| title: "The Beierspace Fraction: Measuring Coordination Fragility in Asset Prices" |
| author: "Gregory Caldwell Beier" |
| institute: "Susarb LLC, Cambridge, Massachusetts" |
| date: "March 2026" |
Computer Science, Mathematics, and Physics
by Gregory Caldwell Beier (forthcoming 2026).}
{
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The Mixing Theorem (Beier, 2026i) establishes that the distinction between physically determined quantities (eigenspace) and belief-constituted quantities (Beierspace) is a continuous spectrum parameterized by maintenance bandwidth. This paper introduces the empirical object the theorem implies: the Beierspace Fraction \(b(p,t)\), the share of an asset's valuation variance attributable to coordination-dependent rather than physically determined value drivers. The fraction is defined through a counterfactual - the share of price variance that would begin drifting toward incoherence if all observation channels were shut off - and operationalized through three measurement proxies. A second quantity, the Fragility Gap \(F(p,t) = b(p,t) - M(p,t)/M^*(p,t)\), measures the deficit between actual and required maintenance bandwidth. The Dissipation Law predicts that assets with \(F > 0\) accumulate coordination decay on unobserved constraint dimensions, invisible in the price until a discontinuous repricing. A cross-sectional asset pricing test is proposed: portfolios sorted on \(F\) should carry a fragility premium in normal conditions that reverses catastrophically during stress events. The framework predicts a secular rise in aggregate \(b\) as constraint complexity outpaces observation bandwidth, consistent with the documented rise in Tobin's \(Q\).
Keywords: Beierspace fraction, coordination fragility, fragility gap, maintenance bandwidth, dissipation, asset pricing, Tobin's \(Q\), intangible assets, rate-distortion theory, structured disclosure
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The companion paper in this series (Beier, 2026i) establishes an axiom - observation of a coordination-dependent structure is inseparable from the maintenance of that structure - and derives from it a dissipation law, a mixing theorem, and an equation linking maintenance bandwidth to minimum coordination loss through the rate-distortion function. The mixing theorem dissolves the eigenspace -- Beierspace partition into a continuous spectrum: any dimension's stability is a function of the maintenance bandwidth directed at it, not an intrinsic property of the quantity being measured.
The theoretical architecture is complete, but it lacks the instrument that would connect it to observable data.
If the eigenspace -- Beierspace distinction is a spectrum, then every asset price sits somewhere on it. A barrel of oil at the wellhead is priced largely by thermodynamics and geology - its eigenspace content is high. A cryptocurrency with no physical collateral is priced entirely by coordination among holders - its Beierspace content approaches one. Most assets fall in between, and their position shifts over time as constraint complexity grows or observation bandwidth changes.
This paper introduces the Beierspace Fraction \(b(p,t)\) as the measurable empirical object that the Mixing Theorem implies. It defines the quantity, proposes measurement proxies, derives the conditions under which high \(b\) produces fragility, designs a cross-sectional asset pricing test, and connects the results to the Disclosure Incentive that runs through the Prolegomena.
The paper's central claim is precise: the fraction of an asset's value that depends on human coordination rather than physical reality is estimable from observable market data, and the gap between that fraction and the maintenance bandwidth directed at it predicts which assets are vulnerable to discontinuous repricing. The paper designs the test; it does not execute it. What follows is the theoretical derivation, the measurement strategy, and the empirical test specification that would confirm or refute the framework's asset-pricing predictions.
For any asset with price \(p\) at time \(t\), the fundamental drivers of its value partition into two classes.
Class E (Eigenspace drivers). Components of value determined by physical, thermodynamic, geological, or engineering reality. These persist without ongoing human observation. A copper mine's ore grade does not change because analysts stop covering the mining company. A power plant's thermal efficiency does not drift because investors lose interest. The eigenspace component of value is anchored in matter and energy, and its variance is driven by physical processes that operate independently of market attention.
