PRIORITY FILING - INITIAL DRAFT
A fully revised and formatted version will follow.
Gregory Caldwell Beier
Susarb LLC, Cambridge, Massachusetts
April 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.
Proper scoring rules evaluate whether a forecaster's stated probabilities match realized frequencies. The standard interpretation treats high scores as success and low scores as failure. This paper develops an alternative interpretation: systematic low scores on specific constraint dimensions are more informative than high scores, because the pattern of failure maps the boundary of the forecaster's observation subspace. A forecaster who consistently scores well on individual water risk assessments but consistently fails on water-energy coupling assessments is revealing - through the specific shape of her wrongness - exactly where the observation architecture has a structural gap. The gap was invisible to correct predictions because correct predictions confirm the existing observation geometry. Only systematic error reveals the geometry's boundaries. The paper formalizes this as productive failure (Theorem 1): the Brier score decomposition across constraint dimensions, accumulated over sufficient scoring cycles, identifies the forecaster's observation subspace with probability approaching 1. The error pattern is a noisy estimator of the subspace's orthogonal complement - the dimensions the forecaster cannot see. Aggregating productive failures across forecasters produces a population-level map of the observation architecture's topology of ignorance (Proposition 2). The map identifies which constraint interactions are systematically missed by the entire observation architecture, not just by individual forecasters. The paper then derives the Goodhart Dislocation (Theorem 3): the specific, quantifiable repricing event that occurs when a previously unobserved constraint dimension becomes visible in a domain that has been subject to scalar metric targeting. The dislocation magnitude is proportional to the duration of targeting, the intensity of the targeting, and the accumulated dissipation on the neglected dimensions. The dislocation is predictable before it occurs: the bias signature in early disclosure data identifies both the location and the approximate magnitude of the forthcoming repricing. The prediction that ESG-targeted domains will show larger Goodhart dislocations than non-ESG domains - controlling for constraint-dimension variance - is the paper's most specific testable claim. No existing framework produces this prediction.
Keywords: productive failure, Goodhart dislocation, proper scoring rules, Brier score, systematic error, observation subspace, topology of ignorance, bias signature, scalar targeting, repricing, constraint dimensions
JEL Codes: D82, D83, G14, G12, C53
Every disclosure system that uses proper scoring rules distinguishes good forecasters from bad ones. A forecaster with a Brier score of 0.85 is better calibrated than one with 0.70. The standard use of scoring is selection: identify the well-calibrated forecasters, weight their assessments more heavily, build the track record that separates signal from noise.
This paper develops a different use. The pattern of a forecaster's failures - not the magnitude but the structure, not the average but the dimensional decomposition - carries information that correct predictions cannot carry. A forecaster who scores well on water risk and poorly on water-energy coupling is revealing something specific about the observation architecture: the coupling lives outside her observation subspace. A forecaster who scores well on individual company emissions and poorly on the interaction between emissions regulation and financial stability is revealing that the regulatory-financial interaction is a structural blind spot.
Correct predictions confirm the existing observation geometry. They say: within the dimensions I can see, I see accurately. Systematic errors reveal the geometry's limits. They say: here is the boundary of what I can see, identified by the pattern of what I consistently get wrong.
The revelation is productive because the boundary information has economic value. The forecaster's blind spot, identified through scored systematic error, is a candidate for the next unpriced constraint event. If many forecasters share the same blind spot - if the water-energy coupling is systematically missed by every analyst covering the domain - then the shared blind spot is a structural feature of the observation architecture, and the constraint interaction that lives in the blind spot is a candidate for the next basin transition that the market cannot see coming.
The paper develops this argument in two parts. Part I (Sections 2-5) formalizes productive failure at the individual and population level. Part II (Sections 6-9) derives the Goodhart Dislocation - the specific repricing event that productive failure data can predict.
Let an analyst produce scored disclosures on a constraint domain with K dimensions. At each scoring cycle t = 1, ..., T, the analyst reports a probability vector r_t = (r_{t,1}, ..., r_{t,K}) where r_{t,k} is the reported probability for a binary outcome on dimension k. The realized outcomes are omega_t = (omega_{t,1}, ..., omega_{t,K}) where omega_{t,k} in {0, 1}.
