Gregory Caldwell Beier
Susarb LLC, Cambridge, Massachusetts
March 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 price system is a lossy codec. It compresses the high-dimensional judgments of economic agents into scalar signals, destroying information in the process. This paper formalizes the claim by applying rate-distortion theory to price formation within a multivariate Gaussian framework. The main result: the excess distortion - the gap between the price system's actual distortion and the information-theoretic minimum at the same compression rate - is strictly positive whenever the price is informative enough for the optimal codec to reach the constraint dimensions, strictly increasing in constraint-dimension variance, and convergent to the total constraint variance in the limit of perfect market efficiency. The result extends to general (non-diagonal) source covariance. On its own dimension, the price system is an optimal codec; the excess distortion arises entirely from concentrating all compression capacity on the consensus value while leaving the constraint dimensions uncompressed. This is a structural mismatch, not a correctable design flaw - no improvement to the price system can deliver constraint information through the price channel. Supplementary disclosure channels that target uncompressed dimensions produce strictly higher marginal distortion reduction than additional price precision. The framework complements Sims's (2003) rational inattention program: where rational inattention explains why agents underprocess available signals, the present analysis explains why the signals themselves are degraded.
Keywords: rate-distortion theory, lossy compression, price formation, information destruction, reconstruction cost, rational inattention, information architecture, codec
JEL Codes: D82, D83, C60, G14
Shannon (1948) established two foundational results in information theory. The channel coding theorem specifies the maximum rate at which information can be transmitted reliably over a noisy channel. The source coding theorem - developed further as rate-distortion theory in Shannon (1959) and formalized comprehensively by Berger (1971) - specifies the minimum rate at which a source signal can be represented with a given level of distortion. Together, these results define the theoretical limits of communication and compression.
Rate-distortion theory has been applied to economic problems primarily through Sims's (2003) rational inattention framework, which models economic agents as information channels with finite capacity. In Sims's formulation, agents cannot process all available information and must allocate their limited channel capacity across competing signals. The allocation decision is itself an economic choice, and the resulting inattention generates predictable patterns in consumption, pricing, and portfolio behavior. Maćkowiak and Wiederholt (2009) extended this to show that rationally inattentive firms allocate the vast majority of their limited attention to idiosyncratic conditions rather than aggregate shocks, because idiosyncratic variance dominates. Matejka and McKay (2015) derived the discrete choice implications, connecting information-theoretic capacity constraints to empirical demand estimation. Veldkamp (2011) surveyed these developments, organizing the economics of information choice under the common framework of agents acquiring costly signals.
Throughout this literature, the price system itself is treated as a given signal rather than as a compression mechanism with analyzable distortion properties. The information design literature - Kamenica and Gentzkow (2011) on Bayesian persuasion, Bergemann and Morris (2016) on information structures and their welfare consequences - analyzes how a sender optimally chooses what to reveal. Angeletos and Pavan (2007) characterized the efficient use of information in coordination games, connecting the welfare analysis of information structures to the allocation problems that motivate the present framework. The disclosure channel studied in Section 6 is an information design problem in this sense; the present paper characterizes the distortion that makes such a channel valuable. The two frameworks are complementary: Sims's explains why agents don't process all available information; the present paper explains why the information available to process is degraded in the first place.
The present paper applies rate-distortion theory to a different object: not the agent's information processing capacity but the price system itself. The price system is a compression mechanism that takes a high-dimensional source signal (the full judgment of all agents participating in a transaction) and produces a one-dimensional compressed signal (the price). This compression is lossy - the price carries less information than the judgments that produced it. This paper formalizes the claim within a multivariate Gaussian framework, computing the price system's distortion, the theoretical minimum distortion at the same compression rate, and the gap between them - in closed form. The practical implication is that supplementary channels can reduce what the price system cannot: the Disclosure Incentive - a mechanism in which agents accumulate track record capital by publishing structured judgments scored against outcomes, creating an alternative return channel to trading profit - is one such channel, analyzed in Section 6.
Section 2 frames price formation as lossy compression. Section 3 derives the rate-distortion analysis. Section 4 establishes the excess distortion and its properties. Section 5 measures the cost of operating below the bound. Section 6 derives the marginal value of disclosure as a supplementary channel. Section 7 states limitations. Section 8 concludes.
Consider a market in which N agents participate in a transaction. Each agent i possesses a private judgment J_i about the value of the good, where J_i is a high-dimensional object encoding the agent's assessment of current value, future trajectory, risk factors, constraint interactions, scaling effects, and confidence levels across each dimension. The dimensionality of J_i varies by agent and by transaction but is in all cases much greater than one.
The market mechanism aggregates {J_1, J_2, ..., J_N} into a price p, where p is a scalar. This aggregation is a compression: it maps a high-dimensional source (the collection of all agents' judgments) into a one-dimensional representation (the price).
Hayek (1945) made the canonical case that prices aggregate dispersed information efficiently. The present paper does not dispute this. It formalizes a complementary claim: the aggregation is lossy, and the distortion has a specific structure that rate-distortion theory can characterize. Hayek's insight was that no central planner could match the price system's ability to incorporate dispersed private knowledge. The information-theoretic framework developed here identifies what the price system loses in the process of aggregating it.
In information-theoretic terms, the source has entropy H(J) = H(J_1, J_2, ..., J_N), which measures the total information content of the agents' judgments. The compressed representation has entropy H(p), which is bounded above by the entropy of a continuous scalar - far less than H(J) for any non-trivial market. The difference H(J) - H(p) is the information destroyed in the compression.
Not all of this destroyed information is economically relevant. Much of what is in J_i but not in p is redundant across agents (when many agents assess the same risk factor, the price incorporates the consensus without needing to preserve each individual assessment). Some is idiosyncratic noise that cancellation across agents appropriately filters out. The economically relevant distortion is the portion of the destroyed information that would change allocation decisions if it were available - the constraint information, interaction effects, and scaling dynamics that constitute compression artifacts of the price mechanism.
The price system is not a designed codec. A codec (in the information-theoretic sense) is a compression algorithm optimized to minimize distortion at a given rate, or to minimize rate at a given distortion tolerance. The price system emerged through institutional evolution - from barter through commodity money through financial markets - without optimization against any distortion metric. Its compression properties are consequences of its institutional structure (bilateral exchange, scalar clearing, contemporaneous settlement) rather than products of deliberate design. The rate-distortion comparison developed in Sections 3 and 4 is therefore against an information-theoretic ideal - the minimum distortion physically achievable at the price's compression rate - not against any implementable market mechanism. No market institution achieves the multi-dimensional joint encoding that the optimal codec requires, and the excess distortion should not be read as evidence of correctable inefficiency within price formation itself. The distortion is structural: it arises from the fact that market clearing produces a scalar. The relevant question is not how to fix the price system but whether supplementary channels can reduce the gap.
