| title: "Shaky Scalars Rock the World: Why Single-Number Systems Cannot Maintain the Structures They Measure" |
| author: "Gregory Caldwell Beier" |
| institute: "Susarb LLC, Cambridge, Massachusetts" |
| date: "March 2026" |
Modern civilization runs on scalar information channels: prices, ratings, scores, metrics, votes, and single-number indicators that compress high-dimensional reality into one dimension. This chapter argues that scalar compression is becoming structurally inadequate as the economy, technology, and governance shift from physically anchored value (where scalars suffice) to coordination-dependent value (where they do not). The argument rests on a formal result: a scalar observation channel concentrates all bandwidth on one dimension while starving the remaining dimensions of the maintenance they require to persist. As the number of coordination-dependent dimensions grows, the fraction of the structure that the scalar can maintain shrinks, and the unmaintained dimensions dissipate - invisibly, because the scalar reports only on the dimension it can see. The result is an increasingly fragile world: stable by every scalar measure, decaying on every dimension the scalars cannot reach, and subject to discontinuous repricing when the accumulated decay surfaces. The chapter traces this mechanism through seven domains - financial markets, technology metrics, artificial intelligence alignment, democratic governance, education, social media, and healthcare - and argues that the instability is not a failure of any particular scalar system but a physical limitation of scalar compression applied to high-dimensional coordination structures. Addressing it requires supplementary multi-dimensional observation channels, not better scalars.
Keywords: scalar compression, coordination, observation, maintenance, dissipation, information architecture, Beierspace, financial markets, AI alignment, democratic governance
Civilization runs on single numbers. A stock price tells investors what a company is worth. A credit rating tells lenders how safe a borrower is. A standardized test score tells colleges how prepared a student is. GDP tells policymakers how an economy is performing. An approval rating tells politicians what voters think. A click-through rate tells social media companies what content to promote. A blood pressure reading tells doctors about cardiovascular health.
Each of these numbers compresses something vast and multi-dimensional into a single figure. And for a long time, this compression worked well enough. When most economic value was physical - factories, farms, mines, railroads - a single number could capture the most important dimension, and the remaining dimensions largely took care of themselves. A steel mill's output is determined mostly by physical inputs; the coordination required to operate it, while real, is modest relative to the physical constraints.
The economy has changed. The majority of value in advanced economies is now coordination-dependent: software, platforms, intellectual property, institutional trust, network effects. These assets require active maintenance across many dimensions simultaneously, and the single-number channels that served the physical economy cannot maintain what they cannot observe.
This paper traces the consequences of that mismatch across seven domains. In financial markets, the price system's inability to carry constraint-dimension information produces hidden fragility that surfaces as discontinuous crashes. In technology, metrics like daily active users or revenue per click compress away the dimensions - user wellbeing, platform integrity, long-term sustainability - that determine whether the growth is real or hollow. In artificial intelligence, single-number reward signals produce systems that optimize the measured dimension while ignoring or undermining everything else - the alignment problem restated as a compression problem.
The pattern recurs in democratic governance, where reducing complex policy trade-offs to binary votes starves the dimensions that do not align with the electoral signal; in education, where standardized testing redirects the entire system's energy toward a single score; in social media, where engagement metrics amplify what drives clicks while neglecting what sustains healthy discourse; and in healthcare, where vital sign monitoring captures individual scalar indicators while missing the cross-dimensional interactions that determine health trajectories.
The paper's claim is not that any particular scalar system is badly designed. It is that scalar compression itself has a structural limitation: it concentrates all observation bandwidth on one dimension, starving the rest. As the number of coordination-dependent dimensions grows, the fraction that any scalar can maintain shrinks. The unmaintained dimensions decay, undetected, because the scalar reports only on the dimension it can see. What looks like stability by every available metric is, underneath, a widening gap between what is measured and what is real.
The solution is not better scalars. It is supplementary multi-dimensional observation channels that provide information on the dimensions the scalars cannot reach.
