MCK Region Transitions: Restricted Realization Mechanisms and Their Limits
Davide Bragetti, Edoardo Palumbo, Carmelo Asaro, Giuseppe Francesco Italiano · Zenodo (CERN European Organization for Nuclear Research) · 2026
The MCK framework partitions artifacts into seven regions by three predicates — meaning, opacity, function — and its region transitions form a group acting on the region set. The result is algebraic: it exhibits the transition between any two regions and supplies no procedure that moves an artifact along one. We study four restricted realization families and fix, for each, what it covers and where it stops. Assimilation reclassifies pure opacity and nothing else; we prove that the residue — the opacity co-present with meaning or with function — is a floor no comprehension strength lowers, so an agent that possesses symbols cannot assimilate its way out of possessing them. Operator admission seats a candidate in the fully intersecting region without extending structural reach, so any extension of reach under a fixed realization witnesses a changed basis rather than a new name. Denotation admits as sources exactly the symbol domain and is not recoverable from the predicates: perfect recovery of a region label leaves denotation undetermined. The reconstruction certificate is strictly weaker than the discovery predicate, the gap being a key-membership condition we show unclosable in general. Each mechanism is studied for a declared observer; no universal artifact-level realizer follows. The realization is ideal and exact; whether a trained representation supports an approximate counterpart is not addressed here, and no infinite-reachability claim rests on finite evidence. Five temporal conjectures over locally declared runs mark open trace-level questions; no monitor is constructed. --- RELATIONSHIP TO "MEANING, OPACITY, AND FUNCTION" (concept DOI 10.5281/zenodo.21772772). FOUNDATIONAL IMPORT. The seven-region partition, the three predicates, the structured ambiguity domain, the suspension convention on the purely functional cell, the semantic horizon, and the denotation map are taken from that paper and used exactly as defined there. They are not re-derived, and no embedding or reduction between the two settings is claimed. EXTENSION. That framework establishes, for any ordered pair of regions, that a unique transition between them exists, and supplies no procedure that carries a given artifact along one. This paper studies four restricted realization mechanisms rather than the general case, and fixes for each what it covers and where it stops. Nothing here closes that gap. NEW RESULTS. The invariance of the symbol domain under assimilation and the residual-opacity floor; the admission of an operator without extension of structural reach; the non-recoverability of denotation from the predicates; the key-membership gap between the reconstruction certificate and the discovery predicate; and the denotation graph with its four modes and fourteen canonical symbol classes, which discharges a deferral the foundations paper makes by name. CONJECTURES. Five statements written with temporal operators over locally declared runs. All are conjectural or explicitly open, none is used in a proof, and no monitor is constructed.