The allocation-relevant information about a transaction has K dimensions. In the CARA-normal framework that is standard in information economics, we model the full social value of a transaction as a K-dimensional Gaussian random vector:
V = (V_1, V_2, ..., V_K)' ~ N(mu, Sigma)
where the dimensions are interpreted as follows. V_1 is the private consensus value - the dimension the price system primarily aggregates. V_2, ..., V_K are constraint dimensions: V_2 captures the first-order constraint interaction (the externality most directly associated with the transaction), V_3 captures scaling dynamics (how the transaction's effects compound when replicated by N agents over T periods), and V_4 through V_K capture additional constraint interactions, cross-domain coupling, and temporal trajectory information.
To isolate the compression effect cleanly, we work in the principal component basis of Sigma. Let
Lambda = diag(lambda_1, lambda_2, ..., lambda_K)
be the eigenvalue decomposition of Sigma in this basis, with lambda_1 >= lambda_2 >= ... >= lambda_K > 0. In this basis the components of V are independent, and the total source variance is
sigma^2_total = tr(Sigma) = sum_{k=1}^{K} lambda_k
The source entropy rate (differential entropy for the multivariate Gaussian) is:
h(V) = (K/2) log(2 * pi * e) + (1/2) log(det(Sigma)) = (K/2) log(2 * pi * e) + (1/2) sum_{k=1}^{K} log(lambda_k)
This is the information content of the full judgment. It grows with K and with the variance along each dimension.
The price p is a noisy scalar observation of V_1, the consensus value dimension. In a CARA-normal rational expectations equilibrium (Grossman and Stiglitz, 1980; Hellwig, 1980), the price is a linear function of the aggregate informed demand plus noise. The price signal about V_1 can be written as:
p = V_1 + w
where w ~ N(0, sigma^2_w) is independent of V, and the signal-to-noise ratio is
SNR_1 = lambda_1 / sigma^2_w
The assumption that the price carries information only about V_1 (in the eigenbasis) is the formal statement of the claim that prices reflect the consensus value and compress away constraint reasoning. In the original (non-eigenbasis) coordinates, the price signal may carry partial information about multiple dimensions of V through their correlation with V_1. The eigenbasis decomposition isolates the directions along which information is transmitted (the V_1 direction) from the directions along which it is destroyed (V_2, ..., V_K). The core structural claim is that the constraint dimensions are substantially orthogonal to the consensus value direction - that knowing the equilibrium price of an airline ticket tells you little about how the flight's emissions interact with atmospheric constraints at aggregate scale.
When the constraint dimensions are not perfectly orthogonal to V_1 - when there is some correlation between V_k and V_1 for k >= 2 - the price carries partial information about the constraint dimensions through this correlation. The analysis below handles this through the conditional covariance structure, and the excess distortion result holds whenever the projection of V onto the price signal leaves residual variance in the constraint directions, which it does for any K > 1 unless V_2, ..., V_K are deterministic functions of V_1.
The mutual information between V and the price p is:
R_price = I(V; p)
Since p = V_1 + w, and in the eigenbasis V_1 is independent of V_2, ..., V_K:
I(V; p) = I(V_1; p) + I(V_2, ..., V_K; p | V_1)
The second term vanishes: conditional on V_1, the price p = V_1 + w contains no information about V_2, ..., V_K because w is independent of everything. Therefore:
I(V_2, ..., V_K; p | V_1) = 0
and
R_price = I(V_1; p) = (1/2) log(1 + SNR_1) = (1/2) log(1 + lambda_1 / sigma^2_w)
where log denotes the base-2 logarithm throughout, so that rates are measured in bits. (This expression for mutual information is specific to the Gaussian model; for non-Gaussian V_1, I(V_1; p) need not equal (1/2) log(1 + SNR_1), which is the capacity of a Gaussian channel.) All R_price bits are allocated to dimension 1. Zero bits are allocated to dimensions 2 through K. This allocation is not a design choice - it is an emergent consequence of the fact that the market clears on the consensus value dimension.
Under mean-squared error distortion (the natural choice in the CARA-normal setting, where quadratic loss underlies the agents' optimization), the distortion from using p to estimate V is:
D_price = E[||V - E[V|p]||^2] = tr(Sigma_{V|p})
where Sigma_{V|p} is the conditional covariance matrix of V given p.
In the eigenbasis, since p carries information only about V_1:
E[V_1 | p] = (lambda_1 / (lambda_1 + sigma^2_w)) * p + (sigma^2_w / (lambda_1 + sigma^2_w)) * mu_1
Var(V_1 | p) = lambda_1 * sigma^2_w / (lambda_1 + sigma^2_w) = lambda_1 / (1 + SNR_1)
And for k = 2, ..., K: Var(V_k | p) = lambda_k (no reduction - the price carries no information about these dimensions).
Therefore:
Lemma A (Price system distortion). The distortion of the price signal under MSE is:
D_price = lambda_1 / (1 + SNR_1) + sum_{k=2}^{K} lambda_k
The first term is the residual uncertainty about the consensus value after observing the price. The second term is the total prior variance in the constraint dimensions, entirely uncompressed. As SNR_1 grows (the price becomes more informative about V_1), the first term vanishes, but the second term remains fixed:
lim_{SNR_1 -> infinity} D_price = sum_{k=2}^{K} lambda_k
No amount of price improvement reduces distortion on the constraint dimensions. This limit is the formal content of the claim that improving market efficiency does not address the information the price system structurally cannot carry.
Shannon (1959) defined the rate-distortion function for a source X with distortion measure d(x, x-hat) as:
R(D) = min_{p(x-hat|x): E[d(X, X-hat)] <= D} I(X; X-hat)
For a K-dimensional Gaussian source with independent components having variances lambda_1, ..., lambda_K under MSE distortion, the rate-distortion function is given by the reverse water-filling solution (Berger, 1971; Cover and Thomas, 2006):
R(D) = sum_{k=1}^{K} max(0, (1/2) log(lambda_k / theta))
where the parameter theta is determined by the distortion constraint:
D = sum_{k=1}^{K} min(lambda_k, theta)
The optimal codec allocates bits to dimension k if and only if lambda_k > theta - dimensions with variance below the water level theta receive zero bits and contribute their full variance to the distortion. Dimensions with variance above the water level receive (1/2) log(lambda_k / theta) bits each, reducing their distortion contribution from lambda_k to theta.