Class B (Beierspace drivers). Components of value constituted by ongoing human coordination. Brand value exists because consumers coordinate on purchasing from a recognized name. Network effects exist because users coordinate on a shared platform. Regulatory regime value exists because market participants coordinate on compliance with rules whose enforcement depends on institutional coordination. Supply chain trust exists because counterparties coordinate on fulfilling contracts whose terms depend on shared expectations about future coordination. These drivers persist only as long as the coordination persists, and the coordination persists only as long as agents observe and maintain it (Axiom 1 of Beier, 2026i).
The partition is not a binary classification of assets. It is a decomposition of value drivers within each asset. An oil company has eigenspace drivers (proved reserves, refinery capacity, pipeline infrastructure) and Beierspace drivers (OPEC coordination, regulatory regime, geopolitical risk pricing, ESG narrative). A software company has almost no eigenspace drivers and is almost entirely Beierspace (user coordination, developer ecosystem, enterprise procurement trust, regulatory classification).
Remark.
The decomposition is cross-cutting with respect to the Campbell -- Shiller (1988) variance decomposition into cash flow news and discount rate news. Cash flow news can be either eigenspace (an oil well runs dry) or Beierspace (a supplier network loses institutional trust and halts shipments). Discount rate news can be either eigenspace (physical depreciation changes the marginal product of capital) or Beierspace (central bank coordination shifts, inflation expectations rotate). The Campbell -- Shiller decomposition cuts variance along the accounting axis; the present decomposition cuts it along the ontological axis.
Remark.
The connection between intangible assets and financial risk is not new. Organization capital - the firm-specific knowledge embodied in key talent - has been shown to carry a cross-sectional risk premium of approximately 4.6
Definition (Beierspace Fraction).
For an asset with price \(p\) at time \(t\), the Beierspace Fraction \(b(p,t) \in [0,1]\) is the share of the asset's total valuation variance attributable to Class B (coordination-dependent) value drivers. Let \(V_E(p,t)\) denote the variance of the eigenspace component and \(V_B(p,t)\) the variance of the Beierspace component:
\[
b(p,t) = \frac{V_B(p,t)}{V_E(p,t) + V_B(p,t)}
\]
The counterfactual interpretation: \(b(p,t)\) is the fraction of the asset's price variance that would begin drifting toward incoherence if all observation channels were shut off today. The eigenspace fraction \(1 - b\) is the fraction that would remain anchored by physical reality.
Remark.
The Mixing Theorem (Beier, 2026i, Theorem 8.1) establishes that any dimension's persistence rate is a continuous, strictly increasing function of its maintenance bandwidth \(m_k\). In the eigenspace limit (\(m_k \to \infty\)), the dimension is invariant to belief perturbations. In the Beierspace limit (\(m_k \to 0\)), the dimension dissipates at its maximum rate. The Beierspace Fraction aggregates this per-dimension characterization to the asset level: an asset with high \(b\) has most of its value concentrated in dimensions near the low-\(m_k\) end of the spectrum.
The Beierspace Fraction connects to observable market quantities through the disagreement structure established in the companion papers. From Proposition 3.1 of Beier (2026i), the steady-state disagreement on dimension \(k\) is \(V_{k,\text{ss}} = \sigma_v^2/m_k\). Dimensions with high Beierspace content (low \(m_k\) relative to their variance) produce high disagreement among agents. Dimensions with high eigenspace content produce low disagreement.
Total valuation disagreement decomposes as:
\[ V_{\text{total}} = \sum_k V_{k,\text{ss}} = \sum_{k \in E} \frac{\sigma_{v,k}^2}{m_k} + \sum_{k \in B} \frac{\sigma_{v,k}^2}{m_k} \]
The eigenspace dimensions contribute little to disagreement (their physical observability provides high effective \(m_k\) even without formal disclosure channels). The Beierspace dimensions contribute most of the disagreement (their coordination-dependent nature means \(m_k\) is determined entirely by the observation architecture, and the price system provides zero bandwidth on constraint dimensions per the Lossy Compression result of Beier, 2026c).
