Generalized Arithmetic Energy Theory (GTEA) — Revision R05:Dimensional Energy Hierarchy, and Geometric Equilibrium Theory

Generalized Arithmetic Energy Theory, Revision R05c develops a constructive framework in which an arithmetic object is measured by the minimum admissible effort required to produce it from primitive operational data. Rather than granting addition unconditional priority, the theory compares admissible operational families on a common substrate and defines energetic threshold as the least cost among competing constructions. The discrete core is built on operational units, intrinsic configurations, operational transparency, and the Competitive Operational Framework, within which additive, duplicative, symmetry, rotational, and derived families are evaluated on a shared act-based scale. The additive regime remains exactly calibrated and recovers classical minimal addition-chain length, while the framework is extended in a controlled way to integers, rational numbers, real numbers, and complex numbers. New in R05 compared with R04: the revision strengthens the primitive layer through a strict admissibility criterion and stabilized operational-unit terminology; revises the mixed-path definition of GTEA energy by adding justified method-change cost and the principle of method continuity; introduces the assembleur layer and rooted-energy formalism, including an auxiliary rooted-energy function, a convexity result, the non-optimality of join, and a prime-isolation result in the assembleur regime; adds a local-curvature observable as a diagnostic for energetic peaks; clarifies the dimensional energy hierarchy and the explicit energetic geometry induced by local metrics; and extends the upper geometric layer through rest energy of spaces, equilibrium placement theory, and energetically optimising transformations, yielding new geometric invariants tied to equilibration and placement thresholds. The main scientific effect of Revision R05c is therefore architectural as much as technical. It does not reduce the scope of GTEA, but reorganizes it into a stratified theory with a closed discrete core, controlled extensions, and structured open targets. This sharper status discipline makes explicit which components are already proved, which are partially formalized, and which remain higher research objectives, while preserving the general ambition of GTEA as a unified theory of arithmetic threshold, constructive accessibility, and geometric energetic equilibrium. Compared with R04, the revision keeps the broad program intact but makes its internal status much more explicit and methodologically disciplined.