The Phase Chosen Domain by Domain: Kibble's Independence Rule and the Density Inside Every Defect Census

In 1976 T. W. B. Kibble founded the cosmology of topological defects: which structures can form at a symmetry-breaking transition in the early universe — domain walls, strings, monopoles — is decided by the homotopy groups of the manifold of degenerate vacua, and that classification is a derivation that stands entire. The paper then estimates how many form, and says in its own words that the estimate rests on "two simple assumptions": the initial choices of the order parameter in different cells are uncorrelated, and where cells meet the field interpolates along the shortest path in the vacuum manifold. Under those assumptions it finds one in four edges forming strings. This paper evaluates the first assumption. The bench reproduces the paper's return exactly under its assumptions — string fraction 0.250000 on the continuous circle, approached from below by exact clock enumerations (0.222222, 0.240000, 0.244898) — and then relaxes independence by one declared correlation: at copying probability r = 1/2 the string fraction falls to 0.093750, a ratio of 0.390625 to the independent value; at r = 3/4 to 0.023438; at r = 9/10 to 0.003750. The topology does not move; the density is the assumption's arithmetic. What is counted here is the derivation status of the correlation structure the estimate uses: 1 independence rule (with its geodesic companion), 0 derivations in 1976 of the phase correlation at formation. Census headline, unchanged: NOT SOLVED / PARTIAL · producing-count 0.