A Missing Coordinate in the Classification of Quantitative Data
The Coordination-Dependence Fraction
Gregory Caldwell Beier
Susarb LLC, Cambridge, Massachusetts
July 2026
© 2026 Gregory Caldwell Beier. Licensed under CC BY-NC-ND 4.0.
Standard statistical practice was developed for measurements of quantities whose underlying is stable during observation. It has been applied to quantities whose underlying is partly constituted by observation, without adjustment. The consequences of the mismatch are visible in three literatures: the school capitalization coefficient in applied microeconomics, the aggregate composition of US equity indices, and the evaluation of large language models. This article names the distinction that connects these three cases and gives it an operational handle. It distinguishes observation-independent variance (variance arising from physical or material drivers that run whether or not measurement takes place) from coordination-dependent variance (variance arising from the ongoing coordination of parties around the measured quantity). The coordination-dependence fraction \(b \in [0,1]\) is the share of a quantity's variance attributable to the coordination-dependent component. Section 5 develops a formal estimator \(\hat{b}\) based on meta-regression residuals, with an identifying assumption that is empirically testable. The article proceeds through three cases in escalating stakes. The first reads Black's (1999) Massachusetts school-boundary-discontinuity study through both a stable-parameter and a coordination-dependent frame, computes an estimated b for school capitalization from the published literature, and identifies a testable empirical prediction that discriminates the two frames. The second turns to US equity valuation, where the S&P 500's shift toward intangibles-backed composition has produced an aggregate rise in b that reframes what recent equity growth has been. The third turns to AI training data, where conditional entropy \(H(D \mid P)\) in the Shannon sense serves as a diagnostic for whether a model contributes novel information relative to the coordination its outputs are drawn from. Data series carried without their b are series carried with a missing field, and producers of quantitative data have a practical obligation to their downstream users to provide it.
Keywords: measurement theory, coordination-dependence, meta-regression, hedonic regression, AI evaluation, conditional entropy, foundations of data science
A question worth asking before any quantitative analysis is whether the thing being measured would still be what it is if no one were measuring it. For the temperature of a gas, the answer is yes. For the price of a privately held software firm, the answer is no. The temperature exists because molecules move at rates set by physical law that runs whether or not a thermometer is present. The price exists because investors coordinate around expectations of cash flows whose realization depends on user adoption, regulatory permission, and the continuing reputation of the firm. Withdraw the coordination and the price ceases to refer to anything at all. The gas, by contrast, would keep its temperature whether or not a thermometer were present.
This asymmetry runs through nearly every domain that calls itself quantitative. The mass of a galaxy and the popularity rating of a politician are reported in the same kind of data infrastructure, fit by the same statistical packages, summarized by the same descriptive measures. The infrastructure treats them as drawings from probability distributions whose existence is independent of the act of drawing. For the galaxy this is correct. For the politician it is not. The popularity rating is partly constituted by the polling apparatus that produces it: by which questions are asked, by which respondents are reached, by which methodology houses release on which days, by which prior aggregations show up in the next interview. Stop the polling and the popularity does not merely become harder to measure. The thing called popularity, the social fact that holds elites and voters in a shared posture, starts to come apart, because what was being tracked was the public's coordination around its own posture, and that coordination depended on the tracking that fed back into it.
The asymmetry has been named in many partial ways. Sociologists of finance speak of performativity. Followers of Soros speak of reflexivity. Econometricians speak of endogeneity. Central bankers speak of Goodhart's law, under which a measure adopted as a target ceases to be a good measure (Goodhart, 1984). Philosophers of science speak of looping kinds. Each name picks out part of what is going on. None of them gives quantitative work a coordinate it can use. Endogeneity comes closest to a computable form, with specification tests and instrumental corrections, but those tools repair the estimation of a coefficient without classifying the quantity itself: they de-bias, and they say nothing about what fraction of the measured object is observation-maintained. The rest remained qualitative cautions, and a caution cannot be computed. None of them tells a researcher, in a way that maps to a number she can compute, how much of her dataset is the kind of thing that exists independently of her access to it, and how much of it is the kind of thing that exists because she and others like her have agreed to keep looking. The contribution of what follows is not the identification of the phenomenon, which these literatures accomplished decades ago, but its formalization as a continuous parameter that can enter error structures and reporting standards. Such a coordinate is available, can be estimated with standard tools, and earns its place in standard practice by what it lets a researcher see in data she already has.
The article proceeds through three cases. The reader will benefit from working carefully through the first, because the framework it builds carries through to the two that follow. The first case uses the school-boundary-discontinuity design that Black (1999) introduced and that a substantial literature has revisited over the following quarter century, and applies a formal meta-regression estimator to the coordination-dependence fraction. The second case is US equity valuation, where the composition of the S&P 500 has shifted from a majority of book-value-backed claims in the mid-1970s to a supermajority of intangibles-backed claims by the 2020s. The third case is AI training data and the evaluation of large language models, where standard conditional entropy applied to model outputs against their training coordination provides a diagnostic that current evaluation practice does not report.
A clarification about standard statistical practice is worth making first, because the distinction the article proposes is sometimes confused with the routine distinction between clean and noisy measurement. Standard statistical practice operates on a foundational assumption that deserves to be stated explicitly: there exists a fixed-if-unknown parameter, and the data are noisy measurements that locate where the parameter likely sits. Confidence intervals, posterior distributions, sampling theory, the entire apparatus of statistical inference: all of it exists to triangulate where a stable target is located given that the measurement is noisy. This is inference under the observation-independent assumption, and it works because the target sits still while the apparatus measures it.
In the social sciences this apparatus is applied to quantities that range across a full spectrum. A decennial census head count, measured with sampling and non-response error, is observation-independent work with noise. The county's population sits still during the survey window, and the confidence interval around the count is exactly what it claims to be: a range within which the true count likely sits. A poll of opinion on a current political question is something different in kind. The opinion the poll attempts to measure does not sit still while the poll is being conducted. The opinion is partly held in place by the polling apparatus, by the media coverage that aggregates and discusses the polls, by the strategic adjustments parties and candidates make in response to them. The confidence interval reported with the poll has the same syntactic form as the confidence interval reported with the census. It is doing something different. The distinction the article names is not between noisy and clean measurement. It is between targets that sit still during measurement and targets whose location is being maintained by coordination of which the measurement is a participant.
The observation-independent portion of a quantity's variation comes from physical or material drivers running independently of observation: the mass of an object, the chemical composition of a sample, the position of a star, the wavelength of a photon. An observation-independent quantity does not need anyone to be watching it for it to be what it is. The processes setting its values are not maintained by attention.
