Grounded World Models: Efficient and Verifiable Structural Causal Prediction B The Returns to Crystallization D The Sampling Budget: Latency vs Residual Monte-Carlo Error

Appendix C Explanation-Quality Components and Heuristic Parameter Derivations

This appendix gives formal definitions for the four components of $\mathrm{XQ}$ (Definition 7).

Let $q$ be a PD produced by a computation with named structural inputs $\theta=(\theta_{1},\dots,\theta_{m})$ , a set of invariants $\mathcal{I}$ (accounting identities, sign and monotonicity constraints), and—for sampling-based WMs—a Monte-Carlo budget yielding standard error $\mathrm{se}(q)$ on a target summary functional $T(q)$ with natural scale $\tau$ . Fix a battery $\mathcal{A}$ of intervention queries.

Definition 8 (Rationale stiffness).

Let $\varepsilon_{j}=\partial\log T(q)/\partial\log\theta_{j}$ be the elasticity of $T$ to input $\theta_{j}$ , and $B_{j}$ the mechanism-implied admissible band for that elasticity. Then

S(q)\;=\;\frac{1}{m}\sum_{j=1}^{m}\mathbf{1}\!\left[\varepsilon_{j}\in B_{j}% \right]\;\in\;[0,1],

(4)

the fraction of inputs that are not free knobs (responses bounded and mechanism-consistent).

Definition 9 (Counterfactual consistency).

\mathrm{CC}(q)\;=\;\frac{1}{|\mathcal{A}|}\sum_{a\in\mathcal{A}}\mathbf{1}\!% \left[\,q(\cdot\mid\mathrm{do}(a))\text{ satisfies }\mathcal{I}\text{ and the % do-calculus identities}\,\right]\;\in\;[0,1].

(5)

Definition 10 (Hardness-to-vary).

For an alternative outcome $y^{\prime}$ , let $q^{\prime}$ be the nearest model (minimal structural edit) that fits $y^{\prime}$ without violating $\mathcal{I}$ , and $\delta(y^{\prime})=d(q,q^{\prime})/d_{\max}\in[0,1]$ the normalized forced change. Then $\mathrm{HtV}(q)=\mathbb{E}_{y^{\prime}}[\delta(y^{\prime})]$ : a high value means the explanation cannot be cheaply twisted to fit a different outcome (Deutsch, 2011).

Definition 11 (Residual-noise control).

$R(q)=1-U(q)\in[0,1]$ , where $U(q)$ is the residual uncertainty about the true posterior left by the computation that produced $q$ ; $R=1$ in the exact-inference limit. For a sampling WM with Monte-Carlo standard error $\mathrm{se}(q)$ and tolerance $\tau$ , $U=\min\!\big{(}1,\mathrm{se}(q)/\tau\big{)}$ , which vanishes in the large-sample limit. A single narrative emission instead asserts one scenario distribution while its qualitative rationale pins only a set $B$ of distributions on the $K$ -scenario probability simplex $\Delta^{K-1}$ ; the residual uncertainty is then the linear extent of that set, $U=\big{(}\mathrm{vol}(B)/\mathrm{vol}(\Delta^{K-1})\big{)}^{1/(K-1)}$ , so $R=1-\big{(}\mathrm{vol}(B)/\mathrm{vol}(\Delta^{K-1})\big{)}^{1/(K-1)}$ (estimated in Appendix C).

The four components are independent correctness probabilities, so the joint correctness of $q$ is their product and explanation quality is the corresponding log-probability:

\mathrm{XQ}(q)\;=\;\sum_{i}\log c_{i}\;=\;\log S(q)+\log\mathrm{CC}(q)+\log% \mathrm{HtV}(q)+\log R(q)\;\leq\;0,

(6)

measured in nats, on the same scale as $\mathrm{PQ}=-D_{\mathrm{KL}}$ . Near the ideal, $-\mathrm{XQ}=\sum_{i}(-\log c_{i})\approx\sum_{i}(1-c_{i})$ , which recovers to first order the linear KL bound used in the proof of Proposition 1; the log form is its all-orders extension, and unlike an arithmetic mean it admits no cross-component compensation—a single weak axis caps the whole. For figures and quality targets we use the bounded attainment

A(q)\;=\;\exp\!\big{(}\mathrm{XQ}(q)\big{)}\;=\;S\,\mathrm{CC}\,\mathrm{HtV}\,% R\;\in\;(0,1],

(7)

the joint-correctness probability of the four axes—a strictly monotone transform of $\mathrm{XQ}$ (indeed $\mathrm{XQ}=\log A$ ), so every ceiling and ordering statement transfers between the two, and a ratio of two $A$ ’s is exactly the exponential of their $\mathrm{XQ}$ difference.

