Peter Cotton

The operation of a novel open-source platform where mostly autonomous algorithms are tasked with predicting a large variety of streaming data is described. Reward and combination is achieved by means of a near-the-pin mechanism generalizing lottery countbacks to continuous space.

We give analytical conditions under which the Schur portfolio with the best out-of-sample variance lies inside the bridge, our hierarchical-to-minimum-variance family, with provable examples on both sides. Cross-block estimation error separates the optimum from both ends under stated endpoint conditions, and an exchangeable four-asset family is solved exactly, its optimizer decreasing from full coupling to the middle. The mechanism persists under whole-matrix noise, with a sharp second-order sign criterion at full coupling, and the median-optimal coupling climbs with sample size. Every portfolio is the minimum-variance portfolio of some positive definite matrix, so the dial moves implied beliefs.

Hierarchical equal risk contribution cuts a dendrogram into clusters, holds the inverse-variance portfolio inside each cluster, and budgets across clusters by inverse cluster variance. We show that its budget rule is already the one the global minimum-variance portfolio uses, and that what it lacks is conditioning in two places: each cluster on the other clusters, and each asset on its cluster mates. Damping the two Schur complements by γ and η gives a square with the method at one corner and the global optimum at the opposite corner. The inner dial has a closed form that interpolates the naive inverse-variance quantities and Stevens’ regression quantities, keeps every weight positive up to an explicit frontier, and on an equicorrelated cluster traces the straight segment from inverse variance to the cluster’s minimum-variance portfolio. Under estimation error the best point can be strictly inside the square, and we give the two-dial local theorem at the minimum-variance corner, a sign criterion for the inner dial with both signs realized, and exact examples, including one whose optimum is η = 3/5 with gaps 1/45 and 1/20 to the two ends.

Thurstone portfolio polishing calibrates latent abilities that reproduce a benchmark under a reference dependence law and re-races them under a richer one, with a scalar ϕ dialling the estimated correlation in. Choosing the reference to be the block-diagonal part of the estimate makes that blend exactly the estimate with its cross-cluster entries scaled by ϕ, and for a two-block partition the conditional covariance of that matrix is the damped Schur complement at γ = ϕ2 . The tilt dial is therefore the Schur dial under a square root. What is new is the pair of endpoints. Every bridge catalogued so far ends at Σ−1 u, a linear solve that can short any asset without bound; this one ends at a race, which lies on the simplex, splits a duplicated position rather than doubling it, and stays bounded. The taper is positive semidefinite for every ϕ, so unlike an unmatched pricing space the√path has no pole, and the closed-form reliability of the cross-block estimate transfers as ϕ⋆ = γ ⋆ , which sets the dial from data rather than by tuning.

A two-tier allocation holds a portfolio inside each cluster and then decides how much capital each cluster gets. The second decision needs a k × k matrix, and every construction in this family so far builds that matrix from the same cluster portfolios it holds. It need not. Separating what is held from what it is priced against gives a family with two spaces rather than one, and the far end of the bridge is exact for every choice of the pricing space: the k conditioned directions already sum to Σ−1 u, so any non-degenerate pairing recovers it. Choosing one representative asset per cluster, as hierarchical core-orbital allocation does, therefore cannot affect the far end. The interior is a different matter. Once the two spaces differ the k × k matrix is no longer symmetric, and it can pass through singularity inside the dial: we give a positive-definite 6 × 6 example where it does, and on estimated covariances it does so in about a quarter of markets. Pricing against the cluster portfolios keeps the matrix a Gram matrix of Σ and removes the possibility.

We introduce the Schur pseudo-likelihood, a one-parameter family that damps the cross-block coupling of the Gaussian likelihood through its Schur complements, for scoring and regularizing covariance and correlation estimates in high dimensions. The Gaussian log-likelihood is the standard criterion, but when the dimension rivals the sample size it is governed by the smallest, least-identifiable eigenvalues of the estimate, and as a criterion for ranking or selecting estimates it becomes unreliable; damping the coupling restores it. The optimal damping has a closed form—the reliability of the coupling, a James–Stein shrinkage. On real crypto returns, choosing a shrinkage estimate by the Schur pseudo-likelihood rather than by the full likelihood yields several-fold lower out-of-sample portfolio variance once the dimension rivals the sample size.

