Correction, 30 October 2026. STC-1' (False positive bound, Gaussian case). Let phi_{k,j} be independent N(mu_G, sigma_G^2), n = K*M - 1, and r = eta^2 mu_G^2 / sigma_G^2 < 1.
STC-2' (Window test as a band-stop filter). Under (CO1) and (CO2), the expected sample variance of K consecutive window means of length T is about 2 sigma^2 tau / T when tau << T, about sigma^2 K T / (3 tau) when tau >> K T, and of order sigma^2 for tau between T and K T. A summary of this correction is on the papers page at https://www.susarb.com/papers; the full correction note, dated 30 October 2026, will be deposited as a version of this record.
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.
The companion paper on symbolic time crystals (Beier, 2026at) 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
Proposition STC-1 (False positive bound for temporal invariance detection). Let D be a data stream generated by a process P. Partition D into K non-overlapping temporal windows W_1, ..., W_K, each of length T. For each window W_k, train M models with independent random initializations, producing models {m_{k,j} : j = 1, ..., M}. Extract a geometric feature phi(m) from each model (a scalar derived from the Fisher information matrix - an eigenvalue, a curvature, an attention decay length).
Define the feature as "apparently invariant" if the coefficient of variation of phi across all K*M models is below a threshold eta:
CV = std({phi(m_{k,j})}) / mean({phi(m_{k,j})}) < eta
Under the null hypothesis H_0 that the feature is an artifact of the model architecture rather than a property of the data-generating process:
(H_0): phi(m_{k,j}) are independent draws from a distribution G with mean mu_G and variance sigma_G^2 > 0, independent of the temporal window.
Under the alternative hypothesis H_1 that the feature reflects a temporally invariant property of the data-generating process:
(H_1): phi(m_{k,j}) = phi_true + epsilon_{k,j} where phi_true is a constant (the invariant's value), and epsilon_{k,j} are independent noise terms with mean zero and variance sigma_epsilon^2 < sigma_G^2.
Then:
P(CV < eta | H_0) <= exp(-K*M * eta^2 * mu_G^2 / (2 * sigma_G^2))
and the false positive rate decays exponentially in the product K*M.
Proof. Under H_0, the sample mean of phi across K*M models has variance sigma_G^2 / (K*M) and the sample standard deviation converges to sigma_G. The coefficient of variation CV = std / mean converges to sigma_G / mu_G as K*M grows.
The event CV < eta requires the sample standard deviation to satisfy std < eta * mean. Under H_0, the sample variance S^2 = (1/(K*M - 1)) * sum (phi_{k,j} - phi-bar)^2 has expected value sigma_G^2. The probability that S^2 < eta^2 * phi-bar^2 is bounded using the chi-squared concentration inequality. Since S^2 * (K*M - 1) / sigma_G^2 follows a chi-squared distribution with K*M - 1 degrees of freedom:
P(S^2 < eta^2 * mu_G^2 (1 + o(1))) <= P(chi^2_{K*M-1} < (K*M-1) * eta^2 * mu_G^2 / sigma_G^2)
By the chi-squared tail bound (Laurent and Massart, 2000):
P(chi^2_n < n - 2*sqrt(n*t)) <= exp(-t)
Setting n = K*M - 1 and matching the target probability, the event S^2 < eta^2 * mu_G^2 requires the chi-squared variable to be below its mean by an amount proportional to K*M * (1 - eta^2 * mu_G^2 / sigma_G^2). When eta^2 * mu_G^2 / sigma_G^2 < 1 (the threshold is tighter than the true CV under H_0), this is a large deviation below the mean, and the probability is bounded by:
P(CV < eta | H_0) <= exp(-c * K * M)
where c = (1 - eta^2 * mu_G^2 / sigma_G^2)^2 / 2 is a constant that depends on the threshold eta and the true coefficient of variation under H_0. By applying the Laurent-Massart bound to the lower tail and taking the leading-order expansion for small eta, the probability simplifies to:
P(CV < eta | H_0) <= exp(-K * M * eta^2 * mu_G^2 / (2 * sigma_G^2))
when eta << sigma_G / mu_G (the threshold is much smaller than the true CV). QED.
Interpretation. For K = 10 temporal windows and M = 5 random seeds per window, K*M = 50. With eta = 0.05 and a true CV under H_0 of sigma_G / mu_G = 0.3, the false positive probability is bounded by exp(-50 * 0.0025 / (2 * 0.09)) = exp(-0.69) = 0.50. This is weak. For K = 20, M = 10 (K*M = 200), the bound becomes exp(-2.78) = 0.062 - a 6.2% false positive rate. For K = 50, M = 10 (K*M = 500), the bound is exp(-6.94) = 0.001 - a 0.1% false positive rate.
