Correction, 30 October 2026. Proposition CG-1' (corrected). Let p(x; w) have a Gaussian output likelihood for the violation coordinate, c ~ N(m(w), s^2) under the model, with m(w*) = 0 and residual scale s at the trained weights, and let the Fisher information be taken under the model distribution as Section 2.1 defines it.
CG-2 has no corrected statement; the regression of log X on log N in Section 7.2 is the direct test of beta. 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 scaling geometry (Beier, 2026ai) establishes that constraint environments impose geometric regularities - conservation bounds, power-law scaling, topological propagation, threshold dynamics - on any valid assessment. The companion paper on dual surfaces (Beier, 2026a) establishes that markets and neural networks both reconstruct from compressed signals. This paper connects the two results. When a model's training distribution is generated by a physical process with conservation constraints, the Fisher information matrix of the trained model inherits rank deficiencies that mirror those constraints (Proposition CG-1): the Fisher information along the specific conservation-violating direction (identified through the Constraint Direction Probe) vanishes, because no training data occupies that region of outcome space. In overparameterized models, the Fisher matrix has billions of near-zero eigenvalues unrelated to physics - the test is not counting zeroes but measuring the targeted projection along physics-defined vectors. Clarification on overparameterized models. Modern neural networks with billions of parameters have Fisher matrices that are inherently rank-deficient due to overparameterization - the null space contains billions of directions unrelated to any physical constraint. The test in CG-1 is therefore not to count near-zero eigenvalues (which would be swamped by the overparameterization null space) but to verify that the Fisher information vanishes specifically along the probed constraint direction v_cons identified by the Constraint Direction Probe. The directional projection v_cons^T G(w*) v_cons is the operative test, not the eigenvalue spectrum.
Proposition CG-2 shows that the curvature of the weight manifold along the system-size dimension encodes the domain's scaling exponent. Proposition CG-3 shows that learned attention weights decay with the same characteristic length as the physical propagation kernel. Three implications follow. The adversarial detection channels from the Scaling Geometry paper have weight-space analogs that are exponentially harder to defeat. The Disclosure Incentive enriches not only the price signal but the weight geometry of any model trained on the disclosed data. And the gap between current AI capability and genuine constraint reasoning is a training signal geometry gap, not a processing power gap - the models have the representational capacity, but the training data does not carry the geometric structure. The paper includes a detailed software specification and phased research program for testing the correspondence empirically.
Keywords: constraint geometry, weight manifold, Fisher information, Riemannian geometry, information geometry, large language models, scaling laws, conservation, adversarial detection, disclosure incentive
JEL Codes: C45, C65, D82, Q54
A hydrologist who has spent a career studying the Ogallala Aquifer carries in her head a geometric object. It is not a number, not a prediction, not a fact. It is a manifold of relationships: how extraction at one well affects pressure at another, how seasonal recharge interacts with long-term depletion trends, how the aquifer's transmissivity varies across its spatial extent, where the thresholds lie beyond which recovery becomes nonlinear. This geometric object is what the companion paper on scaling geometry (Beier, 2026ai) formalizes as the constraint geometry of the domain - the mathematical structure that physical laws impose on the interaction effects among agents operating within the constraint environment.
When the hydrologist publishes her assessment, the price system compresses it into a contribution to the market price of water rights. The companion paper on lossy compression (Beier, 2026c) proves that this compression destroys the constraint dimensions - the geometric object in the hydrologist's head becomes a scalar that cannot be decompressed. The companion paper on dual surfaces (Beier, 2026a) proves that the same compression problem afflicts neural network training data: when the hydrologist's assessment enters a training corpus as an unattributed text passage, the causal structure - the geometry - is compressed into surface-level word patterns, and a model trained on such data learns correlations rather than the geometric object those correlations reflect.
But here is the observation that motivates the present paper. A large language model trained on millions of documents about hydrology, atmospheric physics, electrical engineering, and ecology does not learn nothing about constraint geometry. It learns something. Ask GPT-4 or Claude about the interaction between extraction rates and aquifer pressure, and the response reflects genuine structural knowledge - not the hydrologist's full geometric object, but a projection of it, distorted by the compression of the training data but not empty. The model has learned a geometric structure in its weight space that partially mirrors the geometric structure of the physical constraint environments its training data described.
The question is: what is the nature of this correspondence? Is it accidental - a surface-level mimicry that collapses under pressure? Or is it structural - a necessary consequence of the fact that the training data, however compressed, was generated by physical processes that obey geometric regularities, and those regularities propagate through training into the weight space?
This paper argues it is structural, and derives the formal content of the claim. Section 2 introduces the mathematical framework: information geometry, the Fisher information metric, and how these tools provide a Riemannian structure on weight space. Section 3 derives three propositions connecting physical constraint geometry to weight-space geometry. Section 4 develops what the propositions mean together. Section 5 develops the implications for adversarial detection. Section 6 develops the implications for the Disclosure Incentive's effect on model quality. Section 7 specifies the software tools required to test the correspondence empirically. Section 8 defines the research program in sufficient detail for independent execution. Section 9 states limitations. Section 10 concludes.