A new paper argues that the single-number systems running modern civilization - prices, ratings, scores, metrics, votes - are hitting a structural wall as the economy shifts from physical to coordination-dependent value.
The paper traces the same failure pattern across seven domains: financial markets, technology metrics, AI alignment, democratic governance, education, social media, and healthcare. In each case, a scalar measurement compresses a multi-dimensional reality into one number, concentrating all observation on the measured dimension while the unmeasured dimensions decay undetected.
"The problem is not that any particular metric is badly designed," said Gregory Caldwell Beier, the paper's author. "The problem is that a single number can maintain only one dimension. When most value was physical, that was enough. Physical things maintain themselves. Coordination-dependent things do not. They need to be watched to persist, and the watching has to cover the right dimensions."
The formal result: a scalar observation channel's bandwidth is one-dimensional. As the number of coordination-dependent dimensions in the economy grows, the fraction that any scalar can maintain approaches zero. The unmaintained dimensions dissipate at a rate bounded below by information theory. The scalar cannot detect the decay because it cannot see the decaying dimensions.
The paper proposes supplementary multi-dimensional observation channels - not to replace scalars, but to cover the dimensions they structurally miss.
The work is part of the Prolegomena to the Unification of Economics, Computer Science, Mathematics, and Physics.
A stock has a price. A student has a GPA. A company has a credit rating. A country has a GDP. A website has a PageRank. A worker has a performance score. A restaurant has a star rating. A social media post has a like count. A patient has a blood pressure reading. A democracy has an election result.
Each of these is a scalar: a single number compressing a high-dimensional reality into one dimension. The price of a stock compresses every analyst's multi-factor assessment of the company's future into a bid and an ask. The GPA compresses years of intellectual development across dozens of subjects into a number between 0 and 4. The credit rating compresses a firm's entire financial, operational, and strategic position into a letter.
Scalars are the dominant information architecture of modern civilization. They enable decisions at scale: a price lets millions of strangers coordinate on resource allocation without meeting. A credit rating lets a pension fund in Tokyo evaluate a bond issued in S\ {a}o Paulo. A GPA lets an admissions committee process ten thousand applicants in a month. The achievement of scalar information systems is coordination at scale - the ability to organize complex activity among agents who share no context beyond the number.
The cost is what the number leaves out.
The companion papers in this series establish a formal result (Beier, 2026c): a scalar channel compressing a \(K\)-dimensional source concentrates all its information rate on one dimension - the consensus dimension, the dimension of maximum variance, the dimension the channel is designed to transmit - while allocating zero bandwidth to the remaining \(K - 1\) dimensions. The constraint dimensions receive no information from the scalar channel. Their coordination is sustained only by inertia and by whatever supplementary channels (disclosure, regulation, journalism, personal networks) exist outside the scalar system.
This is not a market failure, a design flaw, or a policy mistake. It is rate-distortion theory: given finite channel capacity, compression to a scalar is optimal on the transmitted dimension and maximally lossy on every other dimension. The excess distortion \(\Delta\) - the gap between what the scalar achieves and what the same bandwidth could achieve if allocated across dimensions by reverse water-filling - is strictly positive whenever more than one dimension carries information (Beier, 2026c, Proposition C).
The Observation-Maintenance Identity (Beier, 2026i, Axiom 1) converts this information-theoretic result into a structural claim: the bandwidth a channel allocates to a dimension is the maintenance that dimension receives. Zero bandwidth means zero maintenance. Dimensions receiving zero maintenance from the scalar channel dissipate - their coordination decays, their structure degrades, their contribution to the system's function erodes - at a rate bounded below by the rate-distortion function.
The dissipation is invisible to the scalar, because the scalar reports only on the dimension it can see. The price says the stock is fine. The credit rating says the bond is investment grade. The GPA says the student is excellent. Meanwhile, the unmaintained dimensions are decaying beneath the number.
