Greg Beier, Susarb LLC, Cambridge, Massachusetts
12 October 2026. Preprint, version 3.3.
Machine learning systems that act in markets, forecasting, and governance learn from records of human judgment, and the most abundant such record is the price series, which compresses many judgments into a scalar and discards who judged, when, and how well each judge has performed before. I state a conjecture about what that compression costs.
Let b in [0, 1] be the coordination-dependence fraction of an outcome, the share of its variance that coordination among the parties around it produces, defined in Beier (2026a) and computed here by ablation of the coordination channel. The Judgment Record Conjecture holds that the gain in model decision quality from training on attributed, timestamped, and later-scored price records rather than on the public record, at matched observations, vanishes as b approaches zero in deep markets, rises with b, and rises with the depth of the scoring record. Two preliminary results fix the sign and the ceiling of the ideal gain, and a two-round beauty-contest model, with b computed by ablation, fixes the condition the conjecture needs: the coordination gain is zero at every b when the next coordinated move depends on the aggregate alone and positive when it depends on the cross-section of positions. In the model the coordination gain tracks a second quantity, c, the share of the next aggregate that the cross-section predicts beyond the aggregate, and b caps the gain where the two components are uncorrelated. The model builds those outcomes in; what it supplies is the accounting and the quantity c, which can be measured in any identified record without an outcome. The conjecture is then a claim that markets have c above zero after the timing channel is removed. The assumption that carries the null, that the public record already holds what is public about an outcome, is the hypothesis under test.
A companion conjecture, Price-AI Coupling, addresses the loop that opens once models set prices as well as learn from them: under blind prices, price informativeness and model quality thin together as the machine share of price formation rises, while attributed records let the human-originated signal be isolated. A test plan across simulation, forecasting and betting records, and market microstructure is set out for registration, and the deposit fixes the date of both conjectures.
Most of what a model learns about how people coordinate, it learns from traces that people left while coordinating. Prices are the densest of those traces. They are struck billions of times a day, they cost something to produce, and they resist casual falsification because money changes hands. The plain form of that last fact is that it is expensive to lie with money. Wash trades and spoofing exist, and a record that keeps who struck each price keeps them too, which is what scoring is for. Hayek's (1945) account treats the price as a sufficient carrier of dispersed knowledge, while the line of argument that runs through Grossman and Stiglitz (1980) shows why it cannot be fully informative. My own earlier deposits develop the second line, and the one this paper rests on is the six-facet record that subsumes the price as its nucleus (Beier 2026b).
What none of this work states in testable form is the consequence for machine learning. The vocabulary I use for the record is from those deposits, and a cold reader needs three terms. The Protocol is a disclosure protocol under which every judgment that moves a price is recorded as a six-facet entry: Identity (who judged), Input (what they looked at), Method (how they reasoned), Output (the judgment itself, with the price as its nucleus and the time it was struck), Scaling (what happens if many act on it), and Transmit (when and to whom it was released). Across the deposits the entry is called a Beier Tensor, and in this paper it is simply the record. The judgment kernel is the hypothesis that one scored record of this form can serve as training substrate across markets, politics, and technology, which prediction L4 tests. A model trained on blind prices inherits the compression. A model trained on records that keep the identity of each judge, the time the judgment was struck and transmitted, and the later scoring of that judgment against outcomes has access to the skill structure that the price erased. The question is how much that access is worth, and where.
The answer cannot be "everywhere." Protein structure, orbital mechanics, and board games have outcomes indifferent to who predicted them, and there is no reason to expect a judgment record to help a model there. The region where b is above zero, where the outcome is partly made by the parties around it, is what my earlier deposits call Beierspace, a name used here for that region and nothing more; outside it, the record has nothing to teach. The coordination-dependence fraction b, introduced in "A Missing Coordinate in the Classification of Quantitative Data" (Beier 2026a, deposited 3 July 2026), gives the boundary a number. Merlet (2026) has since drawn the same boundary qualitatively, arguing that quantitative instruments built for natural systems indifferent to prediction were carried into reflexive markets and opened a "reflexivity gap." Soros (1987) is the common ancestor of both. The contribution here is to make b the cap on what a judgment record can add to a model, and to name the second quantity that decides whether the cap is approached.
Fix a decision domain D with outcome Y, and say what Y is, because it is not the same thing in every domain. In equities Y is a realized fundamental at a horizon, earnings or cash flow, and not a future price, which would be mostly coordination and would leave Y_0 undefined. In sports it is the result. In forecasting it is the resolution of the question. The outcome is always something the world settles, so that a price can be scored against it. Let Y_0 be the outcome as it would be with the coordination channel closed, and let Y_c = Y minus Y_0 be what coordination adds. In an agent population Y_0 is computed by rerunning the population with the channel removed on the same draws. The deposited Missing Coordinate paper (Beier 2026a) defines b and gives an observational estimator for it from meta-regression residuals; the ablation form used here is the computed counterpart that an agent population makes available, and it is this paper's, not that one's. Irreducible noise in the outcome sits in Y_0, since coordination did not produce it. I assume
A0. Y_0 and Y_c are uncorrelated.