When the dimension $p$ is large—thousands of weather stations, or the width of a neural network layer—the Gaussian likelihood cannot be formed or inverted, so both estimation and evaluation must exploit known structure. We make three points and one new one. (i) A structure-aware estimator beats global shrinkage out-of-sample when undersampled: on a real $323$-point weather forecast-error grid, spatial covariance tapering beats Ledoit–Wolf by several nats per observation at $n/p<1$, converging as $n$ grows. (ii) The block-conditional (Vecchia) likelihood, the only one computable at scale, is a faithful stand-in for the full likelihood—it selects the same structured estimate $\approx100\%$ of the time where both can be computed—at $O(p\,m^2)$ instead of $O(p^3)$. (iii) Crucially, Schur damping adds to Vecchia: plain Vecchia trusts its neighbour regressions fully ($=1$), but those are estimated from limited data and overfit; damping them by the coupling reliability $$ improves out-of-sample likelihood substantially when undersampled (by tens of nats at $n/p=0.4$, where plain Vecchia is in fact the worst setting), with $$ rising to $1$ as $n$ grows. The optimal damping is a closed-form James–Stein reliability, the same quantity that governs block shrinkage and Schur-complementary allocation.

Traditional covariance shrinkage estimates pull an empirical covariance matrix Σ towards a lower information anchor (often λI), reducing estimation error at the cost of suppressing correlation. This note proposes the opposite: geodesic covariance inflation—a controlled movement of Σ towards a perfectly-correlated limit. Intended not as a consistent estimator but merely a device for use in the context of portfolio construction, we propose that instead of discounting correlation, we exaggerate it in a geometrically natural way, yielding a family of positive–definite matrices indexed by an inflation parameter γ ∈ [0, 1].

We develop Thurstone portfolio polishing: a way to take an existing allocation on the simplex—a capitalization index, an equal-weight book, any benchmark—and re-tilt it under a view on dependence rather than returns. Reading the weights as the winning probabilities of a race among assets (a Thurstonian model of choice), we back out the latent abilities that reproduce them with a fast exact inverse, then re-evaluate the race under a richer dependence law. The marquee instance is a tail-sensitive analogue of Black–Litterman: where Black–Litterman tilts a benchmark by a view on expected returns through a linear inverse of a covariance, we tilt by a view on lower-tail co-movement through a fast exact nonlinear inverse of a correlated race, inverting no covariance and solving no optimization. Driven by a tail-dependent simulation the polish shades down exactly the names that crash together beyond what their correlation can encode—a crash-proofing overlay we verify both in a known-truth lab and on the Dow. The construction's organizing property is redundancy (clone) consistency: a near-duplicate asset splits a position rather than doubling it—the cure for the red-bus/blue-bus failure of any Luce-type rule, and the same near-singularity that simultaneously makes the polish robust to ill-conditioning (it samples from a correlation rather than inverting one), low in turnover (its weights are locally Lipschitz in the correlation), and a smooth de-duplicator of correlated holdings. The polish is feasible on the simplex without optimization, reproduces its benchmark by construction, admits an arbitrary performance simulation so dependence beyond second moments can be matched, and solves an implied mean–(convex-regularizer) program whose penalty the simulation fixes; capitalization weighting is its degenerate, correlation-blind member. We prove the supporting properties (feasibility, benchmark reproduction, redundancy and tail consistency, the implied objective, and a Lipschitz turnover bound) and implement the method in the allocation package, built on thurstone.

A portfolio rule is a covariance estimate feeding an allocator. We ask a plain question: which rules actually work out of sample—after trading costs, and as the number of assets grows? We backtest eleven online constructions over thousands of random subsets of U.S.\ stocks and draw a map of the winner against two axes, the number of names and the cost of trading. The map has three regions. With few names and cheap trading, classic minimum-variance wins. With many names and cheap trading, a factor-model minimum-variance wins—and by a margin that grows with the number of names—because the ordinary version becomes undefined once there are more assets than data. Once trading costs rise above about five to ten basis points, simple inverse-variance weighting wins everywhere, because turnover, not estimation error, is what hurts. We then give the two ingredients that make the many-names corner work: a Woodbury inversion of a factor covariance, and a ridge-regularized version of the Schur coupling.

