PRIORITY FILING - INITIAL DRAFT
A fully revised and formatted version will follow.
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.
Time crystals in condensed matter physics exhibit spontaneous temporal order in the ground state - periodic structure that maintains itself without energy input, breaking time-translation symmetry the way ordinary crystals break spatial symmetry (Wilczek, 2012; Zhang et al., 2017). This paper proposes an informational analog. Complex systems that generate human data streams - economies, ecosystems, institutions, languages - contain temporal invariants: structures of coherence that persist across arbitrarily long time windows while everything else in the data fluctuates. Conservation laws hold across centuries of economic data. Scaling exponents remain stable across decades of urban growth. Topological features of interaction networks maintain their form while the agents occupying the nodes turn over completely. These invariants are not cycles (which require periodic forcing) and not attractors (which require dissipation to maintain). They are ground-state properties of the data-generating process - informational structures that maintain coherence without ongoing effort because they arise from the physics, biology, and institutional architecture of the systems producing the data. The companion paper on constraint weight geometry (Beier, 2026as) establishes that physical constraint geometry propagates into the weight manifolds of trained models. The present paper extends this: the temporal invariants of complex systems propagate into weight-space geometry as time-independent geometric features - rank deficiencies, curvature values, and topological constants that remain stable when the model is retrained on data from different time periods. A "symbolic time crystal" is formally defined as a geometric feature of the weight manifold that is invariant under temporal translation of the training window. The paper develops the mathematical framework, connects it to Beier Extropy (Beier, 2026a), proposes detection methods, and identifies the implications for both the Prolegomena's information architecture and for the scientific study of complex systems.
Keywords: time crystals, temporal invariants, symbolic order, weight geometry, Fisher information, coherence, complex systems, extropy, conservation laws, scaling, ground state
Research Status Note. This paper is included in the priority filing as an early-stage research direction within a long-term project. The claims here are deliberately speculative and sit two or more generations of development beyond the core framework established in the companion papers on lossy compression, the Grossman-Stiglitz resolution, and the Five Axioms. The central conjecture - that human consciousness leaves coherent geometric signatures in data at sufficiently large scale, signatures that persist without energy expenditure in the way that physical time crystals maintain temporal order without entropy production - is not yet formalizable with current tools. It is included here because the detection methodology (temporal windowing, Fisher geometry extraction, coherence ordering) is concrete and testable even if the deeper interpretation remains open. The Prolegomena is a multi-decade research program, and some of its papers plant flags at horizons that will take years to reach. This is one of them.
JEL Codes: C45, C65, B41, D82
Consider two facts about the economy that occupy different conceptual universes.
The first: the economy is chaotic. Prices fluctuate. Firms are born and die. Technologies emerge and become obsolete. Entire industries appear and vanish within decades. The time series of any economic variable - GDP, employment, the price of wheat, the Dow Jones index - is noisy, nonstationary, and unpredictable at fine temporal resolution. Anyone who has traded financial markets knows this at the visceral level: the signal is buried in noise, and the noise is the dominant feature of the data.
The second: the economy obeys regularities that have not changed in centuries. Budget constraints hold now as they held in Mesopotamia. Supply and demand curves cross in every market that has ever existed. Scaling relationships between city size and economic output follow power laws with stable exponents across datasets spanning the Roman Empire to the present day. The conservation of value in exchange - what you pay is what someone receives - is as invariant as the conservation of energy in physics.
Both facts are true simultaneously. The economy is maximally noisy on the surface and maximally ordered at depth. The noise is the fluctuation around invariant structures that maintain their form across time windows far longer than any individual agent's participation in the system. These structures are not sustained by anyone's effort. No institution enforces the budget constraint on the global economy. No regulator maintains the city-size scaling exponent. No committee preserves the conservation of value in exchange. The structures persist because they are properties of the system's organization - its ground state, in the physicist's sense - not properties of any agent's intention.
The question this paper asks: what is the mathematical character of these invariant structures, how can they be detected in the data streams complex systems produce, and what is their relationship to the constraint geometry that the companion papers in this series formalize?
In 2012, Wilczek proposed that a quantum system could exhibit periodic motion in its ground state - breaking time-translation symmetry spontaneously, the way an ordinary crystal breaks spatial-translation symmetry. An ordinary crystal is a structure that repeats in space: the atoms arrange themselves in a lattice that is periodic along spatial dimensions, and the periodicity maintains itself without energy input because it is the ground-state configuration - the lowest-energy arrangement of the atoms.