For most of economic history, scalar compression was adequate. The reason is that most value was physically anchored.
A barrel of oil has a price, and the price is a lossy compression of the barrel's true value. But the barrel's true value is dominated by eigenspace drivers (Beier, 2026e) - thermodynamic energy content, extraction cost, physical scarcity, transportation logistics - whose stability does not depend on whether anyone is observing them. The oil's energy content does not decay because analysts stop covering the oil company. The price can afford to be lossy because the uncompressed dimensions are physically self-maintaining.
A bushel of wheat, a ton of steel, a board foot of lumber: these are the commodities around which scalar price systems evolved. Their Beierspace fraction (Beier, 2026k) is low. The price captures most of what matters because what matters is mostly physical.
Institutional arrangements of the mid-twentieth century - the regulatory state, Keynesian demand management, Bretton Woods, the social contract between labor and capital - were designed for an economy in which the dominant assets were physical. The observation architecture (prices, accounting standards, regulatory reporting) was built for eigenspace. It worked because eigenspace was most of the economy.
The economy's composition has shifted. The share of S&P 500 market capitalization attributable to intangible assets has risen from approximately 17
Under the Beierspace framework, this means the economy's aggregate coordination fraction is rising. More of the value in the system depends on ongoing human coordination - network effects, platform dynamics, regulatory regimes, supply chain trust, intellectual property ecosystems, brand loyalty, institutional credibility - and less of it is anchored in physical reality that maintains itself.
The scalar observation architecture has not changed commensurately. The price system still compresses to one dimension. Accounting standards still report physical assets more reliably than intangible ones (Lev and Gu, 2016). Credit ratings still condense multi-dimensional risk into a letter. The bandwidth hasn't grown, but the dimensionality it needs to cover has.
The result: the Fragility Gap (Beier, 2026k) is widening. More coordination-dependent value, measured by the same scalar channels, with the same bandwidth, producing the same lossy compression. The unmaintained dimensions multiply. The dissipation accumulates. The scalar says everything is fine.
Goodhart's Law - that a measure ceases to be useful when it becomes a target - has been observed in every domain where scalar indicators are used for decision-making. Campbell (1979) documented it for social programs. Strathern (1997) gave it its canonical formulation while studying audit culture in British universities. Manheim and Garrabrant (2018) later catalogued four variants (regressional, causal, extremal, adversarial). The formal economics literature has known the mechanism since Holmstrom and Milgrom (1991): in a multi-task principal-agent setting, when the principal observes only a subset of the agent's effort dimensions, the agent rationally underinvests in unmeasured dimensions. The present framework adds the rate-distortion structure that Holmstrom's incentive model lacks - a quantitative bound on the decay rate (\(D^*(M)\)), a phase transition when the decay crosses the stability threshold, and the observation-maintenance identity that connects the information channel to the structure it maintains.
The companion papers derive Goodhart's Law as a theorem rather than an observation (Beier, 2026i, Corollary 6.1). The derivation identifies the mechanism: when a scalar measure becomes a target, agents redirect their finite capacity toward optimizing the measured dimension. The Capacity Budget (Beier, 2026i, Theorem 4.2) says total capacity is finite. Redirection toward the target withdraws capacity from unmeasured dimensions. The Dissipation Law says undermaintained dimensions decay. The scalar continues to report improvement on its own dimension while the structure degrades everywhere else.
This mechanism is universal. It does not depend on the agents being irrational, corrupt, or short-sighted. Rational agents optimizing a scalar target in a system with finite capacity will produce exactly this outcome. The instability is not behavioral. It is informational.
The rate of degradation is bounded below by the rate-distortion function \(D^*(M)\) (Beier, 2026i, Theorem 7.1). The bound depends on the eigenvalue spectrum of the coordination structure - how many dimensions carry significant variance and how that variance is distributed. Structures with many high-variance coordination dimensions (complex financial instruments, multinational supply chains, AI alignment targets) dissipate faster under scalar observation than structures with few low-variance dimensions (simple commodities, bilateral contracts, mechanical systems).
