Fix a single observable system and an agent that must predict it. The weakest useful object is a machine that emits predictions; we sharpen “machine” in four steps, each adding a property that a later benchmark will price.
A world model (WM) is any machine that, in a given internal state, samples point predictions of an observable .
A WM is proper if in every internal state its samples are draws from a single coherent predictive distribution (PD): there is a probability measure such that the queried samples are exchangeable with empirical law converging to , and obeys the probability axioms.
A WM that is not proper has no well-defined : repeated queries need not be mutually consistent, and there is no object on which to compute calibration or divergence.
A proper WM is grounded (a GWM) if its PD is a Bayesian posterior obtained by conditioning an explicit, executable mechanism on evidence and exogenous assumptions :
| (1) |
where are structural parameters and supports Pearl’s -operator for a given class of interventions .555We adopt the -calculus as “syntactic sugar” for standard probabilistic queries on a suitably expanded model, as in Mlodozeniec et al. (2025).
A narrative world model (NWM) produces predictions as the output of a language model rather than as a posterior under an explicit mechanism. Its predictive may be read off in either of two ways:
point samples: each point prediction is the final output of (prompt a number), and a PD is estimated by drawing many such points;
wholesale distribution: a predictive distribution is emitted in one pass (prompt a described distribution or set of quantiles).
The point-sample formulation is disfavored in both principle and practice. Any point sample is a fact about ’s decoding process at fixed temperature, with no guarantee of coinciding with the quantiles states when asked directly; and re-querying many times per prediction would in any case defeat the cost comparison to a GWM’s single belief-propagation pass. We therefore work throughout with the wholesale-distribution case: denotes the predictive distribution or quantiles emits in one pass. Neither emission constrains its output to be the posterior of any explicit mechanism; the “model” is implicit in ’s weights and the prompt.
In this language the paper’s claims are statements about these classes. An NWM is a proper WM only in the infinite-sample limit, and only if ’s emission law is stable; it asserts (or estimates) a with no guarantee that it is any posterior. The GWM is the unique class whose every query is, by construction, a posterior under a verified mechanism—the property the rest of the paper exploits.
Equation (1) is realized by a concrete object that we use throughout.
A GWM for a system is a tuple where
is a set of variables with a causal ordering;
is a vector of structural parameters (elasticities, growth rates, margins);
is a vector space of exogenous variables (assumptions: scenario inputs, conventions, policy settings);
is a causal generative model defining the joint distribution over and supporting Pearl’s -operator for interventions (we realize it as a probabilistic program);
is a corpus of evidence items;
is a grounding map taking evidence to a posterior over parameters, .
A query is a triple : a target functional of , an assumption setting (exogenous scenario inputs, policy settings), and an optional intervention supported by .666Two kinds of what-if are distinct: changing re-evaluates the same mechanism under different exogenous inputs (ordinary conditioning—the common case, with ), whereas an intervention overrides the mechanism for one or more variables in —e.g. pinning a latent to a value —and is evaluated with the do-operator rather than by conditioning on . We also note that and may also be specified as parameterized distributions rather than point values. The corresponding prediction is the posterior functional (1) evaluated at .
The clause that distinguishes a GWM from everything else is : a prediction is a posterior under an explicit mechanism. There is no step at which a probability is asserted; every number is the image of evidence under and of structure under .
The “grounding” lives entirely in the construction of and —the first ingesting real-world observations into , the second defining how its outputs will be interpreted in terms of real-world outcomes and actions. There is no philosophical symbol-grounding problem (Harnad, 1990), as no variables in are ascribed any special ontological status; “observables” are merely those variables for which the ingestion and extraction are defined, and which, pragmatically, define ’s context of applicability.777We trace this posture to Quinean holism (Quine, 1951).
The grounding map is the operator that distinguishes a GWM: it turns a corpus of evidence into a posterior over the model by Bayesian updating rather than assertion. It is best read not as a single formula but as a spectrum of conditioning operations of increasing reach, all sharing the property that the output is a posterior under the explicit mechanism —so the Bayesian quality guarantee of Proposition 2 applies throughout.