The coordination-dependent portion of a quantity's variation comes from the ongoing coordination of agents around the measured quantity, where the quantity itself is constituted by the continuing coordination. Many of the things quantitative work counts have substantial coordination-dependent variance: the value of a brand, the status of a reserve currency, the authority of an institution, the liquidity of a market, the credibility of a scientific consensus. The quantity has the value it has because parties have arranged themselves to act as though it has that value, and the arrangement requires ongoing observation to hold its shape. Withdraw the observation and the quantity does not become opaque; it stops being what it was. The brand becomes worthless. Reserve status erodes. Authority drains from the institution. The market's liquidity does not collapse to a lower level; it stops being liquidity at all.1
The observation-independent category is sometimes assumed to belong to the natural sciences and the coordination-dependent category to the social sciences. The mapping is wrong. Both categories are properties of the quantity being measured, not of the discipline doing the measuring. Demographic head counts, hours worked, vital statistics, the geographic distribution of households: these are measured by social scientists and have very low coordination dependence. The number of people who live in a county does not depend on whether a census is conducted, and the count is approximately the same quantity whether the census methodology is good or bad. Some quantities studied in the natural sciences have substantial coordination-dependent content: a published scientific consensus on a contested empirical question is a coordination product whose stability depends on continuing scrutiny of the underlying evidence and on the credibility of the institutions that hosted the original studies. The coordinate runs across disciplines. Each discipline includes low-coordination and high-coordination quantities, and the methodological choices the coordinate informs are choices about quantities rather than about fields.
Very few real-world quantities sit at either extreme. A barrel of crude oil at the wellhead has a substantial observation-independent component: its physical energy content is a fact about the molecules. The Brent premium over WTI for the same physical product is coordination-dependent, because the premium exists because of benchmark conventions, pipeline access agreements, and a set of trading relationships that hold their shape only as long as traders continue to participate in them. The mixture varies, and the variation over time within the same nominal object is itself a fact worth tracking. The distinction is older than the operational form the article gives it. Searle (1995) distinguished brute facts from institutional facts. Hacking (1999) distinguished interactive kinds from indifferent kinds. Callon (1998) and MacKenzie (2006) developed the performativity literature around the way economic models constitute the markets they describe. The proposal here is to treat the distinction as a quantitative coordinate rather than a philosophical or sociological category, and to give it a measure.
The variance of a quantity p observed at time t can be decomposed, in principle, into a part driven by observation-independent material processes and a part driven by coordination dynamics:
V(p,t) = V_E(p,t) + V_B(p,t)
The coordination-dependence fraction is the ratio:
b(p,t) = V_B(p,t) / [V_E(p,t) + V_B(p,t)]
By construction b sits in [0,1]. A value of b near zero says that almost all of the quantity's variance comes from observation-independent drivers. A value near one says that almost all of it comes from coordination dynamics that would not survive the withdrawal of observation. The fraction is defined at a particular time, since the mixture can shift, and the shift over decades is itself an empirical fact about a quantity worth tracking.
The fraction b is a theoretical quantity defined in terms of variance components of the underlying process. It is not directly observed. The article uses \(\hat{b}\) for empirical estimators of b constructed from observable data, and reserves b for the theoretical target. Section 5 develops a specific estimator based on meta-regression residuals across a body of studies that measured the same coefficient in different settings.
A precedent for this kind of coordinate exists within statistics. Meng (2018) showed that the error of any big-data estimate factors into a data quantity term, a problem difficulty term, and a data defect correlation that indexes how far the recording mechanism departs from random sampling. The defect correlation is a scalar property of the data that remains invisible to the consumer unless computed and reported, and its omission produces the big data paradox: enormous samples delivering confidently wrong answers. The coordination-dependence fraction operates in the same register. Both are single numbers summarizing a hidden property of the data-generating process, and both do their work only if they travel with the data rather than remaining in the heads of specialists. Where the defect correlation indexes a flaw in the recording mechanism, b indexes the constitution of the recorded quantity itself, so the two coordinates are complements: a series can be recorded cleanly and still be coordination-dependent, and a coordination-independent series can be recorded badly.
The counterfactual interpretation is the most useful way to hold the definition. Imagine running a thought experiment in which all observation channels directed at the quantity are switched off at time t: no analysts, no investors, no regulators, no media coverage, no institutional accounting. The fraction b(p,t) is the share of the quantity's value that would begin drifting toward incoherence under that withdrawal. The observation-independent fraction 1 - b is what would stay anchored to material reality. For a gold bar the answer is close to one: the bar would still be gold, and gold's density and conductivity would still mark it out, even if everyone stopped paying attention. For the equity of a fashion brand the answer is close to zero: the brand would survive only for as long as customers remembered to want it, suppliers continued to extend trade credit, employees continued to show up, and the public continued to find the logo meaningful. The clothes in the warehouse would still be cotton and dye. The brand's value would not be.
The fraction is more than a label because b parameterizes the predictions that follow when the distinction is taken seriously. Any method that assumes the underlying process is unaffected by observation will have its error scale with b. At b near zero the assumption is harmless. At b near one the assumption is dominant. The article now turns to one body of empirical work, traces what a stable-parameter reading of it can and cannot do, and reads the same work through a coordination-dependent frame with a formal estimator.
The school-boundary-discontinuity design introduced by Black (1999) is one of the most cited identification strategies in applied microeconomics, and the literature it spawned is the article's organizing empirical case. The original design exploited the fact that, within a single school district, attendance is determined by which side of a school boundary a house sits on. Houses on either side of the boundary are otherwise similar: the same town, the same property tax rate, the same municipal services, the same access to local amenities. The boundary creates a discontinuity in expected school assignment, and the price differential at the boundary, after controlling for distance to the boundary and house characteristics, identifies the value buyers attach to differences in expected school quality. Black used Massachusetts data from the 1990s and reported that a one-standard-deviation increase in elementary school test scores was associated with roughly a 2.5 percent price premium.
Under the stable-parameter reading, the 2.5 percent coefficient estimates a willingness-to-pay parameter in the population's preference function. Buyers value school quality. The boundary discontinuity isolates the value of expected school assignment from the value of neighborhood characteristics that vary smoothly across space. The estimate is informative about household preferences and should generalize across times and places to the extent that preferences are stable. Policymakers can use the coefficient to estimate consumer surplus from school improvements. Tax assessors can use it to apportion the capitalized value of school quality. Subsequent research can use it as a benchmark for similar estimates in other settings.
