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https://github.com/potatoinfinity/geo-llama

Creating an AI that is 100x more efficient, has infinite memory, and exhibits mathematically certain reasoning with the utilization of Conformal Manifold by using Geometric Algebra.
https://github.com/potatoinfinity/geo-llama

compute conformal-geometry efficiency future gapu geometric-algebra geometric-algorithms large-language-models llama4 math mathematics memory modern optimizations paradigms resource rust-lang shift topology

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Creating an AI that is 100x more efficient, has infinite memory, and exhibits mathematically certain reasoning with the utilization of Conformal Manifold by using Geometric Algebra.

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# Geo-Llama: Foundational Theory of Structural Intelligence via Conformal Manifolds and $Cl_{4,1}$ Recursive Isometries



**Date:** January 20th, 2026  

**Authors:** Trương Minh Huy, Edward George Hirst  

**Subject:** Geometric Deep Learning, Isotropic Spatio-Temporal Modeling, Structural Latent Manifolds



![Version](https://img.shields.io/badge/version-1.1.0--theoretical-blue) ![Algebra](https://img.shields.io/badge/algebra-Cl(4,1)-red) ![Status](https://img.shields.io/badge/status-Research_In_Progress-orange)



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## Abstract



Contemporary deep learning architectures are fundamentally constrained by their reliance on high-dimensional Euclidean embeddings and the statistical approximation of relationships. These "Flat-AI" models treat data points as isolated coordinates in $\mathbb{R}^n$, necessitating massive parameter counts to emulate structural dependencies that are naturally present in complex physical and hierarchical systems. We present **Geo-Llama** (Geometric Latent Language & Manifold Architecture), a novel framework that moves beyond Euclidean translation to a native **Conformal Geometric Algebra (CGA)** representation. By encoding information into the Clifford Algebra $Cl_{4,1}$, we represent state evolution not as weight-driven shifts, but as **Isotropic Rotations** (Rotors) within a 5D Minkowski-signature manifold. This paper outlines the move from stochastic pattern matching to **structural intelligence**, detailing the advancements in **Geometric Product Attention (GPA)** and the **$O(1)$ Recursive Rotor Accumulator**. This shift enables a new class of foundation models capable of native reasoning in high-fidelity topological environments where spatial, temporal, and hierarchical constraints are primary.



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## 1. The Euclidean Crisis: Topological Entropy



The scaling laws of standard Transformers $O(N^2)$ are reaching a ceiling dictated by the entropy of flat space. This "Euclidean Bottleneck" manifests through:



1.  **Semantic Sparsity:** In high-dimensional flat space, the **Curse of Dimensionality** ensures that tokens remain semantically distant, forcing models to rely on hyper-fine weights to capture structural nuance.

2.  **Contextual Decay (The Memory Wall):** Euclidean memory (the KV Cache) is a non-compressed history—a growing list of points that eventually exceeds computational limits.

3.  **Lack of Grade:** Standard vectors occupy a singular rank, preventing models from natively enforcing hierarchical or containment relationships ($A \subset B$) without exhaustive training data.



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## 2. Theoretical Framework: The $Cl_{4,1}$ Conformal Manifold



Geo-Llama utilizes the **Conformal Model of Geometry**, mapping latent states into the Clifford Algebra $Cl_{4,1}$, generated by a 5D basis $\{e_1, e_2, e_3, e_+, e_-\}$ with the signature $(+,+,+,+,-)$.



### 2.1 Latent Space Manifold Bundling



Rather than a single massive embedding, we treat the latent space as a **Fiber Bundle** of $H$ independent manifolds:



$$ \mathbb{R}^{d_{model}} \xrightarrow{\phi} \bigoplus_{h=1}^{H} Cl_{4,1}^{(h)} $$



Where each "Geometric Head" operates on a 32-dimensional multivector basis. This architecture ensures that structural rotations in one manifold (e.g., spatial orientation) do not corrupt the integrity of another (e.g., logical hierarchy). By utilizing **Grades**, the model inherently encodes containment: a Token-Point can be mathematically proven to reside within a Category-Sphere via the inner product, reducing high-level reasoning to geometric intersection.



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## 3. GPA: Geometric Product Attention



We redefine attention as an algebraic interaction between multivector fields. Instead of the scalar Dot-Product, we employ the full **Clifford Geometric Product**:



$$ \mathcal{A}(Q, K) = Q \cdot K + Q \wedge K $$



*   **Symmetric Part ($Q \cdot K$):** Captures traditional semantic similarity (proximity).

*   **Anti-Symmetric Part ($Q \wedge K$):** Generates a **Bivector**, representing the "Plane of Interaction"—the directed structural tension and rotational relationship between concepts.



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## 4. The $O(1)$ Recursive Rotor Accumulator



The core innovation of Geo-Llama is the transition from "State as Data" to "State as Path." 