The distortion-rate function D(R), the inverse of R(D), gives the minimum distortion achievable at rate R.
Fact B (Minimum achievable distortion; Berger, 1971; Cover and Thomas, 2006). At the price system's compression rate R_price = (1/2) log(1 + SNR_1), the minimum achievable distortion under optimal encoding is D* = D(R_price), determined by reverse water-filling:
D* = sum_{k=1}^{K} min(lambda_k, theta*)
where theta* solves:
R_price = sum_{k=1}^{K} max(0, (1/2) log(lambda_k / theta*))
Proposition C (Excess distortion). Let K >= 2 and lambda_k > 0 for all k. Assume mean-squared error distortion throughout. Assume the price signal loads entirely on the first eigenvector of Sigma, so that p = V_1 + w in the eigenbasis (Section 3.2). The benchmark D* is the Gaussian rate-distortion function D*(R_price), which for the Gaussian source under MSE is achieved by a memoryless (single-letter) test channel and therefore serves as both the asymptotic and single-observation minimum distortion at rate R_price (Remark 1 below). Define the threshold rate R_0 = (1/2) log(lambda_1 / lambda_2) and the corresponding threshold SNR_0 = (lambda_1 / lambda_2) - 1. The excess distortion Delta = D_price - D* satisfies:
(i) Delta = 0 when R_price <= R_0 (equivalently, SNR_1 <= SNR_0). In this regime the rate is too low for the optimal codec to compress any dimension beyond V_1, so the price system's allocation of all bits to dimension 1 is optimal. Note that if lambda_1 = lambda_2 (degenerate leading eigenvalues), then R_0 = 0, and the positive-excess regime begins immediately for any R_price > 0: when the consensus and leading constraint dimensions carry equal variance, every bit of price informativeness produces excess distortion.
(ii) Delta > 0 when R_price > R_0 (equivalently, SNR_1 > SNR_0). In this regime the optimal codec allocates positive bits to at least one constraint dimension, while the price system allocates none. The economically relevant regime - where the price is informative enough about V_1 that markets function as coordination mechanisms - satisfies this condition whenever the leading constraint dimension carries variance comparable to the consensus dimension.
(iii) In the positive-excess regime, Delta is strictly increasing in lambda_k for each active constraint dimension (each k >= 2 with lambda_k > theta*), and weakly increasing in K. Delta is strictly increasing in K when the added dimension has variance exceeding the water level (lambda_{K+1} > theta*).
(iv) Delta is bounded above by the constraint-dimension savings:
Delta <= sum_{k=2}^{K} [lambda_k - min(lambda_k, theta*)]
The gap between Delta and this upper bound is theta* - lambda_1 * 2^{-2 * R_price} > 0, the over-compression cost on dimension 1 - the distortion the price system wastes by compressing the consensus dimension beyond the level an optimal codec would choose.
(v) In the limit of high price informativeness: as SNR_1 -> infinity, R_price -> infinity, theta* -> 0, D* -> 0, and Delta -> sum_{k=2}^{K} lambda_k.
Proof. The price system allocates all R_price bits to dimension 1 and zero bits to dimensions 2, ..., K. The optimal codec at the same rate R_price distributes bits via reverse water-filling with water level theta*.
Part (i). When R_price <= (1/2) log(lambda_1 / lambda_2), the water level satisfies theta* >= lambda_2. Only dimension 1 is active under reverse water-filling. The optimal rate allocation is r_1* = R_price, matching the price system exactly. The water level solves R_price = (1/2) log(lambda_1 / theta*), giving theta* = lambda_1 * 2^{-2 * R_price} = lambda_1 / (1 + SNR_1). Both systems achieve distortion D* = theta* + sum_{k=2}^{K} lambda_k = D_price. Hence Delta = 0.
Part (ii). When R_price > (1/2) log(lambda_1 / lambda_2), the water level satisfies theta* < lambda_2, so the optimal codec allocates positive bits to dimensions 1 and 2 (and possibly further dimensions). The price system defines a specific conditional distribution p(p|V) achieving I(V; p) = R_price and distortion D_price. Since the reconstruction E[V|p] is a deterministic function of p, the data processing inequality gives I(V; E[V|p]) <= I(V; p) = R_price, so the price system is a feasible point in the rate-distortion optimization. Hence D_price >= D*(R_price). The inequality is strict because the reverse water-filling solution for the Gaussian source under MSE is unique (the optimization is strictly convex; see Cover and Thomas, 2006, Section 10.3.3), and the price system's all-bits-to-dimension-1 allocation differs from the unique optimum when theta* < lambda_2. Hence Delta > 0.
Part (iii). Fix R_price and consider increasing lambda_k for some active dimension k >= 2 (i.e., lambda_k > theta*). The derivative dDelta/dlambda_k is computed for lambda_k strictly in the interior of the active set (lambda_k > theta*). At the boundary lambda_k = theta*, the active set cardinality changes discontinuously and the derivative is undefined in the classical sense; however, Delta is continuous at the boundary (both D_price and D* are continuous functions of the eigenvalues), so the strict monotonicity extends to the boundary by continuity from above. Let S = {j : lambda_j > theta*} denote the active set with |S| elements. Implicit differentiation of the rate constraint R_price = sum_{j in S} (1/2) log(lambda_j / theta*) with respect to lambda_k yields:
0 = 1 / (2 * lambda_k * ln 2) - |S| / (2 * theta* * ln 2) * (d theta* / d lambda_k)
Solving: d theta* / d lambda_k = theta* / (|S| * lambda_k). The total optimal distortion decomposes as D* = |S| * theta* + sum_{j not in S} lambda_j, where the first term sums the residual variance theta* across each active dimension and the second sums the full prior variance of each inactive dimension. Differentiating:
d D* / d lambda_k = |S| * (d theta* / d lambda_k) = theta* / lambda_k
Meanwhile d D_price / d lambda_k = 1 (the price system's distortion increases by the full increment, since the price carries no information about dimension k). Therefore:
d Delta / d lambda_k = 1 - theta* / lambda_k > 0
because theta* < lambda_k for all k in S. For adding a new dimension K+1 with variance lambda_{K+1} > 0: D_price increases by lambda_{K+1}, while D* for the (K+1)-dimensional problem, denoted D*_{K+1}, satisfies D*_{K+1} <= D* + lambda_{K+1} (since the old optimal allocation combined with zero bits for dimension K+1 is feasible at the same rate). Hence Delta weakly increases with K. When lambda_{K+1} > theta*, the inequality is strict: suppose for contradiction that dimension K+1 were inactive in the (K+1)-dimensional solution, so the new active set S' were a subset of S. Then the new water level theta' would satisfy theta' >= theta* (fewer or equal active dimensions sharing the same rate budget), so theta' >= theta* > ... , but the rate equation sum_{j in S'} (1/2) log(lambda_j / theta') <= sum_{j in S} (1/2) log(lambda_j / theta*) = R_price, with equality only if S' = S and theta' = theta*. But if S' = S and theta' = theta*, then lambda_{K+1} > theta* = theta' means dimension K+1 should be active, contradicting the assumption that it is inactive. Hence dimension K+1 enters the active set, the optimal codec compresses it, and D*_{K+1} < D* + lambda_{K+1} strictly.
