A Communication Theory of Computation Realization

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.

This paper introduces the foundational object of Computation–Realization Geometry (CRG). A terminalized computation and resource contract define a legal realization family P; typed signatures yield the exact attainable region R, the upper obligation region Γ, and the additive-price-visible body K. Contextual equivalence gives canonical directed interface states in the declared tree model, while circuit–protocol translations and interactive examples show why realization identity must remain distinct from a resource image. Monotone reconstruction and nonnegative support duality identify Γ and K as exact decision-sufficient quotients for their respective objective classes. The Safe-Commitment Law characterizes, in both directions, when a retained family preserves every declared future decision and supplies an explicit regret witness when it does not. A strict engineering-cost separation corollary strengthens the necessity direction to continuous, coordinatewise strictly increasing, coercive costs. The paper also develops hierarchy coherence, resource monotonicity, products, local data processing, and communication synergy.

This paper develops a conditioned scaling theory for Computation–Realization Geometry. A Rent-like profile is defined as the multiscale observation of one legal realization under a declared hierarchy, accounting convention, estimator, and fit window. Stable dilation limits yield regular variation; preferred digital dilations permit periodic hierarchy phase; phase-ratio limits characterize power–slow-variation–phase decompositions; and positive periodic recursions with vanishing relative perturbations yield Perron–Floquet direction cycles. For heterogeneous modes, the local slope is a traffic-weighted mean and its curvature separates between-mode slope variance from within-mode drift. Interface reach, boundary caps, finite-window bias, concentration, and demand–service scale matching are derived within the same framework. Registered public-netlist diagnostics demonstrate hierarchy and estimator sensitivity, exactly close the declared same-hierarchy accounting identity as a software consistency check, and show nontrivial cross-hierarchy transfer failure. The result reclassifies a Rent exponent from a portable intrinsic scalar to a conditioned observation of a larger realization geometry.

This paper develops the design calculus of Computation–Realization Geometry. Technology enters after computation, legality, and attainability through feasible embeddings, service laws, and declared objectives. Finite algebraic registries induce semialgebraic technology phase cells with stable feasibility, active bottlenecks, class winners, ties, and margins. Inner constructive witnesses and scope-matched outer obligations form certified brackets and permit safe pruning. Argmin-hitting characterizes exact retention over a finite technology ledger, while price changes, embedding changes, and capability changes induce distinct reprice, re-embed, and reopen actions. The theory is connected to physical link count, serialization, energy, power, and shoreline laws; matrix multiplication, exact attention, state sharing, and ownership conversion; a working graph-to-certificate representation; and compact routed 2.5D/3D realizations. Computational-complexity limits of exact phase decomposition and minimum-cardinality ledger retention are stated explicitly. The resulting design discipline carries a selected realization together with its witnesses, bounds, margin, dependencies, and reopen conditions.

This paper develops selected exact deterministic fragments of Computation–Realization Geometry rather than claiming closure of the complete nonlinear, randomized, approximate, or physically embedded theory. It establishes circuit–protocol translations under named accounting rules, an exact interactive directional cut-rank orthant for finite-field linear computations, deterministic carrier/network-function-code equivalence, schedule- reachable temporal cut-state conservation, fractional certificate packing, hierarchy coherence, and exact retention statements for monotone and additive-price decision classes. The Safe-Commitment Law is supplemented by a strictly increasing coercive-cost separation result. Regular-variation, positive-system, boundary-collapse, and architecture–technology scaling theorems close additional declared slices. Each result keeps computation, ownership, legal transformations, hierarchy, accounting, and evidence class explicit, so theorems can be composed without transferring conclusions across incompatible contracts.

This paper presents a complete evaluation within a declared finite algebraic registry and typed exclusions for a long-context Transformer superblock. The registry distinguishes dedicated, universal time-shared, split-specialist, hybrid, memory-centric, and vertical realization classes under explicit state, traffic, execution, ownership, and technology contracts. Exact tensor-work equations, executed-pair policies, semantic bridge obligations, state aggregation laws, aligned-layout assignment, conserved timing, package-energy coordinates, integer allocation enumeration, technology phases, and context sensitivity are derived within that model. Standard, advanced 2.5D, and vertical technology contracts select different registered classes. A controlled follow-on retention experiment demonstrates decision-class-aware retention against winner-only and initial two-point Pareto reference policies, while compact routed 2.5D/3D models carry one semantic ownership boundary into physical timing, shoreline, area, temperature, test, yield, and reliability proxies. Exact decimal values identify the registered algebraic model; physical calibration and broader architecture coverage remain separate evidence questions.