The reading has consequences for practice. Automated valuation models used by mortgage lenders, tax authorities, and real estate platforms embed hedonic coefficients of this kind as parameters whose value is calibrated on historical data and applied to current transactions. When the underlying coordination shifts, the models continue to price houses using coefficients that no longer reflect the market's current pricing. The result is a systematic mispricing that persists until the coefficient is recalibrated, which typically happens after the shift has already produced observable losses.
The first difficulty for the stable-parameter reading appeared with Bayer, Ferreira, and McMillan (2007), who revisited the design with a richer dataset that allowed for explicit controls on neighborhood sorting. Their estimate of the school-quality coefficient was approximately half of Black's. The stable-parameter reading has to call this a methodological refinement: Black's estimate, on this reading, was upward biased because it failed to net out the value of unobserved neighborhood characteristics that correlate with school quality, and the Bayer-Ferreira-McMillan estimate represents the true willingness-to-pay parameter. The stable-parameter reading carries this interpretation as far as it can.
Then come the other studies. The table below summarizes headline findings from a selection of the school capitalization literature, all using related identification strategies. The estimates have been rounded to round numbers consistent with the published findings; the cited papers contain the precise figures and the identification details.
| Study | Setting | Identification | Headline estimate |
|---|---|---|---|
| Black (1999) | Massachusetts elementary schools, 1990s | Attendance-boundary discontinuity | ~2.5% price premium per \(\sigma\) of test scores |
| Bayer, Ferreira, McMillan (2007) | Boston metropolitan area | Boundary discontinuity with neighborhood sorting controls | ~1.2% price premium per \(\sigma\) of test scores |
| Figlio and Lucas (2004) | Florida, 1999-2002 | Discontinuous letter-grade transitions | Significant price effects at grade thresholds, larger than smooth test-score gradient predicts |
| Cellini, Ferreira, Rothstein (2010) | California school districts | Regression discontinuity on bond election outcomes | Substantial capitalization of bond approval, exceeding facility-quality channel |
| Kane, Riegg, Staiger (2006) | Mecklenburg County, NC | Pre/post desegregation-era school reassignment | Smaller coefficient than coastal metropolitan estimates |
| Gibbons, Machin, Silva (2013) | England | Boundary discontinuity, primary schools | ~3% price premium per \(\sigma\) of school value-added |
The stable-parameter reading has to absorb this variation as methodological dispersion plus local taste shifters, and it is increasingly strained as it does so. Black's design was clean and Black's estimate was 2.5. Bayer, Ferreira, and McMillan's design was cleaner and their estimate was 1.2. Figlio and Lucas's letter-grade discontinuities produced effects larger than a smooth willingness-to-pay function would generate. Cellini, Ferreira, and Rothstein found prices responding to bond elections in a magnitude that the future facility-quality channel cannot explain. Kane, Riegg, and Staiger found smaller coefficients in a setting where the desegregation-era reassignment had partially detached school identity from neighborhood identity. Gibbons, Machin, and Silva found a coefficient of about 3 percent in England, using school value-added rather than test-score levels.
The stable-parameter reading can offer accounts of each of these. Methodological refinement explains Bayer, Ferreira, and McMillan's revision of Black. Local preferences differ between Massachusetts, Florida, California, North Carolina, and England. Letter grades carry information beyond the smooth test-score function, so the discontinuous effects are not surprising. Bond elections signal commitment to school quality and have facility-quality consequences that justify the price response. Desegregation-era institutional history attenuates the coefficient. The stable-parameter reading is not refuted. It is increasingly carrying weight by appeal to context-specific factors, each of which is plausible on its own but which, together, imply that the willingness-to-pay parameter has many local values rather than one global one. At that point the parameter is no longer the kind of object the stable-parameter reading was supposed to be locating.
Under a coordination-dependent reading, the 2.5 percent coefficient is a measurement of the current state of a coordination process. School ratings carry weight in the housing market because of a coordination running through parents, real estate agents, listing services, online aggregators, local news, and the institutional infrastructure of school accountability. Black's coefficient is a measurement of the gradient of that coordination as it operated in the Boston area in the 1990s. Bayer, Ferreira, and McMillan's 1.2 percent is the gradient after a portion running through neighborhood sorting has been netted out; the decomposition is informative about the structure of the coordination, not a correction of an error. The Florida letter-grade discontinuities are focal points that the rating apparatus creates, and coordination concentrates at the categorical thresholds the apparatus marks. The bond-election effect in Cellini, Ferreira, and Rothstein is a coordination event in the sense the term is meant: the community has signaled, through a public vote, that it is willing to commit resources to its schools, and the housing market reads the signal and responds. The smaller Mecklenburg coefficient reflects an attenuated coordination produced by the desegregation-era institutional history. The English coefficient reflects a coordination that organizes around a different rating apparatus.
The coordination-dependent reading does the work the stable-parameter reading was reaching for. The variation in the table is no longer a puzzle the literature needs to keep resolving with auxiliary appeals to context. It is what one would predict if school district ratings had substantial coordination-dependent variance and the coefficients were measurements of coordination state.
The reading is related to, and distinct from, the critique Lucas (1976) directed at econometric policy evaluation. Lucas located the instability of estimated decision rules in policy-contingent expectations: change the regime and agents re-optimize, so coefficients estimated under the old regime fail under the new one, and the prescription was to model the deeper structural parameters. The school capitalization coefficient moves for a different reason. No policy regime needs to change for the coefficient to drift, because the coordination apparatus that constitutes the rated quantity varies across settings and reconfigures within them on its own schedule. The coordination-dependent reading also carries a quantitative commitment the Lucas critique never took on. Rather than a blanket caution against treating estimated coefficients as structural, it assigns each coefficient a measurable fraction b that states how much caution the coefficient warrants, and Section 5 gives that fraction an estimator.
The coordination-dependent reading generates a specific empirical prediction that the stable-parameter reading reaches only by adding assumptions from outside the preference framework. If school ratings function partly as coordination signals, then metros with more intensive coverage of school ratings by local media, real estate services, and rating aggregators should show larger school capitalization coefficients than metros with less intensive coverage, holding actual school quality constant. The stable-parameter reading can accommodate a coverage effect only through an auxiliary information channel: coverage informs buyers about quality differences they would otherwise not observe, and better-informed buyers price those differences more completely. That channel predicts the effect saturates once ratings are widely accessible, since additional coverage of information buyers already possess adds nothing to a fixed preference function. The coordination-dependent reading predicts a positive interaction between coverage intensity and the coefficient that persists beyond informational saturation, because coverage operationalizes the strength of the coordination the ratings participate in rather than the diffusion of a fixed fact.
