HDC-CBC/Ic Correlational Index Ic, Historical Admissibility, and Executable Domain of Ct(a)

We present HDC–CBC/Ic, the fifth volume of the Supplementary Structural & Interpretative Program (SSIP) within the HDC–CBC framework. This work introduces no new physical dynamics, no new degrees of freedom, and no additional parameters. Its objective is to establish the structural status of the Correlational Index  once correlational coherence, treated in Ω as an effective constant magnitude, is promoted in Ωcₜ to the historical realization .

The starting point of the analysis is the central correlational condition of the framework,

interpreted not as a local equation of motion, but as the structural condition regulating coherence between the basal quantum domain and the projected geometric regime. We show that, once the minimal promotion

is admitted, the architecture of the model requires an explicit, dimensionless, and controlled parametrization of the historical correlational state. That parametrization is the Correlational Index,

which must not be understood as a free fitting parameter, but as the minimal historical form under which the irreducible correlational sector can be described without being artificially reabsorbed into the static hypothesis of Ω.

Building on the results of HDC–CBC/I and HDC–CBC/Ib, we argue that  must be interpreted as an index of historical admissibility. In this reading, the index does not freely parametrize an undifferentiated space of possible histories, but rather organizes access to a domain severely restricted by three cumulative criteria: projective compensation, structural stability, and persistence under coarse-graining. Not every mathematically definable history of  is physically realizable; only a subclass can sustain projectable regimes and, within it, only a part also proves operationally executable.

On this basis, we distinguish between:

the space of mathematically definable histories of the index,

the historically admissible domain,

and

the executable domain under the minimal validation environment of ΩCt/N. The pipeline is thus reinterpreted not as a fitting engine that grants physical meaning to the index from observational success, but as an operational environment that explores what part of the historically admissible domain can be deployed without breakdown under a criterion of structural consistency plus minimal observational compatibility. In this logic, the so-called operational window of  does not constitute the origin of the meaning of the index, but its minimal proof of executability.

The work further establishes a stricter methodological protocol for reading runs. The Planck-like, ICCUB-like, and SH0ES-like regions must not be treated as primary physical inputs of the analysis, but as derived observational basins assigned only after the calculation of , the checklist, and the GLOBAL state of the run. In analogous fashion, the references break-(BAO) and break+(WL) are interpreted not as arbitrary labels, but as operational frontiers of the executable domain of the index.

The main result of HDC–CBC/Ic is therefore a structural reinterpretation of the Correlational Index: neither eliminable, nor freely adjustable, nor ontologically autonomous, but rather the minimal index of historical admissibility and executable domain of correlational coherence within the HDC–CBC/Ωcₜ framework. In doing so, the volume completes the logical step linking irreducibility, admissibility, historical promotion, and operational falsifiability, thereby making explicit the place of  within the general HDC–CBC program.