A0 is an assumption and not a construction. Coordination that amplifies a physical shock, a squeeze forming around a real bottleneck, makes Y_c covary positively with Y_0, and coordination that dampens one makes it covary negatively; the ablation estimator measures the covariance directly, and where it is material b is reported with the covariance term beside it. b is Var(Y_c) over Var(Y) throughout, with Y_c the ablation residual. Where A0 fails, that quantity and the one-order ablation share, Var(Y) minus Var(Y_0) over Var(Y), which equals b plus the covariance term, come apart, and both are reported; where reflexive interaction is multiplicative the split is not unique in any case, the familiar problem of attributing variance among interacting factors, and the coordination channel is always the channel removed so that the quantity is fixed by the order even where it is not fixed by the algebra. This is the definition of the Missing Coordinate paper, V_B over V_E plus V_B with the two components taken as orthogonal, carried over with the orthogonality now stated as A0 rather than assumed silently. Under A0 the coordination-dependence fraction is
\[b_D = \frac{\operatorname{Var}(Y_c)}{\operatorname{Var}(Y)} \in [0,1].\]
In real data no rerun is available, and b is identified instead by whether and how much the aggregate moves the outcome. Where no causal path runs from the aggregate to the outcome, b is zero by structure: the betting line does not play the game, so a sports result has b = 0 whatever its noise, which is what makes sports the low-b domain the test plan needs. Where the path exists, b is estimated from exogenous variation in the aggregate, as the share of outcome variance that the aggregate's own movement explains, and where no instrument exists the domain enters the ordering by its structure alone, with the ordering and its rationale written into the registration before any data are examined, so that L2 is tested on an ordinal b with point estimates where they can be had. b is a property of a domain in a window, and it moves as the mix of physical and coordinated variance moves, so every test below estimates b in the window in which the gain is measured and never carries a b across regimes.
Two records describe the same activity. The Tensor record T is a set of entries, each carrying a price nucleus p, an agent identifier a, a strike time s, a transmit time x, the remaining facets F, and a set L of later Tensors that score the entry against outcomes. Scoring is never a field inside the original entry; it accumulates as separate Tensors struck by anyone, by any method, and linked back. The blind record B is the public record: the price series together with everything public that a market model is ordinarily trained on, volume, quotes and order-book depth where they exist, news, and public covariates. Nobody trains on the scalar alone, and a comparison against the scalar alone would credit ordinary feature richness to attribution. What T adds to B is exactly the identity, strike and transmit times, and scoring facets. Because the price is the nucleus of each entry and the public record is a function of the entries, B = π(T) for a projection π.
Two objects are measured and should be kept apart. Write Q*(R) for the Bayes-optimal decision quality given record R, the object the two propositions concern. For the predictions, fix a model class M and a training budget, write m_R for the best model in M trained on R, and define decision quality as negative expected loss on a held-out future period:
\[Q(R) = -\,\mathbb{E}\big[\ell\big(Y, m_R\big)\big], \qquad \Delta_D = Q(T) - Q(B).\]
Two comparisons are made, and the order between them matters. The primary comparison is at matched observations: T and B cover the same entries over the same period, and the Tensor facets enter as additional features on each entry. This is the standard way to test whether a feature carries information, and it is the comparison the conjecture is stated for. The secondary comparison is at matched bytes, with the volume V fixed and the Tensor record therefore holding fewer entries; it answers a different question, what the facets cost when storage or bandwidth is the binding constraint, and it is reported beside the primary so that the cost is visible. Byte-matching is not primary because starving a model of observations to pay for metadata tests sample density rather than attribution, and the byte ratio is smaller than it looks, since an agent identifier, two timestamps, and a scoring vector compress to a few integers per entry. Write k for the mean scoring depth, the average number of linked scoring Tensors per entry.
Proposition 1 (ordering). For any loss, Q*(T) ≥ Q*(B).
Proof. B = π(T), so every decision rule measurable with respect to B is measurable with respect to T. The minimum risk over the larger class cannot exceed the minimum over the smaller. ∎
This is the data-processing inequality in decision form. It fixes the sign of the ideal gap; real models are not Bayes-optimal, records are finite, and extra fields can hurt a finite-sample learner through overfitting.
Proposition 2 (boundary). Let (Y_0, Y_c, T) be jointly Gaussian and squared-error loss apply. Assume
A1 (price sufficiency). Given the price record, the Tensor record carries no further information about the observation-independent component: Y_0 is independent of T given B.
Then
\[0 \;\le\; \Delta_D^{\mathrm{MSE}} \;\le\; \operatorname{Var}(Y_c \mid B) \;\le\; b_D \operatorname{Var}(Y).\]
Proof. Under joint Gaussianity the optimal predictors are conditional means with constant conditional variances, so the gain equals Var(Y | B) minus Var(Y | T), which by the law of total variance equals Var(E[Y | T] | B). Under A1, E[Y_0 | T] = E[Y_0 | B], a function of B, so E[Y | T] differs from a B-measurable quantity only by E[Y_c | T]. Hence the gain equals Var(E[Y_c | T] | B), which is at most Var(Y_c | B) by conditional Jensen, and at most Var(Y_c) = b_D Var(Y) because conditioning in the Gaussian case cannot raise variance. ∎
The corollary carries the weight: when b = 0 the gain is zero. For outcomes that coordination does not touch, no judgment record can help a model, however rich. The corollary does not need the Gaussian assumption, and Proposition 1 does not need A1. At b = 0 the outcome is Y_0, and A1 says Y_0 is independent of T given B; for any loss and any Bayes-optimal learner the best rule on T then does no better than the best rule on B, because the conditional law of the outcome is the same under both.