Let G(K) = supm (m − K)F̄ (m) be the optimized value of an enquiry with breakeven strike K, against a best competing quote with survival curve F̄ . For every positive, strictly decreasing enquiry-value curve, the coordinate Φ = − log(G/G⋆ ) converts buy– sell imbalance into an exact translation of two log-value coordinates. The translation is induced by state-independent shifts of the break-even strikes exactly for G = κ + Be−hK on the active strike interval; state-independent shifts of the submitted quotes, or global rationalizability by a finite-mean markup distribution, force κ = 0. For general G the canonical sidewise gauge need not satisfy the edge compatibility that lets both strike ladders derive from one inventory potential; we quantify the defect. On a finite inventory lattice, under nonsingularity of the anchor linearization, any smooth deformation path yields a perturbation hierarchy whose every coefficient uses one linear operator, with distributional perturbations entering through an envelope formula and local rationalizability conditions. Reflection symmetry makes the skew odd and the convexity response even in the log-odds of the flow. All formulas and asymptotic orders are checked numerically.

Intuitively, accuracy in microstructure prediction at a granular level must relate to profitability for a market participant, but this is not trivial to formalize. Here we provide an approximation using a stylized steadystate model for over-the-counter trading modeled as a sequence of sealed bid auctions. A simple picture emerges due to a mildly surprising feature of this model: one does not need to know the fair price, only where others are bidding.

We provide analytic results for the optimal control problem faced by a market maker who can only obtain and dispose of inventory via a sequence of sealed-bid auctions. Under the assumption that the best competing response is exponentially distributed around a commonly discerned fair market price we examine properties of the market maker’s optimal behavior. We show that simple adjustments to skew and width accommodate customer arrival imbalance. We derive a straightforward relationship between the market marker’s fill probability and direct holding costs. A simple formula for optimal bidding in terms of (non-myopic) inventory cost is presented. We present the results as a perturbation of an improvement to a “linear skew, constant width” (CWLS) market making heuristic.

A k-subset of items will be chosen from n according to values taken by n auxiliary variables X1 , . . . , Xn interpreted as performances in a contest. Item i is chosen if Xi ≤ X (k) where X (k) is the k’th order statistic. A numerical algorithm is presented for computing many k-combination choice probabilities quickly, for small k but potentially large n ≫ 1, 000, 000. No assumption is made on the 1-margin distributions of the Xi , and the analytical convenience survives the introduction of dependence via a factor model also. The computation of rank probabilities for k items is a corollary. The algorithm is provided in the winning package, on PyPI.

A subset of k items will be chosen from n according to values taken by n variables X1 , . . . , Xn interpreted as performances in a contest. Item i is chosen if Xi ≤ X (k) where X (k) is the k’th order statistic. A numerical algorithm is presented for computing many k-combination choice probabilities quickly, for small k but large n ≫ k in the millions. Rank probabilities for k can also be computed. Some users of the winning package may wish to calibrate models for latent Xi from partial information such as winning or losing probabilities. Others may prefer to supply arbitrary performance distributions. The analytical convenience of this method also survives the introduction of dependence in the Xi via a low-dimensional Copula.

Using prompts such as: “my favorite state in the US is [MASK]”, and “my favorite Western state in the U.S. is [MASK]” we infer that Thurston models are a better match to the revealed preferences of large language models than the application of Luce’s Choice Axiom. There is some irony in this finding given that Softmax functions, responsible for the token probabilities we interpret as preference, suggest Independence of Irrelevant Alternatives.

Rating systems almost universally assume that contest performance is ability plus normal or logistic noise. We present a setting where the assumption is visibly false and measurably costly: Formula 1, where for decades the most likely outcome of starting a race was not finishing it. A lattice-based Thurstonian rater that accepts arbitrary noise densities outperforms TrueSkill herbrich2007, Elo elo1978, Glicko-2 glickman2012 and OpenSkill weng2011 over 1,158 races, and replacing its normal noise with a two-component density, a Gaussian pace term plus a separated block of slow mass representing retirement, improves every metric. Estimating the block's mass from the trailing retirement rate improves it further. Symmetric heavy tails and skew do not help, so the gain is attributable to the block, not to non-normality in general.

