publication.