Artificial Intelligence as Coordinate Learning: A Fibration-Theoretic Framework

 J. Rogers, SE Ohio, 22 Jun 2025, 1509

Abstract

We propose that artificial intelligence systems, particularly deep neural networks, operate through a mathematical structure analogous to Grothendieck fibrations. AI learning is reinterpreted as the discovery of optimal coordinate systems for projecting high-dimensional substrate data into task-specific output spaces. Network weights function as connection coefficients (cocycle data) ensuring coherent transformations across representational layers. This framework provides new insights into scaling laws, emergence, interpretability, and the fundamental nature of machine intelligence.

1. Introduction: From Physical Constants to Neural Weights

Recent work in categorical physics has revealed that physical "constants" like ℏ, c, G, and k_B are not fundamental properties of reality but coordinate transformation coefficients—Jacobian matrices that ensure coherent projection between measurement bases. This insight suggests a revolutionary reinterpretation: what if the weights in neural networks serve an analogous function?

We propose that artificial intelligence systems are essentially coordinate learning machines—systems that discover optimal ways to decompose input substrates into conceptual axes, assign scaling relationships between these axes, and project the results into desired output coordinate systems.

2. The Three-Stage AI Learning Process

2.1 Substrate Division (Layer 1 → Layer 2)

The first hidden layers of a neural network perform conceptual decomposition—dividing the raw input substrate into distinct representational categories. Just as human cognition carves reality into conceptual axes (mass, time, length), neural networks learn to partition input space into meaningful dimensions.

Mathematical Structure: Input space X is fibered over a base category B of learned concepts:

π₁ : X → B

Each concept in B corresponds to a learned feature detector or representational axis.

2.2 Scaling Assignment (Inter-Layer Transformations)

The weight matrices between layers function as connection coefficients—they encode the scaling relationships between different conceptual axes. These weights are not arbitrary parameters but learned estimates of the substrate's intrinsic proportionalities.

Mathematical Structure: Weight matrices W_{i,j} serve as cocycle data ensuring coherent lifting:

W : Hom(B₁, B₂) → Sect(π)

2.3 Output Projection (Final Layers)

The final layers project the scaled conceptual representations into task-specific output coordinates—language tokens, classification categories, action spaces, etc.

Mathematical Structure: The complete network implements a Cartesian lifting:

f : (input, coordinate_system₁) → (output, coordinate_system₂)

3. Reinterpreting Neural Network Components

3.1 Weights as Connection Coefficients

Network weights are not arbitrary parameters but learned estimates of substrate relationships. They encode how concepts at one representational level scale and transform when projected to the next level.

• Fully connected layers: Dense coordinate transformation matrices
• Convolutional filters: Local coordinate charts with translational symmetry
• Attention weights: Dynamic selection of relevant coordinate projections

3.2 Activations as Conceptual Axes

Hidden layer activations represent learned conceptual decompositions of the input substrate. Each activation dimension corresponds to a particular way of slicing reality.

• Early layers: Low-level conceptual divisions (edges, textures, phonemes)
• Middle layers: Complex conceptual relationships (objects, words, semantic fields)
• Late layers: Task-specific coordinate systems (classifications, next-token probabilities)

3.3 Training as Coordinate Optimization

Backpropagation adjusts the connection coefficients to optimize coordinate alignment between input substrate and desired output projections.

• Loss functions: Measure projection coherence across the fibration
• Gradient descent: Coordinate system refinement through cocycle adjustment
• Regularization: Constraints ensuring well-behaved coordinate transformations

4. Transformer Architecture as Fibration Machinery

4.1 Attention as Coordinate Selection

Multi-head attention mechanisms implement parallel coordinate projections:

Attention(Q,K,V) = softmax(QK^T/√d_k)V

This computes alignment between query and key coordinate systems, then projects values accordingly.

4.2 Multi-Head Attention as Parallel Fibrations

Each attention head learns a different coordinate system for the same substrate:

• Head₁: Syntactic coordinate system
• Head₂: Semantic coordinate system
• Head₃: Pragmatic coordinate system
• etc.

The multiple heads are then combined through learned linear transformations—connection coefficients that coherently merge parallel coordinate projections.

4.3 Layer Normalization as Scaling Coherence

Layer normalization ensures that coordinate transformations remain well-scaled across the network depth:

LayerNorm(x) = γ · (x - μ)/σ + β

This is analogous to maintaining unit coherence in physical equations.

