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 lossy compression (Beier, 2026c) derives the rate-distortion analysis of price formation and proves that supplementary disclosure bits produce strictly higher marginal distortion reduction than additional price-precision bits (Proposition D). The companion paper on dual surfaces (Beier, 2026f) defines Beier Extropy E_B = H(D|P) as the information restored to the compressed price signal by structured disclosure. This paper bridges the two results. Proposition EB-1 derives the distortion reduction achievable when E_B bits of disclosure supplement the price signal, as a function of the source covariance structure and the disclosure's allocation across constraint dimensions. Proposition EB-2 establishes the optimal allocation of E_B bits across constraint dimensions under MSE distortion - the disclosure analog of the reverse water-filling solution. Proposition EB-3 computes the distortion gap reduction: the fraction of the excess distortion (from Beier, 2026c, Proposition C) that E_B bits of optimally allocated disclosure eliminate. The result provides a quantitative link between the dual-surface paper's information measure (E_B) and the lossy compression paper's distortion measure (Delta = D_price - D*), closing the formal loop between the two foundational papers.
Keywords: Beier Extropy, rate-distortion, distortion reduction, disclosure allocation, reverse water-filling, excess distortion
JEL Codes: D82, D83, C60, G14
Two companion papers provide the information-theoretic foundations for the Prolegomena's architecture, but they do not connect to each other formally. The lossy compression paper (Beier, 2026c) works in rate-distortion space: the price system compresses a K-dimensional source into a scalar, introducing excess distortion Delta = D_price - D* that is measured in variance units. The dual-surface paper (Beier, 2026f) works in entropy space: Beier Extropy E_B = H(D|P) measures the information disclosure adds to the price signal, in bits.
Both papers assert that disclosure improves the information environment. The lossy compression paper proves this through distortion reduction (Proposition D: disclosure bits produce higher marginal reduction than price bits). The dual-surface paper proves this through the simultaneous improvement proposition (both surfaces improve when I(V; D|P) > 0). But the quantitative relationship between E_B (bits added) and Delta (variance reduced) is not derived in either paper.
This paper derives it.
Adopt the notation of the lossy compression paper (Beier, 2026c). The source is V = (V_1, ..., V_K) with independent components in the eigenbasis, variances lambda_1 >= ... >= lambda_K > 0. The price signal provides R_price = (1/2) log_2(1 + SNR_1) bits about V_1 and zero bits about V_2, ..., V_K.
The disclosure signal D carries E_B bits of information about V beyond what the price carries. The disclosure is structured: the five-field format allocates information across the K dimensions of V. Let r_k denote the number of disclosure bits allocated to dimension k, with sum_{k=1}^K r_k <= E_B.
The key assumption: the disclosure bits are independent of the price bits on the dimensions they share (dimension 1). This holds when the disclosure carries information about constraint dimensions that the price does not (which is the typical case by Proposition C's result that the price allocates zero bits to dimensions 2 through K). When the disclosure also carries information about dimension 1, the bits are partially redundant with the price and the effective information is less than E_B. For the analysis below, we consider the clean case where the disclosure targets only the uncompressed dimensions (r_1 = 0, all E_B bits allocated to dimensions 2 through K), which is the case where disclosure has maximal impact per the logic of Proposition D.
Proposition EB-1 (Disclosure distortion reduction). Let E_B bits of disclosure be allocated across the constraint dimensions V_2, ..., V_K with allocation (r_2, ..., r_K), sum r_k = E_B, r_k >= 0. The distortion after disclosure is:
D_{price+disclosure} = lambda_1 / (1 + SNR_1) + sum_{k=2}^{K} lambda_k * 2^{-2r_k}
The distortion reduction from disclosure is:
Delta_D(E_B) = D_price - D_{price+disclosure} = sum_{k=2}^{K} lambda_k * (1 - 2^{-2r_k})
Proof. Under the Gaussian model, each bit allocated to dimension k reduces the conditional variance from lambda_k to lambda_k * 2^{-2r_k} (the standard rate-distortion relationship for a Gaussian source under MSE distortion; Cover and Thomas, 2006, Chapter 10). The price provides R_price bits on dimension 1 (reducing its variance to lambda_1 / (1 + SNR_1) = lambda_1 * 2^{-2R_price}) and zero bits on dimensions 2 through K (leaving them at their prior variances lambda_k). The disclosure provides r_k additional bits on each dimension k >= 2, reducing each to lambda_k * 2^{-2r_k}. The total distortion is the sum of residual variances across all dimensions. The reduction is the difference between D_price and D_{price+disclosure}. QED.