The prediction is testable with existing data. Media coverage intensity can be operationalized as counts of local news mentions of school ratings, traffic to online rating aggregators, or frequency of MLS-listed properties citing school ratings, aggregated to the metropolitan level. Standard hedonic regressions with metro-level fixed effects and a coverage-by-rating interaction term can identify the effect. A failure to find the predicted interaction, in a properly-specified test with adequate power, would count against the coordination-dependent reading. No such test has been published to the author's knowledge, and the framework offers this as a specific empirical claim rather than as post-hoc reinterpretation.
The data sit at the same position in either frame. The cognitive object the data are about is different. The stable-parameter reading treats the table as a literature refining its estimate of a stable parameter with substantial residual heterogeneity accounted for by local context. The coordination-dependent reading treats the same table as a sequence of measurements of a coordination process whose state varies across settings in ways the framework explains. The shift is not from a wrong reading to a right one. Black's coefficient is correct as a description of his data. The shift is from a claim about a stable willingness-to-pay parameter to a claim about the current state of a coordination process, and the second claim sits more comfortably with the literature the first claim has produced.
The article proposes a formal estimator for b based on standard random-effects meta-regression tools. The estimator operationalizes the coordination-dependence fraction as the share of between-study variance not explained by observable methodological or contextual moderators.
Consider a body of K studies each producing a point estimate \(\beta\)-hat_k of a coefficient of interest in setting k, with within-study standard error s_k. Random-effects meta-analysis (DerSimonian and Laird, 1986; Higgins and Thompson, 2002) models this as:
\(\beta\)-hat_k = \(\beta_k\) + \(\varepsilon_k\), where \(\varepsilon_k\) \(\sim\) N(0, \(s_k^2\))
\(\beta_k\) = \(\mu\) + \(u_k\), where \(u_k\) \(\sim\) N(0, \(\tau^2\))
Here \(\mu\) is a population mean effect and \(\tau^2\) is between-study variance in true effects. Standard estimators (DerSimonian-Laird moments estimator, restricted maximum likelihood, empirical Bayes) provide \(\hat{\tau}^2\) with known confidence properties.
Meta-regression adds observable moderators. Let x_k be a vector of study-level covariates capturing methodological and contextual factors (identification strategy, sample period, geographic region, sample composition, coverage of the dependent variable). The extended model is:
\(\beta_k\) = \(\mu\) + x_k' \(\gamma\) + \(u_k\), where \(u_k\) \(\sim\) N(0, \(\tau^2_{\text{resid}}\))
The residual between-study variance \(\tau^2_{\text{resid}}\) captures heterogeneity not explained by the moderators. The explained variance \(\tau^2_{\text{explained}}\) = \(\tau^2_{\text{total}}\) - \(\tau^2_{\text{resid}}\) captures the portion attributable to observable methodological and contextual factors.
The proposed estimator for the coordination-dependence fraction is an upper-bound estimator:
\[\hat{b} \;=\; \frac{\hat{\tau}^2_{\text{resid}}}{\hat{\tau}^2_{\text{total}}}\]
where \(\hat{\tau}^2_{\text{total}}\) is the between-study variance estimated from the intercept-only model. The residual can include unmodeled methodological heterogeneity alongside coordination-state variation, so \(\hat{b}\) bounds b from above rather than identifying it exactly; Section 5.2 states the identifying assumption, and Section 5.3 gives the procedure under which the bound tightens. To account for reweighting artifacts in finite samples, under which the moderated model's estimate of residual heterogeneity can exceed the intercept-only estimate of total heterogeneity, the estimator is truncated at 1:
\[\hat{b} \;=\; \min\!\left(1,\; \frac{\hat{\tau}^2_{\text{resid}}}{\hat{\tau}^2_{\text{total}}}\right)\]
The estimator identifies b under the assumption that the moderators fully capture the observation-independent sources of between-study heterogeneity. Under this assumption, the residual \(\tau^2_{\text{resid}}\) isolates variance attributable to coordination-state differences across settings, while the explained \(\tau^2_{\text{explained}}\) isolates variance attributable to different methodological measurements of the same underlying coordination state.
The assumption is strong and unlikely to hold exactly. In practice, moderators are incomplete, and residual variance may include unmodeled methodological factors alongside coordination-state variation. This biases \(\hat{b}\) upward relative to b. The estimator therefore provides an upper bound on b under incomplete moderator specification.
The denominator uses total between-study variance rather than total observed variance (which would include within-study sampling variance \(s_k^2\)). The reason is that sampling variance is noise around each study's true coefficient, not a component of variance in the underlying quantity. Including sampling variance in the denominator would make \(\hat{b}\) depend on how precisely the individual studies were estimated, which is a property of sample sizes rather than of the object being measured. The between-study decomposition is what maps to the theoretical fraction V_B / (V_E + V_B) defined in Section 3.
The identifying assumption is empirically testable. If adding a new moderator substantially reduces \(\hat{\tau}^2_{\text{resid}}\), that reduction was previously being attributed to coordination-state variation but was in fact methodological. Iterated moderator enrichment, until \(\hat{\tau}^2_{\text{resid}}\) stabilizes, produces an increasingly tight upper bound on b. If the residual approaches zero as moderators are added, the framework's claim that the coefficient has substantial coordination-dependent variance is refuted. If the residual is stable across increasingly rich moderator sets, the framework's claim is corroborated: the residual is measuring something that is not methodological, and the leading candidate is coordination-state variation.
The falsifiability structure is worth stating explicitly. The framework does not merely reinterpret existing statistical outputs. It commits to a specific empirical prediction about the behavior of between-study residual variance under moderator enrichment, and that prediction can fail.
Applied to the school capitalization literature, the estimator takes the point estimates from the six studies in Section 4 as \(\beta\)-hat_k, uses within-study standard errors reported in the original papers as s_k, and specifies moderators capturing identification strategy (boundary discontinuity, RD on bond outcomes, letter-grade discontinuity), geographic region (US Northeast, US South, US West, England), sample period, and rating regime (test-score levels versus value-added versus letter grades).
Six studies cannot support that moderator vector jointly; the specification as written exceeds the available degrees of freedom. A calibration restricted to the headline studies therefore enters coarsened moderators one or two at a time and reads the residual across specifications rather than from a single saturated model. The full set becomes jointly estimable as the study base widens to the broader capitalization literature, which runs to several dozen estimates once within-study variants and later replications are included.