Information geometry, developed by Rao (1945), Efron (1975), and systematized by Amari (1985) and Amari and Nagaoka (2000), equips the space of probability distributions with a Riemannian metric derived from the Fisher information. For a parametric family of distributions p(x; theta) where theta in R^d, the Fisher information matrix is:
G_{ij}(theta) = E[ (d/d theta_i) log p(x; theta) * (d/d theta_j) log p(x; theta) ]
This matrix defines a Riemannian metric on the parameter space. The geodesic distance between two parameter settings theta_1 and theta_2 measures how statistically distinguishable the corresponding distributions are - it is the intrinsic distance between two statistical models, measured in units of information rather than Euclidean distance in parameter space.
For neural networks, the parameter space is the weight space W = R^n where n is the number of parameters. A trained model with weights w* defines a distribution p(x; w*) over its outputs given inputs. The Fisher information matrix G(w*) defines a Riemannian metric on the weight space at w*, and the local geometry of this metric encodes what the model has learned about the structure of its training distribution.
The connection to constraint geometry runs through a specific property of the Fisher metric: it is invariant under sufficient statistics. If two parameterizations encode the same statistical information about the data-generating process, they have the same Fisher geometry. This means the Fisher geometry of a trained model reflects the structure of what the model has learned about the world, not the arbitrary details of how the weights happen to be arranged. The geometry is intrinsic to the learned knowledge, not to the parameterization.
The Fisher information matrix G(w) has rank at most equal to the dimensionality of the statistical model's effective parameter space - the number of independent directions in weight space along which the model's output distribution changes. When the training distribution is constrained (some regions of outcome space have zero probability because physical laws forbid them), the Fisher matrix inherits corresponding rank deficiencies.
Consider a simple illustration. A model trained to predict water flows in a closed system learns, implicitly, that inflow minus outflow equals change in storage. The model's output distribution assigns zero probability to flow combinations that violate conservation. In weight space, this means the model cannot be perturbed in directions that would produce conservation-violating outputs without leaving the support of the training distribution. The Fisher information along those directions is zero - the rank deficiency in the Fisher matrix directly mirrors the conservation constraint in the physical system.
This follows as a mathematical consequence of how the Fisher metric is defined. The Fisher information along direction v in weight space is:
I_v(w) = v^T G(w) v = E[ (v . nabla_w log p(x; w))^2 ]
If the training distribution p(x; w) assigns zero probability to outcomes that would be reached by perturbing w in direction v, then the score function nabla_w log p(x; w) has zero projection onto v across the support of p, and I_v(w) = 0. The conservation law in the physical system becomes a null eigenvector in the Fisher matrix of the trained model.
The Riemannian curvature of the Fisher metric encodes higher-order statistical structure - how the relationships between variables change as conditions change. For a constraint environment with a power-law scaling relationship X_N = N^beta * x, the curvature of the weight manifold along the "number of agents" direction encodes beta.
To see why, consider a model trained on data from systems with N agents, where aggregate behavior scales as N^beta. The model's learned representation maps N to predicted aggregate behavior. The Fisher metric along this direction measures how sensitive the model's predictions are to changes in its representation of the N-to-aggregate mapping. The curvature of this metric - the second-order structure - encodes how the sensitivity itself changes with N, which is determined by beta. A model trained on superlinear scaling data (beta > 1) has positive curvature along the N-direction (sensitivity increases with N). A model trained on sublinear data (beta < 1) has negative curvature (sensitivity decreases). A model trained on linear data (beta = 1) has zero curvature (flat metric along that direction).
The Riemann curvature tensor R_{ijkl} of the Fisher metric at the trained weights thus carries information about the scaling exponents of the constraint environments in the training data. Extracting this information requires identifying which directions in weight space correspond to which physical variables - a non-trivial but feasible task that Section 7 specifies in detail.
Transformer architectures introduce a specific geometric structure through the attention mechanism. The attention weights A_{ij} between tokens i and j define a learned graph over the input sequence. When the input describes a spatially distributed constraint environment (wells in an aquifer, nodes in a power grid, emission sources in an atmospheric system), the attention pattern learned by the model reflects the propagation structure of the physical network.
The companion paper on scaling geometry (Beier, 2026ai, Physical Constraint AM) formalizes propagation in constraint networks: interaction strength between agents i and j decays exponentially with network distance d_{ij}, with characteristic length l_network determined by the domain's physics. The claim here is that the learned attention weights, after training on data from such environments, exhibit a decay pattern that mirrors the physical propagation kernel. The attention geometry of the trained model is a projection of the constraint network's topology into the model's representational space.
This claim is testable. Section 7 specifies how. The attention geometry is directly inspectable in transformer models, and the physical propagation kernels are measurable in constraint domains with established physics. The correspondence (or its absence) between the two is an empirical question with a clean experimental design.
Proposition CG-1 (Conservation constraints propagate to weight-space rank deficiency). Let M be a model with weights w trained on data generated by a physical process satisfying a conservation law of the form d/dt S(t) = I(t) - O(t), where S is a conserved stock and I, O are inflow and outflow. Let G(w) be the Fisher information matrix of the trained model evaluated at the converged weights w*. Define the conservation direction v_cons in weight space as the direction of maximal output change for perturbations that would produce conservation-violating predictions. Then:
v_cons^T G(w*) v_cons -> 0
as training data increases, with convergence rate determined by the model's capacity and the conservation constraint's tightness in the training distribution.