(i) Forward accumulation. The simplest and cheapest case treats each evidentiary hypothesis as a parameter with a conjugate prior and accumulates source-weighted support and refutation (Eq. 13). Being conjugate, each update touches parameters: the marginal cost of grounding over an ungrounded deep-research pass is one float and one sign per evidence item—negligible in tokens—yet it converts a pile of citations into a calibrated parameter posterior.
(ii) Propagation to latents and forecasts. A posterior over parameters is not yet a prediction. Belief propagation through carries the parameter posterior forward onto the latent states and the queried observable , producing the predictive functional (1)—the “sample” operation priced in Section 3.
(iii) Backward inference. Evidence often lands downstream of the parameters it should move—a realized outcome, an observed margin. Conditioning on such observations inverts , revising upstream parameters and latents by full Bayesian inference rather than local accumulation; conjugacy is lost, but the update remains exact conditioning, approximated by the inference engine.
(iv) Structure learning. In its fullest form updates not only but the structure itself—adding or removing variables and edges in as evidence demands. This accretive model construction lets a GWM grow and evolve to be grounded not only by factual evidence, but by new theoretical or heuristic causal knowledge.
Across all four, is either exact or controlled-approximate Bayesian conditioning on an explicit mechanism; the cases differ only in reach and cost, from the conjugate edit (13) to a full structural revision.
The program is executable and its invariants (accounting identities, non-negativity, monotonicities) are machine-checked.
A single covers a family of instances by varying ; the structure is reused across the family at near-zero marginal cost.
Two GWMs with compatible interface contracts compose into a third (a supplier model feeds a customer model) without fresh search.
Every prediction decomposes into named structural pathways, so a user can ask why and receive a mechanism, not a rationalization.
Knowledge graphs store entities and relations but no executable mechanism; they answer “what is connected to what,” not “what happens if.” They lack and .
Fitted black-box predictors (neural nets, Gaussian processes) are non-narrative: they fit parameters to data, and some—a GP, say—even yield a proper predictive distribution. But their PDs are not interrogable mechanisms, and therefore they only support correlational prediction, not structural causal prediction.
Narrative world models (NWMs of Section 2.1; LLMs prompted for a forecast or a distribution) emit predictions directly, with no , no , and no . They can be fluent and even accurate on average, but their outputs are not constrained to be any posterior, so they are neither auditable nor coherently updatable.
The GWM is the unique object meeting all four properties and supporting both simulate-forward and condition-on-evidence.
The reader will likely be familiar with at least one “canonical” instance of a GWM. Numerical weather prediction—the physics-based weather models run operationally at every major forecasting center—realizes every clause of the contract above. An operational weather model is an explicit causal mechanism—the discretized equations of atmospheric physics—whose state is grounded by continuously assimilating millions of daily observations, and which is run forward under that conditioning to emit a calibrated predictive distribution. Table 1 maps the correspondence element by element.888Learned forecasters like GraphCast (Lam et al., 2023) and Pangu-Weather (Bi et al., 2023) now match physics models on headline scores, but they are the exception that proves the rule: trained on ERA5 reanalysis (Hersbach et al., 2020)—itself the output of the grounded assimilation system—they distill the GWM rather than replace it; lacking enforced conservation laws, they degrade on extremes and out-of-training regimes (Richens and Everitt, 2024); and they neither assimilate new observations nor answer queries. The grounded model remains the substrate; the emulator is a fast approximation of its forward map.
The example of weather models also illustrates why GWMs are not already prevalent across disciplines, and why our proposal for a GWM in the investment domain is noteworthy. In weather prediction the governing equations (or proven heuristics) are known, the observation network is dense, and verification is automatic every few hours, providing a fast, thorough calibration loop. Most decision domains—finance, policy, epidemiology, supply chains—enjoy no such gift; their mechanism is partial, latent, and must itself be constructed and iteratively validated against “soft” and ambiguous evidence. Primordia’s main process innovation lies in automating GWM construction and maintenance, by casting it as an iterative process of program synthesis and verification. We expect this category of automatically-constructed GWMs to achieve preeminence in fields that require structural causal prediction.
We now make “a better prediction” precise, then connect it to quantities measurable on a deployed system.