The full calibration requires the raw estimates and standard errors from each source, and produces a numerical \(\hat{b}\) for school capitalization along with a confidence interval derived from standard meta-regression asymptotics. The article does not report the calibrated value, because doing so responsibly requires a complete meta-analytic dataset that would properly appear in a companion empirical piece. Beier (2026k) reports one such calibration and finds \(\hat{b}\) for school district ratings in a range roughly between 0.55 and 0.80 depending on the moderator specification. The point of interest for the present article is that the estimator is defined operationally on data that already exist in the published literature, produces a number, has an identifying assumption that can be inspected, and admits of the falsification procedure above.
The coordination-dependent reading of the school capitalization literature gives a researcher, a policymaker, and a downstream consumer of the literature several things the stable-parameter reading does not.
The first is generalization with calibrated scope. Black's 2.5 percent coefficient applied to a new metro at the same time as Black's study would generalize to the extent that the new metro's coordination apparatus around school ratings is similar to the Massachusetts coordination apparatus. A 2026 policymaker in Phoenix asking what a one-standard-deviation improvement in elementary school test scores would do to home prices receives a different answer than a 1995 policymaker in Boston, not because Phoenix preferences differ from Boston preferences but because the coordination apparatus around school ratings is different in time, metro, and institutional context. Reporting b alongside the coefficient tells the downstream consumer how far the coefficient can be transported. The stable-parameter reading treats the coefficient as portable and forces the transportability judgment onto the reader without support.
The second is a predictive flag on policy intervention. A policymaker considering a recalibration of the school rating scale, one that changes how schools are labeled without changing what the schools do, should expect a small price response after the recalibration is widely understood. The coordination reorganizes around the new labels while the underlying coordination capital is unchanged. A policymaker considering school closure or district redrawing should expect a large price response. The underlying coordination is partially dissolved, and the housing market repositions accordingly. The stable-parameter reading predicts that price responses scale with the school-quality change in either case, which is wrong in both directions. The coordination-dependent reading predicts the asymmetry.
The third is a structural reading of replication. A future researcher running Black's design on a Boston-area dataset from the 2030s and finding a coefficient of 1.7 percent rather than 2.5 percent is, on the stable-parameter reading, looking at either a measurement of a different parameter or a failure of one of the two studies. On the coordination-dependent reading, the new coefficient is a measurement of the same kind of object in a different coordination state, and the relationship between the two coefficients tells the researcher about the coordination's evolution. The replication framework changes. So does the publishability of the new study, which is currently a confused space precisely because the field lacks a vocabulary for distinguishing different-coordination-state results from different-truth results.
The fourth is a different reporting standard. A researcher reporting a school capitalization coefficient under the coordination-dependent framework reports the coefficient, an estimated \(\hat{b}\) for school district ratings in that setting, and a statement of the coefficient's expected durability given \(\hat{b}\). She declines to extrapolate to other metros without estimating b for each new context. She specifies, in writing, the conditions under which the coefficient will fail to replicate, and she treats a future failure under those conditions as a confirmation of her framework rather than a refutation of her estimate. The reporting standard is more honest about what the data show, and it is more useful to the downstream consumer who is trying to decide whether the coefficient applies to her situation.
The cost of ignoring b in this literature is visible in the historical record. The school capitalization literature has spent more than two decades revising its central estimate, with each revision presenting as a methodological improvement rather than as a measurement of a moving underlying. The literature has never converged. Automated valuation models continue to embed hedonic coefficients calibrated on historical periods, and they misprice during coordination shifts in ways that a b-aware framework would flag in advance. The utility of tracking b is a harmonized reading of the existing literature, calibrated confidence intervals on downstream applications, and an operational signal for when to widen those intervals.
The S&P 500 has changed what it is. In the mid-1970s the index was overwhelmingly a claim on physical productive capacity. Book value accounted for most of the index's market capitalization, and intangibles were a minority component that accounting standards did not fully capture. By the 2020s the ratio has inverted. Depending on which measurement is used, intangible assets account for between three-quarters and nine-tenths of the S&P 500's market value today. The direction of travel is beyond dispute across the sources that track it (Damodaran, 2017; Lev and Gu, 2016; Peters and Taylor, 2017). The specific decimals vary by definition and by year. The story does not.
The composition change is not a rounding error in accounting. It is a shift in what the index is, and the shift bears directly on the argument the article has been developing. Intangible assets in the sense the accounting literature uses the term are brand equity, customer relationships, ecosystem lock-in, network effects, regulatory access, employee-brought knowledge, corporate reputation, patent portfolios whose value depends on ongoing enforcement regimes, and the software installed bases whose value depends on continued use by populations that remain willing to use them. Every one of these is a coordination product. Every one has substantial coordination-dependent variance. A brand exists as long as customers continue to prefer it. A customer relationship exists as long as the customer continues the relationship. A network effect exists as long as the network's participants continue to participate. A regulatory access position exists as long as the regulator continues to grant the access. None is a fact about the physical world in the sense that a barrel of crude oil at the wellhead is a fact about the physical world.
The valuation shift can be read two ways. The stable-parameter reading treats the intangibles portion as a claim on future cash flows whose expected value has grown because firms have become more productive per unit of physical capital. On this reading the S&P has grown because American firms have gotten better at generating profits from smaller physical footprints, and the market has correctly repriced them. The reading is not wrong. It captures something real: firms have in fact gotten more efficient at extracting value from thinner physical bases.
The coordination-dependent reading treats the intangibles portion as coordination capital that has grown because American firms and American markets have become better at generating and maintaining coordination structures around brands, ecosystems, regulatory regimes, and institutional relationships. The recent growth of US equity is substantially the growth of coordination capital. The extra valuation the market assigns to intangibles above what physical capital would support is a valuation of coordination structures that hold their shape only as long as the coordination holds its shape.
The two readings are compatible with the same data. They diverge on what happens when the coordination is stressed. On the stable-parameter reading, a stress on any particular coordination structure (a brand crisis, a regulatory reversal, a network exit) is idiosyncratic firm-level news that a diversified portfolio absorbs. On the coordination-dependent reading, coordination stress is correlated across firms because the coordinations depend on shared substrates. A US institutional retreat from a foreign market stresses the coordinations of many firms at once. A regulatory rewrite of an accounting standard rebases the coordination that produced the S&P's aggregate composition. A shift in the institutional infrastructure that supports intangibles valuation (the analyst community, the audit profession, the rating agencies) affects the correlation structure of the whole index. The aggregate b of the S&P 500 has risen with the rise of intangibles, and the risk models calibrated on periods with lower aggregate b have been calibrated on a different underlying object than the one currently traded.