An observable proxy for \(b\) follows: the fraction of analyst disagreement attributable to non-physical value drivers. High analyst forecast dispersion on coordination-dependent metrics relative to dispersion on physical metrics indicates high \(b\).
A high Beierspace Fraction is not inherently dangerous. A fiat currency has \(b\) approaching 1 yet well-maintained fiat currencies persist for centuries. A publicly traded software platform has \(b\) near 1 yet can trade at stable multiples for decades. The critical variable is not how much of the value depends on coordination but whether the coordination is adequately maintained.
Definition (Fragility Gap).
For an asset with price \(p\) at time \(t\):
\[ F(p,t) = b(p,t) - \frac{M(p,t)}{M^*(p,t)} \]
where \(M(p,t)\) is the actual maintenance bandwidth directed at the asset's Beierspace dimensions and \(M^*(p,t)\) is the bandwidth required to sustain full coordination at the current level of complexity.
When \(F(p,t) > 0\), the asset is undermaintained: its coordination complexity exceeds the observation bandwidth directed at it. The Dissipation Law (Beier, 2026i, Theorem 4.1) predicts that undermaintained dimensions lose coherence at a rate bounded below by \(D^*(M)\). The dissipation accumulates on the constraint dimensions the price system cannot reach (Beier, 2026c, Proposition C), invisible in the price until the accumulated decay crosses the phase-transition threshold (Beier, 2026e, Theorem 5.2).
When \(F(p,t) \leq 0\), the asset is adequately maintained. High \(b\) is sustainable as long as \(F\) remains non-positive.
The required maintenance bandwidth \(M^*\) is not a free parameter. It is determined by the eigenvalue spectrum of the asset's Beierspace dimensions and a chosen distortion tolerance through the rate-distortion function (Beier, 2026i, Theorem 7.1). Under the Gaussian source model, maintaining coordination on Beierspace dimensions at a total dissipation rate no greater than \(D_{\text{target}}\) requires bandwidth:
\[ M^*(p,t) = \sum_{k \in B} \max\left(0, \tfrac{1}{2}\log\frac{\lambda_k}{\theta^*}\right) \]
where \(\theta^*\) is the water level (Lagrange multiplier) solving:
\[ D_{\text{target}} = \sum_{k \in B} \min(\lambda_k, \theta^*) \]
The sum is over Beierspace dimensions only and \(\lambda_k\) is the variance (coordination intensity) of dimension \(k\). Under reverse water-filling, dimensions with \(\lambda_k > \theta^*\) receive positive bandwidth; dimensions with \(\lambda_k \leq \theta^*\) receive none and dissipate at their full intensity. The distortion target \(D_{\text{target}}\) is an economic parameter, not a mathematical constant: it represents the maximum coordination loss the asset can sustain before crossing its local stability threshold. Setting \(D_{\text{target}} = 0\) (zero dissipation, perfect observation of all dimensions) yields infinite \(M^*\) and is not a useful benchmark. A meaningful \(M^*\) uses a finite \(D_{\text{target}}\) calibrated to the asset's stability margin - the dissipation rate at which the stability ratio \(S\) (Beier, 2026e) approaches 1. In practice, a cross-sectional calibration sets \(D_{\text{target}}\) to the distortion level at which assets historically experience discontinuous repricing: the 5th percentile of drawdown-triggering stress, estimated from the distribution of constraint-event losses in the test sample.
The eigenvalue spectrum \(\{\lambda_k\}_{k \in B}\) is estimable from the cross-sectional dispersion of analyst valuations on coordination-dependent metrics - higher dispersion on a metric implies higher \(\lambda_k\). The estimation chain involves classification noise at every step: identifying which dimensions are Beierspace (a judgment call), estimating \(\lambda_k\) from dispersion on those dimensions, choosing \(D_{\text{target}}\), and computing \(M^*\) from the spectrum. Error in the initial classification propagates through the chain, inflating or deflating \(M^*\) depending on whether eigenspace dimensions are misclassified as Beierspace or vice versa. The convergent validation test in Section 5 serves partly to detect systematic misclassification.
