Complete Local Integration (CLI)

• Complete Local Integration (CLI) is a quantitative concept that measures the mutual dependence among components in spatiotemporal patterns using probabilistic and differential geometric frameworks.
• It applies to Bayesian networks, Markov chains, and Lie algebroid integration, identifying irreducible substructures known as ‘iota-entities’ through specific local integration (SLI).
• CLI underpins formal agent theory by providing intrinsic criteria for action and perception, while establishing a symmetry-based hierarchy of structural decompositions.

Complete Local Integration (CLI) is a formal quantitative concept for detecting the internal cohesion of spatiotemporal patterns, introduced for general Bayesian networks and Markov chains to rigorously identify entities—structures whose parts mutually reinforce each other's likelihood—based only on probabilistic interdependencies, as well as in the explicit construction of local Lie groupoids integrating a given Lie algebroid. CLI appears in agent theory for multivariate stochastic systems and in differential geometry for the local globalization of infinitesimal structures.

1. CLI in Multivariate Stochastic Processes

Complete Local Integration is defined for trajectories ω in a Bayesian network or Markov chain, where the nodes V index random variables. For any partition π of V, the Specific Local Integration (SLI) of ω with respect to π is:

$$\mathrm{SLI}_\pi(\omega) := \log\left[\frac{p_V(\omega)}{\prod_{b \in \pi} p_b(\omega_b)}\right]$$

where $p_V(\omega)$ is the joint probability and $p_b(\omega_b)$ is the marginal probability over block $b$. SLI quantifies the degree to which the mutual presence of all variables in ω enhances its probability over what would be expected from independent blocks.

CLI selects the minimal mutual dependence over all non-trivial partitions:

$$\mathrm{CLI}(\omega) = \min_{\pi \neq \text{unit}} \mathrm{SLI}_\pi(\omega)$$

A pattern ω is completely locally integrated if $\mathrm{CLI}(\omega) > 0$, indicating irreducible dependence among its constituent parts under all possible nontrivial decompositions (Biehl, 2017).

2. Structural and Symmetry Properties

CLI induces a hierarchy of partitions—called the refinement-free disintegration hierarchy—where the blocks surviving at each SLI minimum level correspond exactly to completely locally integrated subpatterns, also termed “iota-entities.” The Disintegration Theorem states that all blocks in these partitions have positive CLI and every CLI>0 pattern appears in one such block at a minimal level.

Furthermore, SLI exhibits strong symmetry properties: if a permutation $g$ leaves $p_V$ invariant, then for any ω and π,

$$\mathrm{SLI}_{g\pi}(g\omega) = \mathrm{SLI}_\pi(\omega)$$

Such symmetries manifest as orbits of partitions with identical SLI in the disintegration lattice (Biehl, 2017).

3. CLI and the Formal Theory of Agents

CLI provides an intrinsic, observation-independent criterion for defining entities as spatiotemporal patterns where every part increases the probability of every other. This yields a quantitative entity-set (“iota-entities”) applicable to arbitrary Markov chains, unlike the perception-action loop formalism which only accounts for exhaustive, stationary sets of random variables.

From CLI, formal definitions of action and perception follow. Co-action iota-entities (occupying the same variables at $t$, identical environment, distinct states at $t+1$) embody the essential unpredictability necessary for agency. Similarly, co-perception iota-entities encode branching conditionals over future slices given identical pasts and environments, generalizing beyond fixed variable sets. These constructions underpin intrinsic agent-detection in complex stochastic dynamical systems (Biehl, 2017).

4. Computational Aspects and Examples

CLI can be computed for finite Markov chains by enumerating all partitions and evaluating SLI for each. Small-scale worked examples with two binary variables over three timesteps reveal that only a finite set of SLI values are realized, and the surviving blocks in the refinement-free hierarchy constitute all iota-entities. For systems with noise, these entities diversify to include spatiotemporal compositionality and counterfactual variation in both extent and value. Typical CLI values reflect the statistical strength with which components mutually enhance their joint probability over the product baseline, ranging from near-zero for weakly coupled patterns up to nearly 1 bit for maximally interdependent structures (Biehl, 2017).

5. CLI in Local Integration of Lie Algebroids

Complete Local Integration also denotes an explicit constructive procedure in differential geometry for integrating Lie algebroids to local Lie groupoids. Given a Lie algebroid $A \to M$, a spray vector field $V$ satisfying homogeneity and anchor-lifting conditions is chosen:

• $(m_t)_* V = t V$ for fiber-wise scalar multiplication $m_t$
• $dq(V(a)) = \rho(a)$ for the anchor $\rho$ and projection $q$

The flow $\varphi^t$ of $V$ defines an open neighborhood $G_V \subset A$, with structure maps:

• Source $s(a) = q(a)$
• Target $t(a) = q(\varphi^1(a))$
• Inversion $i(a) = -\varphi^1(a)$

Multiplication arises from solving a realization-form ODE involving a right-invariant Maurer–Cartan form $\theta$:

$\theta(\dot k(t)) = \varphi^t(a), \quad k(0) = b$

The ODE guarantees the existence and uniqueness of localized groupoid multiplication. Reconstruction of groupoid axioms, anchor, and bracket structure proceeds to first order via Taylor expansions, and all axioms are verified by ODE solubility. This process yields a finite-dimensional equivalence between germs of local Lie groupoids and Lie algebroids (Cabrera et al., 2017).

6. Category-Theoretic Equivalence via CLI

The CLI construction enables an explicit proof of the equivalence between the category of local Lie groupoids (objects: groupoids whose structure maps are defined near the units) and the category of Lie algebroids. The Lie functor assigns to each local groupoid its infinitesimal algebroid. Any algebroid morphism integrates uniquely to a germ of a groupoid morphism, and every algebroid is integrated by precisely such a CLI spray-groupoid construction. This equivalence is central to local Lie theory and can be implemented without reference to infinite-dimensional spaces or global integrability (Cabrera et al., 2017).

7. Context and Broader Significance

As a unifying paradigm, Complete Local Integration supplies a principled basis for identifying entities in dynamical systems—whether probabilistic, algebraic, or geometric—by pinpointing substructures whose internal dependencies are irreducible to independent subsystems. In probabilistic systems, CLI quantifies intrinsic unity; in differential geometry, it operationalizes the passage from infinitesimal to local global objects. These rigorous notions clarify intrinsic agency, local structure formation, and the finite-dimensional localization of global geometric data. A plausible implication is that CLI-type criteria may generalize further to describe emergence and structure-preserving morphisms in broader mathematical and computational contexts (Biehl, 2017, Cabrera et al., 2017).