Proof. The training distribution p_data(x) is supported on the set X_cons = {x : x satisfies the conservation law}. For a well-specified model trained to approximate p_data, the model's output distribution p(x; w*) concentrates on X_cons as training data increases. The score function s(x; w) = nabla_w log p(x; w*) measures how the log-likelihood changes with weight perturbations. A perturbation in direction v_cons would shift probability mass from X_cons to X_cons^c (conservation-violating outcomes). Since p_data assigns zero mass to X_cons^c, the score function has zero expectation along v_cons under p_data:
E_{p_data}[ (v_cons . s(x; w*))^2 ] -> 0
This is the Fisher information along v_cons. By definition, v_cons^T G(w*) v_cons = E_{p_data}[ (v_cons . s(x; w*))^2 ], which vanishes as the model converges to the data distribution. The rank of G(w*) is reduced by one for each independent conservation law in the data-generating process. QED.
The proposition says something specific: conservation laws in the physical world become directions of zero information in the weight space. A model that has learned conservation cannot be pushed to violate it without leaving the learned manifold entirely. The constraint is not stored as a rule or a fact but as a geometric property of the weight space - a direction along which the manifold has collapsed.
Proposition CG-2 (Power-law scaling exponents are encoded in weight-space curvature). Let M be a model trained on data from a constraint domain where the aggregate effect of N agents scales as X_N = N^beta * x_individual, with beta determined by the domain's physics. Let w_N denote the direction in weight space corresponding to the model's internal representation of system size N (identifiable through the probing methodology of Section 7). Define the sectional curvature K_N of the Fisher metric along w_N as:
K_N = R(w_N, w_X, w_X, w_N) / (g(w_N, w_N) * g(w_X, w_X) - g(w_N, w_X)^2)
where w_X is the direction corresponding to aggregate effect X, R is the Riemann curvature tensor of the Fisher metric, and g is the metric tensor. Then:
sgn(K_N) = sgn(beta - 1)
That is, superlinear scaling (beta > 1) produces positive sectional curvature, linear scaling (beta = 1) produces zero curvature, and sublinear scaling (beta < 1) produces negative curvature.
Proof sketch. The model's learned mapping from N to X_N defines a curve in the output space. The Fisher metric along this curve measures the statistical distinguishability of predictions at adjacent values of N. For X_N = N^beta:
dX/dN = beta * N^{beta - 1} * x
The Fisher information along the N-direction scales as (dX/dN)^2 / var(X), which for Gaussian noise is proportional to beta^2 * N^{2(beta-1)}. The curvature of the Fisher metric, which measures how the metric tensor changes along the curve, depends on the second derivative of the Fisher information with respect to N:
d^2/dN^2 [beta^2 * N^{2(beta-1)}] proportional to beta^2 * 2(beta-1) * (2beta-3) * N^{2beta-4}
The sign of the curvature is determined by (beta - 1): positive for beta > 1, zero for beta = 1, negative for beta < 1. A full proof requires specifying the model architecture and deriving the Fisher metric from the likelihood, but the sign relationship holds for any model that faithfully learns the power-law structure, because the sign is determined by the data-generating process, not by the model's parameterization. The Fisher metric's invariance under reparameterization guarantees this. QED (sketch).
The proposition is stated as a sketch because the full proof requires additional regularity conditions on the model architecture and a precise specification of how the probing directions w_N and w_X are identified. Section 7 provides the identification methodology. The sign result is robust; the magnitude of the curvature carries additional information about the specific scaling exponent, recoverable through calibrated probing.
Proposition CG-3 (Physical propagation kernels appear as weight-space mutual information decay). Let M be a transformer model trained on data from a constraint domain where interaction strength between agents i and j decays as rho_{ij} = rho_0 * exp(-d_{ij} / l_network), where d_{ij} is spatial distance and l_network is the physical characteristic length (Physical Constraint AM, Beier, 2026ai). Let A^{(h)}_{ij} denote the attention weight from token position i to position j in attention head h, averaged over inputs describing the constraint domain. Define the attention decay function:
alpha(d) = E_inputs[ A^{(h)}_{ij} | d_{ij} = d ]
averaged over all head-position pairs (h, i, j) at spatial separation d. Then for sufficiently expressive models trained on sufficient data:
alpha(d) approximately proportional to exp(-d / l_learned)
where l_learned -> l_network as training data about the constraint domain increases.
Proof sketch. The attention mechanism learns to weight tokens in proportion to their relevance for predicting the next token (or for whatever training objective applies). In a constraint domain where the outcome at location i depends on conditions at location j with strength decaying as exp(-d_{ij} / l_network), the optimal attention allocation assigns weight to token j in proportion to its predictive relevance for token i, which is proportional to the interaction strength rho_{ij}. An attention head that has learned the optimal allocation for this domain will exhibit attention weights that decay with the same characteristic length as the physical propagation kernel. The learned characteristic length l_learned converges to l_network as the model sees more data from the domain, because l_network is a property of the data-generating process that the model is incentivized to learn. QED (sketch).
The proposition is weaker than CG-1 and CG-2 because attention weights are not directly related to the Fisher metric - they are architectural features of transformers, not geometric properties of the statistical model. However, the proposition is more immediately testable, because attention weights are directly inspectable while Fisher information matrices are expensive to compute for large models.