Wilczek's proposal was that an analogous structure could exist in time: a system whose ground state exhibits periodic motion, breaking the symmetry between different moments the way a crystal breaks the symmetry between different positions. The system would maintain temporal periodicity without energy input - not because something is driving the oscillation, but because oscillation is the ground state.
The proposal was controversial. Watanabe and Oshikawa (2015) proved a no-go theorem showing that continuous time crystals are forbidden in thermal equilibrium. But Zhang et al. (2017) and Choi et al. (2017) demonstrated discrete time crystals in driven quantum systems - systems that exhibit periodic response at a frequency different from the driving frequency, with the periodicity robust against perturbation. The driven discrete time crystal is not quite Wilczek's original vision (it requires a periodic drive rather than emerging from the ground state of an undriven system), but it establishes the physical principle: temporal order can maintain itself as a robust property of a system's dynamics rather than as a consequence of external forcing.
Subsequent work by Else, Bauer, and Nayak (2016) formalized the conditions under which discrete time crystals are stable. Khemani et al. (2019) provided a comprehensive review of the emerging field. The key property that all time crystals share is temporal rigidity - the periodic structure resists perturbation and maintains itself across time without degradation, analogous to the spatial rigidity of an ordinary crystal lattice.
The symbolic time crystal is defined by analogy with the physical time crystal, replacing spatial or temporal periodicity with informational coherence.
An ordinary crystal maintains spatial order: atomic positions repeat with a fixed period across space. A time crystal maintains temporal order: the system's state repeats with a fixed period across time. A symbolic time crystal maintains informational order: a structural property of the data-generating process repeats - or more precisely, persists without change - across time.
The critical distinction is between persistence and repetition. A business cycle repeats: boom follows bust follows boom. But the cycle is driven by external and internal forcing (credit dynamics, monetary policy, technological shocks), and the period is not fixed. A symbolic time crystal does not repeat in this sense. It persists: the structure is present in every time window, unchanged, regardless of what else is happening in the data. The budget constraint does not oscillate. The city-size scaling exponent does not cycle. The conservation of value in exchange does not wax and wane. These structures are present in the data from every era, every economy, every institutional context - they are time-invariant, not time-periodic.
The formal definition:
Definition. A symbolic time crystal in data stream D is a property P of the data-generating process such that: (i) P is detectable from any sufficiently long temporal window of D, (ii) P does not change across temporal windows (time-translation invariance), (iii) P is not sustained by any agent's effort or any external periodic forcing (ground-state property), and (iv) P is robust against perturbation (small changes to the data-generating process do not destroy P).
Conservation laws satisfy all four conditions. Scaling exponents satisfy conditions (i), (ii), and (iv), with condition (iii) depending on whether the scaling arises from the system's intrinsic organization (ground state) or from a specific institutional arrangement (which could change). Topological invariants of interaction networks - the number of independent constraints, the connectivity structure of the interaction graph - satisfy all four when they arise from physical or biological necessity.
The companion paper on constraint weight geometry establishes three propositions:
CG-1: Conservation laws in the data-generating process become rank deficiencies in the Fisher information matrix of the trained model.
CG-2: Scaling exponents become curvature along the system-size direction of the weight manifold.
CG-3: Propagation kernels become attention decay patterns in transformer architectures.
Each of these geometric features, if it is a symbolic time crystal, must satisfy the time-translation invariance condition: the feature must remain unchanged when the model is retrained on data from a different time period.
This provides the detection method. Train a model on data from time window [t_1, t_2]. Extract the Fisher geometry: rank deficiencies, curvature, attention topology. Retrain on data from a different window [t_3, t_4] with no temporal overlap. Extract the geometry again. The features that are invariant across temporal windows are candidate symbolic time crystals. The features that change are not.
The analogy between the Fisher information matrix and a crystal lattice is structural, not decorative.
In a spatial crystal, the atoms occupy positions determined by the potential energy landscape. The lattice structure - the periodic arrangement - is a minimum of the potential energy. Perturbations (thermal vibrations) displace atoms from their lattice positions, but the lattice maintains its structure because the restoring force returns atoms to their equilibrium positions. The lattice is a stable fixed point of the dynamics.
