Appendix E A Worked Example: GWM vs NWM on One Case

The tool-backed judge (Appendix C) reconstructs the NWM’s headline numbers with its calculator: the target reconstructs as the probability-weighted price (${\sum }_i{p}_i{P}_i=302.7\approx 298$, within $2\%$), expected return as $298/254.34-1=+17.2\%$, and each scenario upside as $P_i/254.34-1$—all pass. One number does not reconstruct: the asserted direction probabilities $65/30/5$ contradict the scenario table, which implies $40+25+10=75\%$ LONG, $20\%$ HOLD, $5\%$ SHORT. This single free knob fails one of nine static consistency checks (${\mathrm{CC}}_{\mathrm{stat}}=8/9=0.889$) and one of seven stiffness checks ($S=6/7=0.857$). Probing further, the judge re-queried the same NWM under five off-grid interventions—a between-scenario multiple compression, a buyback that accretes EPS, a compound revenue/margin shift, an interpolated revenue/margin/multiple triple, and a breakeven-multiple inversion, none answerable from the scenario table—and every re-forecast tracked the memo’s own EPS$\times$multiple bridge in direction and magnitude (${\mathrm{CC}}_{\mathrm{int}}=1.0$), so $\mathrm{CC}=\min (0.889,1.0)=0.889$. Hardness-to-vary is judged $0.62$ (the soft scenario probabilities and a reversible narrative can be re-spun for a bear case without breaking the arithmetic). Residual control is measured from the probability bands the narrative supports: only $f=0.8\%$ of the five-scenario simplex is compatible, giving $R=1-f^{1/4}=0.705$—high, but below the GWM’s large-sample $0.95$.

The GWM has no such knob to break. Its scenario probabilities sum to one, each scenario upside equals $P_i/\text{spot}-1$, prices are monotone in severity, and—crucially—its direction probabilities are not asserted but read off the same posterior samples as the scenarios, so they cannot disagree with the table. Stiffness and counterfactual consistency are therefore $1$ by construction (Table 4); the residual gap to a perfect explanation is bounded misspecification ($\mathrm{HtV}=0.92$) and Monte-Carlo residual ($R=0.95$), not a free parameter. Table 3 collects the scores.

Table 3 is the $Q=1$ slice, which most flatters the NWM. ACME is in fact queried repeatedly—each what-if and re-forecast is another PD—so its per-case economics are those of Sections 3.2–3.2: ACME’s GWM build is paid once, and every follow-on PD, including the $\mathrm{do}(a)$ battery (re-rating the exit multiple, shifting demand, flipping ValuationReRates), is a flat $c_{\mathrm{PD}}$ call that preserves the invariants above ($\mathrm{CC}\to 1$). A one-shot NWM run costs up to $1.82 and buys no reusable posterior (only a KV cache of the run’s input and output tokens), so every additional PD is another run at a cost that climbs with the XQ bar. The crossover and ceilings are exactly those of Figure 4.