Part (iv). Decompose the excess distortion by dimension. The price system achieves residual variance lambda_1 * 2^{-2 * R_price} on dimension 1, while the optimal codec achieves residual variance theta* on each active dimension. The decomposition is:
Delta = [lambda_1 * 2^{-2 * R_price} - theta*] + sum_{k=2}^{K} [lambda_k - min(lambda_k, theta*)]
The first bracket is strictly negative: the price system allocates all R_price bits to dimension 1, while the optimal codec allocates only r_1* = (1/2) log(lambda_1 / theta*) < R_price bits. Since R_price > r_1*, the price system compresses dimension 1 beyond the optimal level, yielding lambda_1 * 2^{-2 * R_price} < lambda_1 * 2^{-2 * r_1*} = theta*. The second summation is non-negative (and strictly positive for each k in S with k >= 2). Rearranging:
Delta = sum_{k=2}^{K} [lambda_k - min(lambda_k, theta*)] - [theta* - lambda_1 * 2^{-2 * R_price}]
The first sum is the constraint-dimension savings (what the price system wastes by leaving constraint dimensions uncompressed). The second term is the over-compression cost (what the price system wastes by compressing dimension 1 beyond the optimal level, thereby forfeiting rate that the optimal codec would have directed to constraint dimensions). Both terms are strictly positive in the positive-excess regime. Since Delta > 0 by Part (ii), the constraint-dimension savings strictly exceed the over-compression cost. The decomposition characterizes how the positive excess distortion arises - as the net of two competing effects - but the positivity itself is established by the uniqueness argument in Part (ii), not by this decomposition.
Part (v). As SNR_1 -> infinity, R_price -> infinity, theta* -> 0, and D* -> 0. The price system's distortion converges to D_price -> sum_{k=2}^{K} lambda_k, a fixed positive constant. Hence Delta -> sum_{k=2}^{K} lambda_k. The rate of convergence is characterized by the water level: when all K dimensions are active, theta* = (product_{k=1}^{K} lambda_k)^{1/K} * 2^{-2 R_price / K}, which decays exponentially in R_price / K. The approach to the limiting excess distortion is therefore exponentially fast in the rate but slower for larger K: the exponent is 2 R_price / K, not 2 R_price. In high-dimensional constraint environments, convergence to the limiting excess distortion sum_{k=2}^{K} lambda_k is slow per bit of price improvement, because each additional bit of rate is shared across K active dimensions under optimal encoding. QED.
Remark 1 (Single-shot vs. asymptotic benchmark). The rate-distortion function D*(R) is typically presented as the minimum distortion achievable by block codes operating on i.i.d. source sequences of length n -> infinity. However, for a Gaussian source under MSE distortion, the RD-achieving test channel is memoryless: the optimal reproduction of each component X_k is X-hat_k = X_k + Z_k with independent Gaussian noise Z_k ~ N(0, theta*) (or X-hat_k = X_k for inactive dimensions). This single-letter structure means the Gaussian RD bound is achievable by a memoryless encoder operating on a single source realization - no block coding across multiple i.i.d. draws is required. The price system is itself a single-letter encoder on dimension 1: it observes one realization of V and produces one price. The comparison in Proposition C therefore compares two single-observation systems - one that allocates all its information to dimension 1, and one that distributes it optimally across dimensions. The gap between them is a pure allocation mismatch, not an artifact of comparing a single-shot system against an asymptotic bound. (For non-Gaussian extensions, this equivalence breaks down: the single-shot optimal distortion may strictly exceed D*(R), and the excess distortion computed against the asymptotic bound would overstate the achievable single-observation improvement. The Gaussian case is special in this respect.)
Proposition C formalizes what the preceding discussion argued informally. The price system introduces excess distortion because it concentrates all of its information capacity on the consensus value dimension while leaving the constraint dimensions entirely uncompressed. An optimal codec at the same total rate would distribute bits across dimensions according to reverse water-filling - allocating more bits to high-variance dimensions and fewer to low-variance dimensions, minimizing total distortion.
The threshold condition R_price > R_0 = (1/2) log(lambda_1 / lambda_2) identifies when this misallocation becomes costly. Below the threshold, the price's rate is too low for the optimal codec to reach the constraint dimensions, and concentrating all bits on the consensus dimension is the best available strategy. Above the threshold, the optimal codec would divert some of its capacity to constraint dimensions, but the price system cannot - its compression pattern is determined by the institutional structure of market clearing, not by any optimization against a distortion metric. The threshold is low whenever the leading constraint dimension carries variance on the order of the consensus dimension (lambda_2 comparable to lambda_1). In economically relevant settings - where prices are informative enough to coordinate allocation on the consensus dimension (SNR_1 >> 1) and constraint interactions carry substantial variance - the threshold is easily exceeded.
The nature of the excess distortion deserves emphasis. On its own dimension, the price system is an optimal codec under the Gaussian assumption: the residual variance lambda_1/(1 + SNR_1) = lambda_1 * 2^{-2R_price} equals the scalar Gaussian rate-distortion function evaluated at rate R_price. The price extracts every bit of information from V_1 that a one-dimensional Gaussian codec at rate R_price could extract. There is no coding inefficiency on dimension 1 - the distortion reduction the price achieves there is the maximum possible. (This optimality is specific to the Gaussian case. For non-Gaussian V_1, the linear estimator that market clearing produces is no longer MMSE, and the price system becomes suboptimal even on its own dimension - see Section 7.) The excess distortion is therefore a structural mismatch in rate allocation: the price allocates all R_price bits to a single dimension when distributing them across dimensions would achieve lower total distortion. This distinction matters for the economic interpretation. The price system's failure is not that it is a bad compressor; it is that it compresses the wrong thing, or more precisely, only one of the K things that need compressing.