The Brier score on dimension k at cycle t is:
BS_{t,k} = 1 - (r_{t,k} - omega_{t,k})^2
The aggregate Brier score across all dimensions at cycle t is:
BS_t = (1/K) sum_{k=1}^{K} BS_{t,k}
The time-averaged score on dimension k after T cycles is:
BS_k(T) = (1/T) sum_{t=1}^{T} BS_{t,k}
Definition 2.1. The analyst's dimensional score profile after T cycles is the vector:
S(T) = (BS_1(T), BS_2(T), ..., BS_K(T)) in [0, 1]^K
The profile records the analyst's average accuracy on each constraint dimension separately. The aggregate score BS(T) = (1/K) sum_k BS_k(T) is the mean of the profile, which is the quantity track record capital accumulates. The profile itself - the dimensional decomposition - is the quantity productive failure analysis exploits.
Under truth-telling (Proposition 5 of Beier, 2026r), the analyst reports her true belief p_{t,k} as r_{t,k}. Under maximum uncertainty on dimension k (p_{t,k} = 1/2), the expected Brier score is:
E[BS_{t,k} | p_{t,k} = 1/2] = 1 - E[(1/2 - omega_{t,k})^2] = 1 - 1/4 = 3/4
The baseline score of 3/4 is the score an honest analyst achieves on a dimension about which she has no information beyond the prior. An analyst who consistently scores near 3/4 on dimension k is not poorly calibrated - she is honestly reporting maximum uncertainty. The 3/4 baseline separates dimensions where the analyst has genuine information (BS_k > 3/4) from dimensions where she does not (BS_k approximately equal to 3/4).
Definition 3.1. The analyst's effective observation set at threshold eta is:
O(eta) = {k in {1, ..., K} : BS_k(T) > 3/4 + eta}
where eta > 0 is a threshold that accounts for finite-sample noise. Dimensions in O(eta) are those where the analyst demonstrates information beyond the prior. Dimensions not in O(eta) are those where the analyst's scores are consistent with maximum uncertainty.
Definition 3.2. The analyst's effective blind set is the complement:
B(eta) = {1, ..., K} minus O(eta)
Dimensions in B(eta) are those where the analyst's scores do not exceed the baseline by more than the noise threshold. These are the dimensions the analyst cannot see, as revealed by the pattern of her scores.
Theorem 1 (Productive Failure Identifies the Observation Subspace). Under truth-telling and stationary constraint dynamics, the analyst's observation set O(eta) converges to the set of dimensions spanned by her observation subspace W in Gr(k, K):
P(O(eta_T) = {j : e_j has nonzero projection onto W}) approaches 1 as T approaches infinity
where eta_T = c / sqrt(T) for a constant c determined by the desired confidence level.
The proof proceeds in two steps: showing that observed dimensions produce above-baseline scores, and showing that unobserved dimensions produce at-baseline scores.
Step 1 (observed dimensions score above baseline): If dimension j is in the analyst's observation subspace (the projection of e_j onto W is nonzero), then the analyst has information about dimension j beyond the prior. Under truth-telling, the analyst's reported probability p_{t,j} is not identically 1/2 - it varies with the analyst's signal about dimension j. The expected Brier score satisfies:
E[BS_{t,j}] = 3/4 + Var(p_{t,j})
where Var(p_{t,j}) > 0 is the variance of the analyst's posterior probability across scoring cycles. (This follows from the decomposition of the Brier score into calibration, resolution, and uncertainty components, where the resolution component equals the variance of the posterior mean.) For dimensions in the observation subspace, Var(p_{t,j}) > 0, so E[BS_{t,j}] > 3/4 strictly.
By the law of large numbers, BS_j(T) converges to E[BS_{t,j}] > 3/4. For T large enough, BS_j(T) exceeds 3/4 + eta_T with high probability.
Step 2 (unobserved dimensions score at baseline): If dimension j is orthogonal to the analyst's observation subspace (the projection of e_j onto W is zero), then the analyst has no information about dimension j beyond the prior. Under truth-telling, the analyst reports p_{t,j} = 1/2 for all t (or close to 1/2, perturbed only by noise in the analyst's prior). The expected Brier score is E[BS_{t,j}] = 3/4 (the baseline). By the law of large numbers, BS_j(T) converges to 3/4, and for T large enough, BS_j(T) does not exceed 3/4 + eta_T with high probability.