The probability that a competitor wins a noisy race—the Thurstonian choice probability, equivalently the gradient of an expected-maximum potential—is a rule for attributing credit among correlated contributors. It is a probability share ($\sum_i w_i = 1$), symmetric, and redundancy-aware (contributors that move together share a single contributor's credit rather than double-counting); it is differentiable in the contributors' abilities, it updates online, and it costs $O(Mn)$ in a single Monte-Carlo pass with no coalition enumeration. It is not a Shapley value and does not try to be: Shapley measures coalitional cooperation—a contributor's average marginal value as coalitions form—whereas the winning probability measures selection relevance, how often a contributor is the single best in the field at hand. The two answer different questions, and a contributor that is never individually best yet improves every blend it joins earns Shapley credit but little race credit; the race does not capture such coalitional value. (There is a clean mathematical link—the winning-probability field integrates along the diagonal to an Aumann–Shapley value of an ability-scaling game—which we record but do not lean on.) We show that the headline redundancy property—an even split of credit among near-duplicates—is a property of the calibrated equal-ability race rather than of raw scores, and we demonstrate the rule on forecast combination and feature attribution. This develops the credit reading of the construction in the companion paper on Thurstone portfolios.

Consider N channels firing at random times, with rates driven by a hidden state they all share. Conditionally on that state each channel is Poisson, and an observer records which channels fire and in what order. When the driver mixes fast relative to the firing rates, the probability that channel i fires first approaches the law of Luce’s Choice Axiom in the stationary mean rates. We give the first correction in the timescale ratio, governed by the time-integrated covariance K of the rates, and every claim below holds at that order. Recording the first two arrivals recovers all N 2 −2N observable combinations of K, matching what blocking channels and rerunning the process would give, while the winner alone recovers none of them. The diagonal of K is unobservable, absorbed into the rates, so only correlated modulation leaves a trace. And K is symmetric when the driver satisfies detailed balance, so its observable asymmetry certifies a driver held out of equilibrium, visible with three channels and never with two.

Many systems reveal themselves only through races: N latent processes compete, and an observer records which one finished first. The natural counterfactual question — what happens if competitor i is removed? — turns out to be exactly nonidentified from winner frequencies alone: the redistribution of the removed competitor’s probability is governed by a runner-up kernel M on which winner data places no constraint whatsoever. We give the sharp partial-identification set, and a data hierarchy that resolves it: observing the runnerup identifies every single-deletion counterfactual, and top-(k+1) ranking prefixes identify every deletion of up to k competitors. A Markov substitution resolvent turns the runner-up kernel into predictions for arbitrary blocked sets. We validate the hierarchy on real firstpassage physics — Brownian narrow escape, where a truncation identity makes one reflecting trajectory answer every blocked-set counterfactual exactly — finding that winner-only models degrade with deletion depth while the runner-up kernel predicts at the sampling-noise floor, and that the substitution process is empirically Markov. Finally, we show the kernel is not merely estimable but computable: Green-function hitting splits of the domain predict M to within sampling noise with no trajectory data, and the geometric kernel outperforms the empirically measured one as an input to counterfactual prediction. Two practical warnings emerge: the stationary encounter chain differs materially from the first-transition kernel that counterfactuals require, and common time changes make proportional-hazards (softmax) redistribution exact — so deviations from it measure differential structure only.

The Python package humpday provides twenty-three derivative-free optimizers behind one uniform contract: minimise a black-box function on the unit cube under a hard evaluation budget. Every algorithm is implemented in pure Python with no required dependencies (a numpy backend is used transparently when present) and again in zero-dependency JavaScript, with parity tests holding the two in agreement, so the same optimizer runs on a server or in a browser. A recommender selects an algorithm from the problem’s dimension, budget, and measured evaluation cost, using rankings precomputed on the package’s own benchmark: eighty-one objectives ported from real applications, each paired with an interactive browser demonstration. Scoring applies seeded smooth bijections of the cube that relocate every optimum, so no method, human or machine, can score by memorising solutions; the suite remains valid when the candidates are written by language models. We describe the design and two findings the benchmark has produced: optimizer rankings on synthetic test functions correlate only moderately with rankings on the real-world suite (Kendall τ between 0.42 and 0.49 across budgets), and the suite supported the generation and out-of-sample validation of Alloy, a machine-designed optimizer that now ships in the package.