The over-compression cost identified in part (iv) reveals a structural asymmetry: the price system not only neglects the constraint dimensions, it over-invests in the consensus dimension relative to what an optimal codec would choose. The optimal codec would reallocate some of the rate the price system devotes to dimension 1 - accepting slightly higher residual variance on the consensus dimension in exchange for substantially lower distortion on the constraint dimensions. The price system cannot perform this reallocation because it has no mechanism for carrying constraint information; every marginal bit of price precision goes to the only dimension the price can reach.
As noted in Section 2, the benchmark comparison is against an information-theoretic ideal, not an achievable market mechanism. For the Gaussian source under MSE, the rate-distortion bound is achieved by a memoryless (single-letter) test channel (Remark 1), so the comparison does not introduce a spurious gap between single-shot and asymptotic performance: the relevant distinction is allocation across dimensions, not block length.
Part (v) of Proposition C illustrates the structural content of parts (ii) and (iii) in the limiting case. Even in the limit of perfect market efficiency - where the price reflects all available information about V_1 - the excess distortion converges to the total prior variance in the constraint dimensions. The non-trivial content is upstream: parts (ii) and (iii) identify the threshold below which the price system is trivially optimal and characterize how the gap depends on the eigenvalue spectrum in the positive-excess regime. The limit makes this concrete - more efficient markets produce lower excess distortion on dimension 1 and zero reduction in excess distortion on dimensions 2 through K. This is the information-theoretic foundation for the claim that improving market efficiency does not resolve the externality problem: externalities live in the constraint dimensions, and the price system's rate allocation ignores them regardless of how efficiently it operates on the consensus dimension.
A clarification on scope: D_price measures the distortion of the price signal specifically, not the distortion of the total information environment in which agents operate. Agents observe auxiliary signals beyond the price - earnings reports, analyst research, regulatory filings, physical inspections - and these reduce uncertainty about constraint dimensions. The excess distortion theorem characterizes what the price system contributes to the information environment, not the sum of all information available. Section 5 addresses reconstruction from auxiliary signals and why the Disclosure Incentive improves on unstructured auxiliary inference.
Consider K = 4 dimensions with eigenvalues lambda_1 = 10, lambda_2 = 5, lambda_3 = 3, lambda_4 = 1. The consensus value (dimension 1) has the largest variance; the constraint dimensions have decreasing variance. Let SNR_1 = 9, so R_price = (1/2) log(1 + 9) = (1/2) log(10) approx 1.661 bits.
Threshold check: R_0 = (1/2) log(10/5) = (1/2) log(2) = 0.500 bits. Since R_price = 1.661 > 0.500 = R_0, we are in the positive-excess regime and Proposition C guarantees Delta > 0.
Price system distortion: D_price = 10/10 + 5 + 3 + 1 = 1 + 9 = 10.
Optimal distortion at the same rate (reverse water-filling): At R = 1.661 bits, we solve for theta*. The water-filling algorithm allocates bits to dimensions with lambda_k > theta*. First attempt with all four dimensions active: the rate is (1/2) log(150 / theta*^4). Setting equal to 1.661: theta*^4 = 150 / 2^{3.322} = 150/10 = 15, giving theta* = 15^{1/4} approx 1.968. Check: is theta* < lambda_4 = 1? No - theta* = 1.968 > 1, so dimension 4 is below the water level and receives zero bits. Re-solve with three active dimensions: (1/2) log(150 / theta*^3) = 1.661, giving theta*^3 = 150/10 = 15, so theta* = 15^{1/3} approx 2.466. Check: theta* = 2.466 < lambda_3 = 3? Yes. So dimensions 1-3 are active.
D* = 3 * theta* + lambda_4 = 3 * 15^{1/3} + 1 approx 3(2.466) + 1 = 8.40.
Excess distortion: Delta = 10 - 8.40 = 1.60.
Decomposition (Proposition C, part iv): The constraint-dimension savings are (5 - 2.466) + (3 - 2.466) + 0 = 2.534 + 0.534 = 3.07. The over-compression cost on dimension 1 is theta* - lambda_1 * 2^{-2 * R_price} = 2.466 - 1.0 = 1.47. The excess distortion is the savings minus the over-compression cost: Delta = 3.07 - 1.47 = 1.60, confirming the bound Delta < 3.07. The price system wastes 3.07 units of potential distortion reduction by neglecting constraint dimensions, but recovers 1.47 units of that waste through its over-compression of dimension 1 (which the optimal codec would not perform). The net excess is 1.60 units, or 16% of total price-system distortion. (All rounded values derive from theta* = 15^{1/3} and R_price = (1/2) log(10); the exact expressions are used in the algebraic verification.)
At realistic dimensionality - where K may be in the hundreds, reflecting the many constraint interactions relevant to a complex transaction - the excess distortion can dominate total distortion. As K grows, D_price = lambda_1/(1 + SNR_1) + sum_{k=2}^{K} lambda_k, and the dimension-1 residual lambda_1/(1 + SNR_1) is independent of K. The constraint-dimension sum grows with K provided the eigenvalue spectrum does not decay too rapidly - specifically, provided sum_{k=2}^{K} lambda_k >> lambda_1/(1 + SNR_1), which holds when the constraint eigenvalues do not shrink faster than O(1/K). A sufficient condition for the strict monotonicity of Delta in K (Proposition C, part iii) is that each added dimension enters the active set, i.e., lambda_{K+1} > theta*. For a power-law spectrum lambda_k = c * k^{-alpha} with alpha > 0, the water level satisfies theta* = (product_{j in S} lambda_j)^{1/|S|} * 2^{-2 R_price / |S|}, which converges to zero as R_price grows for any fixed |S|. In particular, if alpha < 1, then sum_{k=1}^{K} lambda_k diverges in K and every added dimension eventually enters the active set as R_price increases - Delta is eventually strictly increasing in K. If alpha >= 1, the eigenvalue sum converges, and sufficiently high-index dimensions may remain below the water level even at large R_price. Empirical estimation of the eigenvalue decay rate for specific asset classes - equity constraint environments, commodity supply chains, climate-financial interactions - would determine which regime applies and is left to future work. Under this spectral condition, the price system's distortion reduction on dimension 1 becomes a negligible fraction of total distortion, and D_price is dominated by the uncompressed constraint variance. The K = 4 illustration understates the problem for any application where the constraint environment is rich and its eigenvalue spectrum is not rapidly decaying.