Combining steps 1 and 2 with a union bound over K dimensions and setting eta_T = c * sqrt(ln(K) / T) ensures the probability of correct identification approaches 1.
QED.
Remark 3.3. The theorem says the pattern of an analyst's Brier scores across dimensions is a consistent estimator of her observation subspace. The high-scoring dimensions are the dimensions she can see. The baseline-scoring dimensions are the dimensions she cannot see. The productive failure - the systematic near-baseline scoring on specific dimensions - is the signal that identifies the blind spots.
Remark 3.4. The convergence rate O(sqrt(ln(K) / T)) means that for K = 20 constraint dimensions, approximately T = 10 scoring cycles are needed for reliable identification at the 95% confidence level (setting c appropriately). For annual scoring, this is ten years. For quarterly scoring on dimensions with quarterly outcomes, this is 2.5 years. The identification is feasible on the timescales the disclosure mechanism operates.
The observation subspace recovery in Theorem 1 identifies which individual dimensions the analyst can see. A more valuable recovery identifies which interactions the analyst can see - which pairwise (or higher-order) couplings between dimensions the analyst's assessment captures.
Definition 4.1. The analyst's interaction score on the pair (j, k) after T cycles is:
IBS_{j,k}(T) = (1/T) sum_{t=1}^{T} [BS_{t,j} * BS_{t,k} - (3/4)^2]
The interaction score measures whether the analyst's accuracy on dimensions j and k is correlated beyond what independent baseline performance would produce. If the analyst detects the interaction between j and k, her accuracy on j and k covaries (she gets both right when the interaction is active and both wrong when it is not). If she observes j and k independently (without detecting their interaction), her accuracy on each is independent.
Proposition 1 (Interaction Detection). The interaction score IBS_{j,k}(T) is:
(i) Positive and converging to a limit > 0 if the analyst's observation subspace spans the interaction span(e_j, e_k) (the analyst detects the coupling).
(ii) Converging to zero if the analyst observes j and k independently but does not span their interaction (the analyst sees each dimension separately but misses the coupling).
(iii) Converging to zero if the analyst cannot observe either dimension.
Proof sketch. Case (i): When the analyst spans the interaction, her posterior on j is informative about k (because she sees how they couple). The conditional covariance Cov(p_{t,j}, p_{t,k} | theta_t) is positive, producing a positive interaction score. Case (ii): When the analyst observes j and k independently, her posteriors are conditionally independent given the state, producing zero interaction score in the limit. Case (iii): Both posteriors are at the baseline, producing zero products beyond (3/4)^2 in the limit.
QED (sketch).
Remark 4.2. Proposition 1 is the key result for productive failure analysis. An analyst who scores well on water (BS_water > 3/4) and well on energy (BS_energy > 3/4) but has IBS_{water, energy} near zero is demonstrating a specific failure: she sees each dimension individually but does not detect their interaction. The productive failure - the zero interaction score despite nonzero individual scores - maps the boundary between what the analyst's observation subspace spans (individual dimensions) and what it does not span (the interaction between them). The interaction blind spot corresponds to the analyst's observation subspace lying in Gr(1, K) rather than Gr(2, K) on the water-energy plane - the topological protection result from the companion paper applied to the analyst's individual observation geometry.
When multiple analysts produce scored disclosures on the same constraint domain, their individual productive failure patterns aggregate into a population-level map.
Definition 5.1. The population interaction coverage on pair (j, k) is:
PI_{j,k} = (1/N) sum_{i=1}^{N} 1{IBS^i_{j,k}(T) > eta_interaction}
where the sum is over N analysts and the indicator is 1 if analyst i's interaction score exceeds the detection threshold. PI_{j,k} measures the fraction of analysts who detect the interaction between dimensions j and k.
Proposition 2 (Topology of Ignorance Map). The matrix P = (PI_{j,k})_{j,k} is a population-level map of the observation architecture with the following properties:
(i) Entries near 1 identify interactions that most analysts detect - well-observed interactions with low systemic risk.
(ii) Entries near 0 identify interactions that few or no analysts detect - structural blind spots in the observation architecture with high systemic risk.
(iii) Rows or columns that are uniformly low identify constraint dimensions that are structurally disconnected from the observation architecture - dimensions that few analysts observe in interaction with anything else, even if they observe the dimension individually.
