A Communication Theory of Computation Realization: Resource-Conditioned Geometry of Legal Realizations and the Safe-Commitment Law

A computation specifies a semantic relation, not one inevitable algorithm, hierarchy, architecture, or physical implementation. This article introduces Computation–Realization Geometry (CRG), a computation-first theory of the legal ways a declared computation can become physical. A computation and resource contract define a hierarchy-indexed realization family P; typed signature maps produce the exact attainable region R, the upper obligation region Γ visible to continuous monotone resource decisions, and the convex upper body K visible to additive nonnegative prices. The Safe-Commitment Law gives necessary and sufficient conditions for a retained realization family to preserve every decision in either declared class. A strict engineering-cost separation corollary constructs a continuous, coordinatewise strictly increasing, coercive regret witness whenever the required geometry is lost.

The theory assembles contextual-state, hierarchy, and circuit–protocol constructions from established residual- state and communication-complexity arguments; establishes an exact interactive directional cut-rank orthant for finite-field linear computations; identifies deterministic carrier realizations with zero-error network function computation under matched contracts; and adapts convex support duality, temporal cut-state conservation, regular variation, positive-system contraction, and semialgebraic phase analysis to typed computation realizations. It reclassifies Rent-like laws as outputs of a declared realization, hierarchy, accounting convention, estimator, and fitting window. Technology is applied after legality and attainability, yielding a formal distinction between repricing an unchanged realization family and reopening it when capability or legality changes.

The framework is carried through matrix multiplication, exact attention, layout conversion, a complete evaluation within a declared finite Transformer registry and typed exclusions, a graph-to-certificate reference representation, a controlled retention experiment, and compact routed 2.5D/3D realizations. The work supplies a unified abstraction layer linking semantic obligation, legal realization, decision-sufficient retention, scaling, architecture selection, and physical evidence.