Poset limits of covering growth laws

A covering process grows a finite poset from a unique minimum by adding a new maximum of Hasse in-degree 1 or 2. Three named sequential laws are studied: uniform_legal, port_resolved, and t1_only. Under t1_only the order is a tree, every exclusive interval is a chain, the unsaturated fraction tends to 1/phi, the global poset limit is the empty kernel, and every window limit is the chain kernel. Consequently neither the global nor the window limit is Lorentzian of spacetime dimension at least 2. Those statements concern typical realisations of the sequential process; sequential probabilities on unlabelled covering classes fail label-independence from size 4. For uniform_legal and port_resolved the maximal elements are exactly the two-port vertices. Unconditionally, the residual-port count under uniform_legal is Omega(sqrt(N)). A named incomparability condition (weak A-INC) assumes a uniformly positive density of incomparable unsaturated pairs. Under that condition, uniform_legal has A_N and P_N of order sqrt(N). A first-moment argument, recorded as a proposition with listed caveats rather than as a theorem, sketches an upgrade from weak A-INC to almost-sure incomparability of the frontier for uniform_legal; that upgrade is not claimed as a theorem, and it fails for port_resolved. Conditional on A-INC and a typical-vertex saturation statement, a typical vertex has depth of order sqrt(N) and saturates ancestry in log2(A_N)+O(1) generations; the longest-chain order and the rate of 1-r_N remain the sketch of that picture. On the same picture the global ordering fraction tends to 1, the global limit is the chain kernel, and t(2+2) tends to 0 if the order is o(N^2)-close to a height-threshold interval order. Windows with m/A_N tending to infinity are of unit-interval-order type for both mean-field laws; typical related-pair windows with m = O(A_N) and m/log A_N tending to infinity are of dust type for uniform_legal only; atypical thin chains are not claimed. Combined with a positive 2+2 density for every Lorentzian causal kernel of dimension at least 2, none of these limits is Lorentzian of dimension at least 2. The weak incomparability condition remains a conjecture. Measured height log-log slopes above 1/2 at N at most 8192 are not an exhibit of a change of growth-law type: a finite covering order, rescaled by such a slope, retains the same underlying poset and the same covering-kernel support. Covering histories are finite posets of the mathematical type of finite causal sets. They are not identified with physical spacetime. This note does not claim Lorentz invariance and does not assert that covering structures cannot produce geometry. Other laws remain open. It does not replace the parent programme manuscript, and it is not a new version of Companions 13-38.