Proposition EB-2 (Optimal disclosure allocation). The allocation (r_2*, ..., r_K*) that minimizes D_{price+disclosure} subject to sum_{k=2}^K r_k = E_B is the reverse water-filling solution applied to the constraint dimensions:
r_k* = max(0, (1/2) log_2(lambda_k / theta_D))
where theta_D is determined by the constraint:
sum_{k=2}^{K} max(0, (1/2) log_2(lambda_k / theta_D)) = E_B
Under optimal allocation, the residual variance on each active constraint dimension is equalized at theta_D:
lambda_k * 2^{-2r_k*} = theta_D for dimensions where r_k* > 0
and the distortion is:
D_{price+disclosure}^* = lambda_1 / (1 + SNR_1) + sum_{k: r_k*>0} theta_D + sum_{k: r_k*=0} lambda_k
Proof. This is the standard reverse water-filling solution (Berger, 1971; Cover and Thomas, 2006) applied to the source (V_2, ..., V_K) at rate E_B. The optimization min_{r_2,...,r_K} sum lambda_k * 2^{-2r_k} subject to sum r_k = E_B and r_k >= 0 is a convex optimization whose KKT conditions give the reverse water-filling solution. QED.
Proposition EB-3 (Excess distortion closure). Define the excess distortion from the lossy compression paper as:
Delta = D_price - D* (where D* is the minimum distortion at rate R_price, from Proposition B)
Define the disclosure excess reduction ratio as:
phi(E_B) = Delta_D^*(E_B) / Delta
where Delta_D^*(E_B) is the optimally allocated disclosure distortion reduction from Proposition EB-2. Then:
(i) phi(E_B) is strictly increasing in E_B for E_B in [0, E_B^{max}] where E_B^{max} = sum_{k=2}^K (1/2) log_2(lambda_k / theta_D^{min}) and theta_D^{min} is the water level when the total rate R_price + E_B^{max} achieves D* exactly.
(ii) phi(E_B) = 1 (complete closure) when E_B = E_B^{max}. At this point, the combined system (price + disclosure) achieves the rate-distortion bound that the price alone cannot reach.
(iii) The marginal closure rate is:
d phi / d E_B = (2 * ln(2) * theta_D) / Delta
which is highest when theta_D is largest (early disclosure bits, allocated to the highest-variance uncompressed dimension) and decreasing as theta_D falls (later disclosure bits produce smaller marginal closure because they target lower-variance dimensions or further compress already-partially-compressed dimensions).
(iv) For the first disclosure bit (E_B = 1, allocated to the largest uncompressed dimension lambda_2):
phi(1) = lambda_2 * (1 - 2^{-2}) / Delta = (3/4) * lambda_2 / Delta
Proof. (i) follows from the fact that each additional disclosure bit reduces distortion (Delta_D^* is increasing in E_B by the monotonicity of the rate-distortion function). (ii) follows from the definition: E_B^{max} is the number of bits needed to close the entire gap between D_price and D*. The combined rate R_price + E_B^{max} achieves D* because the optimal allocation of the combined rate distributes bits across all dimensions via reverse water-filling, achieving the same D* that the lossy compression paper computes as the minimum distortion at rate R_price (but recall D* was computed for the optimal allocation at rate R_price, while the combined system has rate R_price + E_B^{max} > R_price, so it achieves a distortion D*_{combined} <= D*; the closure is complete when D_{price+disclosure} reaches D*).
E_B^{max} closes the gap between D_price and D*_{at rate R_price}, not the absolute minimum D* = 0. The full closure means the combined system allocates its total rate (R_price + E_B^{max}) optimally, matching the distortion an optimal codec would achieve at the same total rate. This is achieved when the combined allocation satisfies reverse water-filling across ALL K dimensions at total rate R_price + E_B^{max}.
(iii) The marginal distortion reduction from the marginal disclosure bit is 2 * ln(2) * theta_D (the derivative of the distortion with respect to rate at the current water level). Dividing by Delta gives the marginal closure rate.
(iv) The first disclosure bit targets the largest uncompressed dimension lambda_2 (which has residual variance lambda_2 since the price allocated zero bits to it). One bit reduces the variance from lambda_2 to lambda_2 * 2^{-2} = lambda_2 / 4. The reduction is (3/4) * lambda_2. The excess reduction ratio is this reduction divided by the total excess distortion. QED.
Using the companion paper's example (Beier, 2026c, Section 4.4): K = 4, lambda_1 = 10, lambda_2 = 5, lambda_3 = 3, lambda_4 = 1, SNR_1 = 9.
