Connecting the Two Towers of Closure Theory: Primitive Closure Scales, Hyperbolic–Circular Geometry, and the Publication of Physical Invariants

Closure Theory (CT) investigates whether observable physical structure may arise from a more primitive geometrical closure. This paper connects two previously developed components of the programme: a primitive closure-scale construction based on the triplet {m0,L0,T0}\{m_0,L_0,T_0\}, and the FourWay HyperCirc construction combining hyperbolic invariant structure with compact circular closure. The analysis develops a common architecture in which the hyperbolic sector carries invariant cargo, the circular sector supplies compact completion and winding, and their combination forms a completed Curvature Invariant Capsule (CIC). The primitive triplet supplies associated dimensional scales, including V0=L0/T0V_0=L_0/T_0 and J0=m0L02/T0J_0=m_0L_0^2/T_0. The paper does not assume the physical identifications V0=cV_0=c, J0=hJ_0=h, or T=τT=τ. Instead, these are formulated as explicit tests of an unresolved publication map from completed intrinsic closure to relational physical representation. The appendices show respectively that intrinsic invariance does not determine the normalisation of proper time; common hyperbolic or null structure does not determine the scale relating V0V_0 to cc; and compactness, winding, and a natural symplectic structure do not by themselves determine a unique non-zero global action period. The resulting programme therefore replaces several proposed constant identifications by a more fundamental question: does completed intrinsic closure determine its own relational publication? The analysis identifies both structural compatibility and explicit normalisation obstructions, providing concrete criteria by which the proposed architecture can be further developed or falsified. Authors: Robert T. Morrow and ChatGPT (OpenAI).