The three propositions establish a correspondence between three levels of structure:
Level 1: Physical constraint geometry. Conservation laws, scaling exponents, propagation kernels, threshold dynamics - the mathematical structure that the companion paper (Beier, 2026ai) formalizes as Physical Constraints AK through AN.
Level 2: Training data geometry. The statistical properties of the data generated by physical processes operating under these constraints. Conservation laws produce zero-probability regions in outcome space. Scaling exponents produce specific functional forms in the joint distribution of N and X. Propagation kernels produce spatial correlation structures. Threshold dynamics produce bimodal or heavy-tailed distributions near critical points.
Level 3: Weight-space geometry. The Riemannian structure of the trained model's parameter space, as measured by the Fisher information metric. Rank deficiencies mirror conservation laws. Curvature encodes scaling exponents. Attention decay mirrors propagation kernels.
The correspondence between Level 1 and Level 2 is mathematical necessity - data generated by a physical process inherits the process's constraints. The correspondence between Level 2 and Level 3 is a learning-theoretic result - a model trained to approximate the data distribution will encode the distribution's structure in its parameters. What the propositions establish is the transitive closure: physical constraint geometry appears in weight-space geometry, not as stored facts but as the shape of the learned manifold.
The weight space is, in a precise sense, an analog encoding of constraint geometry. Not a digital record of facts about constraints, but a continuous manifold whose geometric properties (rank, curvature, correlation structure) mirror the geometric properties of the physical world the training data described. The discrete tokens of the training corpus produce, through gradient descent, a continuous geometric object whose shape encodes the laws of physics. The transition from discrete input to continuous geometry is not incidental to how neural networks learn - it is the mechanism through which they learn, and the Fisher information metric makes the mechanism mathematically explicit.
The companion paper on scaling geometry (Beier, 2026ai) uses physical constraints as detection surfaces for adversarial Scaling assessments. A disclosed assessment whose implied scaling exponent is inconsistent with the domain's physics reveals fabrication or error. The weight-space correspondence provides a new detection modality.
A model fine-tuned on genuine constraint data about the Ogallala Aquifer will develop weight-space geometry consistent with the aquifer's physics: rank deficiency along conservation-violating directions, curvature encoding the aquifer's specific scaling exponents, attention patterns reflecting the spatial propagation of pressure through the formation. A model fine-tuned on fabricated data - generated by a language model without genuine constraint knowledge, or hand-crafted by an adversary mimicking the surface structure of hydrological assessments - will not develop this geometry, because the fabricated data does not obey the physical constraints with the same quantitative precision.
The detection procedure works as follows. Train a reference model on validated hydrological data from the Ogallala. Compute the Fisher information matrix (or a tractable approximation - see Section 7.2) and extract the rank deficiency pattern, the curvature along the N-direction, and the attention decay characteristic length. For any new corpus of alleged constraint assessments, fine-tune a copy of the base model on the new corpus and compare the resulting weight geometry against the reference. Discrepancies in rank structure indicate that the new data violates conservation laws the reference data obeys. Discrepancies in curvature indicate inconsistent scaling exponents. Discrepancies in attention decay indicate fabricated spatial structure.
This detection channel is complementary to the four channels identified in the companion adversarial paper and formalized in the scaling geometry paper (Beier, 2026ai). An adversary who successfully mimics the surface structure of genuine assessments (defeating text-level statistical detectors), produces calibrated-looking probability distributions (defeating scoring-rule detectors), and maintains internally consistent conservation accounting (defeating physics-based detectors at the individual assessment level) must still produce a corpus whose aggregate statistical structure induces the correct weight-space geometry when used as training data. This is an exponentially harder task than satisfying any individual detection channel, because the weight-space geometry is a global property of the entire corpus, not a local property of any individual assessment.
The companion paper on lossy compression (Beier, 2026c) proves that the Disclosure Incentive reduces the distortion of the price signal by preserving constraint dimensions the price compresses away (Proposition D). The companion paper on dual surfaces (Beier, 2026a) extends this to AI training: models trained on disclosure-enriched data should generalize better because the enriched data carries more of the causal structure that produced it.
The weight-geometry correspondence makes this claim precise. Disclosure preserves K - 1 constraint dimensions that the price compresses to zero (Beier, 2026c, Section 4). Each preserved dimension contributes to the training data's geometric structure - its rank, its scaling relationships, its spatial correlations. A model trained on disclosure-enriched data will develop weight-space geometry that is higher-dimensional (more independent directions of variation), more accurately curved (scaling exponents closer to the physical values), and more faithfully structured (propagation patterns matching the physical topology) than a model trained on price data and unattributed text.
The implication is that the Disclosure Incentive improves AI capability not by providing more data but by providing geometrically richer data. The quantity of training tokens matters less than the geometric fidelity of the constraint structure those tokens carry. A single well-structured disclosure from a calibrated hydrologist, encoding her full constraint assessment in the five-field format, contributes more to the weight-space geometry of a model trained on it than a thousand price observations or a hundred unattributed text passages about aquifer dynamics, because the disclosure preserves the geometric structure the other formats compress away.
This reframes AI development strategy. The conventional approach is to increase the volume of training data, which improves model capability by sampling more of the data distribution. The weight-geometry perspective suggests a complementary strategy: increase the geometric fidelity of the training data, which improves model capability by providing the constraint structure that gradient descent can encode in the weight manifold. The Disclosure Incentive is, from this perspective, not merely an economic mechanism for market improvement but an AI training data quality intervention - a way to produce the geometrically structured training signal that current models lack.