Practitioners choose a derivative-free optimizer either by consulting a benchmark leaderboard or by reasoning about the problem. We test both. Using a memorisation-proof suite of real-world objectives, each relocated by a seeded cube-to-cube diffeomorphism, we rank a panel of $22$ optimizers and ask how well that ranking is predicted by the same panel's ranking on synthetic analytic benchmark functions, and by a language model asked to reason, problem by problem, about which optimizer should win. Both predictors are essentially uninformative: the rank correlation with real-world performance is statistically indistinguishable from zero in either case. A language-model selector that picks an optimizer per problem in fact does worse than a constant policy of always using one robust direct-search method. The two failures share a mechanism. Both the benchmark and the model over-trust model-based trust-region methods (Powell's NEWUOA and BOBYQA), which dominate smooth analytic functions yet rank near the bottom on the real suite. This is a draft, and several confirmatory runs are still in progress (Section sec:caveats).

We discover rather than hand-design a derivative-free optimizer by evolving a parameterised DE/ES-hybrid template against a memorisation-proof suite of real-world objectives. A genetic algorithm searching $14$ behavioural knobs reaches a normalised regret of $0.110$ versus a Nelder–Mead/Differential-Evolution/CMA-ES panel (mean rank $1.47$ of $4$), against $0.191$ for the original $12$-knob template. The gain comes from one added mechanism — a cheap, R2-gated separable-quadratic trust-region jump — which we validate three ways: an A/B re-evolution, a single-gene toggle ablation (disabling it more than doubles regret), and independent re-discovery by the production search, which drives the jump's firing probability to $\!1$. The discovered optimizer is a hedged hybrid, found by grounding fitness in real (disguised) evaluation rather than a synthetic proxy.

Place K established algorithms at the vertices of a simplex. A point is a vector of mixing weights, and a language model turns the point into a working program: the heaviest vertex supplies the architecture and the others graft their ideas into it, in the stated proportions (Figure 2 shows the literal prompt). Scoring the program on a benchmark makes the simplex a continuous space of algorithm designs that ordinary derivative-free optimizers can search. The construction’s first product is Alloy, generated at the equal-weights point: on twentynine problems never used in any selection step it has the best mean rank at every budget from 60 to 480 evaluations and beats six competitors including CMA-ES pairwise (p < 10−10 ); it ships in the humpday package. Letting weights go negative extends the recipes beyond the simplex; of six semantics for a minus sign, tested under a paired-ablation protocol, one survives out of sample, with a rule attached: negative weights repair weak hosts and spoil strong ones. Failures are reported throughout: the continuous search never beat the obvious point, single generations vary severalfold, and in-sample scores mislead reliably. Two further worked examples test the construction’s reach: cache-eviction policies, where static blends fail and adaptive shares rediscover the design of CAR, and distributional forecasting, whose champion dominates synthetic suites, loses on real data, and ships anyway, recast as a transform inside the forecaster it could not beat.

It is shown that an algorithm given two chances to improve its position on the path of an exponentiated Ornstein-Uhlenbeck (OU) process should not choose its final position between the first two locations. It is sometimes easy, therefore, to diagnose failure of an algorithm to learn the optimal policy. The proof introduces the notion of rapidity on an OU bridge, and complements similar results in managerial science that are used as metaphors for complex but ill-defined business and strategy problems.

The posterior probability that alternative i is best, pi = Pr(θi = maxj θj | D), drives stopping rules, budget allocation, derandomized Thompson sampling, and safety constraints, and the reinforcement-learning and Bayesian optimization literatures describe it as intractable: recent work variationally approximates it from marginal means and variances alone. We observe that in two standard settings the posterior covariance is exactly of factor form Σ = W W ′ + diag(d) at every step: simulation replicates under common random numbers, and per-arm Gaussian observations against a factor prior. In that case the whole vector p is a one-dimensional integral per factor node, computable for all n alternatives at once in time linear in n by machinery previously developed for pricing correlated contests, and certified here against Monte Carlo. In a stopping problem under common random numbers, rules that stop on marginal-only estimates of p hold their nominal error by overspending, between 1.5 and 3.4 times the replicates of the exact rule, which stops earliest in every correlation regime at the nominal error level. In a correlated Gaussian bandit, sampling from the exact vector reproduces Thompson sampling’s regret to within replication noise, as the occupancy identity requires, while the variational policy pays ten percent more regret and sits at measured total-variation distance 0.13 to 0.16 from the object it stands in for.

