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 constraint weight geometry (Beier, 2026as) establishes Proposition CG-1: conservation laws in the data-generating process appear as rank deficiencies in the Fisher information matrix of the trained model. The proposition's proof assumes the training distribution is exactly supported on the conservation-respecting set X_cons. In practice, training data satisfies conservation only approximately - simulation data carries floating-point errors (delta ~ 10^{-15}), and empirical data carries measurement errors (delta ~ 10^{-2} to 10^{-1}). This paper proves the quantitative convergence rate. Proposition NR-1 establishes that the Fisher information along the conservation-violating direction scales as O(delta^2) where delta is the conservation violation magnitude, with an explicit constant that depends on the model architecture and the data distribution's structure near the conservation boundary. Proposition NR-2 establishes the Fisher ratio between conservation-trained and conservation-violating models as a function of delta, providing the quantitative success criterion for the research program's Phase 1 experiment. Proposition NR-3 extends the result to multiple conservation laws, showing that each independent conservation constraint contributes an independent near-zero eigenvalue to the Fisher matrix, and the eigenvalue magnitudes decay as O(delta_k^2) where delta_k is the violation magnitude of the k-th conservation law.
Keywords: Fisher information, conservation laws, noise regime, convergence rate, weight manifold, measurement error, rank deficiency
JEL Codes: C45, C65, D82
Proposition CG-1 of the companion paper states: when a model is trained on data generated by a physical process satisfying a conservation law, the Fisher information matrix of the trained model has a rank deficiency along the conservation-violating direction. The proof shows that the Fisher information I_{v_cons}(w*) = E[(v_cons · nabla_w log p(x; w*))^2] vanishes when p_data assigns zero mass to conservation-violating outcomes.
The proof is correct in the limit of exactly conservation-respecting data. Real data violates conservation by a small amount delta, and the question is how the Fisher information depends on delta. This paper provides the answer.
Let X_cons = {x : |C(x)| = 0} denote the exact conservation surface, where C(x) is the conservation constraint function (for mass conservation: C(x) = inflow - outflow - delta_storage). Let X_cons^delta = {x : |C(x)| < delta} denote the delta-thickened conservation surface - the set of outcomes that violate conservation by at most delta.
The training distribution under noise is p_data^delta, supported on X_cons^delta. In the limit delta -> 0, p_data^delta concentrates on X_cons and we recover the companion paper's setting. For delta > 0, the distribution assigns mass to outcomes that violate conservation by amounts up to delta.
Let the model with weights w define output distribution p(x; w). The Fisher information along direction v in weight space is:
I_v(w) = E_{p_data^delta}[(v · nabla_w log p(x; w))^2]
We seek the behavior of I_{v_cons}(w*) as a function of delta, where v_cons is the conservation-violating direction defined in the companion paper and w* are the converged weights.
The key is to decompose the expectation into contributions from the near-conservation region and the conservation-violating region.
Partition X_cons^delta into two regions: the core C_epsilon = {x : |C(x)| < epsilon} for some epsilon << delta, and the boundary B_{epsilon,delta} = {x : epsilon <= |C(x)| < delta}. The core is where the data approximately satisfies conservation; the boundary is where the violations are concentrated.
I_{v_cons}(w*) = E_{x in C_epsilon}[(v_cons · s(x; w*))^2] + E_{x in B_{epsilon,delta}}[(v_cons · s(x; w*))^2]
In the core, the score function's projection onto v_cons is small because the data is close to the conservation surface and the model has learned to assign high probability to conservation-respecting outcomes. In the boundary, the score projection can be large because the model is poorly fit to the rare conservation-violating examples.
Proposition NR-1 (Quadratic convergence). Let p_data^delta be a training distribution supported on X_cons^delta with the following properties:
(A1) The marginal density of conservation violation magnitude |C(x)| is bounded: f_{|C|}(c) <= M for c in [0, delta] and some constant M.
(A2) The model p(x; w*) achieves bounded KL divergence from p_data^delta: KL(p_data^delta || p(x; w*)) <= D_KL for some constant D_KL.
(A3) The score function is Lipschitz in x along the conservation direction: |v_cons · s(x; w*)| <= L * |C(x)| for x near X_cons, where L depends on the model architecture and the data distribution's curvature at the conservation boundary.
Then:
I_{v_cons}(w*) <= L^2 * M * delta^3 / 3
and more tightly, for distributions where the conservation violation is approximately uniform on [0, delta]:
I_{v_cons}(w*) = L^2 * delta^2 / 3 + O(delta^3)
The leading-order behavior is O(delta^2).
Proof. By (A3), |v_cons · s(x; w*)|^2 <= L^2 * |C(x)|^2. Therefore:
I_{v_cons}(w*) = E_{p_data^delta}[|v_cons · s(x; w*)|^2] <= L^2 * E_{p_data^delta}[|C(x)|^2]
The expectation E[|C(x)|^2] is the mean squared conservation violation in the training data. By (A1):
E[|C(x)|^2] = integral_0^delta c^2 * f_{|C|}(c) dc <= M * integral_0^delta c^2 dc = M * delta^3 / 3
For the uniform case f_{|C|}(c) = 1/delta on [0, delta]:
E[|C(x)|^2] = (1/delta) * integral_0^delta c^2 dc = delta^2 / 3
Combining: I_{v_cons}(w*) = L^2 * delta^2 / 3 + O(delta^3). QED.