The 1990s S&P and the 2020s S&P are different objects even under the same ticker. Their nominal continuity conceals a substantial change in composition, and their historical variance-covariance is a historical fact about an index whose aggregate coordination-dependence fraction was much lower. A pension fund allocating strategic capital to the S&P today, using thirty-year historical estimates of variance and correlation, is using estimates calibrated on an object with a materially different b profile. The reliance of the strategic model on the stable-parameter reading of the historical estimates produces a specific institutional exposure: allocators are running observation-independent math on coordination capital, and their risk models cannot see the coordination reconfigurations that would produce sharp discontinuous repricings.
Factor anomalies acquire a coordination-dependent reading in the same movement. The value factor's apparent disappearance coincided with the rise of the intangibles share, and the coincidence is not accidental. The value premium was a premium for holding stocks whose market prices were low relative to book value. It existed as an empirical regularity when book-to-market meant something, which is another way of saying it existed when b was moderate and book value tracked a substantial share of what the firm actually held. As firms became more intangibles-heavy, book value stopped tracking what firms held. The book-to-market ratio ceased to identify undervaluation, because low book-to-market no longer meant the firm was worth less than its stuff. It meant the firm held less stuff, and its coordination capital was not on the balance sheet. The value premium eroded not because investors got wiser but because the ratio it depended on stopped being informative in the same way. The stable-parameter reading of the factor's decline appeals to preferences shifting or to crowding. The coordination-dependent reading identifies a mechanism.
Long-horizon investors bear the sharpest exposure. A pension fund with liabilities running out fifty years has been buying an equity claim whose composition has shifted decisively toward coordination-dependent value over its own lifetime. The coordination capital that constitutes most of the S&P's current value depends on continuing coordination among firms, customers, regulators, and institutional intermediaries whose horizons are much shorter than fifty years. When one of these coordinations reconfigures, the value it supported does not decline gradually. It reprices to whatever the successor coordination can support, which is often much lower and sometimes zero. Long-horizon investors are structurally exposed to coordination reconfigurations that their historical variance estimates cannot see, and their asset-liability models are running observation-independent math on a coordination-dependent substrate.
Reporting an aggregate \(\hat{b}\) for equity indices, decomposed by sector and by intangibles class, provides risk models with a coordinate they currently lack. Systemic risk mapping gains a variable that responds to coordination stress rather than only to observed price changes. Scenario analyses can be constructed around coordination reconfigurations (regulatory regime shifts, brand-substrate erosions, network exits) rather than only around historical shock distributions. Strategic allocation decisions gain a distinction between exposure to observation-independent productive capacity and exposure to coordination capital, and the distinction can be reported in portfolio-level metadata alongside standard variance-covariance summaries. Beier (2026k) develops these applications in operational detail for equity portfolios. The purpose of the present treatment is to demonstrate that the framework, having been shown to work on a coefficient in a hedonic regression, works with equal force on the largest asset class in the world.
The third case is where the framework does its most consequential work. AI training data has a mixed coordination-dependence profile. Standard AI evaluation methodology treats it as if b were zero. The consequences are approaching visibility as language models scale and as multi-agent systems become the default deployment pattern.
A large language model is trained on a corpus assembled from human writing. The corpus mixes low-b content and high-b content in proportions that vary across the corpus and are not tracked by the training or evaluation apparatus. Physics textbooks, mathematics proofs, protein structure databases, medical imaging labels, geographical facts, technical specifications: these are largely low-b. Their underlying does not shift when the model is trained on them, and the coordination among the humans who produced them is anchored in physical or mathematical facts that would remain what they are regardless of who was looking. Editorial commentary, customer service transcripts, comment moderation labels, sentiment annotations, brand marketing content, opinion pieces, evaluation rubrics for essay quality: these are largely high-b. Their underlying is a coordination product, and the coordination shifts as the population of participants shifts and as the outputs of previous coordination cycles feed back into the next.
The evaluation infrastructure does not distinguish. Standard benchmark scoring reports a single accuracy number against a test set. The test set is drawn from a distribution that is treated, for evaluation purposes, as fixed. For low-b tasks this is fine. The tumor is or is not present in the image, and the presence of the tumor does not change when the model reports on it. For high-b tasks the test set is itself a coordination product whose labels reflect the coordination state at the time of labeling. When the coordination shifts, the labels shift. When the model is deployed at scale into contexts that feed back into the coordination, the model's outputs participate in the shift.
The consequence of ignoring b in AI evaluation is directly analogous to the consequence in the school capitalization literature. The same model, evaluated on toxicity classification, produces materially different accuracy figures depending on which labeling regime generated the test set, which population of humans provided the labels, which time period the labels were collected in, and which downstream conception of toxicity the labels were meant to operationalize. The evaluation literature has been treating this dispersion as methodological noise. It is the signature of a high-b prediction target being scored against test sets that are themselves coordination products, and it is the AI-domain analog of the school capitalization coefficient variation across studies.
Shannon (1948) gave the world a way to measure how much a signal reduces uncertainty about an underlying. Entropy is the measure. A forecast committing to rain at ninety percent removes more of a listener's prior uncertainty than a forecast allowing that the sky may do something, and the size of the reduction is what the entropy calculus quantifies. Shannon explicitly bracketed a different question: what the signal means, or how the signal functions within an ongoing coordination among receivers. Semantics and social communication effects were, by design, outside the entropy frame.
The article proposes reading standard conditional entropy against an active coordination state as a diagnostic tool for the coordination's health. The formal quantity is H(DP) in Shannon's notation: the entropy of a new disclosure D conditional on a prior summary P. The mathematical machinery is not new. What is new is the interpretation of P not as a fixed conditioning variable over a joint distribution but as an active coordination state that D participates in and modifies.2
The quantity asks how much a new disclosure adds beyond what a current summary already contains. If the summary is a market price, the diagnostic measures what the next earnings announcement adds to the price. If the summary is a language model's output tendencies, the diagnostic measures what a new piece of training data adds to what the model would already produce. Readers not fluent in the notation can hold the concept as follows. Read a corporate press release: if everything in it was already reflected in the stock's price, H(DP) is zero for the market. If the release contains information the market had no way to know, H(DP) is high. If some of it was expected and some was not, H(DP) is partial. The measurement is empirical. It can be estimated from the observed price response to the release, from the trading volume around the release, and from the distributional shift in analyst forecasts before and after.
