Executable Ecology: Multi‑Sovereign Admissibility and the Geometry of Co‑Generated Reality

This work establishes the fourth stage of Executable Geometry by extending the theory from single‑sovereign systems to multi‑sovereign reality. Executable Geometry I–III demonstrated that physical existence is determined by admissibility‑driven fixed‑point consistency under a single self‑induced enforcement operator $\mathrm{\Xi }$. The present study generalizes this formulation to describe real‑world systems in which multiple independent execution structures coexist.

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In the single‑sovereign regime, a realized configuration $S^{\ast }$ satisfies:

$${\mathrm{\Xi }}_{S^{\ast }}(S^{\ast })=S^{\ast },$$

and induces its own admissible structure:

$$M_{\text{adm}}\equiv M_{\text{adm}}(S^{\ast })\mathrm{.}$$

Within this closed structure, existence, persistence, and geometry are determined by a single sovereign enforcement operator. However, this closure implicitly assumes that admissibility is governed by a unique sovereign domain.

Here, we show that biological systems—exemplified by plant–microbe–metabolite interaction networks—operate in a fundamentally different regime in which multiple independent execution operators coexist:

$${\mathrm{\Xi }}_{\text{gene}},{\mathrm{\Xi }}_{\text{microbe}},{\mathrm{\Xi }}_{\text{metabolite}}\mathrm{.}$$

Empirical analysis using multi‑omics integration, machine learning (Random Forest, XGBoost), and game‑theoretic attribution (SHAP) reveals three invariant signatures of this regime:

(1) Competing Sovereignties Admissibility emerges from the coexistence of multiple independent enforcement structures rather than from a single operator.

(2) Sovereignty Switching Environmental conditions induce discontinuous transitions in the dominant operator governing admissibility. Genetic sovereignty dominates under stable conditions, whereas metabolic and microbial sovereignty dominate under stress.

(3) Non‑Linear Co‑Generation Realization is determined not by individual variables but by non‑linear cross‑domain combinations, indicating that admissibility is defined by geometric consistency across operators rather than additive contributions.

These observations cannot be explained by single‑sovereign admissibility and therefore require a formal extension of Executable Geometry.

We introduce multi‑sovereign admissibility:

$$M_{\text{adm}}=\bigcap _i{M}_{\text{adm}}^{(i)},$$

where each admissible manifold $M_{\text{adm}}^{(i)}$ is induced by an independent execution operator ${\mathrm{\Xi }}_i$. A configuration $S^{\ast }$ is realized if and only if:

$$\mathrm{\forall }i,{\mathrm{\Xi }}_i(S^{\ast })=S^{\ast },$$

which defines the intersection fixed‑point condition:

$$S^{\ast }\in \bigcap _i\text{Fix}({\mathrm{\Xi }}_i)\mathrm{.}$$

This generalizes the fixed‑point principle from self‑recirculation to co‑generated consistency across sovereign domains.

Furthermore, we establish the principle of environment‑dependent sovereignty switching, showing that external conditions reweight admissibility geometry and induce discontinuous shifts in the governing execution structure.

These results lead to the fourth foundational law of Executable Geometry:

Reality is the fixed point of multi‑sovereign admissibility.

Accordingly, Executable Ecology replaces single‑agent causality with co‑generated admissibility across interacting systems. Biological, ecological, and distributed systems are governed not by a single internal logic but by the geometric consistency of multiple competing execution structures.

EG‑IV — Multi‑Sovereign Geometry: Reality is determined not by a single admissibility structure, but by the intersection of admissibility domains.