### 4.1 From Memory Lists to Spinor States



In standard models, history is a database. In Geo-Llama, history is a **Rotor** (an element of the $Spin(4,1)$ group). Every incoming piece of structural information is transformed into a specialized rotor $R_i$, and the global context $\Psi$ is updated through a recursive sandwich product:



$$ \Psi_{t+1} = R_{t} \Psi_{t} \tilde{R}_{t} $$



*   **Recursive Isometry:** Because $R$ is a rotor, the geometric integrity is preserved. The system state is "rotated" by the meaning of new data, achieving infinite theoretical context in a fixed (32-float) memory footprint per head.



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## 5. Hardware Implications: The GAPU



To realize these theoretical gains, we identify the need for specialized hardware: the **GAPU (Geometric Algebra Processing Unit)**.



*   **Parallel Cayley Streams:** Unlike standard GPUs, the GAPU processes multiple manifolds in parallel through bit-masked algebraic instructions ($e_1 e_2 = e_{12}$).

*   **Efficiency:** By loading algebraic rules into the instruction set rather than calculating them as matrices, the GAPU achieves nearly 100% compute occupancy for multivector products.



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## 6. Empirical Validation



To validate the theoretical advantages of the $Cl_{4,1}$ manifold, we performed a series of controlled experiments comparing Geo-Llama against standard Euclidean Transformers and domain-specific SOTA architectures.



### 6.1 Recursive Spatio-Temporal Modeling (N-Body Gravity)

In a 5-body gravitational simulation, we measured the model's ability to learn physical laws and respect conservation principles (Energy/Momentum) without explicit enforcement over a 100-step autoregressive rollout.



| Model | MSE (Motion) | Energy Drift (Physics) | Notes |

|-------|--------------|------------------------|-------|

| Standard Transformer | 18.63 | 214.31 | Unstable |

| GNS (Relational) | 23.83 | 1261.01 | Suffers from coordinate drift |

| HNN (Energy-based) | 11.12 | **61.86** | Great physics, average motion |

| **GeoLlama (Rotor RNN)** | **5.45** | **66.13** | **Best performance on balance** |



*   **Observation:** Geo-Llama achieved the lowest trajectory error and matched the stability of Hamiltonian NNs—which are mathematically forced to conserve energy—proving that conformal rotors naturally stay on the physical energy manifold.



### 6.2 Topological Complexity (Maze Connectivity)

We tested the model's ability to solve connectivity tasks where simple statistical "counting" (e.g., magnetization) is impossible.



*   **Scaling Collapse:** As grid size increased to 32x32, standard Transformers (even with 30x more parameters) collapsed to **34% MCC**, whereas Geo-Llama maintained **100% MCC**.

*   **Curriculum Ladder:** Logic learned by the geometric "brain" on an 8x8 grid was successfully "transplanted" to 32x32 grids with near-instant convergence, whereas Euclidean models failed to generalize across spatial scales.



### 6.3 Hierarchical Context Retention (Dyck-N)

To test the $O(1)$ Recursive Rotor Accumulator, we used the Dyck-N language task (balanced parentheses) as a proxy for structural containment.



*   **Performance:** Geo-Llama achieved **99.7% accuracy** at nesting depths of 100 (sequence length 200).

*   **Implication:** This confirms the "Infinite Context" capability, where the global state $\Psi$ maintains structural integrity over deep recursive levels in a fixed memory footprint.



### 6.4 Computational Efficiency

By engineering custom fused kernels in Triton and MLX, we reduced the materialization overhead of the Cayley table. **Geometric Linear layers now operate with only 1.2x the latency of standard PyTorch Linear layers**, making the architecture viable for large-scale deployment.



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## 7. Implications and Future Directions



The transition from Euclidean vectors to Conformal multivectors marks a move toward **Topological AI**. 



### 7.1 Mathematical Robustness

By grounding AI operations in the rigorous rules of Geometric Algebra, we mitigate common failure modes such as "Hallucination" (which we characterize as a deviation from the structural manifold). Contradictions in data manifest as destructive interference in the Bivector plane, providing a native mechanism for logical "red-teaming."



### 7.2 Scaling via Geometry

We posit that future scaling won't come from increasing parameters, but from increasing the **Dimensionality of the Manifold**. A model that understands the geometry of its domain requires orders of magnitude less data and power to achieve structural certainty.



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## 8. Conclusion



The history of AI has been a race toward "brute-force" statistics. Geo-Llama introduces a pivot toward **Human-Centric Geometry.** By embedding knowledge in a $Cl_{4,1}$ conformal manifold, we provide the AI with a sense of "space," "permanence," and "logical hierarchy," enabling transformative applications in neural physics engines, robotic control, and protein folding.



**Current Status:** Theory validation and kernel optimization for specialized structural benchmarks.



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