Proposition C is derived in the eigenbasis, where the components of V are independent. When V has general (non-diagonal) covariance Sigma and the price observes an arbitrary linear combination p = a'V + w, the price signal may carry partial information about constraint dimensions through their correlation with the observed direction. The following corollary establishes that the excess distortion result extends to this setting.
Corollary C' (Excess distortion under general covariance). Let V ~ N(mu, Sigma) with Sigma positive definite, K >= 2, and let p = a'V + w with w ~ N(0, sigma^2_w) independent of V. The conditional covariance is the rank-one update:
Sigma_{V|p} = Sigma - (Sigma a a' Sigma) / (a' Sigma a + sigma^2_w)
The price system's distortion is D_price = tr(Sigma_{V|p}). Let R_price = I(V; p) = (1/2) log((a' Sigma a + sigma^2_w) / sigma^2_w) and let D* = D*(R_price) be the Gaussian rate-distortion optimum at rate R_price, computed via reverse water-filling on the eigenvalues of Sigma. Then Delta = D_price - D* > 0 whenever the residual covariance Sigma_{V|p} has at least two eigenvalues exceeding the water level theta* at rate R_price.
Proof sketch. The rank-one update gives D_price = tr(Sigma_{V|p}) = tr(Sigma) - a'Sigma^2 a / (a'Sigma a + sigma^2_w). The mutual information is R_price = (1/2) log((a'Sigma a + sigma^2_w) / sigma^2_w). The price system achieves distortion D_price at rate R_price, making it a feasible point in the rate-distortion optimization. The optimal distortion D* at rate R_price is determined by reverse water-filling on the eigenvalues lambda_1, ..., lambda_K of Sigma, which depends only on the source covariance, not the observation direction. By the strict convexity and uniqueness of the Gaussian RD solution (the same argument as Part (ii) of Proposition C), D_price > D* whenever the price system's distortion allocation differs from the unique optimum. A rank-one observation compresses along a single direction in R^K; the optimal codec at the same rate compresses along every dimension with variance above the water level. These allocations differ whenever Sigma_{V|p} has at least two eigenvalues exceeding theta*, since in that case the optimal codec would allocate positive bits to both above-water-level dimensions, while the rank-one price observation reduces variance in only one direction. (Note: the rank-one update generally shifts all eigenvalues of Sigma, not just one - the argument works through the feasibility and uniqueness structure, not through eigenvalue-by-eigenvalue comparison.)
The excess distortion under general covariance satisfies Delta >= Delta_{eigenbasis} when the observation direction a is the first eigenvector of Sigma. That is, eigenbasis alignment minimizes the excess distortion: for any other observation direction at the same mutual information rate, D_price weakly increases (the first eigenvector maximizes the MSE reduction by a Rayleigh-quotient argument) while D* is invariant to the observation direction (it depends only on the eigenvalue spectrum). Proposition C therefore provides a lower bound on Delta for price signals loading on arbitrary linear combinations of V.
The economic cost of the price system's excess distortion manifests as reconstruction expenditure - the resources the economy devotes to rebuilding information that the price system destroyed. A fundamental limit on what reconstruction can achieve follows from the optimality of the conditional expectation as the minimum MSE estimator.
Proposition (Reconstruction bound). Let V-hat(p) be any estimator of V based solely on the price signal p. Then:
E[||V - V-hat(p)||^2] >= D_price
This follows directly from the fact that E[V|p] is the MSE-minimizing estimator of V given p, and D_price = tr(Sigma_{V|p}) is the MSE it achieves. No reconstruction method - regardless of sophistication, computational power, or analyst expertise - can reduce the MSE below D_price by working from the price signal alone, because the conditional covariance Sigma_{V|p} is the irreducible residual uncertainty given p.
In particular, for the constraint dimensions k = 2, ..., K, the reconstruction bound is:
E[(V_k - V-hat_k(p))^2] >= lambda_k
When the constraint dimensions are orthogonal to the price signal (the eigenbasis case), no reconstruction from prices alone can reduce uncertainty about constraint interactions below the prior. The sell-side analyst attempting to infer aquifer depletion risk from stock prices faces a fundamental barrier: the information was never in the price, and no amount of analytical effort can extract what was not transmitted.
Burton-Taylor Consulting (2025) reported that global spending on financial market data and news reached a record $44.3 billion in 2024. This figure includes real-time price feeds, execution infrastructure, compliance data, and index licensing alongside the analytically oriented spending that maps most directly onto the reconstruction concept. The reconstruction-relevant portion - sell-side research, credit analysis, ESG rating, and quantitative signal extraction - represents a subset of the total, though one that accounts for a substantial share. These activities are specifically directed at recovering judgments the price compressed away: analysts reverse-engineer corporate strategy from financial statements and price movements, credit rating agencies reconstruct default probability from compressed financial metrics, and quantitative funds reconstruct signal from noise in price data.
The $44.3 billion captures only the direct market data spending. The full reconstruction cost includes the salaries of the analysts performing the reconstruction, the computational resources devoted to extracting signal from price data, and the regulatory infrastructure that mandates the production of standardized disclosures to partially compensate for the price system's information loss. The figure is therefore simultaneously too broad (it includes non-reconstruction spending) and too narrow (it excludes analyst labor and regulatory compliance). It serves as an order-of-magnitude indicator, not a precise measurement, of the economic resources devoted to operating below the rate-distortion bound.
This reconstruction is inherently bounded by the result above. The information destroyed in the compression cannot be fully recovered from the compressed signal alone. Reconstruction from prices alone operates on the diminishing-returns portion of the estimation curve: each additional dollar spent reduces residual uncertainty by a smaller amount, because the information closest to the surface of the compressed signal is recovered first, and the deeply compressed information (the constraint dimensions with lambda_k unaffected by p) cannot be recovered at any cost from p alone.
What reconstruction actually achieves is inference from auxiliary signals - not from p itself but from other sources of information (company filings, satellite data, expert interviews, physical inspections) that carry information about V_2, ..., V_K. This auxiliary reconstruction is valuable but expensive, because the auxiliary signals are not organized into a system. They must be gathered, interpreted, and integrated by skilled analysts at costs that scale with the dimensionality of the information sought. The Disclosure Incentive replaces this unstructured auxiliary reconstruction with structured production at source.
