This trilogy establishes a unified physical framework in which generation, persistence, and geometry are not independent domains, but successive eliminations of external assumptions, culminating in a single structural law:
Reality is the fixed point of its own admissibility.
Executable Geometry I — The Failure of Reachability-Based Engineering
demonstrated that physical realization cannot be explained through trajectories, continuity, or process. The concept of reachability collapses under observation: structures are not produced by motion, but by the selection of admissible configurations satisfying E(S) > 0. Realization is therefore not a temporal process, but a non-sequential condition of existence.
Executable Geometry II — Condition-Based Engineering
extended this result by showing that persistence is not survival through time, but invariance under self-induced admissibility. A realized configuration S* satisfies
Ξ_{S*}(S*) = S*,
excluding all non-admissible states prior to realization. Persistence is thus not maintained dynamically, but defined structurally as recirculation—the fixed-point invariance of admissibility itself.
Executable Geometry III — Reality as Self-Induced Geometry
completes the framework by eliminating background geometry. Geometry is not an external manifold, but the admissible structure induced by the realized configuration:
M_adm ≡ M_adm(S*).
Configuration, admissibility, and geometry form a co-generated system satisfying the consistency condition:
(S*, Ξ, M_adm) = 𝒞(S*, Ξ, M_adm).
Existence is therefore equivalent to fixed-point consistency under co-generation.
Across the trilogy, the primitives of classical physics—space, time, dynamics, and causality—are successively removed, not by assumption, but by logical necessity. What remains is a structural ontology in which:
Generation is admissibility-driven selection,
Persistence is admissibility-locked invariance,
Geometry is the self-induced admissible structure.
These are not independent mechanisms, but three inseparable expressions of a single principle. Space is not a background in which reality exists; it is the structure induced by admissibility. Time is not a fundamental dimension; it is an observational ordering across admissible configurations. Dynamics is not primary; it is an effective description of relations between fixed-point structures.
Executable Geometry therefore replaces motion, continuity, and external geometry with admissibility-driven consistency as the basis of physical reality. Existence is not defined by evolution, but by the condition under which realization is permitted.
This establishes the fundamental physical law:
Existence ≡ Fixed-point consistency under self-induced admissibility.
Reality is the fixed point of its own admissibility.
