Geofinitism: When Geometric Finitism Reconstructs the Cognitive Foundation of Artificial Intelligence

Geofinitism (Geometric Finitism) proposes a brand-new AI paradigm, whose core argument is that language is a geometric system rather than a probabilistic system, and the current autoregressive Transformer architecture has fundamental flaws (it cannot represent semantics or reconstruct the geometric manifold of language). This theory is systematically elaborated by the School of Machine Intelligence, which proposes the MARINA alternative, bringing about a philosophical shift from probabilism to geometricism and multiple impacts on AI safety, data efficiency, etc.

The traditional AI field is accustomed to using probability to describe model behavior (such as token prediction probability, hallucination statistical frequency), but Geofinitism believes this may misunderstand the nature of language. Its core view points out that language is a geometric system rather than a probabilistic system, and through the systematic elaboration of the School of Machine Intelligence on the fundamental flaws of current AI architectures, it promotes the paradigm shift from probabilism to geometricism.

Geofinitism points out through mathematical proof that the Transformer architecture is based on three wrong assumptions:

1. Static Embedding Layer Defect: The mutual information between static word vectors and word meanings is zero, so they cannot carry semantic information;
2. Failure of Autoregressive Training Objective: There are three major problems: non-rigid embedding (no fixed coordinates), uniform history processing (insufficient multi-scale delay), and information loss (non-injective mapping);
3. Accidental Correctness but Imperfection of Attention Mechanism: Attention calculation delays coordinate inner product (consistent with Takens embedding), but lacks fixed coordinates, principled delay, and information integrity.

Geofinitism proposes the MARINA (Takens-Based Transformer) four-subsystem architecture, which explicitly represents geometric structure:

1. Exponential Delay Embedding: Adopts a logarithmic space delay design (e.g., [e_t, e_{t-1}, e_{t-2}, e_{t-4}...]) to cover multiple Lyapunov time scales;
2. Adaptive Manifold Projection: Dynamically adjusts the representation layer to capture the topological structure of the semantic manifold, achieving 100% accuracy in basin separation tasks;
3. Channel Theory and Memory Fibers: Introduces two attractor mechanisms (tubular attractors for fact retrieval, wide basins for creative generation), defining AI safety as dynamic topology maintenance.

MARINA's experiments reveal key evidence:

1. Basin Separation Task Performance: Achieves 100% accuracy in this task, verifying its ability to capture semantic manifolds;
2. Double Data Paradox: When training data is doubled, the validation accuracy increases by 84%, which becomes an indicator for diagnosing geometric learning (rather than statistical memory)—if the model does not have this improvement, it may only memorize statistical patterns instead of the underlying manifold structure.

Geofinitism represents a profound philosophical shift: from probabilism asking "what is possible" to geometricism asking "what is the shape". Its impacts include:

• AI Safety: Shifting from rule-constrained black boxes to designing geometrically stable systems;
• Data Efficiency: MARINA demonstrates O(log N) complexity, which is better than the exponential data demand of traditional Transformers;
• Interpretability: The geometric structure of semantic manifolds provides an intuitive understanding framework. This theoretical system (from P09 to P01) not only criticizes existing architectures but also provides a roadmap for future AI development, reminding us to re-understand the nature of language—reconstructing its geometric manifold rather than just predicting the next word.