Interpretation. The Lipschitz constant L is the key architectural parameter. It measures how sharply the model's output distribution changes per unit of conservation violation - how steep the "cliff" is in the model's learned representation at the conservation boundary. Deep models with sharp decision boundaries (ReLU activations, high depth) tend to have large L, while shallow models with smooth boundaries have small L. For a given delta, larger L means the model is more sensitive to conservation violations, which produces larger Fisher information along v_cons - a model that has learned a sharp conservation boundary is more "aware" of violations, which is a feature for detection purposes (it detects violations more sensitively) but means the Fisher information is not as close to zero.
Proposition NR-2 (Fisher ratio). Let I_{v_cons}^{cons}(w*) denote the Fisher information along v_cons for a model trained on conservation-respecting data (with violation magnitude delta), and I_{v_cons}^{noisy}(w*) denote the Fisher information for a model trained on data with arbitrary conservation violations (violation magnitude up to sigma >> delta). The Fisher ratio is:
FR = I_{v_cons}^{noisy} / I_{v_cons}^{cons} = (sigma / delta)^2 * (1 + O(delta/sigma))
For simulation data (delta ~ 10^{-15}) versus noisy data (sigma ~ 10^{-1}):
FR ~ (10^{-1} / 10^{-15})^2 = 10^{28}
For empirical data (delta ~ 10^{-2}) versus noisy data (sigma ~ 10^{-1}):
FR ~ (10^{-1} / 10^{-2})^2 = 100
Proof. From Proposition NR-1, I_{v_cons}^{cons} = O(delta^2) and I_{v_cons}^{noisy} = O(sigma^2). The ratio follows directly. QED.
Implication for the research program. The Phase 1 success criterion is FR > 100. For simulation data (delta ~ 10^{-15}), this is satisfied by a margin of 10^{26} - the test is a sanity check. For empirical data (delta ~ 10^{-2}), FR ~ 100 is achieved when sigma ~ 10 * delta, meaning the noisy comparison model must have conservation violations roughly ten times larger than the conservation-respecting model. This is achievable by training on data where conservation noise has been deliberately amplified, but it means the Phase 4 experiment requires careful control of the noise levels in both the conservation-respecting and conservation-violating training corpora.
Proposition NR-3 (Multiple conservation laws). Let the data-generating process satisfy K independent conservation laws C_1(x) = 0, ..., C_K(x) = 0, each violated with magnitude delta_k in the training data. Let v_{cons,k} denote the conservation-violating direction corresponding to the k-th law. Then the Fisher information matrix G(w*) has K near-zero eigenvalues, with the k-th eigenvalue satisfying:
lambda_k(G) = O(delta_k^2)
and the eigenvalues are associated with directions that are asymptotically orthogonal to each other as delta_k -> 0 for all k.
Proof. By Proposition NR-1 applied to each conservation law independently:
I_{v_{cons,k}}(w*) = O(delta_k^2)
The directions v_{cons,1}, ..., v_{cons,K} are asymptotically orthogonal because each corresponds to a distinct constraint function C_k, and the score function's projection onto v_{cons,k} depends on the violation of C_k but not on the violation of C_j for j != k (in the limit of small violations where cross-terms are higher order).
The Fisher matrix therefore has K eigenvalues of order O(delta_k^2) associated with approximately orthogonal eigenvectors. The remaining eigenvalues are O(1), corresponding to directions in weight space along which the model's output distribution changes substantially - the non-conservation directions where the model encodes the actual variation in the data.
The eigenvalue gap between the K conservation eigenvalues and the remaining eigenvalues is:
gap = min_{j > K} lambda_j(G) - max_{k <= K} lambda_k(G) >= O(1) - O(max_k delta_k^2)
which is large whenever the conservation violations are small relative to the data's natural variability. QED.
Implication for detection. Proposition NR-3 provides a quantitative test for whether a model has learned the conservation structure of a physical domain. Count the number of near-zero eigenvalues (eigenvalues below a threshold proportional to delta^2). If the count matches the number of independent conservation laws in the domain's physics, the model has learned the conservation structure. If the count is higher, the model may have overfitted to spurious regularities. If the count is lower, the model has failed to learn some conservation laws - identifying which ones are missing by examining the corresponding eigenvectors.
For the research program's stretch goal (testing across domains): an aquifer model should have one near-zero eigenvalue (mass conservation). An electrical grid model should have two (Kirchhoff's current law at each node reduces to one independent constraint per loop, but the number depends on the network topology). An atmospheric model should have one for carbon mass conservation and additional near-zero eigenvalues for energy conservation and other thermodynamic constraints. The eigenvalue count is domain-diagnostic.
Assumption (A3) - the Lipschitz condition on the score function - is the most restrictive. The Lipschitz constant L depends on the model architecture, and for models with very sharp decision boundaries (deep ReLU networks with many layers), L can be extremely large, which loosens the bound. In the worst case, L ~ exp(depth), and the O(delta^2) bound becomes O(delta^2 * exp(2 * depth)), which for deep networks and small delta could produce Fisher information that is not as close to zero as the leading-order analysis suggests. The bound is tightest for shallow networks with smooth activations and loosest for deep networks with sharp activations.
The convergence rate O(delta^2) is a worst-case bound. In practice, the Fisher information may converge faster than delta^2 if the model's loss landscape has favorable geometry near the conservation direction - for instance, if the loss surface is locally flat along v_cons (a valley rather than a ridge), the score function's projection onto v_cons is even smaller than the Lipschitz bound predicts. Empirical measurement of the convergence rate (by training models on data with controlled delta values and plotting I_{v_cons} versus delta) would provide a tighter estimate.
The proposition assumes the model has converged to a region of weight space where the training distribution is well-approximated. For large models with complex loss landscapes, the relationship between the training distribution and the Fisher geometry at the converged weights depends on the training trajectory (which local minimum the model reached). The propositions describe the expected Fisher geometry averaged over training runs, not the geometry of any specific run.