Gaussianity is used only for the quantitative cap at b > 0, where it supplies constant conditional variances. In the heavy-tailed regimes where markets live the cap is not available in this form, the gain is unbounded by this argument, and the test plan treats the size of the gain there as an empirical quantity; the compositional results in Beier (2026e) extend the qualitative ordering beyond the Gaussian case but not the cap. Under A1 the corollary is close to definitional: if the outcome has no coordination component and the record adds nothing about the observation-independent component beyond the price, the record adds nothing. The proposition's value is that it makes the null at b = 0 follow from one named assumption rather than from intuition, so that the null prediction below is a test of A1 and not of the algebra. A1 is where the economics enters, and it is a market-efficiency assumption, not a fact: it says that whatever is public and physical about the outcome is already in the price, so who judged matters only for the part of the outcome that judgment itself helps to make. It is separate from the question of whether coordination touches the outcome at all, which is b. A1 can fail in two ways that the test plan must keep apart. It fails by attribution when the identity and scoring history of a judge carry information about Y_0 that the price does not. That happens wherever the aggregate is inefficient, and the forecasting literature has found it repeatedly: weighting forecasters by their contribution to past accuracy beats the simple consensus (Budescu and Chen 2015), and nothing in that result depends on coordination touching the outcome; the gain comes from the inefficiency of the aggregate and appears wherever the aggregate is thin. That gain is well documented, and it is not the conjecture's gain. The conjecture's gain runs through Y_c and requires A1 to hold; an attribution gain at low b is the aggregation gain, and the test plan measures it as such. It fails by timing when a Tensor is struck before the price has absorbed a physical fact that one agent already holds, so that the record acts as faster news rather than as a record of judgment. The second case is the ordinary condition of markets (Kyle 1985), and a gain from it would appear at low b as a general failure of A1 when it is nothing of the kind. The test plan therefore ablates identity and timestamp separately and adds a lagged-Tensor control, described in Section 7, so that a timing gain and an attribution gain cannot be confused.
The bound is an upper bound on an ideal gain, and it is loose: nothing in it says the realized gain approaches b Var(Y), and nothing in it says the gain rises with b. The bound is also on Var(Y_c | B), not Var(Y_c), and that matters: where the coordination component is itself a function of the aggregate, a self-fulfilling price, Var(Y_c | B) is small exactly where b is large and the record is redundant where the conjecture needs it most. Whether the gain rises or falls with b therefore turns on what the coordination component depends on, and Section 4 shows both cases in one model. For finite learners the extra facets can hurt through overfitting where they carry nothing, and cost observations in the byte-matched comparison, so the conjecture states the null at b = 0 as approximately zero rather than exactly zero.
Whether the gain rises or falls with b can be looked at in a small model before any data are touched. The model is a two-round beauty contest in the family of Morris and Shin (2002). N agents hold private signals x_i = θ + ε_i about a fundamental θ, with heterogeneous noise, so that some agents are skilled and some are not. In round one each agent acts on its signal. In round two each agent's action is a persistence-weighted mix of its round-one action and a fresh signal plus a sentiment term η, with persistence varying by agent. The outcome is Y = (1 minus r) θ + r ā_2, where ā_2 is the round-two average action and r is the weight of the aggregate in the outcome. Two records of round one are available: the public record, the round-one average, and the attributed record, every agent's round-one action under its identity. Both learners are ordinary least squares at matched observations, scored out of sample, and every gain is a share of outcome variance.
The sentiment term is the coordination channel, and it has three settings. With no channel, η is noise. In the aggregate setting, η follows the round-one average, so agents respond to each other only through the mean. In the cross-section setting, η follows the round-one positions of an identifiable tenth of the agents, so what coordination does next depends on who holds what. b is computed by ablation, as Section 2 sets out: the same draws are rerun with the channel closed, Y_0 is the outcome of that rerun, and b is Var(Y minus Y_0) over Var(Y). The gain splits the same way. Where A0 fails the one-order ablation share is reported beside b. The gain measured on Y_0 with the same draws is the aggregation gain, which exists because the simple average is an inefficient aggregate of heterogeneous signals and A1 fails by attribution; the remainder is the coordination gain. A last quantity, c, is measured without reference to the outcome at all: the incremental out-of-sample R² of the round-two average on the round-one cross-section beyond the round-one average, which asks directly whether positions beyond the mean predict the next coordinated move. Each cell is twenty seeds, each with its own draw of skills, persistences, and leaders, with 20,000 training and 5,000 test episodes per seed; the half-width shown is two standard errors.