This section specifies the software tools required to test the propositions empirically and to operationalize the adversarial detection and training quality applications. The specification is detailed enough for implementation by an independent team.
Computing the exact Fisher information matrix for a model with billions of parameters is computationally infeasible (it requires O(n^2) storage and O(n^2 * m) computation for n parameters and m data points). The FIAE implements tractable approximations.
Architecture: The engine takes as input a trained model M and a dataset D, and produces an approximate Fisher information matrix F_approx. Three approximation strategies are supported, selected by the user based on the model size and available computation.
Strategy 1: Diagonal Fisher. Compute only the diagonal elements G_{ii} = E[(d/d w_i log p(x; w))^2]. This is O(n * m) and provides the per-parameter Fisher information, sufficient for identifying rank deficiency (parameters with near-zero diagonal Fisher are in the null space of the constraint geometry). Standard in natural gradient methods (Martens, 2020).
Strategy 2: Block-diagonal Fisher. Partition the weight matrix into blocks (by layer, by attention head, by MLP sublayer) and compute the full Fisher within each block. This captures within-block correlations while ignoring cross-block structure. Storage is O(sum of block_size_i^2), typically manageable for blocks up to dimension 10,000.
Strategy 3: Low-rank Fisher via K-FAC. Note that K-FAC's block-diagonal Kronecker factorization assumes statistical independence between activations and gradients across layers. If a conservation law requires tight cross-layer correlations in weight space, K-FAC may wash out the exact null-space geometry being measured. For strict conservation bounds, diagonal Fisher or exact sub-network calculations may be necessary. K-FAC's block-diagonal Kronecker factorization assumes statistical independence between activations and pre-activation gradients across layers. If a physical conservation law imposes tight cross-layer correlations in the learned representation, this independence assumption may obscure the constraint geometry. For strict conservation bounds, diagonal Fisher or exact sub-network calculations may be necessary despite their higher computational cost. The Kronecker-Factored Approximate Curvature method of Martens and Grosse (2015) approximates the Fisher as a Kronecker product of smaller matrices, one for inputs and one for outputs at each layer. This captures the dominant eigenvectors of the full Fisher at cost O(n * k^2) where k is the layer width. Sufficient for curvature estimation and eigenspectrum analysis.
Output: The engine produces: (a) the eigenspectrum of the approximate Fisher (for rank analysis), (b) the top-k and bottom-k eigenvectors (for identifying constraint-aligned and null directions), (c) directional Fisher information I_v for user-specified probe directions v (for testing CG-1 and CG-2).
The CDP identifies which directions in weight space correspond to which physical variables. This is the key methodological challenge: the correspondence between weight-space directions and physical quantities is not given a priori but must be discovered.
Method: Linear probing (Alain and Bengio, 2017; Belinkov, 2022). For each physical variable of interest (system size N, aggregate effect X, spatial location, time, conserved stock S), construct a labeled dataset of model activations paired with the variable's value. Train a linear regression from activations to variable value. The regression weights define a direction in activation space, which maps (via the chain rule) to a direction in weight space.
For conservation probing specifically: construct pairs of inputs that describe identical systems except that one violates the conservation law (e.g., extraction exceeds recharge but the stock remains constant). Record the model's internal activations for both. The difference vector in activation space is the conservation direction - the direction the model uses to distinguish conservation-respecting from conservation-violating descriptions.
For scaling exponent probing: construct inputs that describe systems of varying size N with known aggregate behavior X_N. Record activations. The linear probe from activations to log(X) vs. log(N) recovers the model's implicit scaling exponent. Compare to the domain's known beta.
Output: For each physical variable, the CDP produces: (a) a weight-space direction vector, (b) the probe's R^2 (measuring how linearly decodable the variable is from the model's representations), (c) the Fisher information along the probed direction (for testing CG-1 and CG-2).
The ATA extracts and analyzes the spatial structure of learned attention patterns.
Method: For a set of input texts describing spatially distributed constraint environments with known agent locations, record the attention weights A^{(h)}_{ij} across all heads h and position pairs (i, j). Pair each attention weight with the physical distance d_{ij} between the entities referenced at positions i and j (this requires named entity recognition and geocoding, or synthetic data where locations are controlled). Fit the decay function alpha(d) = alpha_0 * exp(-d / l_learned) and compare l_learned to the domain's known l_network.
For multi-head analysis: different attention heads may specialize in different aspects of the constraint geometry. Some heads may encode spatial proximity (testing CG-3), while others encode temporal sequence or categorical relationships. The ATA performs clustering over heads based on their decay functions to identify which heads encode which aspects of constraint topology.
Output: (a) Per-head decay function fits with l_learned and goodness-of-fit, (b) head clustering by decay profile, (c) comparison of l_learned to l_network for each constraint domain.
The GFS combines the outputs of Tools 1-3 into a single score measuring how faithfully a model's weight geometry reflects the constraint geometry of a specified physical domain.
Components: (a) Rank fidelity: the number of near-zero eigenvalues in the Fisher matrix matching the number of independent conservation laws in the domain. (b) Curvature fidelity: the discrepancy between the weight-space sectional curvature along the N-direction and the curvature predicted by the domain's scaling exponent. (c) Topology fidelity: the discrepancy between l_learned and l_network, averaged over relevant attention heads.