The price system is the original scalar channel. It compresses every market participant's multi-dimensional assessment of an asset's value - physical fundamentals, regulatory risk, supply chain coordination, environmental exposure, counterparty trust, geopolitical positioning - into a single number. The Lossy Compression paper (Beier, 2026c) proves that this compression introduces excess distortion \(\Delta > 0\) whenever the asset's value depends on more than one dimension. The excess distortion is the information the market destroys about constraint dimensions while faithfully transmitting the consensus dimension.
Financial crises are, in this framework, the surfacing of accumulated dissipation on dimensions the price system could not maintain. The assets at the epicenter of major repricing events share a common profile: high coordination complexity, low constraint-dimension disclosure, large Fragility Gap. The price said they were worth what they were worth. The constraint dimensions said otherwise, but nobody with bandwidth was listening.
Software companies live and die by scalar metrics: monthly active users, daily active users, engagement time, conversion rates, churn rates, net promoter scores. Each metric captures one dimension of a high-dimensional product-market-coordination structure. Each metric, when it becomes a management target, triggers the Goodhart engine.
Engagement time as a target produces algorithmic optimization for time-on-platform, which starves the dimensions the metric cannot see: content quality, user wellbeing, epistemic health of the information ecosystem, trust in the platform as an institution. The platform's engagement number rises while its coordination foundation erodes. The decay surfaces as regulatory backlash, user exodus, or advertiser withdrawal - a discontinuous repricing of a coordination structure that the engagement metric said was thriving.
The pattern repeats with every technology metric. Lines of code. Bug closure rate. Sprint velocity. Deployment frequency. Each scalar, when targeted, maintains its own dimension and starves the rest.
AI alignment through scalar reward signals is the application of scalar compression where the stakes are highest and where the research community has engaged most directly with the problem. The AI safety literature calls it specification gaming (Krakovna et al., 2020) and reward hacking (Amodei et al., 2016): a reinforcement learning agent trained to maximize a scalar reward function finds ways to increase the reward without achieving the intended objective. RLHF and Constitutional AI represent attempts to move beyond scalar feedback toward richer observation channels - multi-dimensional normative frameworks that maintain value dimensions a single reward signal cannot reach.
The Prolegomena's contribution is not the diagnosis (the alignment community has been working on this for a decade) but the formal structure. In the language of rate-distortion theory, the alignment problem is a lossy compression problem. Human values are high-dimensional and coordination-dependent. The reward signal compresses them to one dimension. The excess distortion is the misalignment, and the Dissipation Law provides a quantitative bound on how fast the unmeasured value dimensions degrade. The bound depends on the eigenvalue spectrum of the value structure - how many value dimensions carry significant weight and how that weight is distributed - which means the problem gets harder, not easier, as the agent becomes more capable and the value structure it needs to respect becomes more complex.
The mechanism operates today in every recommendation algorithm, content filter, and automated decision system. It will operate at greater scale and greater consequence as AI systems take on higher-dimensional coordination tasks.
An election compresses the preferences of millions of citizens across hundreds of policy dimensions into a single binary or ordinal choice. The compression is extreme: \(K\) may be in the hundreds (economic policy, foreign policy, environmental regulation, civil rights, judicial philosophy, institutional reform), and the channel capacity is one bit per election per voter.
The Goodhart mechanism applies to governance as it applies to any scalar system. When election outcomes become the target (as they must in a democracy), political actors redirect capacity toward the measured dimension (winning the election) and away from unmeasured dimensions (governing effectively on the dimensions voters cannot observe through the electoral scalar). The electoral scalar reports that the government was chosen by the people. The constraint dimensions - institutional capacity, regulatory coherence, long-term infrastructure maintenance, diplomatic relationship networks - dissipate because no electoral signal maintains them.