The practical requirement is that detection of symbolic time crystals requires many temporal windows and many random seeds. This is computationally expensive but feasible: the training runs can be parallelized, and the temporal windows can be constructed from any data stream with sufficient length.
The bound's key property is the exponential decay in K*M, which guarantees that the false positive rate can be made arbitrarily small by increasing the number of windows and seeds. A feature that survives the windowing test at K*M = 500 is almost certainly a property of the data-generating process rather than an artifact of the model or the training procedure.
Proposition STC-2 (Coherence ordering from temporal windowing). Let P be a data-generating process with two structural features F1 and F2 that manifest as geometric features phi_1 and phi_2 in the weight space of trained models. Suppose:
(CO1) F1 has characteristic timescale tau_1: the autocorrelation of F1's manifestation in the data decays as rho_1(h) = exp(-h / tau_1) for lag h.
(CO2) F2 has characteristic timescale tau_2 < tau_1: rho_2(h) = exp(-h / tau_2).
(CO3) The temporal windows have length T satisfying tau_2 < T < tau_1.
Then phi_1 is invariant across temporal windows (CV_1 < eta for sufficiently large K*M) while phi_2 is not invariant (CV_2 > eta), and the windowing test correctly orders the coherence times: the feature that passes the invariance test has the longer characteristic timescale.
Proof. The geometric feature phi(m_{k,j}) reflects the model's learned representation of the data in window W_k. When a structural feature F has characteristic timescale tau, the feature's manifestation in data from window W_k of length T depends on the average value of F within the window.
For F1 with tau_1 > T: the feature changes negligibly within any single window, so its manifestation is approximately constant across windows. The between-window variance of phi_1 reflects only the estimation noise (the random seed variation and finite-sample effects), not genuine variation in F1. The coefficient of variation CV_1 converges to the noise-only CV as K*M grows, which is small by construction (estimation noise is controlled by window length T and model capacity).
Formally, the between-window variance of phi_1's true value is:
Var_{between}(phi_1) = Var(F1(W_k)) approximately 2 * sigma_{F1}^2 * tau_1 / T * (1 - tau_1/T * (1 - exp(-T/tau_1)))
For T << tau_1, this simplifies to Var_{between} approximately sigma_{F1}^2 * T / (3 * tau_1), which vanishes as tau_1 / T grows.
For F2 with tau_2 < T: the feature changes substantially within each window, and different windows sample different values of F2. The between-window variance reflects genuine variation in F2:
Var_{between}(phi_2) approximately sigma_{F2}^2 * (1 - 2 * tau_2 / T)
which is O(sigma_{F2}^2) - a substantial fraction of the total variance. The coefficient of variation CV_2 reflects this genuine variation and remains large even as K*M grows.
The ordering follows: CV_1 < CV_2 whenever tau_1 > T > tau_2, because the between-window variance is negligible for F1 (the feature is constant on the window timescale) and substantial for F2 (the feature varies on the window timescale). The windowing test at threshold eta passes F1 and rejects F2 for any eta between the noise-only CV (lower bound) and the genuine-variation CV (upper bound). QED.
Interpretation. The proposition establishes that the windowing test is a coherence-time filter: features with coherence time longer than the window length pass, features with shorter coherence time fail. By varying the window length T, the researcher can probe different levels of the coherence hierarchy described in the companion paper. Short windows (T = months) detect features invariant on the monthly timescale. Long windows (T = decades) detect features invariant on the decadal timescale. Features that pass at every window length are the deepest symbolic time crystals - the structures with the longest coherence times.
Proposition STC-3 (Hierarchical extropy decomposition). Let the constraint state V decompose as:
V = V_surface + V_intermediate + V_deep
where V_surface has characteristic timescale tau_S (days to months), V_intermediate has timescale tau_I (months to decades), and V_deep has timescale tau_D (decades to permanent). Let D be a disclosure about V, carrying total Beier Extropy E_B = H(D|P). Define:
E_surface = I(V_surface; D | P)
E_intermediate = I(V_intermediate; D | P, V_surface)
E_deep = I(V_deep; D | P, V_surface, V_intermediate)
Then:
(i) E_B >= E_surface + E_intermediate + E_deep (the decomposition is a lower bound on total extropy due to potential interactions).
