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

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. Ferlic, “Class-discriminant codebook construction for single-token signal compression,” Zenodo, 2026, doi:10.5281/zenodo.20788187. 2 R. J. Ferlic and K. K. Ferlic, “Residual vector quantization within a class-discriminant subspace,” Zenodo, 2026, doi:10.5281/zenodo.20802826. 3 A. van den Oord, O. Vinyals, and K. Kavukcuoglu, “Neural discrete representation learning,” NeurIPS, 2017. 4 D. Griffin and J. Lim, “Signal estimation from modified short-time Fourier transform,” IEEE Trans. ASSP, vol. 32, no. 2, pp. 236–243, 1984. 5 T. M. Cover and J. A. Thomas, Elements of Information Theory, 2nd ed. Wiley, 2006. 6 C. E. Shannon, “Coding theorems for a discrete source with a fidelity criterion,” IRE Nat. Conv. Rec., vol. 7, pt. 4, pp. 142–163, 1959. 7 N. Tishby, F. C. Pereira, and W. Bialek, “The information bottleneck method,” Allerton, 1999, pp. 368–377. 8 D. Gündüz et al., “Beyond transmitting bits: Context, semantics, and task-oriented communications,” IEEE JSAC, vol. 41, no. 1, pp. 5–41, 2023. 9 L.-Y. Duan, J. Liu, W. Yang, T. Huang, and W. Gao, “Video coding for machines: A paradigm of collaborative compression and intelligent analytics,” IEEE TIP, vol. 29, pp. 8680–8695, 2020. 10 S. Talukder, Y. Yue, and G. Gkioxari, “TOTEM: Tokenized time series embeddings for general time series analysis,” TMLR, 2024. 11 W. A. Smith and R. B. Randall, “Rolling element bearing diagnostics using the CWRU data: A benchmark study,” MSSP, vol. 64–65, pp. 100–131, 2015. 12 E. Bechhoefer, “A quick introduction to bearing envelope analysis,” MFPT bearing fault dataset, 2013. 13 A. Bagnall et al., “The UEA multivariate time series classification archive, 2018,” arXiv:1811.00075. 14 L. Seabra Lopes and L. M. Camarinha-Matos, “Feature transformation strategies for a robot learning problem,” Springer, 1998. 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.