| Channel | N | r | b | gain (2 s.e.) | aggregation gain | coordination gain | c |
|---|---|---|---|---|---|---|---|
| none | 50 | 0 | 0.000 | 0.045 (0.003) | 0.045 | 0.000 | 0.016 |
| none | 50 | 1 | 0.000 | 0.016 (0.001) | 0.016 | 0.000 | 0.016 |
| none | 500 | 0 | 0.000 | 0.005 (0.000) | 0.005 | 0.000 | 0.001 |
| none | 500 | 1 | 0.000 | 0.001 (0.000) | 0.001 | 0.000 | 0.001 |
| aggregate | 50 | 0.5 | 0.046 | 0.017 (0.001) | 0.017 | 0.000 | 0.007 |
| aggregate | 50 | 1 | 0.121 | 0.007 (0.000) | 0.007 | 0.000 | 0.007 |
| aggregate | 500 | 0.5 | 0.047 | 0.002 (0.000) | 0.002 | 0.000 | 0.000 |
| aggregate | 500 | 1 | 0.125 | 0.001 (0.000) | 0.001 | 0.000 | 0.000 |
| cross-section | 50 | 0.5 | 0.068 | 0.098 (0.007) | 0.025 | 0.072 | 0.236 |
| cross-section | 50 | 1 | 0.219 | 0.235 (0.013) | 0.013 | 0.222 | 0.236 |
| cross-section | 500 | 0.5 | 0.071 | 0.074 (0.001) | 0.003 | 0.071 | 0.231 |
| cross-section | 500 | 1 | 0.229 | 0.231 (0.003) | 0.001 | 0.230 | 0.231 |
All gains are shares of outcome variance, out of sample, means over twenty seeds. In the aggregate setting A0 fails, with the covariance term between Y_0 and Y_c running from 0.33 to 0.45 of Var(Y), so that the one-order ablation share is 0.38 to 0.58 against the b of 0.05 to 0.13 shown; in the other settings the term is within 0.01 of zero. The c of 0.016 in the no-channel rows comes from persistence heterogeneity, since agents who hold their positions carry round-one information into round two. The script, seeds, and complete raw output accompany this deposit.
The table's headline rows are built in, and should be read as such: the aggregate channel is a function of the round-one average, which the public record holds, so its coordination gain is zero by construction; the cross-section channel is a linear function of identified positions, so a linear learner given the positions recovers it; and the channel is standardized, so its strength is fixed whatever N is, and the depth result in the cross-section rows is an assumption of the setting rather than a finding. What the model is for is the accounting those constructions permit. First, with no channel, b is zero and the whole gain is aggregation gain, which falls roughly tenfold as depth rises from 50 to 500 agents: this is the aggregation gain and the depth effect that L1 states. Second, in the aggregate setting b is well above zero and the coordination gain is zero to four decimals at every depth and every r. The round-one average is a sufficient statistic for what coordination does next, the public record already carries it, and the attributed record adds nothing to it; the objection raised against Proposition 2 is realized, and the total gain falls with r because the outcome loads more on an aggregate the public record already holds. Third, in the cross-section setting the coordination gain is positive and rises with r, and at fixed channel strength it holds across depth, 0.222 at N = 50 and 0.230 at N = 500 at r = 1, while the aggregation gain beside it falls tenfold as before. The coordination gain equals b to within the precision of the decomposition, between 1.00 and 1.06 of it, the excess over 1 lying inside the two-standard-error band; in this linear model the record recovers the whole coordination component, which is the ideal case and not a general one. Fourth, c separates the settings without reference to the outcome: near zero where the channel is absent or runs through the mean, 0.23 where it runs through the cross-section. A mixture setting, with the channel a weighted mix of the aggregate and the cross-section, shows what c does and does not do. The weight on the aggregate rises from 0 to 1 at N = 500, at r = 0.25, 0.5, and 1, ten seeds each with 8,000 training and 2,000 test episodes. c, which does not depend on r, falls 0.231, 0.116, 0.045, 0.010, 0.000 with two standard errors of 0.010 or less. The coordination gain falls with it at every r, and the coordination gain divided by c is nearly constant within each r: 0.08 to 0.11 at r = 0.25, 0.31 to 0.34 at r = 0.5, 1.0 at r = 1. That constancy is close to algebra, since the outcome is linear in the round-two average and c is that average's incremental R², so the gain is approximately c times a function of r. What the mixture shows is that c is graded and that the gain tracks it; it does not show that c measures distance to a cap. The cap of Proposition 2 is defined under A0, and A0 holds in the mixture only at the cross-section endpoint, w = 0: wherever the aggregate component is present it covaries with Y_0, including at w = 1, where the Section 4 table already shows the failure, b is non-monotone in the weight (0.070, 0.040, 0.029, 0.032, 0.047 at r = 0.5) while the one-order ablation share rises with it (0.07 to 0.38), and the coordination gain as a share of b runs 1.02, 0.93, 0.52, 0.11, 0.00 while as a share of the ablation share it runs 1.01, 0.24, 0.06, 0.01, 0.00. Both denominators are reported; neither is a cap where A0 fails, which in this table is every row but the first at each r.
The model therefore fixes the condition the conjecture needs, and it is a condition about the world rather than about the record. The attributed record earns a coordination gain where what coordination does next depends on the cross-section of positions and not only on their aggregate; where the aggregate is a sufficient statistic for the next coordinated move, the public record already carries it and the gain is zero whatever b is. b caps the coordination gain where A0 holds. c is the quantity the gain tracks, and it can be measured in any identified record without an outcome at all, which makes it the first test to run in any domain. The model is linear and the result is a demonstration and not a theorem: if an outcome contains a function of identified positions that the mean hides, a learner given the positions wins. What the model adds to that sentence is the ablation accounting, the separation of the two gains, and c as a graded, direct measurement, and it reframes the empirical tests below around one question, whether in a given domain the next move of coordination depends on who holds what.