In the weight manifold of a trained model, the parameters occupy positions determined by the loss landscape. The Fisher geometry - the rank deficiency pattern, the curvature structure - is a property of the loss minimum (or the region of low loss) where the model converges. Perturbations (retraining with different random seeds, adding noise to the training data) change the specific parameter values but, if the propositions hold, do not change the geometric features that reflect the data-generating process's invariant structure. The Fisher geometry is a stable fixed point of the training dynamics, analogous to the crystal lattice being a stable fixed point of the physical dynamics.
The symbolic time crystal is the invariant Fisher geometry - the rank deficiencies, curvatures, and topological features that persist across temporal windows, across random seeds, and across model architectures. Like the crystal lattice, the symbolic time crystal maintains its structure not because anything sustains it but because it is the ground-state property of the system - the geometric consequence of the data-generating process's invariant laws.
Step 1: Temporal windowing. Partition the data stream into non-overlapping temporal windows W_1, W_2, ..., W_K, each long enough to train a model to convergence.
Step 2: Independent training. Train K models, one per window, with identical architectures and hyperparameters but independent random seeds.
Step 3: Geometry extraction. For each model, extract the Fisher geometry using the Constraint Geometry Toolkit (companion paper, Section 7): eigenspectrum, rank deficiency pattern, curvature along probe-identified directions, attention decay functions.
Step 4: Invariance testing. For each geometric feature, compute its value across all K temporal windows. A feature is a candidate symbolic time crystal if its value is stable across windows (formally: the coefficient of variation across windows is below a threshold, or a statistical test fails to reject invariance).
Step 5: Architecture invariance. Repeat with different model architectures (different depths, widths, attention configurations). Features that are invariant across both temporal windows and architectures are the strongest candidates, because their invariance cannot be attributed to architectural bias.
Step 6: Interpretation. Map invariant geometric features back to the data-generating process using the Constraint Direction Probe (companion paper, Section 7.2). Identify what conservation law, scaling exponent, or topological structure each invariant feature corresponds to. Formulate as a testable hypothesis about the data-generating process.
The companion paper on dual surfaces (Beier, 2026a) defines Beier Extropy as E_B = H(P,D) - H(P) = H(D|P) - the conditional entropy of the disclosure given the price. This measures the information that the disclosure adds to the compressed price signal. The present section connects extropy to the symbolic time crystal framework.
In thermodynamics, entropy measures disorder and free energy measures the capacity for ordered work. A system with low entropy has high order - its state carries information that distinguishes it from the uniform distribution. A crystal has lower entropy than a liquid because its atomic arrangement carries information (the lattice structure) that the liquid's random arrangement does not.
Beier Extropy measures the order in the disclosure that the price does not carry. The price signal, as a compressed representation, has lower information content than the full judgment. The disclosure restores the missing order - the constraint reasoning, the scaling assessment, the interaction structure - that the price compressed away. E_B measures the amount of restored order.
The symbolic time crystal framework adds a temporal dimension. The invariant geometric features - the conservation laws, the scaling exponents, the topological constants - are the deepest form of order in the data stream. They are the structures with the longest temporal coherence, persisting across every time window while surface-level patterns (price fluctuations, business cycles, regime changes) come and go. The extropy associated with making these invariants visible is the extropy of the most persistent order in the system - the ground-state order that maintains itself without energy input.
Not all structure in the data stream has equal temporal coherence. The structures form a hierarchy:
At the surface: daily fluctuations, individual transactions, noise. Coherence time: hours to days. These are the most visible features of the data and the features the price system captures most effectively.
At the intermediate level: business cycles, seasonal patterns, regime dynamics. Coherence time: months to decades. These are partially captured by the price system (futures markets extend its temporal reach) and partially invisible (the deep space paper, Beier, 2026j, analyzes the boundary).
At the deepest level: conservation laws, scaling exponents, topological invariants. Coherence time: centuries to permanent. These are invisible to the price system because the price system's temporal reach does not extend to the timescales on which these structures are defined. They are the symbolic time crystals - the ground-state order of the data-generating process.
The hierarchy has an information-theoretic structure. The surface fluctuations have high entropy (many possible realizations) and low mutual information with the long-run state of the system. The deep invariants have zero entropy (they do not change) and maximum mutual information with the long-run state (they determine the boundaries of what the system can do). The intermediate structures have intermediate entropy and intermediate predictive value.