This yields a computable, if noisy, estimate of \(M^*\): the total bandwidth required scales with the number of active Beierspace dimensions and the log-ratio of their variance to the negligibility threshold. An asset with many high-variance Beierspace dimensions (a complex financial instrument, a multinational platform) requires far more maintenance bandwidth than an asset with few low-variance Beierspace dimensions (a single-product commodity firm with minor regulatory exposure).
An alternative operational proxy avoids the eigenvalue estimation entirely: define \(M^*\) as proportional to \(b(p,t)\) times the asset's total dimensionality \(K\), estimated from the number of distinct value drivers analysts identify. This coarser proxy replaces the rate-distortion derivation with a counting argument but preserves the key feature that \(M^*\) increases with coordination complexity. Because \(b\) appears in both the definition of \(F\) and this proxy for \(M^*\), however, the coarse formulation creates mechanical correlation that inflates \(F\) for high-\(b\) assets regardless of actual maintenance deficit. The rate-distortion derivation, where \(M^*\) depends on the eigenvalue spectrum rather than on \(b\) directly, avoids this problem and should be preferred when the dispersion data support eigenvalue estimation.
The Beierspace paper (Beier, 2026e) establishes a phase transition at \(S = 1\). At the asset level, the Fragility Gap is the local analog of the system-level stability deficit: it measures how close a specific asset's coordination structure is to the point where the ordering force (maintenance bandwidth) is insufficient to sustain the structure against Beier drift.
Assets with large positive \(F\) are individually near their local phase transition. A portfolio's aggregate Fragility Gap measures its exposure to coordinated collapse. During stress events, when aggregate maintenance bandwidth drops, assets with high \(F\) cross their local thresholds first, producing the cascade of repricing that characterizes financial crises.
Three complementary approaches estimate the Beierspace Fraction from observable data.
Proxy 1: Analyst disagreement decomposition. Decompose cross-sectional dispersion of analyst valuations into physical-fundamental disagreement (reserve estimates, capacity, unit costs, replacement value) versus coordination-dependent disagreement (growth rates, market share, pricing power, regulatory outlook, network effects). Estimate \(b\) as the ratio of coordination-dependent dispersion to total dispersion. The proxy connects to the theoretical structure through \(V_{k,\text{ss}} = \sigma_v^2/m_k\): dimensions with low maintenance bandwidth produce high dispersion. The proxy overestimates \(b\) for assets with high eigenspace uncertainty - exploration-stage miners (geological uncertainty), early-stage biotech (binary FDA outcomes) - where high dispersion reflects physical stochasticity (\(\sigma_v^2\) is large) rather than low maintenance bandwidth (\(m_k\) is small). The convergent validation test (Section 5) serves partly to detect this: if Proxy 1 disagrees systematically with Proxy 2 for a class of assets, that class likely has high eigenspace uncertainty being misclassified as Beierspace content. The analyst dispersion anomaly documented by Diether, Malloy, and Scherbina (2002) - that stocks with higher forecast dispersion earn lower subsequent returns - raises an apparent tension with the fragility premium predicted by Hypothesis 1. DMS attribute their anomaly to short-sale constraints producing overpricing that corrects over horizons of one to twelve months. The fragility premium operates on a different timescale: it compensates investors for holding assets through potential phase transitions, a risk that materializes over crisis horizons (quarters to years). The two effects can coexist if high-dispersion assets are overpriced in the short run (the DMS effect) while high-\(F\) assets carry a long-run risk premium for coordination fragility. The empirical test should examine returns at both horizons and distinguish the mispricing correction (short) from the risk compensation (long).