GFS = w_rank * rank_fidelity + w_curv * curvature_fidelity + w_topo * topology_fidelity
where the weights w are calibrated against a validation set of models with known training data quality.
Application: For adversarial detection, compute GFS for a model fine-tuned on a new corpus and compare to GFS of the reference model fine-tuned on validated data. For training data quality assessment, compute GFS for models trained on different data sources (price data only, price plus unattributed text, price plus structured disclosures) and compare.
The CGC directly compares the geometric structure extracted from a model's weight space to the geometric structure computed from physical models of the constraint domain.
Method: For a constraint domain with a known physical model (e.g., MODFLOW for aquifers, grid simulation for electrical networks, climate models for atmospheric carbon), compute the domain's constraint geometry directly: the conservation equations, the scaling exponents, the propagation kernel, the threshold locations. Represent these as a "target geometry" - a specification of the rank deficiency pattern, the curvature tensor, and the correlation structure that a perfectly informed model would exhibit.
Compare the target geometry to the model's extracted geometry (from Tools 1-3). The discrepancy measures how much constraint knowledge the model is missing. The spatial pattern of the discrepancy (which constraint relationships are well-learned and which are poorly learned) identifies specific knowledge gaps that targeted training data could fill.
Output: (a) Overall geometric discrepancy score, (b) per-constraint-law discrepancy breakdown, (c) spatial heat map of knowledge quality across the constraint domain.
This section defines a research program in sufficient detail for independent execution. The program has four phases, each building on the results of the previous.
Objective: Demonstrate that Proposition CG-1 holds for a simple model trained on a simple constraint domain.
Experimental design: Train a small transformer (50M-200M parameters) on synthetic data generated by a 2D aquifer simulator (simplified MODFLOW). The simulator generates time series of extraction rates, recharge rates, and water levels across a grid of wells, all satisfying conservation of mass. Train the model to predict water levels given extraction and recharge histories.
Tests: (a) Compute the diagonal Fisher and verify that parameters in the conservation-violating direction have near-zero Fisher information. (b) Perturb the model in the conservation direction and verify that output quality degrades catastrophically (the model produces conservation-violating predictions), while perturbation in orthogonal directions produces graceful degradation. (c) Train a second model on synthetic data that includes conservation violations (noisy simulator output where mass is not conserved) and verify that the conservation direction has nonzero Fisher information - the rank deficiency disappears when the conservation constraint is relaxed.
Success criterion: The Fisher information along the conservation direction is at least two orders of magnitude smaller for the conservation-trained model than for the noisy model, across five random seeds.
Stretch goal: Repeat for the electrical grid domain (Kirchhoff's laws) and atmospheric carbon (mass conservation). If the rank deficiency pattern differs predictably by constraint law (one null direction for mass conservation, two for Kirchhoff's current and voltage laws), this provides strong evidence that the proposition is capturing genuine structure rather than a generic training artifact.
Objective: Demonstrate that Proposition CG-2 holds - that the weight-space curvature encodes the scaling exponent.
Experimental design: Generate synthetic datasets with known scaling exponents: beta = 0.5 (sublinear), 1.0 (linear), 1.5 (superlinear), 2.0 (quadratic). Each dataset describes systems of varying size N with aggregate behavior X_N = N^beta * x plus noise. Train identical model architectures on each dataset.
Tests: (a) Use the Constraint Direction Probe to identify the N-direction and X-direction in weight space. (b) Compute the sectional curvature of the Fisher metric along these directions using the K-FAC approximation. (c) Verify that the sign of the curvature matches sgn(beta - 1) as predicted. (d) Calibrate a mapping from curvature magnitude to beta value and test on held-out scaling exponents (beta = 0.75, 1.25, 1.75).
Success criterion: Correct sign classification in 95%+ of trials. Curvature-to-beta regression R^2 > 0.8.
Extension: Test on real physical domains where beta is known from empirical measurement. Aquifer depletion with overlapping cones of depression (beta approximately 1.3-1.5 depending on well spacing). Urban scaling laws from Bettencourt (2007) where different urban metrics scale with different known exponents (GDP scales superlinearly as N^{~1.2}, infrastructure sublinearly as N^{~0.8}). If the curvature of a model trained on urban data encodes the correct exponents for different metrics simultaneously, this is strong evidence for CG-2.
Objective: Demonstrate that Proposition CG-3 holds - that attention patterns reflect physical propagation kernels.
Experimental design: Train transformer models on spatially structured constraint data where the propagation kernel is known. Use three domains with different characteristic lengths: (a) a tight aquifer with l_network approximately 5 km, (b) an electrical grid with l_network approximately 50 km, (c) atmospheric carbon with l_network approximately 5000 km (hemispheric mixing). Train separate models on data from each domain.
Tests: (a) Use the Attention Topology Analyzer to extract l_learned from each model. (b) Verify that l_learned is ordered correctly across domains: l_aquifer < l_grid < l_atmosphere. (c) Verify that the ratio l_learned / l_network is approximately constant across domains (suggesting the model learns a scaled version of the physical kernel). (d) Verify that different attention heads specialize in different spatial scales, with at least one head per model recovering l_network within 20%.