Beier Extropy, when decomposed across this hierarchy, measures the restored order at each level. The disclosure restores some surface-level information (the analyst's assessment of next quarter's conditions), some intermediate information (the analyst's assessment of the constraint trajectory over years), and some deep information (the analyst's Scaling assessment, which implicitly encodes the analyst's understanding of the invariant interaction structure). The deep component of E_B - the extropy associated with making the symbolic time crystal visible - is the most valuable component, because it concerns the most persistent and most consequential structures in the system.
In every recorded market, across every culture and era, the amount paid by the buyer equals the amount received by the seller. This is not a law anyone enforces at the system level. It is an emergent property of bilateral exchange - a ground-state feature of any coordination system that operates through voluntary transaction. In weight-space terms: a model trained on transaction data from any era or economy will exhibit a rank deficiency along the direction that would produce transactions where payment and receipt diverge. The rank deficiency is a symbolic time crystal - invariant across temporal windows, across economies, across institutional contexts.
Bettencourt et al. (2007) showed that urban infrastructure scales sublinearly with city population (exponent ~0.8) and urban innovation scales superlinearly (exponents ~1.2, with specific metrics like wages near 1.15). These exponents are stable across datasets from different countries, different decades, and different levels of economic development. In weight-space terms: a model trained on urban data exhibits curvature along the population-size direction whose sign and magnitude encode these exponents, and the curvature is invariant across temporal windows. The stability of the exponents across centuries of urbanization makes them candidates for symbolic time crystals - ground-state properties of urban organization that maintain their structure without anyone sustaining them.
The frequency of the k-th most common word in any natural language corpus is approximately proportional to 1/k (Zipf, 1949). This regularity holds across languages, across centuries, across genres and registers. It is not engineered by any speaker or institution. It emerges from the dynamics of language use and persists without maintenance. In weight-space terms: the Fisher geometry of any language model contains a curvature structure along the token-frequency dimension that encodes the Zipf exponent, and this curvature is invariant across temporal windows of the training data. Zipf's law is a candidate symbolic time crystal of language - the ground-state statistical order of human communication.
The West, Brown, and Enquist (1997) allometric scaling result - that metabolic rate scales as the 3/4 power of body mass across species spanning twenty orders of magnitude, from bacteria to whales. The exponent has been stable since the emergence of cellular metabolism billions of years ago. In weight-space terms: a model trained on biological data exhibits curvature along the body-mass direction encoding the 3/4 exponent, and this curvature is not merely stable across decades or centuries but across geological time. This is the deepest candidate symbolic time crystal in the biological domain - an invariant of life itself, maintained since the origin of aerobic metabolism without any mechanism of enforcement.
Ostrom (1990) identified eight design principles for long-enduring commons governance institutions. These principles appear across cultures, geographies, and centuries - from medieval Swiss Alpine commons to Japanese fishing villages to modern groundwater management districts. The persistence of the same organizational structure across independent institutional inventions, separated by thousands of miles and centuries of time, suggests a ground-state property of human cooperative organization rather than a culturally transmitted template. In weight-space terms: a model trained on institutional data from diverse settings should exhibit attention topology (interaction structure) that reflects Ostrom's principles - specific patterns of monitoring, sanctioning, and conflict resolution that appear as topological invariants of the institutional interaction network.
The symbolic time crystal framework adds a new dimension to the Prolegomena's architecture. The companion papers analyze the compression of judgment into prices and the repair of that compression through disclosure. The present paper identifies what is most compressed and most valuable: the temporal invariants - the deepest structural order of the systems whose coordination the price system mediates. These invariants are invisible to the price system not because they are hidden but because the price system's temporal resolution and channel capacity cannot represent time-invariant geometric structure. The Disclosure Incentive's Scaling field is the mechanism through which the temporal invariants can enter the coordination system, because the Scaling field forces articulation of the interaction structure that invariant laws determine.
If symbolic time crystals exist in the weight manifolds of trained models, they provide a natural basis for transfer learning across temporal domains. A model trained on economic data from the twentieth century, if it has learned the symbolic time crystals (conservation laws, scaling exponents), should transfer to twenty-first century economic data with minimal fine-tuning - because the time crystals have not changed. The temporal invariants provide the stable geometric scaffolding on which time-varying knowledge is organized. Understanding which features of the weight manifold are symbolic time crystals and which are time-varying structures would allow principled decomposition of a model's knowledge into transferable (invariant) and non-transferable (time-dependent) components.