Proxy 2: The physical-to-intangible value ratio. The ratio of replacement cost to market price provides a broadly available estimate. When Tobin's \(Q = 1\), all value is physical and \(b \approx 0\). When \(Q \gg 1\), \(b \approx 1 - 1/Q\). The proxy is coarse (Tobin's \(Q\) conflates several intangible categories) but useful for broad cross-sectional sorts and the time-series prediction.
Proxy 3: Options-implied constraint-dimension variance. The volatility skew (OTM put IV premium over ATM IV) estimates the market's implicit assessment of coordination fragility - the risk of discontinuous repricing the Fragility Gap predicts. This proxy measures not \(b\) itself but the market's pricing of the fragility that high \(b\) with low \(M\) produces.
Proxy 1: Constraint-dimension disclosure density. The volume of structured reporting on non-financial constraint dimensions (environmental, social, governance), weighted by specificity and verifiability. Unstructured narrative disclosure receives low weight; structured, quantitative, outcome-scored disclosure receives high weight.
Proxy 2: Dedicated analyst coverage on constraint dimensions. The proportion of research reports addressing constraint-dimension risk versus consensus-dimension analysis. Assets covered exclusively on consensus dimensions have low \(M\) on their Beierspace components regardless of total coverage.
At each rebalancing date, estimate \(F(p,t)\) for each asset using the proxies from Section 4. The baseline specification uses Proxy 1 (\(b\) from analyst disagreement decomposition) for assets with sufficient coverage (three or more analysts with decomposed valuation models) and Proxy 2 (\(b\) from Tobin's \(Q\)) otherwise. \(M\) is estimated from Proxy 1 (constraint-dimension disclosure density). Robustness checks report results for each \(b\) proxy separately and for a composite \(b\) constructed as the first principal component of the three proxies where all three are available. Committing to the aggregation rule before seeing data prevents specification searching. Sort into quintile portfolios by \(F\). Compute value-weighted returns.
Hypothesis (The Fragility Premium).
In normal macroeconomic conditions, the high-\(F\) portfolio earns a positive premium over the low-\(F\) portfolio, compensating for the risk of discontinuous repricing on undermaintained coordination dimensions.
Hypothesis (The Catastrophic Reversal).
During macroeconomic stress events, the high-\(F\) portfolio underperforms the low-\(F\) portfolio by a margin exceeding what standard risk factors predict. The excess underperformance is the signature of coordination collapse on undermaintained Beierspace dimensions.
The test must distinguish the Fragility Gap premium from known factor premiums. Controls include Fama -- French factors (Fama and French, 1993), a liquidity factor (Pastor and Stambaugh, 2003; Amihud, 2002), and analyst coverage (Hong, Lim, and Stein, 2000). For Hypothesis 2 (the catastrophic reversal), the natural benchmark is Chen, Hong, and Stein (2001), who find that crash risk relates to trading volume and past returns. If the Fragility Gap predicts crash risk beyond what their variables capture, that is strong evidence for the framework; if it does not, that is informative too.
The key identification argument: an asset can be heavily covered on the consensus dimension yet have zero coverage on its constraint dimensions. Such an asset has low information asymmetry on \(V_1\) and high Fragility Gap on \(V_2\) through \(V_K\). Sorting on \(F\) rather than total information asymmetry separates the two effects.
A within-industry placebo test sharpens identification: within a single industry (holding sector composition, regulatory environment, and business model constant), do firms with higher constraint-dimension disclosure (higher \(M\), lower \(F\)) exhibit lower crash risk than firms with similar total analyst coverage but lower constraint-dimension coverage? Crash risk is measured by the negative coefficient of skewness of firm-specific weekly returns (following Chen, Hong, and Stein, 2001), supplemented by the frequency of firm-specific returns below \(-2\) standard deviations during identified stress periods. This test holds many confounds constant that the full cross-sectional sort does not.