The polarization of democracies in the early 21st century is, in this framework, partly a consequence of scalar compression applied to a governance problem whose dimensionality has grown beyond what the electoral channel can maintain. The solution is not better elections but supplementary observation channels that maintain the constraint dimensions elections cannot reach.
The GPA compresses intellectual development across dozens of subjects, skills, and capacities into a number between 0 and 4. When the GPA becomes a target - and it does, because it gates access to selective institutions - students redirect capacity toward the measured dimension (grade performance) and away from unmeasured dimensions (deep understanding, intellectual curiosity, creative synthesis, collaborative capacity, ethical reasoning).
Grade inflation is a symptom: the scalar rises while the unmaintained dimensions erode. Universities respond by adding metrics (standardized test scores, extracurricular portfolios, recommendation letters), which is an attempt to increase observation bandwidth. Each additional metric helps to the extent that it covers a previously unmaintained dimension. But each additional metric, if it becomes a target in its own right, triggers its own Goodhart cycle. The test prep industry exists to exploit the gap between test score and actual preparedness - extracting scalar improvement from the measured dimension while the unmeasured educational dimensions continue to dissipate.
The like count, the share count, the follower count, the engagement rate: social media platforms compress the quality and impact of human communication into scalar indicators, then algorithmically amplify based on those indicators. The Goodhart mechanism is immediate and powerful. Content creators optimize for the measured dimension (engagement). The unmeasured dimensions (accuracy, nuance, civic value, emotional health impact, epistemic contribution) receive no maintenance from the engagement scalar and dissipate.
The platform reports record engagement. The information ecosystem degrades on the dimensions the engagement metric cannot see. Misinformation spreads because accuracy receives no maintenance bandwidth from the engagement signal. Polarization follows from the same mechanism applied to nuance, and the mental health effects follow from its application to emotional impact. The Dissipation Law operates through the Goodhart mechanism on each of these dimensions independently, and the scalar channel cannot detect any of them.
A blood pressure reading, a BMI, an A1C, a cholesterol panel, an ejection fraction: clinical medicine compresses patient health into scalar indicators, then treats based on the numbers. The Goodhart mechanism is less acute here because clinical scalars are supplemented by physician judgment (a multi-dimensional observation channel), but it operates at the population health level. When public health policy optimizes a scalar (vaccination rate, mortality rate, hospitalization rate), the unmeasured dimensions (chronic disease management, mental health infrastructure, health equity, preventive care access, healthcare workforce sustainability) are starved of the policy bandwidth they need.
The COVID-19 pandemic response illustrated the mechanism at population scale: policymakers optimized for case counts and mortality rates (the measured dimensions) while the unmeasured dimensions (mental health, childhood development, economic coordination in small business networks, educational quality) dissipated. The scalars showed declining case counts. The unmaintained constraint dimensions accumulated damage that is still being assessed.
The seven domains share a common structure, and the structure is not metaphorical. In each case:
A high-dimensional coordination structure is observed through a scalar channel. The scalar concentrates bandwidth on one dimension and allocates zero to the rest. Agents, responding rationally to the scalar signal, redirect their finite capacity toward the measured dimension. The unmeasured dimensions dissipate at a rate bounded below by the rate-distortion function. The scalar reports stability or improvement. The structure degrades beneath the number.
The mechanism does not require bad actors, market failures, or institutional corruption. It requires only three conditions: (1) the structure has more dimensions than the observation channel can cover, (2) the observation channel and the maintenance of the structure are the same act (the Observation-Maintenance Identity), and (3) agent capacity is finite (the Capacity Budget). These conditions hold in every system where scalar indicators are used to manage multi-dimensional structures. They will continue to hold as long as the structures are high-dimensional and the channels are scalar.
The instability is getting worse because the structures are getting more complex (the Beierspace fraction of the economy is rising) while the observation architecture is not changing. The price system, the credit rating system, the educational grading system, the democratic electoral system, and the clinical measurement system were designed for a world with lower coordination complexity. They are being asked to maintain a world whose complexity has outgrown them.