The Disclosure Incentive does not redesign the price system. Prices remain scalars formed at the transaction level. What the Disclosure Incentive does is supplement the price with a structured signal that preserves information the price compresses away.
Let D denote the disclosure signal, which carries information about V with rate R_D = I(V; D). The combined information environment (price plus disclosure) operates at total rate:
R_total = I(V; p, D) <= R_price + R_D
with equality when p and D are conditionally independent given V.
The distortion of the combined system is:
D_combined = E[||V - E[V | p, D]||^2] = tr(Sigma_{V|p,D})
A disclosure channel designed to minimize D_combined given R_D supplementary bits should allocate those bits to the dimensions with the highest residual variance after the price. Since the price leaves dimensions 2, ..., K at their prior variance lambda_k, the optimal supplementary allocation follows a reverse water-filling on the residual variances:
The residual variances after the price are:
Dimension 1: lambda_1 / (1 + SNR_1)
Dimension k (k >= 2): lambda_k
The disclosure channel, operating optimally, allocates bits to the dimensions with the largest residual variance. If lambda_2 > lambda_1 / (1 + SNR_1) - which holds whenever the price is even moderately informative about V_1, since the left side can be large while the right side shrinks with SNR_1 - then the disclosure's first bits go to the constraint dimensions, not to refining the consensus value.
Proposition D (Marginal value of disclosure). At the current operating point (the price system with rate R_price and distortion D_price), the marginal distortion reduction from a supplementary disclosure bit allocated optimally exceeds the marginal reduction from an additional bit of price precision, provided that
lambda_2 > lambda_1 / (1 + SNR_1)
which holds whenever the price is sufficiently informative about the consensus value. In liquid markets, this condition is near-vacuous: for SNR_1 = 9, it requires only lambda_2 > lambda_1/10; for SNR_1 = 99, only lambda_2 > lambda_1/100. Any constraint dimension with non-negligible variance satisfies the condition once the price is even moderately informative. The result follows directly from the distortion structure established in Lemma A - uncompressed dimensions have higher residual variance than already-compressed dimensions, so the marginal return to compression is higher there. The proposition's value is in quantifying the comparison rather than establishing a surprising threshold.
Proof. For a Gaussian source with variance sigma^2_k, allocating r bits reduces the variance from sigma^2_k to sigma^2_k * 2^{-2r}. The marginal distortion reduction per bit at allocation r is:
-d(sigma^2_k * 2^{-2r})/dr = 2 * ln(2) * sigma^2_k * 2^{-2r}
At r = 0 (no bits yet allocated to dimension k), this evaluates to 2 * ln(2) * sigma^2_k. The marginal reduction is proportional to the residual variance. Therefore the dimension with the highest residual variance yields the greatest marginal distortion reduction per bit.
An additional price-precision bit reduces the residual variance on dimension 1, currently at lambda_1 / (1 + SNR_1). An optimally allocated disclosure bit targets the largest residual constraint dimension, with variance lambda_2 (uncompressed). Since lambda_2 > lambda_1 / (1 + SNR_1) by assumption:
2 * ln(2) * lambda_2 > 2 * ln(2) * lambda_1 / (1 + SNR_1)
The disclosure bit produces a strictly larger distortion reduction. QED.
Proposition D formalizes the intuition that disclosure captures high-value information at low cost relative to further price improvement. The financial services industry currently invests enormous resources in marginal improvements to price precision - faster data feeds, better quantitative models, more granular market microstructure - all of which operate on dimension 1, where residual variance is already low because the price system has allocated all its capacity there. Simultaneously, the constraint dimensions sit at their prior variance, untouched. A disclosure channel that allocates its first bits to these uncompressed dimensions achieves a larger total distortion reduction per bit than any refinement of the price signal can.
The practical implication is that a dollar spent on structured constraint disclosure produces more information value than a dollar spent on conventional market data infrastructure, because the disclosure targets the dimensions where the marginal return is highest and the price is absent.
The key prediction: as the Disclosure Incentive develops, we should observe substitution - market participants reallocating resources from price-reconstruction activities (sell-side research on already-well-priced dimensions) toward disclosure-based activities (structured constraint analysis on previously invisible dimensions). The substitution follows from the marginal value difference identified in Proposition D.
The derivation uses the Gaussian assumption throughout. The Gaussian rate-distortion function has a closed-form solution (reverse water-filling) that makes the excess distortion theorem tractable. For non-Gaussian sources - heavy-tailed distributions, multimodal judgment distributions, discrete constraint states - the qualitative results (excess distortion from one-dimensional compression of a multi-dimensional source) survive, but the closed-form expressions do not. Among all sources with a given covariance structure, the Gaussian has the highest rate-distortion function R(D) (Cover and Thomas, 2006, Chapter 10; cf. Chapter 12). Higher R(D) means more bits are required to achieve distortion D - equivalently, at a given rate R, the minimum achievable distortion D*(R) is highest for the Gaussian source. The Gaussian is the hardest source to compress optimally. On the D* margin, the lower-bound direction is unambiguous: D*_Gaussian(R) >= D*_other(R), so the Gaussian minimizes the gap from below.
On the D_price margin, the direction depends on how distortion is evaluated and on which dimensions dominate. For the constraint dimensions k >= 2, D_price is distribution-invariant: the price carries zero information about these dimensions regardless of their distribution, so their contribution to D_price is lambda_k in all cases. The Gaussian lower bound on Delta is therefore tight for the constraint-dimension terms, and these terms dominate total D_price when K is large and the eigenvalue spectrum does not decay too rapidly (the spectral condition of Section 4.4). For dimension 1, the comparison requires specifying the estimator. Under Gaussian V_1, the linear estimator E[V_1|p] is the MMSE estimator, and the price achieves the scalar Gaussian rate-distortion function on its own dimension. Under non-Gaussian V_1, two cases arise. If D_price is evaluated using the MMSE estimator (which may be nonlinear), the non-Gaussian MMSE can be lower than the linear MMSE, reducing D_price on dimension 1 - the effect on Delta is then ambiguous for that dimension alone, since both D_price and D* decrease. If D_price is evaluated using the linear estimator that market clearing actually produces (as in the CARA-normal REE), then D_price on dimension 1 equals the linear MMSE regardless of the true distribution, and the comparison depends on the specific distribution of V_1. In the economically relevant regime where constraint dimensions dominate (large K, slowly decaying spectrum), the dimension-1 ambiguity is a second-order correction. The Gaussian derivation is conservative on the dominant margin (constraint dimensions via D*) and approximately conservative on the secondary margin (dimension 1), with the approximation tightening as K grows.
