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 Prolegomena uses the term "Beier Tensor" throughout to describe a Deep Price - a price augmented with structured Disclosure. The term promises specific mathematical content: a tensor is a geometric object with defined transformation properties under change of coordinates. This paper delivers that content. The five fields of Beier Disclosure (Input, Method, Output, Identity, Scaling) define a coordinate basis in a five-dimensional information space. The track record weight provides a metric tensor that defines the inner product between disclosures. The transformation law under change of analytical framework (from a fund manager's coordinates to a regulator's coordinates to a farmer's coordinates) preserves the information content of the disclosure while changing its representation - the defining property of a tensor. The paper proves three results. Proposition BT-1 establishes the coordinate structure: the five fields span a vector space whose dimension equals the number of independent information channels the disclosure carries, and this dimension is invariant under change of framework. Proposition BT-2 establishes the metric: the track record weight defines a positive semi-definite bilinear form on the disclosure space that is framework-invariant, making the inner product between two disclosures (their informational agreement) independent of who is reading them. Proposition BT-3 establishes the connection to welfare: the Blackwell improvement (BE-3) is equivalent to the statement that the Beier Tensor carries strictly more information than the scalar price in every coordinate system - the welfare result is the tensor's invariance property expressed in decision-theoretic language. The formalization resolves the gap between the term the book uses and the mathematical content it promises, and provides the foundation for the geometric treatment of Deep Prices developed in Volume VIII.
Keywords: Beier Tensor, Deep Price, coordinate transformation, metric tensor, track record, information geometry, Blackwell improvement
A Blinding Price is a scalar p in R. It carries one real number. Every participant reads the same number.
A Deep Price - a price augmented with Beier Disclosure - carries more than a number. It carries the five fields (Input, Method, Output, Identity, Scaling), each containing structured information about the judgment that produced the price. Different participants read the same Deep Price from different analytical frameworks. A fund manager reads it for investment risk. A regulator reads it for compliance exposure. A farmer reads it for constraint trajectory. Each extracts different components. Each sees different aspects of the same underlying object.
The question is: what kind of mathematical object has this property? What object carries information that transforms systematically when the reader's framework changes, while preserving something invariant across all frameworks?
The answer in physics and differential geometry is: a tensor. A tensor is defined by its transformation law. The components change under coordinate transformation. The geometric content - the thing the tensor represents - does not.
This paper defines the Beier Tensor by specifying the vector space, the coordinate systems, the transformation law, and the metric. The result is a mathematical object that formalizes what the book means by "Deep Price" and what the companion papers mean when they say disclosure carries information that the price does not.
Definition 1 (Disclosure space). A Beier Disclosure D is an element of the vector space V = R^K, where K is the number of independent information channels the disclosure carries. In the standard five-field format, K = 5 and the basis vectors correspond to: e_1 = Input, e_2 = Method, e_3 = Output, e_4 = Identity, e_5 = Scaling.
Each component D_k = I(theta; field_k | P, field_1, ..., field_{k-1}) measures the mutual information (in bits) that field k carries about the constraint state theta beyond what the price P and the preceding fields carry. The disclosure is therefore:
D = sum_{k=1}^{5} D_k * e_k
with D_k >= 0 for all k.
Definition 2 (Analytical framework). An analytical framework F is a linear map from the disclosure space V to a decision-relevant representation space. A fund manager's framework F_fund extracts the components relevant to investment risk. A regulator's framework F_reg extracts the components relevant to compliance. A farmer's framework F_farm extracts the components relevant to constraint trajectory. Each framework is represented by a matrix M_F that maps the disclosure vector to the framework-specific representation:
D_F = M_F * D
where D_F is the disclosure as read in framework F.
Proposition BT-1 (Coordinate invariance of information content). The total information content of a Beier Disclosure is invariant under change of analytical framework. Formally:
I_total = sum_{k=1}^{K} D_k
is the same whether computed in the fund manager's coordinates, the regulator's coordinates, or the farmer's coordinates. The total is the chain-rule decomposition of I(theta; D | P), which is a scalar property of the disclosure and the constraint state, independent of the analytical framework through which the disclosure is read.