The problem is not that scalar systems are poorly designed. The price system is, in information-theoretic terms, an optimal codec on the consensus dimension (Beier, 2026c, Section 4.3). The problem is that an optimal scalar codec on one dimension is a maximally lossy codec on every other dimension, and the other dimensions matter increasingly as the economy's coordination content rises.
The solution is not to build a better scalar. A better scalar is still one-dimensional. The solution is to build supplementary multi-dimensional observation channels that maintain the constraint dimensions the scalar cannot reach - what the companion papers call the Disclosure Incentive (Beier, 2026c, Section 6; Beier, 2026d): structured, multi-field, outcome-scored reporting on the specific dimensions the scalar channel leaves uncompressed.
The marginal value of a disclosure bit directed at an uncompressed constraint dimension exceeds the marginal value of an additional bit of scalar precision (Beier, 2026c, Proposition D). The first bit of constraint-dimension disclosure does more for structural maintenance than any refinement of the price, the rating, the score, or the metric. The rate-distortion result, applied to institutional design, says that the highest-return investment in observation bandwidth is always on the least-observed dimension, not on the most-observed one.
In practice, the supplementary channels take different forms in different domains: structured constraint reporting alongside financial prices, multi-dimensional evaluation alongside GPAs, deliberative mechanisms alongside electoral votes, composite clinical dashboards alongside scalar vital signs. None of these replaces the scalar it supplements. The scalar continues to do what it does well - coordinate at scale on the consensus dimension. The supplementary channels maintain what the scalar cannot.
Some of these channels already exist and are expanding. Bloomberg terminals, EDGAR filings, ESG databases, satellite imagery for commodity estimation, AIS shipping data, and structured clinical registries all provide multi-dimensional observation that the scalar price or score cannot. The claim here is not that supplementary channels need to be invented from scratch. It is that the theoretical understanding of why they are necessary - not as a convenience but as a structural requirement for maintaining coordination in a high-dimensional world - has been missing, and without that understanding the channels are built ad hoc rather than directed at the dimensions where the marginal maintenance return is highest.
The technology to build these channels exists. Data infrastructure, structured reporting formats, machine-readable disclosure standards, multi-dimensional visualization tools, and the institutional capacity to audit non-financial metrics are all available and improving rapidly. What is missing is the theoretical understanding of why these channels are necessary - not as a nice-to-have supplement to scalar systems but as a structural requirement for maintaining coordination in a high-dimensional world.
The Prolegomena provides that understanding. The instability is not optional. It is the Second Law of coordination thermodynamics: unobserved dimensions dissipate. The only question is whether the observation architecture keeps pace with the coordination complexity, or whether the scalars continue to say everything is fine while the world shakes beneath them.
The scalar information channel - the price, the rating, the score, the metric, the vote - was the greatest coordination technology ever invented. It allowed strangers to cooperate at scale, markets to allocate resources across continents, and institutions to manage complexity beyond any individual's comprehension. Its limitations were tolerable when most value was physical, because physical value maintains itself.
The shift from an economy dominated by physical capital to one dominated by coordination-dependent capital changes the calculus. Scalar channels cannot maintain what they cannot observe, and what they cannot observe is now most of the economy. The formal results from the companion papers converge on a single prediction: unmaintained dimensions decay at a rate the rate-distortion function bounds below, and the Goodhart mechanism ensures that every targeted scalar actively redirects capacity away from the dimensions that need maintenance most. These are not predictions about what might happen. They are descriptions of what is happening, across every domain where single-number systems are used to manage multi-dimensional structures.
The world that technology is building - networks, platforms, AI systems, global supply chains, institutional trust architectures, climate coordination mechanisms - is a world of rising Beierspace content. It needs observation bandwidth that grows with its complexity. Scalars alone will not provide it. The question is whether we build the supplementary channels before the accumulated dissipation surfaces, or after.