Let be a WM’s PD for a query given evidence . Its prediction quality is , where the reference is either (i) the true data-generating process (the M-complete reference) when it is well-defined, or (ii) the predictive distribution of the ideal unbounded Bayesian predictor over all computable hypotheses given the same (the AIXI/Solomonoff reference (Hutter, 2005; Solomonoff, 1964)) in the M-open setting where no candidate model is the truth (Bernardo and Smith, 2000).999Working in the M-open setting means we cannot verify from within the formalism that our model class contains (an adequate approximation to) ; checking this against data is itself outside strict Bayesian coherence (Gelman and Yao, 2021). We return to this tension at Assumption 2 in Appendix G.
Even with in hand and exact conditioning, PQ is capped by the system’s intrinsic predictability: for a chaotic process the attainable decays with forecast lead time regardless of model or compute (the Lorenz horizon (Lorenz, 1963)). “Optimal PQ” is thus always relative to a horizon; a GWM’s claim is to reach that horizon-limited ceiling, not to abolish it.
PQ is the right target but is unmeasurable: is unknown (M-complete) or uncomputable (M-open). We therefore define observable explanation-quality functionals on and show they proxy PQ.
Informally, these functionals measure how high a prediction climbs a groundedness ladder—from asserted (a number stated with no support), through sourced (each figure cited to evidence but not reconciled into a model) and derived (figures reconstructed from a consistent set of inputs), to a full Bayesian posterior (every quantity the image of evidence under the grounding map on an explicit mechanism, Eq. (1)). The four properties below certify the upper rungs; only a model that carries a mechanism to condition reaches the top, so a narrative WM can be pushed up the lower rungs at rising cost but is bounded away from a posterior (Proposition 3).
Each of the four observable properties below is scored as an attainment of the computation that produced . Treating them as independent correctness probabilities, explanation quality aggregates them multiplicatively as a log-probability, (Eq. (6)), on the same scale as ; its bounded image —the joint-correctness probability—serves as the quality-target axis in Section 3.2. The properties:
rationale stiffness—the elasticity of to perturbing each named structural input is bounded and mechanism-consistent (small, sourced moves; no free knobs);
counterfactual consistency— queries satisfy the model’s invariants and the do-calculus identities;
hardness-to-vary (Deutsch, 2011)—the rationale cannot be locally edited to fit a different outcome without breaking an invariant;
residual sampling noise—for sampling-based WMs, the Monte-Carlo variance of at the reported budget (zero in the exact-inference limit).
Consider a toy GWM for tomorrow’s local temperature : a two-parameter causal model , where is the assimilated regional temperature anomaly (an observed input grounding via a fit to the historical station network) and is sampled at inference. Answering “what if the regional anomaly is C instead of C” means re-evaluating the same mechanism at the new input, and each XQ component reads directly off that mechanism: (i) stiffness , because the only input the answer can depend on, , is the named structural variable—there is no other knob the forecast could secretly be tracking; (ii) counterfactual consistency , because is a fixed coefficient, so every query returns exactly what the mechanism implies; (iii) hardness-to-vary , because were themselves estimated, so a sufficiently adversarial re-fit on slightly different data could still nudge the prediction—bounded misspecification, not a free knob; (iv) residual control climbs toward as the estimation/Monte-Carlo standard error on falls below the decision-relevant tolerance (say C). The product —and its logarithm, —both follow from these four numbers alone: no access to the true weather process is required.
Now contrast a narrative WM: an LLM asked to forecast tomorrow’s temperature from a paragraph describing today’s map. Its answer is not the output of any fixed : doubling the stated regional anomaly and re-asking may move the forecast by an amount no single would produce (so a battery of re-queries scores ); the forecast may lean on unstated priors about the season or the model’s own uncalibrated intuition alongside the cited anomaly (extra free knobs, ); a mildly adversarial rephrasing of the prompt can pull the same forecast toward a different outcome without the model flagging any contradiction (low ); and repeat queries at nonzero sampling temperature typically disagree with no principled bound on the spread (low ). Each quantity above is measured directly from the model’s outputs (Appendix C), without ever knowing the true —which is precisely the sense in which is an observable proxy for the unmeasurable of Definition 6.