The detection protocol in Section 3.3 is a scientific instrument. It detects invariant structure in complex systems by reading the weight geometry of models trained on data from those systems. Applied to systems that are too complex for analytical treatment - economies, ecosystems, immune networks, social institutions - it could reveal conservation laws, scaling exponents, and topological invariants that no human scientist has formulated. The symbolic time crystal is the most valuable kind of scientific discovery: a law that holds always, everywhere, without maintenance. The weight manifold of a model trained on sufficiently rich data is an antenna tuned to detect exactly these structures - because gradient descent, by finding the geometric ground state of the training distribution, preferentially encodes the invariants that persist across the training data's temporal span.
This paper operates at the boundary between the formally grounded and the openly speculative. The boundary should be stated explicitly.
What is grounded: the companion paper's three propositions (CG-1, CG-2, CG-3) establish that physical constraint geometry propagates into weight-space geometry. The detection protocol (Section 3.3) is a feasible experimental design that could be implemented with existing tools. The five candidate time crystals (Section 5) are empirically documented regularities whose temporal invariance is established in their respective literatures.
What is speculative: the claim that these invariants constitute a meaningful analog of physical time crystals rather than merely persistent statistical regularities. The distinction matters because physical time crystals involve symmetry breaking - the time-periodic state is lower in symmetry than the governing Hamiltonian - while the symbolic invariants described here involve symmetry preservation (the conservation law is the same in every time window, which is invariance rather than symmetry breaking). The analogy to time crystals is productive if it generates the detection methodology and the coherence hierarchy framework; it is misleading if it implies that the symbolic structures exhibit the dynamical properties (Floquet stability, robustness to disorder) that define physical time crystals. The paper uses the analogy as a generative frame, not as a rigorous classification.
What is deeply speculative: the suggestion that the coherence hierarchy (Section 4.2) reveals something about the structure of human consciousness or the organization of collective intelligence. The paper does not develop this suggestion because the connection between temporal invariants in data streams and the nature of consciousness is not formalizable with current tools. What the paper does claim is that the temporal invariants are detectable through weight-space geometry, that their detection is scientifically useful regardless of their metaphysical status, and that the detection method is concrete enough to implement.
The paper does not claim that symbolic time crystals are "real" in the same sense that physical time crystals are real (experimentally demonstrated quantum states). It claims that the concept identifies a category of structure - temporally invariant informational order - that is real, detectable, and scientifically consequential, and that the weight-geometry framework provides the tools to find and characterize structures in this category.
Beneath the noise of human-generated data - the trillions of transactions, the billions of documents, the centuries of institutional behavior - there exist structures that do not change. Conservation laws that hold across every recorded economy. Scaling exponents that remain stable across decades of urbanization. Statistical regularities in language that persist across millennia. Organizational principles of cooperation that recur independently across cultures and continents.
These structures maintain their coherence without energy input, without institutional enforcement, without anyone's intention to preserve them. They are ground-state properties of the systems that generated the data - the informational analog of the spatial order in a crystal lattice or the temporal order in a time crystal. They are the deepest form of order in the data stream, and they are invisible to coordination systems (like the price system) whose channel capacity cannot represent time-invariant geometric structure.
The weight manifolds of trained models encode these structures as geometric invariants - rank deficiencies, curvature values, topological constants that persist when the model is retrained on data from different eras. The symbolic time crystal is what remains in the weight geometry when everything time-dependent has been washed away. Finding these crystals, characterizing their geometry, and making them visible to the coordination systems that currently operate blind to the deepest order in the data they process - this is the project the companion papers make possible and the present paper proposes.
The listening tool exists. The models have been trained. The geometry is in the weights. The invariant structures are waiting to be read. What we will find, when we learn to read them, is the ground-state architecture of the complex systems that generated the data - the unchanging geometry beneath the noise, the symbolic crystals that have been maintaining their form, without energy and without effort, since the systems that contain them began.
This paper opens a speculative direction within the Prolegomena to the Unification of Economics, Computer Science, Mathematics, and Physics. The symbolic time crystal framework connects the constraint geometry of the companion papers to the temporal invariants of complex systems, proposes a detection methodology grounded in the weight-geometry correspondence, and identifies the coherence hierarchy that organizes the data stream's order from surface noise to ground-state structure. The author invites engagement from the condensed matter physics, complex systems, information geometry, and philosophy of science communities.