De-patching the TUCSL–CST bridge Diagnostics, scaling limits, and parameter discipline for the BD/BDG sum-over-causal-sets programm

We address critical "patch-susceptible" junctions in the Transactional Unimodular CSL (TUCSL) synthesis with Causal Set Theory (CST) introduced our previous papers. Four potential vulnerabilities are converted into falsifiable diagnostics: (i) the diffusion/semigroup bridge between BD/BDG causal-set dynamics and the TUCSL q-heat kernel is reformulated as explicit convergence criteria with computable error functionals E_diag(s) and spectral dimension flow d^C_s(s); (ii) the Poisson-to-Gaussian passage is proven as Theorem 4.1 via cumulant expansion, with the dimensionless control parameter χ := kV_cell tracking the high-intensity/small-mark scaling limit; (iii) the apparent freedom in event density k and transactional energy E_⋆ is eliminated by operational definitions tying k to the semigroup diagonal and bounding E_⋆ through laboratory CSL constraints; (iv) the sum-over-causal-sets measure is implemented as a microcanonical admissibility prior Ω^TUCSL_N (support restriction) rather than phenomenological weighting, avoiding new continuous couplings.

We provide an explicit assumption ledger (A1–A5), a "no-hidden-knobs" parameter accounting (Table 1), and four decisive failure modes (F1–F4) that would falsify the bridge. The correlation cell volume V_cell(s;x) := [K_2s(x,x)]^(-1) emerges as the unique non-geometric operational notion linking TUCSL coarse-graining to causal-set sprinkling density. A roadmap to "hard results" specifies: numerical computation of BDG diagnostics on manifoldlike causal-set ensembles, extraction and comparison of discrete vs continuum V_cell, and propagation of experimental bounds through the Paper XIV closure formula to derive tight E_⋆ intervals.