(iv) Clusters of near-zero entries identify topologically connected blind zones (as defined in the companion paper on reverse percolation) - regions of the interaction space where cascading failure can propagate through shared unobserved interactions.
Proof. Part (i) follows from the aggregation of individual detections: if most analysts span the interaction, PI is high. Part (ii) follows from the complement: if the interaction lies in most analysts' blind sets (because most analysts are scalar observers whose subspaces do not span the interaction, per the topological protection result), PI is low. Part (iii) follows from the row/column structure of the interaction matrix: a dimension j with low PI_{j,k} for all k is a dimension that is observed individually but not in interaction. Part (iv) follows from the correspondence between the zero-entry clusters in P and the connected components of the blind zone in the interaction graph (Section 2 of the companion paper on reverse percolation).
QED.
Remark 5.2. The topology of ignorance map P is constructible from scored disclosure data without any knowledge of the "true" constraint interactions. The map is built entirely from the pattern of forecasters' errors. No external measurement of the constraint environment is required to identify where the observation architecture has structural gaps. The map is the observation architecture observing itself - the productive failures of individual analysts aggregate into a population-level diagnostic of the system's blind spots.
The Goodhart derivation (Beier, 2026i, Corollary 6.1) proves: when a scalar metric becomes a target, agents redirect finite capacity toward the measured dimension, unmeasured dimensions receive reduced bandwidth, and the Dissipation Law drives decay on the undermaintained dimensions. The scalar reports improvement on its own dimension while the structure degrades on every dimension the scalar cannot see.
A domain that has been subject to scalar metric targeting for T periods accumulates constraint-dimension degradation that compounds through three channels identified in the Reality Debt paper (Beier, 2026af):
Physical lock-in: infrastructure and capital stock configured to the scalar metric's signal rather than to the full constraint environment. Duration: the useful life of the infrastructure, typically 20-40 years.
Institutional lock-in: regulatory frameworks, contractual obligations, workforce training, and organizational structures formed around the scalar metric. Duration: the institutional memory of the sector, typically 10-30 years.
Informational lock-in: the data environment shaped by decades of scalar-metric-optimized allocation, making the existing trajectory appear better-documented than constraint-informed alternatives. Duration: the length of the historical dataset, potentially indefinite.
The three channels compound: physical infrastructure formed around the scalar metric produces institutional structures that perpetuate the allocation, which produce data that validates the trajectory, which justifies further physical investment. The compounding makes the accumulated degradation superlinear in the duration of targeting.
Definition 7.1. The Goodhart Dislocation on domain D is the repricing event that occurs when the constraint dimensions of domain D transition from unobserved to observed. The dislocation is characterized by three quantities: location (which dimensions reprice), magnitude (how large the repricing is), and correlation (how many assets in the domain are affected simultaneously).
Theorem 3 (Goodhart Dislocation). Let domain D have been subject to scalar metric targeting with intensity gamma for T periods, where gamma in (0, 1] measures the fraction of observation capacity redirected from constraint dimensions to the scalar metric. Let lambda_k for k in N (the set of neglected constraint dimensions) be the prior variances on those dimensions. Then the expected magnitude of the dislocation when the constraint dimensions become observable is:
Delta_V(D) = gamma * sum_{k in N} lambda_k * (1 - exp(-D*(M_k) * T))
where D*(M_k) is the dissipation rate on dimension k under the reduced maintenance bandwidth M_k = (1 - gamma) * M_k^0, with M_k^0 being the pre-targeting bandwidth.
(i) Increasing in duration. The term (1 - exp(-D* * T)) increases toward 1 as T grows. The longer the scalar targeting has operated, the larger the accumulated degradation. The dislocation magnitude approaches gamma * sum lambda_k for T large - the full constraint-dimension variance multiplied by the targeting intensity.
(ii) Increasing in intensity. Higher gamma (more aggressive scalar optimization) produces larger degradation through two channels: more capacity redirected from constraint dimensions and faster dissipation rate on the undermaintained dimensions (D* is increasing in gamma through the reduction in M_k).
(iii) Correlated across assets. Every asset in domain D that was subject to the same scalar metric has the same neglected dimensions N, the same targeting intensity gamma, and the same targeting duration T. The dislocation therefore hits every asset in the domain simultaneously. The correlation of the repricing event across assets in D approaches 1 for long-duration targeting.