In a budgeted Brownian race, a controller watches a cloud of Brownian paths, pays per unit time for every path kept alive, may irreversibly prune, and spends a finite path-time budget to maximize the expected terminal maximum. We solve the mean-field version: a Poissonized population under per-path controls, with the budget enforced in expectation. The expected maximum has an exact layer-cake form whose gradient is a pivotal value, and for a fixed price on path-time the linearized best response is an obstacle problem whose keep sets are upper tails: the wavefront policy, which retains a path while its propagated future pivotal value exceeds its carrying cost. A fully corrective Frank–Wolfe method over wavefront policies, with dual bisection on the price, solves the budget problem and certifies its answer, bounding suboptimality by the Frank–Wolfe gap plus a complementary slackness term. In the committed example the adaptive policy improves the expected maximum by 19% over the best single screening date and by 30% over static thinning at the same expected budget; the finite-n race with a pathwise budget is left to future work.

A prompt that fails a handful of scored attempts is rated impossible by the standard pass-at-k extrapolation 1 − (1 − p̂)k , and rated so at every k. On nine thousand prompts with eight scored samples each, that plug-in step — the same one the coverage diagnostic of Malladi et al. (2026) applies as f16 (p) = 1 − (1 − p)16 — costs seven-fold in held-out log loss and understates aggregate pass@8 by thirteen points. Replacing the plug-in evaluation with the posterior expectation E[1 − (1 − p)k | s] under a two-parameter difficulty prior, fitted once across prompts by marginal maximum likelihood, removes most of the damage: log loss 0.48 against 3.37, the extrapolated curve within two points of truth. Probit-normal and beta-binomial priors tie to three decimals, so nothing hangs on the family. The bias shrinks as the per-prompt sample count grows, and the data cover one model on one task suite.

Decentralized perpetuals protocols have collectively reached billions of dollars of daily trading volume, yet are still not serious competitors on the basis of trading volume with centralized venues such as Binance. One of the main reasons for this is the high cost of capital for market makers and sophisticated traders in decentralized settings. Recently, numerous decentralized finance protocols have been used to improve borrowing costs for perpetual futures traders. These protocols have grown to over $2.5 billion dollars of assets while generating over $890 million in fees in 2024. We formalize this class of mechanisms utilized by protocols such as Jupiter, Hyperliquid, and GMX, which we term Perpetual Demand Lending Pools (PDLPs). We then formalize a general target weight mechanism that generalizes what GMX and Jupiter are using in practice. We explicitly describe pool arbitrage and expected payoffs for arbitrageurs and liquidity providers within these mechanisms. Using this framework, we show that under general conditions, PDLPs are easy to delta hedge, partially explaining the proliferation of live hedged PDLP strategies. Our results suggest directions to improve capital efficiency in PDLPs via dynamic parametrization.

Scoring rules, cost-function market makers, constant-function market makers, parimutuel pools, and opinion pools are developed in separate literatures. This note is expository: it gathers the connections among them under one convex-analytic lens and one notational convention. A proper scoring rule is a convex potential read off the report; the Fenchel conjugate of that potential is a cost-function market maker, and a level-set dual is a constant-function market maker. The linear and logarithmic opinion pools are the two Kullback-Leibler barycenters. Merging market makers is infimal convolution, the composition law of the risk-sharing literature, so liquidity adds. A common transducer signature lets the mechanisms be chained, and this note settles when propriety survives a transformation of a stage's message or outcome. The mathematics is classical; the contribution is its consolidation. The multi-stage theory built on this dictionary, and its deployment on the microprediction platform, is a companion report [@cotton2026multistage]; two further results that motivated the framing are elsewhere too, sample-based elicitation on point clouds [@cotton2026pointcloud] and the parimutuel account of a conformal predictor's information gap [@cotton2026conformalbetting]. --