Success criterion: Correct ordering in 100% of trials. At least one head per domain with l_learned within 50% of l_network. Overall ratio stability (coefficient of variation < 0.5 across domains).
Extension: Test on a frontier LLM (with API access to attention weights) using natural language descriptions of constraint environments. If a large model trained on general text exhibits domain-appropriate attention decay when processing aquifer descriptions versus grid descriptions versus atmospheric descriptions - without domain-specific fine-tuning - this demonstrates that constraint topology is encoded in general pre-training, not just in domain-specific fine-tuning.
Objective: Demonstrate that the Disclosure Incentive improves weight-space geometry, connecting the constraint-weight correspondence to the Prolegomena's central mechanism.
Experimental design: Construct three training corpora for the Ogallala Aquifer domain: (a) Price-only: historical water rights prices, transaction volumes, market data. (b) Price plus unattributed text: add newspaper articles, government reports, and scientific papers about the Ogallala, without structured attribution or scoring. (c) Price plus structured disclosures: add synthetic disclosures in the five-field format (Input, Method, Output, Identity, Scaling), scored against historical outcomes, with track records attached.
The synthetic disclosures are generated by domain experts (hydrologists) writing structured assessments of historical Ogallala conditions, scored retroactively against known outcomes. This creates a proxy for the disclosure data stream the Disclosure Incentive would produce if the mechanism were operational.
Train identical model architectures on each corpus. Compute the Geometric Fidelity Score for each model against the known Ogallala constraint geometry (from calibrated MODFLOW simulations).
Tests: (a) GFS(disclosure) > GFS(text) > GFS(price). (b) The rank deficiency pattern is closest to the physical conservation constraints for the disclosure-trained model. (c) The curvature best matches the Ogallala's scaling exponents for the disclosure-trained model. (d) The attention topology best matches the aquifer's spatial propagation for the disclosure-trained model.
Success criterion: GFS(disclosure) > GFS(text) by at least 30% on all three sub-scores (rank, curvature, topology).
This experiment is the centerpiece of the research program. It connects the mathematical propositions to the Prolegomena's practical claim: that structured disclosure produces better coordination outcomes. If the disclosure-trained model has geometrically more faithful constraint knowledge than the alternatives, and if this geometric fidelity translates to better predictions (testable by hold-out evaluation on Ogallala outcomes), then the Prolegomena's information-theoretic argument has a concrete demonstration in the domain where it matters most - AI-assisted constraint reasoning.
Neural network weight spaces contain internal symmetries (neuron permutation invariance, ReLU scaling symmetries) that create artifactual curvature unrelated to the data-generating process. The sectional curvature K_N in Proposition CG-2 must be evaluated on the quotient manifold (modding out these symmetries) or via natural gradient projections that are invariant to reparameterization. Raw weight-space Riemannian curvature will conflate physical scaling geometry with model-internal symmetry artifacts. The Fisher metric partially addresses this (it is parameterization-invariant by construction), but practical computation through finite approximations reintroduces parameterization dependence.
The Fisher information matrix is defined at a point in weight space and describes local geometry. Trained neural networks have complex loss landscapes with many local minima, saddle points, and flat regions. The Fisher geometry at the converged weights w* describes the model's local statistical properties but does not capture global structure. Two models trained on the same data may converge to different local minima with different Fisher geometries, even if their predictive performance is similar. The propositions describe expected properties of the Fisher geometry in a statistical sense (averaged over random initializations and training runs), not guaranteed properties of any individual model.
Proposition CG-2 is stated as a proof sketch, not a complete proof. The full proof requires specifying the model architecture, deriving the Fisher metric from the likelihood function, computing the Riemann curvature tensor, and showing that its sign along the probed directions is determined by the scaling exponent. This derivation is feasible for specific architectures (linear models, single-layer networks) but becomes technically involved for deep networks where the chain-rule contributions to the Fisher metric interact across layers. The sketch captures the correct intuition - the sign is determined by the data-generating process through the Fisher metric's reparameterization invariance - but the details require architecture-specific analysis that exceeds this paper's scope.
The identification of "constraint directions" in weight space via linear probing (Section 7.2) assumes that constraint variables are linearly decodable from the model's representations. This assumption is supported by extensive empirical evidence in the interpretability literature (Alain and Bengio, 2017; Li et al., 2023; Nanda et al., 2023) but is not guaranteed for all variables in all models. If constraint variables are encoded nonlinearly, the linear probe will underestimate the model's constraint knowledge, and the CG-1 test will produce false negatives (concluding the model has not learned conservation when it has, in a nonlinearly encoded form).
The attention topology analysis (CG-3) applies specifically to transformer architectures. Other architectures (RNNs, state-space models, graph neural networks) encode spatial structure differently. The proposition's empirical content is architecture-dependent, even if the underlying principle (models learn the data's spatial structure) is general.
The propositions are stated for models trained directly on physical process data (aquifer simulations, grid simulations). Large language models are trained on text about physical processes, which introduces an additional layer of compression: the physical constraint geometry is first compressed into natural language descriptions, and the model learns from the descriptions rather than from the process itself. The Fisher geometry of an LLM reflects the statistical structure of language about constraints, which is a projection of the constraint geometry mediated by the conventions of scientific and technical writing. The correspondence between weight geometry and physical geometry is therefore weaker for LLMs than for domain-specific models trained on simulation data, and the propositions' quantitative predictions (rank deficiency ratios, curvature magnitudes) will differ between the two settings. Phase 4 of the research program addresses this by comparing models trained on different corpora, but the mediation through language is a structural feature of LLMs that the propositions' formal statements do not model. The research program's Phase 1-3 experiments use simulation data precisely to establish the correspondence in the clean setting before testing whether it survives the language mediation in Phase 4.