Proof. The total information content of the disclosure about theta is:
I(theta; D | P) = sum_{k=1}^{K} I(theta; field_k | P, field_1, ..., field_{k-1})
by the chain rule for mutual information. This quantity depends on the disclosure and the constraint state, not on who is reading the disclosure. A fund manager and a regulator observing the same disclosure D about the same constraint state theta receive the same total information, even though they extract different components for their respective decision problems.
The analytical framework determines which components are decision-relevant, not how much total information the disclosure carries. The framework M_F is a projection (it selects the relevant components) followed by a rotation (it re-expresses them in domain-specific terms). The projection may discard components that are irrelevant to the framework's decision problem, but the full disclosure carries the same total information regardless.
Formally: the transformation from one framework to another is D_{F'} = M_{F'} * M_F^{-1} * D_F (when M_F is invertible) or more generally through the transition matrix T_{F' F} = M_{F'} * M_F^+ where M_F^+ is the pseudoinverse. The total information I(theta; D | P) is a scalar - a quantity that does not depend on the coordinate system - and is therefore invariant under framework change. QED.
Interpretation. This is what makes a Deep Price a tensor rather than a spreadsheet with extra columns. The spreadsheet's entries change when you change the column headers. The tensor's information content is preserved. The fund manager sees different numbers from the regulator, but both see the same underlying object. The "same underlying object" is the judgment that produced the price - the constraint assessment that the disclosure preserves and the scalar price compressed away.
The disclosure space V needs more than a coordinate system. It needs a way to measure the distance between disclosures and the quality of individual disclosures. In differential geometry, this is the metric tensor - the inner product that defines length and angle in the space.
Definition 3 (Track record metric). Let agent i have track record weight w_i in [0, 1], where w_i is determined by the agent's cumulative proper scoring rule performance. (In the general case, w_i should be a diagonal matrix W_i of field-specific track record weights, reflecting that an agent may be well-calibrated on some fields and poorly calibrated on others. The scalar w_i is a simplifying assumption for isotropic calibration.) (Propositions 5-7 of the companion paper, Beier, 2026r). The track record metric on the disclosure space is the bilinear form:
g(D_i, D_j) = w_i * w_j * sum_{k=1}^{K} D_{i,k} * D_{j,k}
This is the inner product between disclosure D_i (from agent i) and disclosure D_j (from agent j), weighted by both agents' track record quality.
Proposition BT-2 (The track record metric is framework-invariant and positive semi-definite). The metric g satisfies:
(i) g is positive semi-definite: g(D_i, D_i) >= 0 for all D_i, with equality iff D_i = 0 or w_i = 0.
(ii) g is symmetric: g(D_i, D_j) = g(D_j, D_i).
(iii) g is invariant under change of analytical framework: if D_i and D_j are re-expressed in a new framework through an orthogonal transformation O, then g(O * D_i, O * D_j) = g(D_i, D_j).
Proof. (i) g(D_i, D_i) = w_i^2 * sum D_{i,k}^2 = w_i^2 * ||D_i||^2 >= 0. Equality holds iff w_i = 0 (agent has no track record) or D_i = 0 (disclosure carries no information). QED.
(ii) Immediate from the commutativity of multiplication. QED.
(iii) For an orthogonal transformation O (representing a change of analytical framework that preserves information content):
g(O * D_i, O * D_j) = w_i * w_j * (O * D_i)^T * (O * D_j) = w_i * w_j * D_i^T * O^T * O * D_j = w_i * w_j * D_i^T * D_j = g(D_i, D_j)
since O^T * O = I for orthogonal O. QED.
Interpretation. The metric does three things simultaneously.
It weights disclosures by track record quality. A disclosure from an agent with w = 0.9 (strong track record of accurate predictions) has more "length" in the disclosure space than the same information from an agent with w = 0.2 (weak track record). This is not a subjective weighting - it is the objective consequence of proper scoring rule evaluation against realized outcomes (Propositions 5-7).