The bound says the gain is capped by b, and the toy model says the coordination gain reaches the cap when the next coordinated move depends on the cross-section and is zero when it depends on the aggregate alone. Whether markets are of the first kind is the empirical content, and two reasons say why I expect it. A price carries the expected value of an outcome but not its conditional structure, which the attributed record restores (Beier 2026e). And a scored record compounds calibration across a population as scoring accumulates, which is the mechanism behind the scoring-depth term k: each scoring Tensor sharpens the skill estimate of the judge it scores, and the sharpened estimate reweights every later judgment that judge makes.
The Judgment Record Conjecture. In domains where models learn from records of human judgment and the coordination component depends on the cross-section of judgments, the attribution gain from Tensor records over the public record, at matched observations and as a share of outcome variance, behaves as
\[\frac{\Delta_D}{\operatorname{Var}(Y)} = g(b_D, c_D, k, d), \qquad g(0,c,k,d) \to 0 \ \text{as} \ d \to \infty, \qquad g(b,0,k,d) \to 0 \ \text{as} \ d \to \infty, \qquad \frac{\partial g}{\partial b} > 0 \ \text{where} \ c > 0, \qquad \frac{\partial g}{\partial k} \ge 0 \ \text{with diminishing returns},\]
where d is the depth of the aggregate and c_D is the cross-sectional share of Section 4, the incremental out-of-sample explanatory power of the current cross-section for the next aggregate beyond the current aggregate. Four predictions follow, labeled for the test plan.
(L0) Cross-section. In the domains where the conjecture is claimed, positions beyond the current aggregate predict the next aggregate after the timing channel is removed: c, measured with positions entered at the reference lag of Section 7 and in excess of the level a b = 0 control market shows, exceeds a threshold c_min fixed in the registration as the minimum detectable effect. Unlagged, every market passes, because informed traders' positions predict the next price for the reason Kyle gives, and that is the timing channel and not coordination; in the toy model c is clean only because fresh signals are independent. L0 is measured without an outcome, and a domain with lagged c below c_min is one where the conjecture predicts no coordination gain whatever its b, and where the remaining predictions are not tested.
(L1) Null in deep markets. For observation-independent outcomes, the attribution gain falls toward zero as the aggregate's depth rises, and is indistinguishable from zero in deep markets. Where the aggregate is thin the gain is the aggregation gain, and L1 predicts that it does not survive depth; for coordination-dependent outcomes the gain is predicted to survive depth. The interaction of b with depth is the prediction, and sports betting markets, where b is zero by structure and prices exist at every depth, are its natural test. That test has a known adversary. Sportsbooks limit and profile winning customers and move lines on their bets, which is commercial evidence that bettor identity predicts outcomes. The industry's own measure of a sharp bettor is closing line value, the edge of a bet's price over the closing price, which is a timing gain and is removed by the lag control. L1 predicts that a persistent edge against the closing line of a deep exchange is zero; a persistent edge after the close would falsify it, with one named exception, since a fixed match is coordination around the outcome and carries b above zero.
(L2) Scaling. Across domains with lagged c above c_min, ordered by b estimated in the same window and at matched c, the normalized gain increases. The normalized gain is the gain as a share of outcome variance, the scale of Proposition 2's cap, fixed in advance; a share of baseline loss would not do, since in the toy model the aggregation gain holds its share of baseline loss as depth rises while its share of outcome variance falls tenfold. Skill dispersion among participants is the likeliest driver of any identity gain and is carried as a covariate, so that L2 is an ordering on b at matched dispersion.
(L3) Scoring dose. Within a domain, the gain grows as linked scoring Tensors accumulate, and flattens.
A fourth prediction tests whether one substrate can serve markets, politics, and technology together, which is the strongest claim the Protocol makes and the premise of the judgment kernel (Beier 2026d).
If L4 fails, the shared-substrate claim remains an interpretation rather than a result.
The first conjecture treats prices as an input to models. Increasingly, models are also an input to prices. Pricing algorithms set retail prices, bid in auctions, and trade, and Calvano, Calzolari, Denicolò and Pastorello (2020) found that reinforcement-learning pricing agents learn supra-competitive, collusive strategies without instruction. Regulators have warned that wide adoption of similar models may synchronize behavior and add systemic fragility (Gensler and Bailey 2020). Separately, models trained recursively on generated data lose the tails of the original distribution and degrade (Shumailov et al. 2024). The analogy to prices is partial, because an algorithmic price still loads on fundamentals where a recursively generated sample need not, and the loop stated below is correspondingly weaker than full recursion: a thinning of the human signal in proportion to α, not its replacement.
