Correlated Unimodular Cells, Quadratic Transactional Scaling, and the Pre-Inflationary Size of the Observable Patch in TUCSL

We study a quantitative open problem in the Transactional Unimodular ContinuousSpontaneous Localization programme (TUCSL): the size of the correlation correction δβcorrrequired to lift the effective inflation-selection coefficient from its non-correlated baselineto the value preferred by cosmic microwave background phenomenology. We formulate aminimal but mathematically explicit “primordial correlated block” model for unimodular cellscrossing the bandage region into the first Lorentzian record. The key result is a quadraticscaling theorem: under exchangeability, dense pairwise coherence, and second-cumulantdominance—the latter being natural in the Gaussian/Markov regime already favoured bythe TUCSLkernel backbone—the connected contribution scales asδβ_corr(M ) = μ_2*[M (M − 1)]/2 ,with M the number of coherently correlated unimodular cells and μ2 determined by thesecond derivative, with respect to the inflation-selection variable r, of the pairwise connectedcorrelator. We then solve for M in terms of the required residual ∆β, convert the resultinto a physical radius by packing M Planck volumes into a sphere, and show that a targetof order 1032 to 1036 typically points to radii in the corridor 10−30 m to 10−29 m, i.e. thesame order of magnitude often associated with the pre-inflationary patch that later becameour observable universe. We emphasize throughout what is proved, what is conditional onexplicit assumptions, and what remains to be derived from first principles