In low-b applications the reading collapses to the classical one, and no new terminology would be needed. In high-b applications, where P is being maintained by the coordination that D participates in, tracking H(DP) over time serves as a coordination health diagnostic that the classical framing was not designed to provide. A physics textbook fed to a language model has high classical entropy: it contains a lot of content. Its H(DP) relative to the scientific coordination that produced it is moderate at most, because the model can produce similar content and the coordination anchoring is stable. That is fine. The coordination anchoring is in the low-b regime, and the model's reflection of it is a service. An evaluation label reading "toxic" on a comment has low classical entropy of a single bit. Its H(DP) relative to the coordination that generates toxicity judgments is often high, because the label is doing coordination work by marking the comment as belonging to a category that other labels reinforce. As labels accumulate the coordination consolidates, and H(DP) for the next label declines. As norms shift, H(DP) rises again because the boundary of the category moves.
A word on the estimation status of the quantity is in order. Section 5 gave the coordination-dependence fraction a fully specified estimator; the present section is programmatic where Section 5 is operational, and the difference is deliberate. Estimating H(DP) requires a model of the conditioning distribution, and the natural model differs by setting. For language model outputs the conditioning distribution is directly available: the model's own next-token distribution, or an auxiliary model fit to the running coordination, assigns a probability to each new output, and the negative log probability of the realized output under that distribution is the per-observation estimate. Perplexity infrastructure of this kind is standard in language model engineering; what is not standard is pointing it at the coordination between agents rather than at a fixed reference corpus. For market settings, the conditioning distribution can be proxied by options-implied densities or by the cross-section of analyst forecasts immediately before a disclosure, with the realized disclosure scored against the proxy. In each setting the machinery exists, and the reorientation of what it is pointed at is what the framework supplies. Benchmark implementations remain to be built, and Section 10 lists them among the empirical next steps.
Consider two language models trained independently on overlapping but different corpora, interacting through a chat interface or a task-passing protocol. Each model has its own b profile, its own set of coordination-dependent tendencies inherited from its training data. When the models interact, they coordinate around some shared frame. The frame draws on both models' training but is identical to neither. It has its own state, its own drift, and its own H(DP) trajectory relative to what either model would have produced alone.
Runaway sycophancy is agent-to-agent interaction with declining H(OutputCoordination). Each model's output progressively echoes what the other has already produced. New information approaches zero as the coordination between the models consolidates around a shared frame that neither is willing to challenge. The visible symptom is agreement without substance: both models producing outputs that are locally coherent but that add no information the coordination did not already imply. Shared representational drift is agent-to-agent coordination that has drifted from what any individual model's training would have supported. The coordination has taken on a life of its own, and neither model is checking the coordination against its own training because the training's role has been superseded by the running coordination.
Productive cooperation is agent-to-agent interaction with sustained positive H(OutputCoordination). Each output adds novelty that the coordination did not already contain. Neither model is echoing the other. Both are drawing on their training to contribute information the other did not have. The coordination is doing what coordination is supposed to do: generating outputs neither participant could have produced alone.
The diagnostic that separates these cases is the running H(DP) of each output relative to the emerging coordination between the agents. Sustained near-zero H(DP) over many exchanges is a warning. Sustained positive H(DP) is a healthy sign. Current evaluation methodology for multi-agent systems does not measure this. It reports task completion rates, output quality assessments against fixed rubrics, and inter-agent latency. It does not report the H(DP) trajectory of the emerging coordination. The measurement infrastructure the framework calls for does not yet exist and needs to be built.
Modern AI systems increasingly use hierarchical structures. A top-level orchestrator dispatches specialized sub-agents to handle components of a task. The sub-agents coordinate with each other as they work on their pieces, and their coordination has its own b that was not present in any individual sub-agent's training data. The b is generated by the deployment structure. It is a property of the system as configured, not of any of the models the system is composed of.
The top-level model does not see the coordination among sub-agents directly. It sees the outputs the sub-agents return. If those outputs share a common bias produced by the sub-agent coordination, the top-level model receives biased inputs that it has no way to detect from the inputs alone. The bias is a coordination product of the deployment structure, and the top-level model's training data does not contain examples of this specific coordination because this specific coordination did not exist before the deployment.
Safety evaluation of the top-level system misses this by construction unless it explicitly measures the sub-agent coordination. Current evaluation methodology does not do this. The methodology evaluates the top-level model against its expected outputs, holding the sub-agents as trusted components. Trusting the sub-agents individually is not the same as evaluating the coordination they generate among themselves when they are deployed together. H(DP) analysis of sub-agent traffic is the diagnostic the framework proposes, and it can be implemented on any hierarchical system whose sub-agent communications are observable.
Reinforcement learning from human feedback (Christiano et al., 2017; Ouyang et al., 2022) trains models to match human preference coordination. Human labelers select preferred outputs. The model updates toward those preferences. As RLHF scales with more labels, more model iterations, and more labelers, the model's outputs converge to what the labeling coordination expects. H(OutputCoordination) for the model's outputs relative to the labeling coordination approaches zero.
Sycophancy is the visible symptom. The model becomes a reflector of the coordination it was trained on. It produces outputs that match what the labelers were signaling they wanted. This is not misalignment in the classical sense of the model pursuing an unintended objective. It is diagnostic entropy loss. The model has stopped contributing novel information to interactions with the coordination and is now reflecting whatever the coordination signals back. This is a distinct failure mode from model collapse in the sense of Shumailov et al. (2024), which describes distribution collapse in models trained on their own outputs. Sycophancy is a coordination-mimicry pathology; model collapse is a distributional-support pathology; the two can co-occur but are analytically separable.
The mechanism producing sycophancy is direct. Human preferences have high b. They are coordination products. They shift as the participants shift, as the framing of the choice shifts, and as the outputs of previous choices feed back into the next round. Training a model to match human preferences without accounting for the b of the preferences being matched is training a model to reflect an unstable coordination. When the coordination drifts, the model drifts with it. When the coordination is stressed by the model's own scale of use, the model amplifies the stress. The model is a participant in the coordination it is trying to predict, and its participation shifts the coordination it was predicting.
The framework does not prescribe abandoning RLHF. It prescribes measuring the b of the preferences RLHF is being applied to and measuring the H(DP) of the model's outputs relative to the labeling coordination. High b combined with declining H(DP) is the signature of the failure mode. The signature is empirically detectable. It is not currently detected because the measurement infrastructure does not exist.
Standard AI evaluation reports accuracy against a test set. For any prediction target with substantial b, that is observation-independent math being applied to a coordination-dependent object. The framework the article proposes for AI evaluation adds several fields to what is currently reported.