Put these together and a loop appears that, to my knowledge, has not been stated in this form, though three literatures each hold a piece of it. The evidence on the first clause is divided and both sides belong here. Hendershott, Jones and Menkveld (2011) found that algorithmic trading improves liquidity and makes quotes more informative, and Brogaard, Hendershott and Riordan (2014) found that high-frequency traders' liquidity-taking orders move with permanent price changes and against pricing errors, so that algorithmic participation improves price discovery at short horizons. Weller (2018), measuring informativeness about future earnings rather than short-horizon efficiency, found across 54,879 stock-quarters that higher algorithmic trading activity is associated with less information in prices, on the order of nine to thirteen percent per standard deviation. The two findings are not in conflict: algorithms absorb existing information faster and may reduce the acquisition of new information. The coupling conjecture is about the second effect, and it is the second that bears on training data. On stability, Daníelsson, Macrae and Uthemann (2022) argue that homogeneous AI models in financial markets generate endogenous risk, the mechanism Daníelsson and Shin (2003) had earlier located in uniform human risk models. And the recommender-systems literature has the same feedback loop in another market: Chaney, Stewart and Engelhardt (2018) show by simulation that a recommender trained on behavior its own recommendations shaped homogenizes users without raising their utility. The broader work on price informativeness under abundant data treats data technology and informativeness as jointly determined (Farboodi and Veldkamp 2020; Dugast and Foucault 2018). None of these treats the price as training data for the models that set it, which is the loop stated here. Let α be the machine share of price formation in a market. A model trained on that market's blind prices learns, in proportion to α, from prices that models produced. My deposit on the machine-origination boundary (Beier 2026c) shows that a model fed only a record and its own output carries no new information about how open questions resolve. Through the price channel, the same mechanism operates without anyone intending it.
Price-AI Coupling. In a market where models participate in price formation, the informational health of prices and the decision quality of models trained on them are jointly determined. Under blind prices both degrade as α rises. Under attributed records, the human-originated subset of entries is identifiable by its agent facet, and the information it carries about outcomes should not degrade with α.
A human judgment struck in a market where machines set most prices is itself conditioned on machine-set prices, so the human subset is cleaner than the blind series but not independent of the loop. What loosens the dependence is the Input facet: a Tensor records what the judge looked at, so entries can be ordered by how much of their declared input was primary observation and how much was prevailing prices, screens, and model output. Human and machine origination is a gradient rather than a partition, since a human who trades off an algorithmic screen has machine logic in the loop, and the test plan treats declared machine dependence as a continuous covariate rather than a label. Two things keep the facet honest. Scoring is done by others, so an entry whose declared inputs could not have produced its content accumulates a record that does not hold up under conditioning on those inputs; and the differential filtration audit in Beier (2026c), in the form used here, is a test for undeclared dependence rather than for excess insight. Judgment that predicts outcomes beyond its declared inputs is what skill is, and an audit that dropped it would strip the low-dependence subset of its best judges. The audit asks the other question: whether an entry's content is predictable from machine-generated signals, prevailing prices, screens, and model outputs, that the entry did not declare, estimated in held-out data. An entry whose content tracks undeclared machine signals has an incomplete declaration, whether from a hidden screen or from carelessness, and is dropped; an entry that beats its declared inputs without tracking anything undeclared is kept, and is the skill the subset is for. The audit can err in both directions. A weak auditing model misses excess content, and in finite samples an overfitted one manufactures it, with the false-positive rate rising with the number of entries tested, so the audit is cross-fitted, run on held-out data, and corrected for multiple testing, and its false-positive rate is estimated on entries whose inputs are known to be complete. The conjecture's second clause should be read for entries with low declared dependence, net of that audit, and the test plan applies both.
The prediction is concrete. Holding the market fixed, the out-of-sample value of the blind price series as training data for outcome prediction should fall as the algorithmic share of order flow rises, while the value of the identified low-dependence subset of an attributed record should not. Two things keep that falsifiable. α moves with liquidity, regulation, and time, so its effect is identified from staggered and plausibly exogenous changes in algorithmic participation, the design Hendershott, Jones and Menkveld (2011) used with the NYSE's 2003 autoquote rollout, rather than from a trend. And gaming is not an escape hatch: it is diagnosed by the audit in Section 7 before the gain is examined, and if the low-dependence subset degrades with α while the audit finds no excess content, the second clause has failed.
If the coupling conjecture holds, it has a consequence for AI safety that is worth stating as a consequence and not as a result: as α grows, a price system that cannot say who struck a price stops being a reliable teacher for the machines that increasingly strike it. The health of the price system and the health of machine judgment trained on it become one engineering problem.
All tests share a standard. Hypotheses, metrics, and analysis code are registered on the Open Science Framework before data are examined, and the registration link is added to this deposit's next version; until then "preregistered" names the commitment, not a completed act. Test 3 has no named data source yet and is registered when one is secured. Records are compared at matched observations, with the byte-matched comparison reported beside. Evaluation uses a held-out future period with no look-ahead. b is estimated before the gain is measured, from point-in-time rather than time-averaged series, because averaging manufactures autocorrelation (Working 1960), a point Merlet (2026) makes the anchor case of his monograph, where a naive variance-ratio rejection of the random walk in twelve of twelve cases falls to one of twelve against a null that reproduces the monthly averaging. Code, seeds, and outputs are published. Until the registration is filed this section is a protocol, and it is written to be filed as one.
Test 1, agent populations. Simulate a population of agents with heterogeneous private signals and a tunable coordination channel, the design of Section 4 extended to strategic agents. Varying the channel sweeps b from zero to near one. Each run is recorded twice, as Tensors and as prices, and identical models are trained on each. Measure the gain against b (L1, L2), against scoring depth (L3), and under strategic agents who misreport (robustness). Test 1 cannot test A1. The simulator builds the information structure, so A1 holds or fails there by construction; the simulation calibrates effect sizes for L1 through L3 and only Tests 2 and 3 can test the assumption against the world. For the coupling conjecture, add model agents that set prices, sweep α, and measure the training value of each record, with declared machine dependence from the Input facet carried as a covariate and the misreporting agents used to test whether the audit detects a gamed facet. A simulation shows that the conjecture is coherent and indicates effect sizes; it cannot show that the world behaves this way.