Before the portfolio sorts, the three proxies for \(b\) (Section 4.1) should be validated against each other. If the Beierspace Fraction is a real quantity, the three proxies - which measure different observable reflections of it - should correlate positively in the cross section. The pairwise rank correlations between (i) analyst disagreement ratio, (ii) Tobin's \(Q\)-derived \(b\), and (iii) volatility skew premium provide a convergent validity test. If the correlations are low or negative, the proxies are measuring different things and the theoretical construct lacks empirical coherence. If the correlations are moderate and positive, the proxies capture a common latent factor consistent with \(b\).
Sort assets independently on \(b\) (high vs.\ low) and \(M\) (high vs.\ low). Form four portfolios:
Assets with high \(b\) and high \(M\) are the stable quadrant - coordination-dependent value receiving the bandwidth it needs. Normal returns, low crash risk. The critical quadrant is high \(b\) with low \(M\): undermaintained coordination complexity sitting near its local phase transition, carrying a premium in calm markets that reverses when stress withdraws the marginal maintenance bandwidth. Low \(b\) with high \(M\) represents a resource mismatch - excessive observation bandwidth directed at physically anchored assets that don't need it. Low \(b\) with low \(M\) is the physically robust baseline: assets whose value is anchored in matter and energy, requiring little coordination maintenance.
The interaction effect - that crossing \(b\) with \(M\) predicts crisis returns better than either alone - is the joint test of the Dissipation Law and the Observation-Maintenance Identity. Neither standard risk factors nor behavioral finance's "sentiment" variable (De Long, Shleifer, Summers, and Waldmann, 1990) can produce this interaction, because neither distinguishes between coordination complexity and observation bandwidth.
Remark.
The Fragility Gap \(F\) compresses a two-dimensional object (\(b\), \(M/M^*\)) into a scalar. Two assets with identical \(F = 0.3\) may have very different risk profiles: one with \(b = 0.8\) and \(M/M^* = 0.5\) (high complexity, moderate maintenance, many dimensions dissipating) and another with \(b = 0.4\) and \(M/M^* = 0.1\) (lower complexity, almost no maintenance, fewer but completely neglected dimensions). The Dissipation Law's predictions depend on both the gap and the level of \(b\). The joint test - sorting independently on \(b\) and \(M\) - is the more informative specification precisely because it preserves the two-dimensional structure that \(F\) collapses. The quintile sort on \(F\) is the simpler, higher-powered test; the joint sort is the diagnostic.
The framework predicts that aggregate \(b\) is rising secularly. Each new technology, financial instrument, and cross-border dependency adds Beierspace dimensions without commensurate observation bandwidth.
The evidence is consistent. Tobin's \(Q\) for U.S.\ nonfinancial corporations has risen from approximately 0.6 in the early 1980s to approximately 1.5 in the 2020s. The share of S&P 500 market capitalization attributable to intangible assets has risen from approximately 17
Under the present framework, this secular rise reflects a genuine shift toward coordination-dependent value creation - software, networks, platforms, intellectual property, institutional trust. The shift is correct as long as \(M\) keeps pace with \(b\). The systemic risk is that \(M\) has not kept pace.
Remark.
The aggregate figures are confounded by index composition change. The S&P 500 in 1975 was heavy in manufacturing, energy, and utilities (low \(b\)); in 2020 it is dominated by technology, healthcare, and financials (high \(b\)). The aggregate \(b\) rose partly because the index changed its constituents, not solely because individual assets became more coordination-dependent. A sharper test tracks the same industries over time. Did the oil industry's Beierspace fraction increase between 1975 and 2020? Plausibly yes - OPEC coordination dynamics, the ESG narrative, and carbon regulation are all Beierspace dimensions that did not exist or mattered less in 1975. Did banking's increase? Almost certainly - structured finance, derivatives, and shadow banking are massively Beierspace. The within-industry time series, holding composition constant, is the clean test of the secular-rise hypothesis.
The aggregate Fragility Gap at each point in time should predict the severity of subsequent stress events. Any framework can retrodict known crises after the fact; the value of the Fragility Gap is as a forward-looking measure. The asset class with the highest aggregate \(F\) at time \(t\) should be the epicenter of the next repricing event after \(t\).