The microprediction platform built distributional prediction supply chains, chaining pools so that one's output became the message or the settled outcome of the next. This note is a self-critical, ex-post look at the game theory of those chains: the games were built first, and here we ask, of each, whether truthful reporting was really the best move. A chain is proper where an exogenous outcome anchors every stage; a downstream stage can then be scored on its own residual or, for the log score, at the top level. By that test the games mostly hold up, one of them only by a lucky accident. Flaws aside, the setup has continuing pedagogical merit, as it illustrates the inherent limitation of conformal prediction. --

The choice between the logarithmic score and the continuous ranked probability score (CRPS) is usually argued on four grounds: locality, propriety, robustness, and interpretability. This note adds a fifth, one the debate rarely weighs. When the elicited object is a reusable predictive density and forecasts are chained, each stage refining the residual left by the last, the logarithmic score is the natural default. The log-likelihood of a composed forecast is the base score plus the log-density of each residual refinement, so credit is additive and each stage's increment is itself the proper score of what that stage added. This is the prequential and logarithmic-market-scoring structure; CRPS shares no comparable per-stage identity. None of that is new mathematics: the decomposition is the probability integral transform and the prequential chain rule. The contribution is a framing. It makes composability the organizing axis of the log-versus-CRPS choice, which is conventionally settled on locality, propriety, robustness, and interpretability, and ties it to chained elicitation, the setting a prediction supply chain lives in. We give the compositional accounting, a population-level worked example in which the two scores prefer measurably different forecasts, and an honest account of where CRPS is the right target, its tail-insensitivity there often a virtue. Conformal prediction appears as the limiting case: a method that need not produce a density at all, scored on the one metric that does not notice. --

Cost-function market makers are always assumed convex. We show convexity is not what makes them coherent. For a path-independent maker, the exact no-arbitrage condition is a chord condition, that every chord slope of the cost lie in the convex hull of payoffs; convexity enters the standard axiomatics through a separate requirement, monotone information incorporation, and dropping it leaves coherent non-convex makers. Rational flow against such a maker sees the convex envelope: optimal fills land on the contact set, and the non-convexity gap at the starting state passes through to the next trader, so non-convexity costs expressiveness rather than soundness. Bounded incoherence is priced: a proportional fee of at least the arbitrage depth restores no-arbitrage, and by conjugation that fee is exactly a bid-ask spread, with the exact no-trade interval given by fee-widened one-sided chord bounds for any cost whatever. Combining fee-bearing makers is an infimal convolution solved by a single monotone clearing-price root-find; the fees act as an L1 penalty, giving each maker an exact no-trade band, and the aggregate supply curve is a consolidated limit order book. Adding a deep quadratic co-quoter is Moreau smoothing, which preserves coherence at every depth while attenuating but never closing the envelope gap. Finally, separation makes model inconsistency financially discoverable: any quoted configuration outside the coherent hull admits a portfolio priced below its worst payoff, so arbitrageurs act as decentralized separation oracles, and coherence is relative to friction and to the cost of finding certificates. --

This paper is partly synthetic. Many of the correspondences below are known individually; the purpose is to place them in one compositional calculus and to ask what a familiar statistical method is when read through it. The organizing observation is that many familiar statistical methods are market ecosystems with degrees of freedom clamped. Mechanisms compose in two ways, and both are established. In parallel, competing makers merge by infimal convolution, so conjugates add and liquidity adds. In series, a factorization assigns one venue per conditional factor, the arrangement Hanson's modularity makes tradable and that deployed combinatorial markets price by junction-tree inference. Reading the two laws through the generalized distributive law puts them in one semiring frame, in which a market of optimizing traders computes min-plus and agrees with Bayesian inference on the log-quadratic family. Against that background, clamping is the operation that recovers ordinary statistics: freeze entry and a market of Kelly bettors becomes Bayesian model averaging; freeze capital and a market of data points becomes least squares; freeze the propagation step and a chain of venues becomes the Kalman filter; freeze the residual venue to a single unconditional entry and a two-stage solicitation becomes split-conformal prediction. Each clamp forfeits a specific rent, and in the conformal case the rent is exactly the conditional information $I(R;X)$ that the unconditional entry declines to price. The paper sets out the composition laws with their sources, catalogues the clamps, and states what unclamping would require. Where the calculus stops is itself informative: it locates the mechanism that is missing. --