Test 2, forecasting and prediction-market records. Forecasting tournaments, prediction markets, and sports betting exchanges that publish bet-level data with persistent user identifiers are proto-Tensor records: each forecast has an author, a time, and a later resolution. Sports supplies the low-b end with prices at every depth, which L1 needs, and depth varies within it, which identifies the depth effect at b = 0; geopolitical and economic questions supply the high end, and there depth is thin, so b and depth are collinear across Test 2 as a whole. The b by depth interaction is identified in Test 1, where both are set, and in sports, where depth varies at fixed b; across domains L2 is tested only at matched depth, and the deep high-b case, which Test 2 cannot reach, is what Test 3 is for; b for each category is assigned as Section 2 sets out, zero by structure for sports and ordinal by structure elsewhere, with point estimates where exogenous variation in the aggregate exists; depth is measured by volume and spread. The first measurement in each market is c, L0: whether individual positions beyond the current price predict the next price move, out of sample, with positions entered at the reference lag so that the timing channel is removed, which needs no resolution data and is run before any gain is estimated. c_min is not a round number. It is the minimum detectable incremental R² at the registered power for the sample in hand, computed before the data are examined, so that a registered null means the market was tested and not that the bar was set where no market could reach it. The lag removes fast absorption and not slow diffusion, and positions can predict later prices through informed trading that takes days to clear, so lagged c is not pure coordination. Sports, where b is zero by structure, is the negative control: its lagged c is information only, and the quantity of interest in every other domain is the excess of lagged c over the sports level at matched depth. It should be said plainly that real markets may clear none of this. Lagged, excess c may be small everywhere, and the registered result would then be that the conjecture is empty in the data available, which is a result and is reported as one. Then train one model on the market price or consensus series and another on the individual-level record with each forecaster's scoring history, over the same questions and forecasts, and compare on held-out questions. Prior work finds that past accuracy predicts future accuracy strongly within geopolitical forecasting: an item-response model of latent forecasting skill, adjusted for question difficulty and time to resolution, explained 57 percent of the variance in later accuracy once substantial history was available, against 41 percent for a hierarchical model and 34 percent for simple averaging (Himmelstein, Atanasov and Budescu 2021). Cross-domain transfer was not tested there. Ordering question categories by estimated b gives L2; splitting categories gives L4. One confound belongs in the specification: who chooses to record a forecast is itself informative, since confident and informed forecasters self-select onto questions, and that violates A1 through participation rather than through coordination. Every Test 2 model carries a participation covariate (the forecaster's rate and timing of entry across questions) so that the attribution gain is measured net of selection.
Test 3, market microstructure. Order-level data with participant identifiers, compared against the public record, is the strongest test of both conjectures and the hardest to obtain, since audit-trail data are restricted. Kyle (1985) is the theoretical anchor for why trader identity carries information. The algorithmic share of order flow supplies α for the coupling test, identified as in Section 6.
Ablations and controls. The identity facet and the timestamp facets are removed in separate ablations, never together. A lagged-Tensor control is run alongside. Sampling the blind record at the Tensor strike times would not do, because the price at a strike time still lacks the fact the entry carries, which is the whole timing effect. The control therefore runs the other way: each Tensor entry enters the training set only once the price has absorbed its information, at a lag applied from the entry's transmit time. No market has a fixed absorption horizon, since liquidity, volatility, and the kind of information stretch and compress it, so the lag is not a constant.
It is swept, from zero to well past any plausible horizon, with the horizon's scale estimated independently for each market and regime from the impulse response of the price to order flow (Hasbrouck 1991). The gain of the lagged Tensor record over B is then reported as a function of the lag, over a lag grid fixed in advance. Reading a plateau off a noisy curve by eye would hand the analyst a free parameter, so the attribution gain is defined before any gain is computed: it is the gain at a reference lag equal to three times the estimated time for the price's cumulative impulse response to reach ninety percent of its long-run value, with the multiplier, the threshold, and the estimation window all fixed in the registration and the full curve published beside the point estimate. A secondary estimate fits a decay form fixed in the registration, a constant plus an exponential in the lag, and reports the constant with its confidence interval; the two estimates should agree, and where they do not the reference-lag estimate governs.
If the gain at the reference lag is indistinguishable from zero, the whole gain was timing. If it is positive, that is the attribution gain, and the difference between the unlagged gain and it is the timing gain.
Only the attribution gain counts toward the conjecture, and the separation should be read for what it is. In price formation, attribution and timing are one mechanism: a market maker who can see who is trading absorbs information faster (Kyle 1985), and a market that ran on Tensors would clear faster for exactly that reason. The lag does not treat that gain as noise. It sets it aside as a claim about markets rather than about training data, measurable on its own as the timing gain, so that what the conjecture claims, that identity and scoring history improve a model's judgment of outcomes after the price has caught up, is tested on its own. If the timing gain is large and the attribution gain is zero, the conjecture as stated fails and the Tensor's value lies in price formation; the paper would say so.