D_price = 10 (from the companion paper).
D* = 8.41 (from the companion paper).
Delta = 1.59.
First disclosure bit (allocated to dimension 2, the largest uncompressed dimension):
phi(1) = (3/4) * 5 / 1.59 = 3.75 / 1.59 = 2.36.
This exceeds 1, which means one disclosure bit allocated to dimension 2 reduces distortion by more than the entire excess distortion Delta. This does not violate Shannon's bounds: the combined system operates at total rate R_price + 1, not at R_price. The ratio phi(1) > 1 reflects the massive marginal efficiency of the first disclosure bit when applied to a completely uncompressed dimension - not an impossibility, but a measure of how badly the scalar codec misallocates its bandwidth.
Why? Because the excess distortion (1.59) is the gap between D_price and the OPTIMAL allocation at the SAME rate R_price. One disclosure bit adds rate BEYOND R_price, and the first bit at the most underserved dimension produces a massive return because the dimension was completely uncompressed. The disclosure doesn't just close the gap - it surpasses the optimal-allocation benchmark because it's adding rate that the original benchmark didn't have.
The meaningful comparison is: how much disclosure is needed to match the optimal codec at total rate R_price + E_B? The answer from Proposition EB-2 is: the disclosure should be allocated via reverse water-filling on the constraint dimensions, and the distortion achieved is:
For E_B = 1: D_{price+disclosure} = 1 + 5/4 + 3 + 1 = 6.25. The combined system at total rate R_price + 1 = 2.66 bits achieves D = 6.25. The optimal codec at 2.66 bits (reverse water-filling across all four dimensions) achieves lower D because it can reallocate some of the price's bits from dimension 1. But the disclosure's constraint-dimension targeting is nearly optimal because the marginal return on dimension 2 dominates the marginal return on dimension 1 at the current operating point (by Proposition D of the companion paper).
Proposition EB-1 connects E_B (the dual-surface paper's measure) to Delta_D (the lossy compression paper's measure). The conversion formula is:
Delta_D = sum_{k=2}^K lambda_k * (1 - 2^{-2r_k})
where the r_k are determined by the disclosure's information content about each constraint dimension. For optimally allocated disclosure:
Delta_D^* = sum_{k: r_k*>0} (lambda_k - theta_D)
This is the bridge between the two foundational papers. The dual-surface paper says: disclosure adds E_B bits of information to the signal, and both surfaces benefit when those bits carry constraint information. The lossy compression paper says: each bit allocated to an uncompressed constraint dimension produces a specific distortion reduction. The bridge says: E_B bits, optimally allocated, close a specific fraction of the excess distortion, and that fraction is computable from the source covariance structure and the disclosure allocation.
The bridge also connects to the reconstruction burden (Beier, 2026a). The reconstruction cost is bounded below by D_price (the data processing inequality). Each bit of disclosure reduces D_price toward D*, reducing the reconstruction bound. The relationship between disclosure investment (producing D) and reconstruction savings (reduced D_price) is now quantifiable: E_B bits of disclosure produce Delta_D^* units of distortion reduction, which maps to a specific reduction in the lower bound on reconstruction cost.
The analysis assumes the Gaussian framework throughout. Under non-Gaussian sources, the rate-distortion relationship lambda_k * 2^{-2r_k} does not hold exactly, and the optimal allocation differs from reverse water-filling. The qualitative result - that disclosure bits targeted at uncompressed dimensions produce higher marginal distortion reduction than additional price bits - survives (it follows from the concavity of the rate-distortion function, which holds for all sources, not just Gaussian). The quantitative formulas are Gaussian-specific.
The assumption that disclosure targets only constraint dimensions (r_1 = 0) is a simplification. In practice, some disclosure content carries information about the consensus value dimension (an analyst's target price is V_1 information). Disclosure bits allocated to dimension 1 are partially redundant with the price and produce lower marginal distortion reduction than the analysis predicts. The analysis provides an upper bound on disclosure's effectiveness by assuming all bits target the highest-return dimensions.
The optimal allocation of Proposition EB-2 requires knowledge of the source covariance structure (the eigenvalues lambda_k). In practice, the discloser does not know the covariance structure exactly and cannot optimize the allocation. The five-field disclosure format serves as a heuristic allocation mechanism: by structuring the disclosure around five dimensions (Input, Method, Output, Identity, Scaling), the format distributes information across the constraint space in a way that, while not optimal in the rate-distortion sense, avoids the price system's pathological allocation of zero bits to constraint dimensions.