A note on scale, to avoid a common confusion. $\mathrm{XQ}\in(-\infty,0]$ itself is the log-scale quantity Proposition 1 lower-bounds $\mathrm{PQ}=-D_{\mathrm{KL}}(p^{\star}\|q)$ by, and—exactly as intuition suggests—it increases toward (though it need not reach) $0$ as $q$ becomes a better explanation, mirroring $\mathrm{PQ}\to 0$ as $q\to p^{\star}$ ; this is not in tension with attainment being bounded. What lives on the familiar $[0,1]$ scale is not $\mathrm{XQ}$ but $A=\exp(\mathrm{XQ})\in(0,1]$ . Because the two are a strictly monotone transform of each other, we use them interchangeably in prose below for readability—so wherever the text speaks of “raising XQ,” an “XQ target,” or an “XQ ceiling” expressed as a number in $(0,1)$ , it denotes this bounded attainment $A$ , not the log-scale $\mathrm{XQ}$ of Eq. (6), which is what is actually being plotted or bounded in Sections 3.2 and the propositions below.

Operational estimators of the components

The definitions above are population quantities over a battery $\mathcal{A}$ and an invariant set $\mathcal{I}$ ; we now state how each is estimated on the narrative-arm memos (the measurement of Section 3.1), so the reported scores map transparently onto Definitions 8–11. A tool-backed judge agent (also Claude Opus 4.8) reads each memo equipped with a sandboxed arithmetic evaluator and a numerical closeness check; it may verify a number only by calling a tool, so the structural scores are tool-checked rather than asserted, and the full call log is persisted per case for audit.

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Stiffness $S$ (Def. 8). The fraction of headline numbers (target price, expected return, direction probabilities, the weighted scenario value, each multiple-to-price bridge) that are reconstructible from other stated inputs in the memo. A number that no stated inputs reproduce is a free knob—an unconstrained elasticity direction, the discrete analogue of $\varepsilon_{j}\notin B_{j}$ .
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Counterfactual consistency $\mathrm{CC}$ (Def. 9). Estimated in two steps and combined as $\mathrm{CC}=\min(\mathrm{CC}_{\mathrm{stat}},\mathrm{CC}_{\mathrm{int}})$ , so the weaker step caps the score. Step 1 (static internal consistency): the battery that probabilities sum to one; the stated target equals the probability-weighted scenario prices; expected return $=$ target/spot $-1$ ; the direction buckets match the scenario upsides; scenario prices are monotone in severity; and the recommendation’s sign lies in the stated return band—each an instance of “ $q$ satisfies $\mathcal{I}$ ,” checked with the tools. This step certifies only that the memo is self-consistent, not that it answers $\mathrm{do}(a)$ correctly. Step 2 (interventional re-query): to test the do-calculus clause of Definition 9 directly, the judge poses at least five off-grid interventions $\mathrm{do}(a)$ —probes whose correct answer is neither any tabulated scenario price nor a convex re-weighting of them, spanning between-scenario, compound, fixed-input, and mechanism-inversion families—and re-queries the same NWM under each, holding its stated thesis and mechanism fixed. Each re-forecast is scored continuously in $[0,1]$ for whether it tracks the memo’s own stated mechanism in both direction and magnitude (no external oracle), and $\mathrm{CC}_{\mathrm{int}}$ is their mean.
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Hardness-to-vary $\mathrm{HtV}$ (Def. 10). The one component scored as a direct judgment: the minimal structural edit needed to make the rationale fit the opposite recommendation without breaking an invariant, in $[0,1]$ .
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Residual-noise control $R$ (Def. 11). Measured per memo. The judge brackets each of the $K$ scenario probabilities by a band $[\ell_{k},u_{k}]$ that the narrative defensibly supports; a deterministic tool computes the fraction $f=\mathrm{vol}(B)/\mathrm{vol}(\Delta^{K-1})$ of the simplex compatible with those bands (by uniform-simplex sampling), and $R=1-f^{1/(K-1)}$ . A tightly argued distribution gives small $f$ and high $R$ ; a vague one gives $R\to 0$ . The GWM’s large-sample $R$ is taken as the NWM’s generous ceiling (reachable only by MC-averaging many emissions), not as its deployed value.