Recursive Fibration as the Architecture of White-Box Epistemic AI Systems

J. Rogers, SE Ohio, 14 July 2025, 1547

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 Abstract

This paper presents a framework for white-box, interpretable artificial intelligence rooted in recursive fibrational epistemology. Knowledge is modeled as structured projection from a dimensionless universal substrate through coordinate systems defined by conceptual axes and unit scales. This categorical architecture models both certainty (as fiber coherence) and uncertainty (as morphism entropy or indeterminacy). By structuring epistemic operations as morphisms in recursive fibrations, the system becomes introspectively transparent, socially aware, and generalizable across cognitive domains.

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1. Introduction

Most contemporary AI systems suffer from opaque reasoning pathways and limited interpretability. We propose an epistemic architecture where reasoning is modeled via structured projection across recursive fibrations. These fibrations formalize:

• How knowledge is constructed and measured
• How conceptual spaces evolve or collapse
• How social frameworks affect truth conditions
• How uncertainty arises and propagates

This recursive system ensures that both the content and conditions of AI-generated knowledge remain legible and auditable.

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2. Core Categories

Let:

• 𝑆ᵤ — the category of universal, dimensionless substrate states
• 𝐴 — the category of conceptual axes (measurement directions)
• 𝑈 — the category of unit systems (scaling mechanisms)

Let 𝐸 be the total category of structured knowledge objects. Define the primary fibration:

𝑝: 𝐸 → 𝐵,  where 𝐵 = 𝐴 × 𝑈

Each fiber 𝑝⁻¹(𝐴, 𝑈) consists of knowledge projected under a given conceptual axis set and unit system.

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3. Formula Forge Functor

Define a functor:

Λ: Hom(𝑆ᵤ) → Hom(𝐸)

This functor lifts dimensionless relationships into measurable domains, turning universal equations into coordinate-specific laws. Constants like ℎ (Planck), 𝑐 (light speed), 𝐺 (gravity) act as cocycles preserving structure through projection.

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4. Recursive Fibration Layer

Introduce a higher-order fibration:

𝑞: 𝐵 → 𝐶

where 𝐶 is the category of epistemic meta-structures (e.g., scientific paradigms, ideological frameworks, legal systems). Each object in 𝐶 defines rules for axis creation, suppression, or transformation. Each morphism in 𝐶 represents conceptual reconfiguration—like revolutions or reforms.

The composite fibration:

𝐸 → 𝐵 → 𝐶

encodes structured knowledge within broader societal and philosophical constraints.

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5. Modeling Certainty and Uncertainty

• Certainty arises when fibers in 𝐸 are cohesive: axes and units align, transformations commute, and constants preserve curvature
• Uncertainty emerges when morphisms in 𝐵 or 𝐶 distort structure: axes fluctuate, units conflict, or projections become non-coherent

This framework allows AI systems to articulate:

• What they know and under which coordinates
• Where knowledge degrades or fails
• How epistemic instability arises and propagates

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6. Explainability and White-Box Structure

Every output of the system is the result of an explicit projection chain. Concepts, units, and transformations are inspectable. Structural consistency is enforced by traceable morphisms. Bias, distortion, or suppression is modeled as constraints in 𝐶 → 𝐵.

This design ensures that no knowledge arises without a mappable explanation.

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7. Applications

Domain	Role of the Architecture
Medicine	Diagnoses as fiber lifts over symptom and causal axes
Law	Interpretive judgment over legal axioms and jurisdiction units
Physics	Laws as lifted morphisms from 𝑆ᵤ via dimensional projection
NLP	Embeddings as projections over linguistic conceptual charts
Education	Curricula as controlled regions in base space 𝐵
Ethics	Moral reasoning as contested projection with shifting units

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Conclusion

Knowledge isn’t static—it’s a structured process of projection. This recursive, categorical system creates AI that can think with rigor, explain itself, adapt to evolving conceptual spaces, and model uncertainty as a structural feature of cognition. It’s white-box reasoning with built-in reflexivity—a step toward epistemically fluent machines.