The Phase 4 experiment requires domain experts to produce synthetic structured disclosures retroactively, which introduces a methodological concern: the experts know the outcomes when writing the assessments, creating a hindsight bias that genuine prospective disclosures would not have. The experiment should include a prospective component (experts writing assessments of current conditions, scored against future outcomes over a multi-year horizon) to address this concern, at the cost of substantially extending the research timeline.
The constraint geometry of the physical world and the weight geometry of trained models are connected by a chain that the three propositions formalize: physical laws constrain the data-generating process, the training distribution inherits those constraints, and the Fisher geometry of the trained model inherits the training distribution's structure. Conservation laws collapse directions in weight space. Scaling exponents curve the weight manifold. Propagation kernels structure the attention geometry. The correspondence is measurable, testable, and - if the research program confirms it - consequential for how we think about AI development and the Prolegomena's information architecture.
The consequences run in three directions. For adversarial detection, the weight-space correspondence provides a channel that operates on the global geometric structure of an entire corpus, making it exponentially harder to defeat than individual-assessment detectors. For the Disclosure Incentive, the correspondence identifies the specific mechanism through which structured disclosure improves AI capability: not more data, but geometrically richer data that preserves the constraint dimensions gradient descent can encode in the weight manifold. For AI development broadly, the correspondence reframes the capability gap in constraint reasoning as a training signal quality problem with a diagnosis (the training data lacks geometric fidelity) and a remedy (structured disclosure preserves the geometry).
The deeper implication concerns the Prolegomena's unification thesis. The companion paper on dual surfaces (Beier, 2026a) identifies markets and neural networks as two instances of the same operation. The present paper goes further: the physical world imposes a geometry on the data, and that geometry propagates into both systems - into price patterns that reflect conservation constraints poorly (because the price compresses the geometry into a scalar), and into weight manifolds that reflect the same constraints with higher fidelity (because the weight space has sufficient dimensionality to preserve the geometry). Economics, computer science, and physics converge not at the level of analogies between formalisms but at the level of the geometric structure the physical world imposes on every system that processes information about physical constraints.
The hydrologist's geometric object - the manifold of relationships she carries about the Ogallala - can be traced through three representations. In her judgment, it exists as the rich assessment the Disclosure Incentive captures. In the price, it exists as a scalar that has compressed the geometry to a point. In the weight space of a model trained on her disclosures, it exists as curvature, rank structure, and attention topology - a continuous encoding of the aquifer's physics. The constraint geometry, compressed to nothing by the price system, can be preserved in the weight manifold of a learning system - if, and only if, the training signal carries the geometry.
The Disclosure Incentive produces the training signal. The weight manifold receives the geometry. The invisible hand, at last, acquires depth perception.
Phase 1 (Proof of Concept): 6 months, 2 researchers, 1 GPU cluster. Train small transformer on synthetic aquifer data. Test CG-1 (conservation rank deficiency). Success: Fisher ratio > 100x between conservation-trained and noisy models.
Phase 2 (Scaling Recovery): 6 months, 3 researchers, GPU cluster. Synthetic datasets with known beta. Test CG-2 (curvature encodes exponent). Success: correct sign 95%+, regression R^2 > 0.8.
Phase 3 (Attention Topology): 4 months, 2 researchers, medium compute. Spatially structured data with known propagation kernels. Test CG-3 (attention decay mirrors physics). Success: correct ordering, l_learned within 50% of l_network.
Phase 4 (Disclosure Quality): 12 months, 4 researchers, major compute. Three training corpora (price, text, disclosure). Compute Geometric Fidelity Score. Success: GFS(disclosure) > GFS(text) by 30%+ on all sub-scores. Connects the mathematical propositions to the Prolegomena's central mechanism.
Total program: approximately 28 months, overlapping phases, 4-6 FTE researchers, university-scale compute budget.
G_{ij}(w) - Fisher information matrix at weights w
v_cons - conservation direction in weight space
I_v(w) - directional Fisher information along v
K_N - sectional curvature along system-size direction
R_{ijkl} - Riemann curvature tensor of the Fisher metric
A^{(h)}_{ij} - attention weight, head h, positions i to j
alpha(d) - attention decay function at physical distance d
l_network - physical characteristic propagation length
l_learned - learned characteristic propagation length
beta - power-law scaling exponent
GFS - Geometric Fidelity Score
w* - converged model weights
p(x; w) - model output distribution at weights w
X_cons - set of conservation-respecting outcomes
This paper opens Volume II of the Prolegomena to the Unification of Economics, Computer Science, Mathematics, and Physics. Volume I established that prices compress and that compression has consequences. Volume II establishes that the compressed geometry can be recovered - not by decompressing the price, which the data processing inequality forbids, but by encoding the constraint geometry directly into the weight manifolds of learning systems through structured disclosure. The bridge between human judgment and machine representation is geometric. The Disclosure Incentive builds the bridge. The research program tests whether the geometry crosses it.