TCV-ϕ IV: cohesive non-singular black holes from static cores to slow rotation in the canonical EFT

This fourth paper opens the strong-gravity sector of the canonical TCV-ϕ EFT previously developed in Papers I-III. The scope is deliberately conservative: static spherical configurations, regularity diagnostics at the core, asymptotic matching to Schwarzschild-like exteriors, and a controlled slow-rotation extension. The paper formulates a reproducible framework for testing non-singular cohesive cores without changing the canonical field content. In the benchmark static scans, three cohesive families are generated and shown to produce smooth regular-center profiles with finite diagnostic invariants. The manuscript also introduces a slow-rotation template branch in the range 0 <= a/M <= 0.2 and reports conservative template-level observables, including a photon-ring proxy shift of about -9.1 percent. All quantitative claims are explicitly restricted to generated benchmark outputs, while full high-spin closure, waveform modelling, and data-level inference are deferred to later work.