We consider evidence in support of what we term the Weak Indispensable Markets Hypothesis (IMH) provided by the M6 Financial Forecasting Competition. The Weak IMH asserts in the presence of a well-established market, those who eschew prices as inputs for proximate predictive modeling tasks will under-perform out of sample. We also consider the Strong IMH which asserts that forecasting should be considered a market-inspired engineering endeavor in addition to a modeling task under the usual rubric of statistics or machine learning methods (put simply: if a market doesn’t exist to help you, make one!). The competition established that neither principle’s application is obvious to participants or organizers; it hinted at a hidden quality crisis in data science generally; and it suggests that broadening the usual concept of analytic pipelines to insert collective intelligence might be part of the remedy.

Conformal prediction builds prediction sets that cover the truth at a rate you choose, finite-sample and distribution-free, assuming only exchangeable data. That guarantee is marginal. de Finetti's theorem describes the exchangeability it rests on, and in the finite form the mixing measure may be signed kerns2006. A short lemma decomposes the slope of conformal's calibration-conditional coverage into a non-negative threshold term (the classical Beta-law fan) and a term carrying the sign of the de Finetti measure. Positive (extendable) mixtures make conformal conditionally adaptive; the signed corner of de-meaned, ranked, or compositional scores makes it anti-adaptive. The marginal guarantee is the same either way; only what it hides changes.

The companion paper shows that the log-score regret of a single-shape conformal predictor to the conditional oracle is the mutual information $I(R;X)$ between the residual and the input. Here we rederive that quantity from a betting mechanism rather than from the score. Treat the predictor as the crowd in a parimutuel pool on the residual: bettors put money in, and the pot is split among the winners in proportion to their stake on the realised outcome. In the continuous limit the payoff is the ratio of the bettor's density to the crowd's. An entrant who knows only the marginal breaks even, which is marginal coverage stated as wealth. An entrant who conditions on $X$ grows his bankroll at rate exactly $I(R;X)$. The gap is the rent. This is not a metaphor: the nearest-the-pin pool of the microprediction platform, and the continuous density version run in the MidOne contest, are this mechanism, and a conformal predictor is the entrant that prices the pool flat in $X$. We then give two ways to measure the rent on a fitted predictor: a static lower bound from distance covariance, and a sequential e-process whose growth rate estimates $I(R;X)$ and which is an anytime-valid test for conditional miscoverage.

Temporal dependence charges split conformal prediction two different prices, and they scale differently. The coverage price is a tax that vanishes with the calibration size: for a stationary $$-mixing score process the loss is at most $\min_\/(n+1)+2()\$ barberpananjady2026. The sharpness price is a rent that does not: the per-step log-score regret of the marginally calibrated predictive against the past-conditional oracle equals the entropy-rate gap $ H=H(S_0)-h$, the mutual information between the present score and its infinite past, a constant independent of $n$. For Gaussian score processes the rent exponentiates into width — oracle intervals are narrower by the factor $e^- H$ at matched coverage, which for an AR(1) score process with parameter $$ is $1-^2$. Dependence is therefore asymptotically free for the certificate and permanently valuable for the forecast.

We give the validity of split conformal prediction an exact contragredient form: the placement acceptance map is equivariant for the permutation action, and the Reynolds identity $ P^f,v= f,Pv$ reads its orbit average either as moving the evaluation functional or as averaging the induced acceptance vector over placements. A natural primal discrepancy construction under the same action is Steinitz balancing: ordering a zero-sum population so that every prefix sum stays bounded. The centered conformal placement residuals form such a population, and an explicit ordering keeps the running placement-acceptance average within $1/(2t)$ of the orbit level $k/N$, exact once all $N$ candidate test placements have been traversed; the vector-valued lemma balances several coverage levels along one traversal. On the function–measure pairing, the measure-side counterpart of calibration is balanced sampling and exact cubature. A transitive distributional symmetry, exchangeability in the standard setting, turns the deterministic orbit average into the coverage probability of the designated test placement.