Higher explanation quality implies higher prediction quality: is monotonically related to up to a bounded slack. Crucially, for three of the four channels—counterfactual consistency, residual noise, and (under a regularity condition) rationale stiffness—this relation is a result, not an assumption: a model that satisfies the do-calculus identities and has vanishing sampling error simply is closer to on the queries checked. Only hardness-to-vary and the extrapolation from finitely many checks to all queries rest on a mild representativeness condition. The full statement, its assumptions, and the proof are in Appendix G.
Why this matters to a decision-maker. Because XQ lower-bounds PQ, a rational agent trusts a prediction in proportion to how well it is explained: a higher-XQ prediction warrants a larger, better-calibrated action, and a lower-XQ one warrants caution. Explanation quality is thus not a cosmetic property but the very quantity on which a rational principal conditions its trust—and its actions.
A GWM has a well-defined that is non-increasing in expectation under further conditioning: . No such guarantee holds for a narrative WM, whose output is not a posterior and need not even define a fixed .
Intuition. For a GWM, conditioning is Bayesian updating, and the expected log-loss of a Bayes predictor is non-increasing in information (a martingale / model-averaging argument (Hoeting et al., 1999)); the divergence is finite under mild non-degeneracy (Appendix G). An NWM asserts (or estimates it from samples) with no update operator constrained to be Bayesian, so it need not converge to any posterior and does not inherit the guarantee. Proof in Appendix G.
The negative half is in fact stronger than “no guarantee”: a narrative WM is improperly updated, and the defect compounds.
Let an NWM absorb evidence through an update operator realized by a language model —either accumulation in the context window (in-context learning) or retrieval from a knowledge base—and let be its implied predictive after evidence . Then (i) does not preserve posteriors: even if coincides with at one stage, generically after the next evidence item, so propriety—if ever attained—is lost almost immediately; (ii) consequently need not converge to , and there exist evidence streams and queries on which is arbitrarily large. The sole exception is when ’s parameters both encode the exact mechanism of and route each conditioning step to it exactly—an event whose prior probability decreases with the domain’s structural complexity and with (Appendix H).
Intuition. The map from pre- to post-evidence output is whatever the attention stack, context-window truncation and compression, and retrieval policy happen to compute; nothing constrains it to equal Bayes’ rule. There is thus no martingale structure for Lemma 3 to exploit, and a finite context can evict the very evidence a later query depends on.
To reliably produce a grounded predictive, an LLM cannot be relied upon to be the posterior; it must call an explicit, verified GWM that conditions—exactly the Mode-3 architecture of Section 4.1. Continual learning does not help, and arguably hurts: replacing in-context updates with gradient steps does not make the update Bayesian conditioning on either, and it adds a free knob—the training objective must itself be tuned to the domain to even approximate the target (Corollary 3). Proof in Appendix G.
We have argued the GWM’s advantages along two axes: efficiency (cost and latency) and quality. On the efficiency finding, the key characteristic is parsimony: because explicit theory, decomposability, and linear-time inference carry the world’s state directly, a GWM needs far fewer free parameters than a model that must learn that structure from data. Parsimony in turn governs how cost scales. Three operations dominate a GWM’s lifetime—authoring (create or maintain a GWM’s structure), update (absorb evidence), and sample (draw a prediction)—and structure makes all three cheap: authoring is an infrequent, amortizable program synthesis task, while update and sample are an conjugate edit and a single belief-propagation pass whose cost is flat in model size—against a NWM’s re-read of its entire context on every inference.
This advantage is compounded by the GWM’s advantage on the quality axis, which we have already extensively detailed above. In summary, a GWM’s prediction quality is fundamentally bounded only by the system’s intrinsic predictability and observability; given a source of mechanistic hypotheses to be incorporated into the causal generative model, it is capable of asymptotic Bayes-optimality. And because of the parsimony identified above, this asymptotic quality is not bought at extraordinary expense, but realized at startlingly economical budgets.
In what follows, we demonstrate how these theoretical advantages translate into empirical dominance.