Stated prediction at low b. In the primary comparison the attribution gain where b is zero is expected to be positive in thin markets, as the aggregation gain, and to fall to indistinguishable from zero as depth rises. In the secondary comparison at matched bytes it is expected to be negative in deep markets, because the attributed record holds fewer observations, and a negative result there is read as the cost of the facets rather than as evidence against the conjecture.
Falsification. The Judgment Record Conjecture fails, in domains with lagged c above c_min, if the attribution gain is flat or falling in b at matched depth and matched c, if it is materially positive where b is zero in deep markets after the lagged-Tensor control (which would mean A1 fails by attribution where coordination is absent, and the gain has another source), or if removing the identity facet alone at matched observations leaves the attribution gain unchanged. A domain with lagged c below c_min neither confirms nor refutes the conjecture; if no market with identified positions has lagged c above c_min, the conjecture is empty, and that result is reported as such. Price-AI Coupling fails if the training value of blind prices does not fall with identified changes in α, or if the low-dependence subset degrades with α while the audit finds no excess content.
The coordination model behind Section 4 is the beauty contest of Morris and Shin (2002), with the asset-market form in Allen, Morris and Shin (2006) and the welfare treatment of public information in Angeletos and Pavan (2007); what those models fix is how much an outcome loads on the average action, and what they leave open, and the toy model switches on, is whether the next average depends on who holds what. Woodford (2003) is the macroeconomic case nearest to the question: with imperfect common knowledge, aggregate outcomes are driven by higher-order beliefs about what others believe. In the standard linear Gaussian versions of that model and its relatives, though, the hierarchy of average higher-order expectations is a function of the signals and the average is close to sufficient, which places them near the aggregate setting of Section 4, where c is zero. A c above zero needs heterogeneity in who holds what, which those models mostly set aside, and that is the claim this paper makes and they do not. The same literature carries the sharpest objection the Protocol must answer, that more public information can lower welfare when agents overweight a signal because everyone sees it (Morris and Shin 2002; Angeletos and Pavan 2007); an attributed cross-section is not one public signal, and whether it escapes that result is an open question the deposits on the Protocol take up. Prediction markets as aggregators are surveyed in Wolfers and Zitzewitz (2004) and Arrow et al. (2008). The closest empirical analogue to a scored identity is the fund-manager literature on skill against luck, where Kosowski, Timmermann, Wermers and White (2006) find a skilled minority by bootstrap and Fama and French (2010) find the cross-section of net returns close to what luck alone produces; the Tensor's scoring history is that test run on every participant rather than on funds. The closest machine-learning precedent is the modeling of who produced a label. Dawid and Skene (1979) estimated per-annotator error rates jointly with true labels, and the "learning from crowds" approach of Raykar and coauthors (2010) folded annotator reliability into classifier training. Recent perspectivist work in natural-language processing finds that annotator identity matters in socially grounded tasks such as toxicity and less consistently elsewhere, with mixed evidence overall (Xu and Jurgens 2026). That pattern is a qualitative cousin of L1 and L2. What is missing from it, and supplied here, is a graded variable that predicts where identity should matter and a bound that sets the gain to zero at one end.
Performative prediction (Perdomo, Zrnic, Mendler-Dünner and Hardt 2020; Hardt and Mendler-Dünner 2025) studies models whose predictions shift the distribution they are evaluated on, quantified by a sensitivity parameter on the distribution map. b differs in object and in use. Performativity measures how much a deployed model moves its data; b measures how much of an outcome's variance arises from coordination among the parties around it, whether or not any model is deployed. Price-AI Coupling sits where the two meet, since a rising α makes prices performative with respect to the models that learn from them.
Forecast aggregation research established that weighting by track record improves on the simple consensus (Budescu and Chen 2015), and Himmelstein and colleagues quantify the stability of forecaster skill. A fair reading of the test plan is that its instruments are aggregation tests, since each asks whether skill-weighted individual data beats the consensus. What the conjecture adds is b as the variable that predicts where that gain is an aggregation gain, which depth removes, and where it is a coordination gain, which depth does not; and the trained model is not confined to aggregating. A weighting scheme carries skill to later questions in the same domain, which is what contribution weighting does; what it does not do is carry it across domains, and that is L4. The Tensor generalizes the scoring history from tournaments to every price. In finance, the information content of order flow is established theory (Kyle 1985) and an active empirical field; that literature uses order flow to predict prices, not identified and scored judgment as training data for general decision models. On the reflexive side, Merlet's (2026) 77-page monograph is the nearest recent treatment: it reconstructs the lineage of finance's quantitative instruments from calendars and life tables to the random walk, argues that their migration into reflexive markets opened a gap institutions must manage, and proposes a five-dimension provenance audit applied to machine learning in finance. Its audit concerns where an instrument came from, whereas the provenance carried by a Tensor concerns who made each judgment and when; the two are complementary, and b supplies the graded variable his qualitative boundary lacks.
Both conjectures concern decision quality measured against outcomes. General safety is a further argument. They are also independent of any forecast about the pace of AI capability. If capability rises, models need better ground truth about human judgment; if it stalls, they need better data. The conjectures hold or fail on the tests above in either case. Wider claims rest on these results without being part of them: claims about the Protocol as a whole, about a Free Market Price Economy, by which I mean an economy in which prices are struck as Tensors rather than as scalars, and about the governance of technology. I develop those in separate deposits.