It measures agreement between disclosures. The inner product g(D_i, D_j) is large when both disclosures carry similar information (their vectors point in similar directions) and both agents have strong track records. It is small when the disclosures are orthogonal (carrying information about different dimensions of the constraint state) or when either agent has a weak track record. This makes the metric a natural measure of disclosure consensus - high inner product means multiple credible agents are saying similar things about the constraint state.
It is framework-invariant. A fund manager and a regulator computing the agreement between two disclosures get the same answer, even though they're reading the disclosures in different coordinate systems. The agreement is a property of the disclosures and their track records, not of who is assessing them. This is the defining property of a metric tensor: it converts coordinate-dependent components into coordinate-independent measurements.
Definition 4 (The Beier Tensor). A Beier Tensor T is the ordered triple:
T = (p, D, w)
where p in R is the scalar price, D in R^K is the disclosure vector, and w in [0,1] is the track record weight. The tensor lives in the product space R x R^K x [0,1], equipped with the track record metric g from Definition 3.
The Blinding Price is the degenerate case: T_blind = (p, 0, 0). The disclosure vector is zero (no information beyond the price) and the track record weight is zero (no accountability for the judgment behind the price). The Blinding Price is a tensor of rank zero - a scalar.
A Deep Price is a tensor with D != 0 and w > 0. The disclosure vector carries information the price does not, and the track record weight attaches accountability to that information. The Deep Price is a tensor of rank one - it has directional structure in the disclosure space.
Definition 5 (Tensor transformation law). Under change of analytical framework from F to F', the Beier Tensor transforms as:
T' = (p, M_{F'F} * D, w)
where M_{F'F} is the transition matrix from framework F to framework F'. The price p is a scalar (invariant under framework change - a dollar is a dollar). The track record weight w is a scalar (invariant - the agent's track record doesn't change because the reader changed). The disclosure vector D transforms as a vector - its components change but its information content is preserved.
This transformation law is what makes the Deep Price a tensor rather than a data structure. The transformation is linear, invertible (when the frameworks span the same space), and preserves the metric (when the transformation is orthogonal). The object is geometric.
Proposition BT-3 (The Blackwell improvement is the tensor's invariance property). The following statements are equivalent:
(a) The information structure (P, D) is strictly more Blackwell-informative than (P) alone (Proposition BE-3 of the companion paper).
(b) The Beier Tensor T = (p, D, w) with D != 0 carries strictly more information about the constraint state theta than the scalar price p in every analytical framework.
(c) For every decision problem and every analytical framework F, the decision-maker who observes T achieves weakly higher expected utility than the decision-maker who observes p alone, with strict improvement for at least one decision problem.
Proof. (a) implies (b): By BE-3, (P, D) is strictly more Blackwell-informative than (P) when I(V; D | P) > 0. This means D carries information about V beyond P. Since the information content is framework-invariant (BT-1), D carries this additional information in every framework. Therefore T carries more information than p in every framework. QED.
(b) implies (c): By Blackwell's (1953) theorem, a strictly more informative signal structure produces weakly higher expected utility for every decision problem, with strict improvement when the optimal action changes. Since T carries more information than p in every framework (by (b)), the decision-maker in any framework achieves weakly higher utility from T than from p alone. QED.
(c) implies (a): If every decision-maker in every framework weakly prefers (P, D) to (P), then by the converse of Blackwell's theorem, (P, D) must be weakly more Blackwell-informative. The strict improvement for at least one decision problem ensures the ranking is strict. QED.
Interpretation. The Beier Tensor is the geometric object whose invariance property is the welfare theorem. BE-3 says: disclosure improves welfare regardless of preferences and distributions. BT-3 says: this is because the tensor's information content is preserved under every framework change. The two statements are the same statement expressed in different mathematical languages - decision theory and differential geometry respectively.