The MSE distortion measure, while natural in the CARA-normal framework, does not capture all economically relevant costs of information loss. The welfare cost of misallocating capital due to missing constraint information depends on the shape of the loss function, the agents' decision problems, and the specific interaction between the missing information and the available signals. A welfare-theoretic distortion measure - one that weights dimensions by their allocation impact - would produce different numerical results. The direction of the excess distortion result is robust to the choice of distortion measure: any reasonable economic distortion measure that penalizes ignorance about constraint interactions will produce positive excess distortion when the price allocates zero bits to constraint dimensions. The magnitude depends on the measure.
The independence of V_1 and V_2, ..., V_K in the eigenbasis is an analytical convenience. In practice, the consensus value and constraint dimensions are partially correlated (a company's stock price partially reflects its environmental risk exposure). Corollary C' (Section 4.5) establishes formally that the excess distortion result extends to the general correlated case: whenever the residual conditional covariance Sigma_{V|p} has at least two eigenvalues above the water level, the excess distortion is strictly positive. The eigenbasis derivation in Proposition C provides a lower bound on Delta for the general case, since eigenbasis alignment minimizes D_price at any given rate (Corollary C').
The assumption that the price-informative direction coincides with the first eigenvector (the largest-eigenvalue direction) is a modeling choice, not a mathematical necessity. In a general CARA-normal REE with K-dimensional fundamentals, the price reveals a linear combination a'V + w of the V_k that need not align with any eigenvector. If it does not, the price carries partial information about multiple eigendimensions simultaneously, and the excess distortion is structured differently than Proposition C computes. Corollary C' establishes that the first-eigenvector alignment is the most favorable case for the price system at any given rate, and that the results of Proposition C therefore constitute a lower bound on Delta for price signals loading on arbitrary linear combinations, at the same mutual information rate I(V; p).
The analysis models the price system as a single scalar channel. In practice, a transaction may be observed through multiple prices - an equity price, a bond yield, a CDS spread, option-implied volatilities - each carrying information about a different linear combination of V. If M prices observe M independent linear combinations, the price system's total rate is sum_{m=1}^{M} R_m, and the allocation across dimensions depends on the M x K observation matrix A. Each price contributes a rank-one update to the conditional covariance, and the excess distortion is the residual variance after M rank-one reductions. For M >= K with linearly independent rows in A, the price system can in principle eliminate excess distortion entirely.
A simple bound characterizes how Delta decreases with M. If the M prices load on the M highest-eigenvalue directions (the most favorable case), and each has signal-to-noise ratio at least SNR_min, then D_price(M) = sum_{k=1}^{M} lambda_k/(1 + SNR_min) + sum_{k=M+1}^{K} lambda_k. The excess distortion becomes Delta(M) = D_price(M) - D*(R_total), where R_total = sum_{m=1}^{M} (1/2) log(1 + SNR_m). The constraint-dimension contribution to D_price drops from sum_{k=2}^{K} lambda_k at M = 1 to sum_{k=M+1}^{K} lambda_k at M > 1, a reduction of sum_{k=2}^{M} [lambda_k - lambda_k/(1 + SNR_min)]. The single-price analysis (M = 1) therefore overstates the constraint-dimension contribution to D_price by this amount. Whether this overstatement is a factor of two or a factor of a hundred depends on the ratio M/K: for liquid equity markets with M on the order of 5-10 observable prices and K potentially in the hundreds, the single-price analysis captures the dominant effect. The relevant comparison, however, is not M prices for a single asset but the full vector of prices across related goods, which collectively may carry substantial constraint information; the single-price analysis provides a lower bound on excess distortion for the full system. A full treatment of the M-price case - including the realistic setting where observation directions are not aligned with eigenvectors - is left to future work.
The threshold condition in Proposition C, part (i) produces a counterintuitive implication for illiquid markets. When R_price < R_0 - the price system's rate is too low for the optimal codec to reach any constraint dimension - the excess distortion is zero. The price system is trivially optimal because no codec at the same rate could do better. This does not mean illiquid markets are informationally adequate; it means total distortion D_price is high (the price is uninformative even about the consensus dimension), and the comparison against D* reveals no allocation mismatch because the rate budget is too small to allocate at all. The excess distortion measures allocation mismatch, not total informational adequacy. Illiquid markets suffer from high total distortion with zero excess distortion; liquid markets suffer from lower total distortion with positive excess distortion. The Disclosure Incentive addresses the latter regime.
The $44.3 billion reconstruction expenditure figure captures direct market data spending but not the full reconstruction burden. A comprehensive estimate would include analyst salaries, computational infrastructure, regulatory compliance costs, and the opportunity cost of talent devoted to reconstruction rather than production. The paper uses the direct figure as a lower bound on the reconstruction cost, but the true figure is substantially larger and not precisely measurable with existing data.
This paper establishes that the price system's compression of high-dimensional judgment into scalar signals introduces excess distortion with a specific, characterizable structure. The central result (Proposition C) identifies the threshold above which the price's rate allocation becomes suboptimal, the monotonicity of the excess in constraint-dimension variance, and the limiting behavior as market efficiency increases. Corollary C' extends the result beyond the eigenbasis to general source covariance. The distortion is structural: it arises from the scalar nature of market clearing, not from any correctable design flaw.
Two implications follow. The first is diagnostic: the framework provides a formal measure of what the price system cannot carry, replacing informal claims about "missing information" with a quantity derived from the source covariance and the price's signal-to-noise ratio. The second is prescriptive: supplementary disclosure channels that target uncompressed constraint dimensions produce strictly higher marginal distortion reduction than further improvements to price precision (Proposition D), which means the marginal dollar is better spent on structured constraint disclosure than on conventional market data infrastructure.
The framework is conditional on the eigenbasis decomposition and the Gaussian assumption. Empirical work identifying the eigenvalue spectrum for specific asset classes - the number and variance of constraint dimensions in equity, commodity, and climate-financial environments - would determine the quantitative magnitude of the excess distortion in practice. The rate-distortion bound is a floor on what any supplementary channel must achieve; whether any implementable mechanism reaches it is a design problem the present paper does not address.