(ii) E_surface + E_intermediate + E_deep = E_B when V_surface, V_intermediate, and V_deep are conditionally independent given P (no interactions across coherence layers beyond what the price carries).
(iii) For an allocation decision with horizon H, the value per bit of each component satisfies:
value(E_deep) / E_deep > value(E_intermediate) / E_intermediate > value(E_surface) / E_surface
whenever H > tau_I.
Proof of (i). By the chain rule for mutual information:
I(V; D | P) = I(V_surface; D | P) + I(V_intermediate; D | P, V_surface) + I(V_deep; D | P, V_surface, V_intermediate) + I(interactions; D | P, V_surface, V_intermediate, V_deep)
The interaction term captures the information D carries about the dependencies among the three layers. This term is non-negative (mutual information is non-negative), so:
E_B = I(V; D | P) >= E_surface + E_intermediate + E_deep. QED.
Proof of (ii). When V_surface, V_intermediate, and V_deep are conditionally independent given P, the interaction term is zero. The decomposition becomes an equality. QED.
Proof of (iii). Consider an allocation decision at horizon H > tau_I. The decision's quality depends on the allocator's posterior about V at time t + H.
At horizon H, V_surface(t + H) is uncorrelated with V_surface(t) (because H >> tau_S), so any information about the current surface state is worthless for the allocation decision. E_surface has zero value per bit for decisions at horizon H.
At horizon H > tau_I, V_intermediate(t + H) is weakly correlated with V_intermediate(t) (because H is comparable to tau_I), so information about the current intermediate state has positive but diminished value.
At horizon H > tau_I, V_deep(t + H) is strongly correlated with V_deep(t) (because H << tau_D), so information about the deep state retains nearly full value at the decision horizon.
The value of information about each component for the allocation decision is proportional to the correlation between the current value and the value at the decision horizon:
value(E_k) proportional to E_k * rho_k(H)
where rho_k(H) = exp(-H / tau_k) is the autocorrelation at lag H. For H > tau_I:
rho_deep(H) = exp(-H / tau_D) >> rho_intermediate(H) = exp(-H / tau_I) >> rho_surface(H) = exp(-H / tau_S) approximately 0
Therefore value per bit:
value(E_deep) / E_deep proportional to rho_deep(H) > rho_intermediate(H) > rho_surface(H)
which gives the ordering. QED.
Interpretation. The proposition formalizes the companion paper's claim that the deepest layer of the extropy decomposition is the most valuable for long-horizon decisions. The deep component of E_B - the information about temporal invariants (conservation laws, scaling exponents, topological structure) - retains its value at horizons where the surface and intermediate components have decayed to irrelevance. For a pension fund making a thirty-year allocation decision, the deep component is the information that determines the decision's quality: what are the invariant constraints the portfolio must navigate for three decades? The surface component (next quarter's earnings) is noise at that horizon. The intermediate component (business cycle position) is partially relevant. The deep component (aquifer depletion trajectories, atmospheric carbon dynamics, infrastructure resilience thresholds) is maximally relevant because it changes on timescales longer than the decision horizon.
This is the information-theoretic foundation for the companion paper on deep space economics (Beier, 2026j): the most consequential information for long-horizon decisions is the information with the longest coherence time, and the Disclosure Incentive's Scaling field is specifically designed to elicit this deep-layer information.
Proposition STC-1's bound requires the features under H_0 to be independent across temporal windows. If the model architecture induces systematic biases that are correlated across windows (all models trained on all windows converge to similar artifacts due to architectural inductive biases), the bound underestimates the false positive rate. The M random seeds control for initialization-dependent artifacts but not for architecture-dependent artifacts. Testing across multiple architectures (as the companion paper's Section 3.3 recommends) addresses this but is not captured in the formal bound.
Proposition STC-2 assumes exponential decay of autocorrelation, which is the simplest model of temporal dependence but not universal. Features with long-range dependence (power-law decay of autocorrelation) or oscillatory dependence (periodic components) have different windowing properties. The ordering result holds for any monotonically decaying autocorrelation function but the specific relationship between window length T and detection sensitivity depends on the functional form.
Proposition STC-3's value ordering depends on the decision horizon H exceeding tau_I. For short-horizon decisions (H < tau_S), the ordering reverses: surface information is most valuable per bit. The ordering holds for decisions whose horizon falls in the deep range, which is the regime relevant for the Prolegomena's constraint reasoning focus.