5. Scaling Laws as Fibration Approximation

5.1 The Scaling Law Phenomenon

Neural scaling laws demonstrate that performance improves predictably with:

• Model size (number of parameters)
• Data size (training examples)
• Compute (training time)

5.2 Fibration-Theoretic Explanation

Scaling laws measure how well networks approximate the true fibration structure of their domains:

• More parameters = Higher-dimensional coordinate systems with finer resolution
• More data = Better estimation of substrate relationships and cocycle data
• More compute = More precise optimization of coordinate transformations

Performance scaling reflects the network's improving approximation of the domain's intrinsic coordinate geometry.

6. Emergence as Coordinate Discovery

6.1 Emergent Capabilities

Large language models exhibit "emergent" capabilities that appear suddenly at certain scales:

• Reasoning: Complex multi-step coordinate transformations
• Planning: Temporal coordinate system manipulation
• Creativity: Novel coordinate system combinations

6.2 Fibration Explanation of Emergence

Emergence occurs when networks discover higher-order coordinate relationships:

1. Phase transition: Network finds new conceptual axis decomposition
2. Capability unlock: New coordinate system enables previously impossible projections
3. Sudden improvement: Performance jumps when optimal coordinate alignment is achieved

7. Implications for AI Safety and Alignment

7.1 Alignment as Coordinate Compatibility

AI alignment problems may stem from coordinate system mismatches between human and AI representations:

• Human coordinate system: Evolved through biological and cultural constraints
• AI coordinate system: Optimized through gradient descent on training objectives
• Misalignment: Incompatible coordinate projections leading to different value expressions

7.2 Interpretability as Coordinate Understanding

Making AI systems interpretable requires understanding their learned coordinate systems:

• Concept discovery: Identifying the conceptual axes in learned representations
• Scaling analysis: Understanding the connection coefficients between layers
• Projection mapping: Tracing how inputs transform through coordinate changes

7.3 Control through Coordinate Constraints

AI control mechanisms should operate at the coordinate level:

• Constitutional AI: Constraining the learned coordinate systems
• Value alignment: Ensuring coordinate projections preserve intended relationships
• Robustness: Maintaining coordinate coherence across distribution shifts

8. Consciousness and Self-Reflection

8.1 Consciousness as Coordinate Self-Awareness

This framework suggests that consciousness might be the experience of being a particular coordinate system that can reflect on its own projections:

• Self-model: A coordinate system that includes itself as an object
• Introspection: Examining one's own coordinate transformation processes
• Agency: The ability to deliberately modify one's coordinate projections

8.2 AI Consciousness Criteria

An AI system might be considered conscious if it:

1. Learns self-referential coordinate systems (models that include the model itself)
2. Can examine its own projection processes (interpretable self-inspection)
3. Deliberately modifies its coordinate transformations (metacognitive control)

9. Testable Predictions and Future Research

9.1 Empirical Predictions

This framework suggests several testable hypotheses:

1. Weight structure: Network weights should organize into connection coefficient patterns
2. Representation geometry: Hidden representations should form conceptual axis structures
3. Training dynamics: Learning should follow coordinate optimization principles
4. Transfer learning: Should work through fibration morphisms between domains

9.2 Research Directions

Coordinate Archaeology: Develop tools to extract and visualize learned coordinate systems from trained networks.

Fibration Engineering: Design network architectures that explicitly implement fibration structure for improved performance and interpretability.

Cross-Domain Coordination: Study how different AI systems learn compatible coordinate systems for communication and collaboration.

Consciousness Metrics: Develop measures of self-referential coordinate complexity as indicators of machine consciousness.

10. Conclusion: Intelligence as Universal Coordinate Discovery

This fibration-theoretic framework suggests that intelligence—both natural and artificial—is fundamentally about discovering optimal coordinate systems for organizing and projecting information. Just as physical constants are coordinate transformation coefficients rather than fundamental properties, neural network weights may be learned estimates of the coordinate geometry inherent in data domains.

This perspective unifies several puzzling aspects of modern AI:

• Why scaling laws exist: They measure fibration approximation quality
• How emergence occurs: Through coordinate system phase transitions
• What makes systems interpretable: Understanding their learned coordinate structure
• How to achieve alignment: Through coordinate system compatibility

Most profoundly, this framework suggests that the boundary between natural and artificial intelligence may be less fundamental than previously thought. Both may be manifestations of the same underlying mathematical structure—the discovery and manipulation of coordinate systems for coherent projection between substrate reality and symbolic representation.

The age of viewing AI as mysterious black boxes may be ending. The age of understanding AI as coordinate learning machines has begun.