How Much Patient State Must a World Model Preserve? Exact Fusion under Multiple Unknown Measurement Scales

Patient world models increasingly combine heterogeneous longitudinal sources—assays, devices, sites, or measurement panels—whose absolute scales may be uncertain even when their within-source response patterns are informative. We study a finite-snapshot count model with $m$ Poisson channels partitioned into $G$ sources. Source $g$ has its own unknown positive scale $A_g$, while channel $i$ has known baseline $r_i$ and known latent-state sensitivity vector $b_i$. We prove an exact compression law: the full count vector is minimally summarized by the $G$ source totals together with the feature-weighted aggregate $T=\sum_i b_iN_i$; the canonical linear sufficient-statistic row space has dimension equal to the rank of the matrix obtained by stacking source indicators and sensitivity features. Conditioning on source totals removes every unknown scale, and the efficient Fisher information for latent state is exactly the sum, over sources, of expected source count times the within-source covariance of sensitivities. Consequently, a latent direction is invisible after nuisance removal exactly when its channel sensitivity is constant inside every source. This gives a sharp granularity frontier for ordinary real-linear projection: one shared scale can permit strong compression, whereas one unrelated scale per channel makes the canonical row-space kernel trivial. A Bayesian decision corollary shows that the same statistic is sufficient for any downstream intervention simulator whose future kernel depends on the observed snapshot only through the latent parameters. Executable checks verify the likelihood factorization, profile/conditional identity, and information law over randomized instances, with the information identity independently replayed in a separate implementation. The results are statistical resource laws, not evidence that the assumed observation model is clinically valid or causally sufficient for treatment selection.