This equivalence is why the term "tensor" is correct rather than merely suggestive. A tensor is defined by the property that its content is preserved under coordinate transformation. The Deep Price's content (its information about the constraint state) is preserved under framework transformation (from fund manager to regulator to farmer). The preservation is not a design feature. It is a theorem (BE-3/BT-3). The Deep Price IS a tensor in the precise mathematical sense.
When multiple disclosures about the same constraint state are available, they must be combined. The metric tensor provides the natural aggregation.
Definition 6 (Weighted disclosure aggregate). Given N disclosures D_1, ..., D_N about constraint state theta with track record weights w_1, ..., w_N, the aggregate disclosure is:
D_agg = (sum_{i=1}^N w_i^2 * D_i) / (sum_{i=1}^N w_i^2)
This is the track-record-weighted average, where agents with stronger track records contribute more to the aggregate. The weighting is quadratic in w_i (from the metric's bilinear structure) rather than linear, which means the aggregation gives disproportionate influence to the most accurate agents - a property that proper scoring rules are designed to produce.
The aggregate disclosure has total information content:
I(theta; D_agg | P) = I(theta; {D_1, ..., D_N} | P)
which by the chain rule for mutual information is at least as large as the information from any individual disclosure. The aggregation preserves or increases information. It does not destroy it.
This is the formal content of the cascade enrichment proposition (CE-1 from the companion paper): the disclosure cascade is information-enriching because the tensor aggregation preserves information, while the herding cascade is information-destroying because the scalar aggregation (prices averaging away the reasoning) compresses it.
The Beier Tensor resolves the measurement problem that the Prolegomena identifies.
The scalar price carries R_price bits of information about V (Proposition A of the companion paper). The Beier Tensor carries R_price + E_B bits, where E_B is the Beier Extropy (the additional information the disclosure provides). The distortion under the tensor is:
D_tensor = D_price - Delta_D(E_B)
which is strictly less than D_price whenever E_B > 0 (Proposition EB-1).
The Beier Tensor is therefore the mathematical object that REDUCES the reconstruction burden. The scalar price forces every downstream user to reconstruct the judgment that produced it - the $44.3 billion annual reconstruction industry. The tensor preserves that judgment in structured, attributed, scored form. The reduction in reconstruction burden is:
Delta_reconstruction = D_price - D_tensor = Delta_D(E_B)
which is computable from the source covariance structure and the disclosure's information content (Proposition EB-1), convertible to welfare gain (Proposition WT-2), and convertible to TFP improvement (Proposition TFP-1).
The Beier Tensor is not the price with metadata appended. It is the mathematical object that carries economic judgment through the coordination system without compressing it into a scalar, whose information content is preserved under every analytical framework, whose quality is measured by the track record metric, and whose welfare improvement is proved by the equivalence of invariance and Blackwell dominance.
The orthogonality assumption in BT-2(iii) (framework changes are orthogonal transformations) is a simplification. In practice, some framework changes are projections that discard irrelevant components rather than rotations that preserve all components. A regulator's framework may lack the dimensionality to represent the fund manager's risk assessment in full. Under projection (non-orthogonal transformation), the metric is not exactly preserved - the inner product in the projected space is less than or equal to the inner product in the full space. The invariance is approximate rather than exact when frameworks have different dimensionality.
The five-field basis (Input, Method, Output, Identity, Scaling) is treated as fixed. In practice, the information content of each field varies by domain, by agent, and by disclosure. A farmer's Scaling field may carry far more information than their Method field (the farmer's method is "I looked at my wells"), while a quantitative analyst's Method field may carry more than their Scaling field. The fixed-basis treatment assigns equal dimension to each field, which is a simplification of the variable-dimensionality reality.
The track record weight w is treated as a scalar, but in practice it should be field-specific - an agent may have a strong track record on Output predictions and a weak track record on Scaling assessments. A richer formalization would replace the scalar w with a diagonal matrix W = diag(w_1, ..., w_K) that weights each field independently. This produces a richer metric g(D_i, D_j) = D_i^T * W_i * W_j * D_j but complicates the aggregation. The scalar track record is the first approximation.