Exponential path-entropy families, sector-balanced local growth, Dilworth-Turan coarse quotients, and symplectic Berry holonomy on covering-process orders

A covering process grows a finite poset from a unique minimum by adding a new maximum of Hasse in-degree 1 or 2. Companion 37 isolates the unique zero-parameter path-entropy measure on the unlabelled process poset W_n, with |L_8|=11222. The present note tilts that measure by a chemical potential conjugate to the arity-two count n_T2. The resulting law is a full one-parameter exponential family, strictly log-convex for n>=4, with origin identically Companion 37 Theorem 11 (Z_8(0)=11222, theta_8(0)=16393/78554). The sharp covering support is 0<=n_T2<=n-3. Stepwise atom-uniform growth is not a fixed-theta member of the sequential sector-balanced family. A Bernoulli sector draw prior to uniform pair selection within Hasse distance at most 2 ell yields a constitutive T2 probability theta, forbids long-range T2 shortcuts, and retains the closed-cone 1+1 diamond lattice as a legal trajectory. A parameter-free dyadic frontier horizon cuts finite-size ordering-fraction drift from 0.1049 to 0.0516 (512 to 1024) and 0.0296 (1024 to 2048) while remaining inside r in [0.15, 0.31] with Fishburn p₂₊₂>=0.02. Flux-thresholded order-convex quotients are acyclic posets; tau>=2 drops width-1 shortcuts; handshake counting supplies a necessity bound on coarse link density; same-rank antichain slices have zero internal links; every finite poset of Dilworth width w satisfies R 2 requires mean_dplus_1 > 2 T_band/(T_band-1). Keeping r<=0.50 requires a low single-step spatial fraction phi = mean_dplus_1/S_sec of about 0.31 or less, hence S_sec>=9. Partitioning T_band=6 time bands by the first two nontrivial Hasse-Laplacian eigenvectors into a 3x3 spatial grid (|Q|=54) concentrates 86.3 percent of adjacent-layer generators in the Moore 8-neighbourhood and jointly inhabits R(Q_2)=2.306>2, max dpm=6.125>2, r(Q_2)=0.396 in [0.10,0.50], and p₂₊₂(Q_2)=0.0670>=0.02 at N=1536 (and R=2.375, r=0.386, p₂₊₂=0.0531 at N=1024). That joint inhabitation is seed-versus-null geography, not leftover interpolating type and not a derivation of spacetime. Covering histories are finite posets of the mathematical type of finite causal sets. They are not identified with physical spacetime. The degree-normalized GUE statistic is a spectral property of the unlabelled covering Hasse graph; it is not a derivation of quantum mechanics, spacetime, or an algebra homomorphism between programme spines. This note does not derive spacetime, a unique sequential growth law, or an interpolating continuum. It does not replace the parent programme manuscript, and it is not a new version of Companions 13-37.