
Conceptual update:
- Simplified PMF incorporated in the Structural Intelligence Framework
- Entity is the common managed-object base class.
- Host, Sensor, Signal, Event, Knowledge, Agent, Interface, and Log are Entity subclasses..
Open object-oriented models for accountable AuI

Gradient Modulation (∇) is the dynamic salience engine of mind and SGI. If Π organizes structure and Μ defines geometry, ∇ determines which structures and regions come alive under the pressures of life, thought, context, and novelty.

Functoriality (ℱ) is the structural requirement that mappings between Worlds preserve relational integrity among polarities, hierarchies, and contextual modulations. It balances rigidity with flexibility and exactness with approximation. Failures include drift, misalignment, over/under-fitting, and non-invertibility. Across philosophy, psychology, society, and SGI, ℱ is the principle that sustains mutual intelligibility between distinct Worlds.

Multi-Axis Interaction (Μ) governs how multiple σ-axes combine to generate the multidimensional structure of a World. It balances independence with coupling, additivity with emergence, and local with global coherence. Failures include axis collapse, pathological coupling, under-coupling, malformed influence patterns, and volatility.

Recursion is the principle that any expressed polarity (σ) can become the basis for further differentiation, generating new sub-axes that inherit, refine, and extend the structure of their parent polarity.

A Polarity System (Π) is the multidimensional network of σ-axes that generates the differentiated architecture of a World (Wᵢ). Π organizes oppositions into coherent structure, defining a World’s expressive capacity, semantic depth, and modes of intelligibility. Through Π, individual polarities combine, constrain, and transform one another, producing the patterned field within which meaning, reasoning, emotion,…

Mapping (A9) is the operator that makes cross-world relationality possible. It translates, adapts, preserves, and harmonizes structural meaning across contexts and Worlds. Failures include reduction, over-differentiation, misalignment, gradient collapse, and instability. In philosophy of mind, mapping grounds perspective-taking, empathy, narrative identity, and meaning repair. In SGI, mapping is essential for task coordination, concept transfer, user…