The Beierspace Fraction and the Fragility Gap provide the empirical anchor the Prolegomena's policy apparatus has lacked. The Disclosure Incentive (Beier, 2026c, Section 6; Beier, 2026d) increases \(M\) on Beierspace dimensions, reducing \(F\) and pulling the asset back from its local phase transition. The value of the Disclosure Incentive for a specific asset is measurable: the change in \(F\) produced by the disclosure, multiplied by the fragility premium the market charges for the pre-disclosure level of \(F\).
The cross-sectional test (Section 5) provides the fragility premium estimate. The disclosure intervention provides the change in \(F\). The product is the dollar value of structured constraint disclosure for a specific asset, a specific portfolio, or the financial system as a whole.
The fragility premium is estimated from sorts on \(F\), and the disclosure intervention changes \(M\) and therefore \(F\), so the inputs to the valuation are interdependent. Standard identification strategies apply. Natural experiments from disclosure mandates (Regulation FD in the U.S., the EU Corporate Sustainability Reporting Directive) provide exogenous variation in \(M\): the mandate forces disclosure regardless of the asset's current \(F\), breaking the simultaneity. Regression discontinuity designs around reporting thresholds (market capitalization cutoffs for mandatory ESG disclosure, index inclusion criteria triggering analyst coverage) provide local identification. These strategies are standard in the empirical disclosure literature and should be applied to the Fragility Gap context when the data are assembled.
The measurement proxies are noisy. Analyst disagreement decomposition requires sufficient coverage and identifiable valuation models. Tobin's \(Q\) conflates multiple sources of intangible value. Options-implied variance captures market perception of fragility, not fragility itself. The eigenspace/Beierspace decomposition of value drivers is a judgment call, not an algorithm - reasonable analysts will disagree about whether a specific driver (a pharmaceutical patent, a mineral lease) is closer to eigenspace or Beierspace.
The cross-sectional test requires that the Fragility Gap is not fully captured by existing risk factors. If the market already prices coordination fragility through other channels, the \(F\) premium may be small or absent. The test's power depends on the Lossy Compression result being empirically relevant: the price system must actually fail to transmit constraint-dimension information for the Fragility Gap to persist as unpriced risk.
The time-series prediction is confounded by concurrent trends (financialization, globalization, technological change) that affect Tobin's \(Q\) through channels unrelated to the Beierspace framework. Isolating the Beierspace mechanism requires either a structural model controlling for confounds or quasi-experimental variation.
The Beierspace Fraction is the Mixing Theorem's empirical shadow. Every asset price occupies a position on the eigenspace -- Beierspace spectrum, and that position determines its vulnerability to the Dissipation Law. The Fragility Gap measures the deficit between coordination complexity and observation bandwidth. Where the gap is large, constraint dimensions are decaying beneath a price that cannot see them.
The sharpest test is cross-sectional: portfolios sorted on \(F\) should carry a premium that reverses catastrophically in stress events, and the joint sort on \(b\) and \(M\) should isolate an interaction effect that no existing risk factor produces. The time-series prediction - that aggregate \(b\) is rising and aggregate \(F\) predicts crisis severity - is harder to test cleanly because of the composition confounds identified in Section 6.1, but the within-industry version offers a path. The disclosure connection closes the loop back to the Prolegomena: if reducing \(F\) through structured constraint disclosure measurably reduces the fragility premium, the framework converts from theory to engineering.
Each prediction, if it fails, locates a specific break in the chain. If high-\(F\) portfolios carry no premium, the Dissipation Law may not reach asset pricing in the way the framework claims. If the secular trend in \(b\) is absent within industries (not just across changing index composition), the dimensional growth asymmetry needs rethinking. And if structured disclosure fails to reduce \(F\) and the associated premium, the challenge reaches deeper - to the Observation-Maintenance Identity at the framework's root. The framework is now specified precisely enough that data can reach it.