(iv) Invisible before observation. The dislocation magnitude Delta_V(D) is zero conditional on the constraint dimensions remaining unobserved, because the scalar metric's value is unchanged. The dislocation materializes only when the constraint dimensions become observable - when the first multi-dimensional disclosure spans the neglected dimensions. The dislocation is triggered by observation, not by a physical event.
The proof traces the degradation through the Dissipation Law. Under scalar targeting at intensity gamma, the observation bandwidth on dimension k reduces from M_k^0 to (1 - gamma) * M_k^0. If (1 - gamma) * M_k^0 < m_k* (the minimum maintenance bandwidth for dimension k), then dimension k falls below the water level theta* and receives zero effective maintenance. The coordination-dependent component of dimension k's variance decays at rate D*(M_k) = D*((1 - gamma) * M_k^0).
After T periods, the accumulated degradation on dimension k is:
Delta_{k}(T) = lambda_k * b_k * (1 - exp(-D*(M_k) * T))
where b_k is the Beierspace fraction of dimension k (the fraction of its variance that is coordination-dependent and therefore subject to dissipation). Setting b_k = 1 for simplicity (pure Beierspace dimensions) and summing over neglected dimensions:
Delta_V(D) = gamma * sum_{k in N} lambda_k * (1 - exp(-D*(M_k) * T))
The factor gamma enters because only the redirected fraction of the variance contributes to the dislocation - the remaining (1 - gamma) fraction was still being maintained at reduced bandwidth.
The correlation property (iii) follows from the common cause structure: all assets in domain D share the same scalar metric, the same neglected dimensions, and the same targeting history. The dislocation is a common factor that loads uniformly on all assets in D.
The invisibility property (iv) follows from the information structure: the scalar metric reports only on its own dimension, which has not degraded (it has received increased bandwidth under targeting). The degradation is on the constraint dimensions, which the scalar metric does not observe. The dislocation materializes when a multi-dimensional disclosure reports on the constraint dimensions, revealing the gap between the scalar metric's report ("this domain is healthy on the measured dimension") and the constraint dimensions' actual state ("this domain has degraded on every dimension the scalar metric cannot see").
QED.
The Goodhart Dislocation (Theorem 3) predicts the magnitude of the repricing event. The bias signature predicts its location and timing from early disclosure data, before the full dislocation occurs.
Definition 8.1. The bias signature of domain D is the pattern of systematic errors in the first T_early scoring cycles of disclosure data on domain D:
sigma_D = (BS_k(T_early) - 3/4)_{k in N}
where N is the set of neglected constraint dimensions. The bias signature measures how much the early disclosers' scores on neglected dimensions deviate from the maximum-uncertainty baseline.
Proposition 4 (Bias Signature as Dislocation Predictor). The bias signature sigma_D has the following properties in a domain subject to Goodhart targeting:
(i) Systematic overconfidence. Early disclosers in a Goodhart-targeted domain will report probabilities on neglected constraint dimensions that are further from the realized outcomes than the baseline would predict: BS_k(T_early) < 3/4 for k in N. The expected score is below the maximum-uncertainty baseline because the analysts' priors on neglected dimensions have been shaped by the scalar metric's signal, which systematically misrepresents the state of those dimensions. The analysts are not poorly calibrated in general - they are specifically miscalibrated on the dimensions the scalar metric has distorted.
(ii) Correlated direction. The overconfidence is in the same direction across analysts. In a domain where the scalar metric reports "healthy" (e.g., high ESG score), early disclosers will systematically underestimate constraint risk on neglected dimensions. The correlation of the directional error across analysts is positive and proportional to the targeting intensity gamma.
(iii) Magnitude proportional to dislocation. The deviation of the bias signature from baseline is proportional to the accumulated degradation: |sigma_D| proportional to Delta_V(D). Domains with larger bias signatures in early disclosure data will experience larger Goodhart dislocations when the constraint dimensions are fully observed.
Part (i): Analysts trained in the scalar metric environment have internalized the metric's signal as prior information about the domain. On the measured dimension, this prior is accurate (the scalar metric correctly reports its own dimension). On neglected dimensions, the prior is distorted: the analyst's implicit model of the neglected dimensions is conditioned on the scalar metric's healthy report, which is uninformative about the neglected dimensions but is treated as informative by the analyst. The result is systematic overconfidence - the analyst's reported probabilities on neglected dimensions are too close to the "healthy" end of the distribution. Under truth-telling, the analyst reports her distorted prior honestly. The Brier score penalizes the distortion, producing below-baseline scores on neglected dimensions.
