Formal Foundations for Symbolic Time Crystals

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**Formal Foundations for Symbolic Time Crystals**

**Three Propositions on Temporal Invariance in Learned Representations**

Gregory Caldwell Beier
Susarb LLC, Cambridge, Massachusetts
March 19, 2026

From the Prolegomena to the Unification of Economics, Computer Science, Mathematics, and Physics by Gregory Caldwell Beier (forthcoming 2026).

© 2026 Gregory Caldwell Beier. Licensed under CC BY-NC-ND 4.0.

**Abstract**

The companion paper on symbolic time crystals (Beier, 2026, Volume II) defines temporal invariants in human-generated data streams and proposes a detection methodology through weight-space geometry. The definition and detection protocol are informal. This paper provides three formal results. Proposition STC-1 (Temporal Windowing Validity) bounds the probability that a geometric feature appearing invariant across K non-overlapping temporal windows with M independent random seeds is an artifact of the model rather than a feature of the data-generating process. The bound decays exponentially in K*M, providing a quantitative false-positive rate for the detection protocol. Proposition STC-2 (Coherence Ordering) proves that if a geometric feature F1 is invariant across windows of length T while F2 is not, then F1 corresponds to a structure in the data-generating process with strictly longer characteristic timescale than F2, under regularity conditions on the data process. Proposition STC-3 (Extropy Decomposition) proves that Beier Extropy E_B decomposes into three components corresponding to surface, intermediate, and deep temporal coherence layers, and that the deep component has the highest value per bit for allocation decisions with horizons exceeding the intermediate coherence timescale. The three propositions convert the symbolic time crystal concept from a qualitative framework into a testable statistical methodology with quantitative error bounds.

**Keywords:** symbolic time crystals, temporal invariance, Fisher geometry, windowing, coherence ordering, extropy decomposition

**JEL Codes:** C45, C65, D82