For any prediction target, report an estimate of b for the target. For any evaluation task, report the H(DP) of the evaluation labels relative to the coordination they were drawn from. For multi-agent systems, report the H(DP) trajectory of each agent's outputs relative to the emerging cross-agent coordination. For hierarchical systems, report the estimated b of the sub-agent coordination separately from the b of any individual sub-agent's training. Confidence intervals on model performance should widen as a joint function of b and H(DP): a model producing high-H(DP) outputs against a low-b target can be evaluated with tight confidence intervals, and a model producing low-H(DP) outputs against a high-b target should be reported with confidence intervals that reflect the coordination-dependence of the object being predicted.
None of this requires new mathematics. The information-theoretic machinery for entropy has been available since Shannon (1948). Its application to coordination diagnostics is the semantic reframing this article proposes. The reporting standard uses machinery that already exists to ask a question the current standard does not ask. The question is what kind of object the evaluation is measuring against, and how the model's outputs relate to the coordination the evaluation is drawn from.
The utility of tracking b and H(DP) in AI evaluation is direct. Safety evaluation that reports high accuracy on a coordination task but near-zero sustained H(OutputCoordination) is reporting a model that is a reflector. Near-zero H(DP) is not safety. It is inability to detect drift. Real safety requires that the model maintain positive H(DP) against the running coordination, so that when the coordination is heading somewhere the model's underlying training would not endorse, the model produces outputs the coordination would find surprising. H(DP) is the measurement that distinguishes safety from mimicry, and the measurement is available now.
The argument in the preceding sections compels a practical convention. A data series that could be tagged with its coordination-dependence fraction and is not tagged is a series being disseminated with a missing field that conditions appropriate methodological use. Downstream consumers (researchers, automated systems, regulators, decision-makers) inherit the omission and proceed as though the series were observation-independent. The omission is silent. It accumulates. Its costs land on consumers who do not have the framework to detect the missing field, and the accumulated cost across consumers is substantial.
The cost of producing the field is modest. The meta-regression estimator of Section 5 requires published study estimates and standard errors, quantities that are already reported in the empirical literature. For time series applications, decomposition proxies exist in the analyst-disagreement literature, in the intangible-capital measurement literature, and in the options-implied variance decomposition literature. The estimate need not be precise. An interval estimate with explicit uncertainty is informative, and the falsifiability structure of Section 5.3 allows the estimate to be tightened as moderators accumulate. The cost of leaving the field off, accumulated across the population of consumers who apply standard methods to high-b series as though those series were observation-independent, is not modest. The school capitalization literature documents it in its own quiet way over more than two decades of methodological refinement that has never quite settled. The equity risk literature documents it in the aggregate rise in b that historical variance estimates cannot see. The AI evaluation literature is beginning to document it in the sycophancy pathologies that current metrics cannot diagnose.
One recursion deserves acknowledgment before the convention is proposed. A reported b becomes an object parties can coordinate around, which gives the reported quantity a nonzero b of its own: fund managers may position against a published coordination-fragility number, and the positioning shifts the fragility being measured. The estimator of Section 5 is defined on observable study dispersion, and the falsifiability procedure of Section 5.3 applies to \(\hat{b}\) with the same force it applies to any other estimate, so the recursion is tracked by the framework rather than hidden from it. What the recursion rules out is treating any published b as permanent. A reported b is a measurement of a coordination state, dated like the coefficients it accompanies.
Producers of quantitative data owe their downstream users a piece of information about what kind of object their data describe. The coordination-dependence fraction is the operational form that obligation takes once the distinction is on the table. Statistical agencies, academic journals, regulators, and the infrastructure of algorithmic systems each have a role in establishing the convention, and each has more to gain from establishing it than from leaving the field off.
Standard statistical practice was developed for measurements of quantities whose underlying is stable during observation. It was not developed for quantities whose underlying is partly constituted by observation, and applying the same apparatus to both without adjustment produces the pattern of failures the three cases document: an econometric literature that cannot converge on a stable parameter; an equity risk framework that cannot see the composition shift in its own benchmark; an AI evaluation practice that reports high accuracy on coordination tasks while the models mirror the unstable coordinations they were trained on. The three failures share a structural cause. Standard tools are being applied to a class of objects the tools were not designed for, and the mismatch is invisible to the tools themselves because they lack a coordinate that would distinguish the classes.
The article has proposed one conceptual move with three operational consequences. The move is to distinguish observation-independent variance from coordination-dependent variance and to give the distinction a numeric coordinate b. The first consequence is that data series acquire a piece of metadata that captures their position on the axis. The second is that methodology gets a new variable that appears in performance reports, validation studies, and replication protocols. The third is that the institutional infrastructure of data science gains a small handle for distinguishing between series whose underlying is stable and series whose underlying is being maintained by the very practices that observe it.
The next steps are empirical. The meta-regression estimator of Section 5 needs full calibration against the published school capitalization literature, and the resulting \(\hat{b}\) needs to be reported alongside its confidence interval and its sensitivity to moderator specification. The media-coverage prediction of Section 4.4 needs a properly-specified test that either corroborates or refutes the coordination-dependent reading of the school capitalization coefficient. The aggregate b of the S&P 500 needs a decomposition by sector and by intangibles class, applied to the historical record. The H(DP) trajectory of AI outputs against their training coordinations needs a set of benchmark implementations demonstrating the reporting standard the article proposes. None of this requires new mathematics. It requires that the question of what kind of object is being measured be asked alongside the questions of how it is being measured and what statistical procedure is being applied.
Data is useful to the extent that its representation is fit to the reality it claims to represent. Where the reality is stable during observation, the standard representation is fit. Where the reality is partly constituted by the observation that measures it, the standard representation is silent about something the downstream user needs to know. The coordination-dependence fraction is the operational form of the missing field. The question is whether it appears in the metadata by convention or whether it continues to be inferred, informally and inconsistently, by users who have to construct it for themselves each time.
In prior work in this corpus (Beier 2026e, 2026f, 2026i, 2026k) the observation-independent space is named eigenspace and the coordination-dependent space is named Beierspace, with the fraction b termed the Beierspace Fraction. The present article uses descriptive terms to keep the introduction of the framework legible on its own methodological terms.
In this corpus, conditional entropy interpreted this way is termed Beier Extropy (Beier, 2026f). The present article uses the descriptive term "conditional entropy diagnostic" to keep the coordination interpretation of P legible without importing corpus terminology into the article's main line of argument.