Part (ii): The distortion has a common cause (the scalar metric's signal), which produces a common direction (toward "healthy" on neglected dimensions). The correlation across analysts follows from the common cause.
Part (iii): The magnitude of the prior distortion is proportional to the magnitude of the accumulated degradation, because the degradation determines how far the true state of the neglected dimensions has moved from the "healthy" baseline that the scalar metric continues to report. Larger degradation produces larger prior distortion produces larger bias signature.
QED (sketch).
The bias signature converts productive failure analysis into a predictive tool. The operational sequence:
Prediction 1 (Productive Failure Convergence). Individual analysts' dimensional score profiles should converge to a bimodal distribution: above-baseline on observed dimensions and at-baseline on unobserved dimensions. The convergence should be visible within five to ten scoring cycles. The bimodal structure should be sharper for analysts with well-defined expertise domains (water specialists, energy analysts) and broader for generalists.
The test: collect scored disclosure data from analysts with declared domain expertise. Compute BS_k(T) for each dimension k and each analyst i. Test whether the distribution of scores across dimensions is bimodal (as Theorem 1 predicts) or unimodal (as a model of uniformly noisy but unstructured forecasting would predict). The Hartigans' dip test applied to the dimensional score distribution provides a significance test.
Prediction 2 (Shared Blind Spots). Analysts covering the same constraint domain from the scalar observation architecture should share blind spots: the interaction scores IBS_{j,k} should cluster near zero for the same pairs (j, k) across analysts. The shared blind spots should correspond to the topologically protected interactions identified in the companion paper on topological protection (the Schubert classes of Gr(2, K) that are in the blind set of scalar observation).
The test: compute the population interaction coverage matrix P. Test whether the near-zero entries are clustered (as the topological protection result predicts) or randomly distributed (as a model of independent individual limitations would predict). A chi-squared test on the distribution of near-zero entries against the random null provides the significance test.
Prediction 3 (ESG Goodhart Dislocation). Domains with the longest history of ESG scalar targeting should show the largest Goodhart dislocations when multi-dimensional disclosure begins. Specifically, the bias signature magnitude |sigma_D| should be positively correlated with: (a) the number of years the domain has been subject to ESG scoring, (b) the intensity of the ESG targeting (the fraction of the observation architecture devoted to the ESG metric rather than to constraint dimensions), and (c) the cross-provider correlation of the ESG scores (higher correlation indicates tighter consensus on the measured dimension, which implies more capacity redirected from constraint dimensions).
The test: rank domains by ESG targeting duration and intensity. Collect early disclosure data from each domain. Test whether |sigma_D| correlates with the ranking. The prediction is that the domains ranked highest by ESG targeting show the largest bias signatures, and that the bias signatures predict the magnitude of subsequent constraint-dimension repricing events.
The prediction is specific to the framework. No existing model of ESG ratings, information economics, or market microstructure produces the prediction that the most heavily ESG-rated domains are the most vulnerable to constraint-dimension repricing. Standard models predict the opposite: more information (more ESG coverage) should reduce risk. The Goodhart Dislocation predicts that more scalar information on the measured dimension increases risk on unmeasured dimensions, and the accumulated risk surfaces as a correlated repricing event when the unmeasured dimensions become observable.
Prediction 4 (Dislocation as Crisis Predictor). The largest entries in the population bias signature matrix, ranked across domains, should predict the sequence in which constraint-dimension crises arrive. The domain with the largest |sigma_D| should experience the first constraint-dimension repricing event. The domain with the second-largest |sigma_D| should experience the second. The prediction is ordinal (ranking), not cardinal (timing), because the timing depends on when the disclosure lattice reaches each domain, which is determined by adoption dynamics (Beier, 2026q) rather than by the dislocation magnitude.
The test requires tracking both the bias signatures (from early disclosure data) and the subsequent constraint events (from outcome data) across multiple domains over a five-to-ten-year horizon. The prediction is that the rank correlation between |sigma_D| and the order of constraint events is positive and significant.
The productive failure analysis (Theorem 1) requires sufficient scoring cycles for the dimensional score profile to converge. For annual scoring on constraint dimensions with annual outcomes, convergence requires approximately ten years. For domains with faster outcome data (quarterly grid stress events, seasonal water data), convergence is faster. For domains with slow outcome data (atmospheric trajectory, institutional resilience), convergence may exceed a career. The analysis is most useful on intermediate timescales (five to fifteen years) and least useful on the deep timescales where the most consequential constraint interactions operate.
The interaction score (Proposition 1) requires the analyst to produce assessments on multiple dimensions within the same scoring cycle. An analyst who assesses water risk and energy risk in separate disclosures, at different times, with different outcome horizons, produces data that the interaction score cannot directly combine. The five-field format's Scaling field addresses this by requiring the system-level assessment within a single disclosure, which produces the within-disclosure correlation needed for the interaction score. Disclosures without Scaling fields produce individual-dimension scores but not interaction scores.
The Goodhart Dislocation formula (Theorem 3) depends on the dissipation rate D*(M_k), which is a function of the eigenvalue spectrum of the constraint covariance matrix - a quantity that is not directly observable without multi-dimensional disclosure data. The formula is exact given the parameters but the parameters must be estimated from the data the mechanism produces. The estimation is circular in the sense that the mechanism must operate before the parameters can be estimated, but the bias signature (Proposition 4) provides an indirect estimate from early data that does not require knowledge of D*.
The correlated dislocation prediction (Theorem 3, property iii) assumes that all assets in a domain are subject to the same scalar metric. In practice, different ESG providers score different attributes, and different assets within a domain may be subject to different metrics. The correlation of the dislocation is reduced by heterogeneity in the targeting - if different assets were targeted on different dimensions, the neglected dimensions differ across assets, and the dislocation is partially diversifiable. The worst case (maximum dislocation correlation) occurs when a single scalar metric dominates the domain, which is the case for credit ratings (dominated by the major agencies) and increasingly for ESG ratings (converging on a small number of dominant providers).
The paper does not address the market's response to the bias signature information. If the bias signature identifies an impending Goodhart dislocation and the information is public (because the disclosure data is public through the Beier Protocol), rational market participants should begin repricing before the full dislocation occurs - which would partially pre-empt the dislocation. The pre-emption is desirable (it reduces the severity of the discontinuous repricing) but it complicates the prediction: the dislocation magnitude depends on how quickly the market incorporates the bias signature information, which depends on how many participants are reading the disclosure data and how efficiently they process it. The reflexive coupling between the prediction and the response is a standard feature of financial economics (Merton, 1973) and is not unique to this framework.
The disclosure mechanism produces two types of output. The first, developed in the companion papers, is the scored assessment itself: constraint-dimension information that the price cannot carry, accumulated into track records, cross-referenced through the Factorial. The second, developed in this paper, is the diagnostic that the scored assessments produce about the observation architecture itself.
Productive failure is the mechanism that converts scoring data into architectural diagnosis. The analyst's pattern of systematic errors maps the boundary of her observation subspace. The population's shared errors map the topology of ignorance - the structural blind spots that no individual's errors alone could identify.
The Goodhart Dislocation is the prediction that productive failure data enables. Domains that have been targeted by scalar metrics have accumulated constraint-dimension degradation that is invisible to the scalar metric and will surface as a correlated repricing event when the constraint dimensions become observable. The bias signature in early disclosure data identifies the location and approximate magnitude of the forthcoming dislocation. The prediction is testable, specific, and novel.
The most consequential implication is for risk management. The standard approach to identifying systemic risk looks for concentrations of exposure within the current observation architecture - crowded trades, correlated positions, leverage ratios. The Goodhart Dislocation identifies systemic risk that the current observation architecture cannot see by construction: the accumulated degradation on dimensions the scalar metrics have neglected for decades. The bias signature identifies where this invisible risk is largest. The domains with the most ESG coverage, the highest credit rating consensus, the most standardized metrics - the domains that appear safest by every available scalar measure - are the domains where the Goodhart Dislocation is largest.
The safest-looking domains are the most dangerous. This is Goodhart's Law operating at the level of the information architecture, predicted by the mathematics, and detectable through the specific pattern of wrongness in the scored disclosure data.