Projective Geometric Frameworks for Semantic Structures

In the paper's words

Current critical literature often dismisses Large Language Models (LLMs) as statistical parrots operating via mere stochastic approximation, yet their capacity for cross-lingual zero-shot inference suggests the internalization of deep, invariant topological rules. This study establishes that semantic memory inherently relies on ultrametric topologies, and that true comprehension requires invariant structural mappings. To resolve this tension, we developed a computational methodology utilizing synthetic hierarchical semantic vectors, subjected to Ward’s minimum variance clustering and subsequent continuous Möbius transformations.

The abstract, verbatim, and its opening sentence is the claim. Every line below is quoted from the text.

Numbers it reports

1. Conversely, classic cognitive literature demonstrates that semantic memory inherently relies on ultrametric topologies to compress and retrieve abstract concepts efficiently (Parga & Virasoro, 1986).

   from 1.1

2. This cooperative inference requires invariant reference frames, ensuring that the cross-ratio—a measure of equivalence in projective geometry—remains perfectly constant (Wang et al., 2019).

   from 1.4 Projective

3. We will prove this via simulated cross-ratio extraction, validating that algorithmic cores remain invariant under severe transformation (Schiffman, 2026).

   from 1.7 Thesis Statement

4. Early neural networks modeled memory as static attractors, wherein cortical capacities fundamentally grouped similar concepts together to maximize storage efficiency (Boboeva et al., 2018).

   from 2.2 Classic

5. Ultrametric organization naturally emerges in these Hopfield-like systems as the most energy-efficient method for resolving overlapping memory traces (Parga & Virasoro, 1986).

   from 2.2 Classic

6. In projective geometry, the relationship “A is to B as C is to D” is strictly defined as an invariant cross-ratio equivalence class (Wang et al., 2019).

   from 2.5

How it is done

1. Ultrametric organization naturally emerges in these Hopfield-like systems as the most energy-efficient method for resolving overlapping memory traces (Parga & Virasoro, 1986).

   from 2.2 Classic

2. Generation Protocol To test structural invariants without the computational constraints of massive live models, we developed a protocol to generate a highly controlled synthetic semantic corpus.

   from 3.1 Synthetic Corpus

3. We compute both Euclidean and Cosine baselines utilizing Python’s scipy.spatial.distance to extract the geometric null-space representation of the data (Yadid et al., 2026).

   from 3.2 Metric

4. We utilized Ward’s minimum variance method via scipy.cluster.hierarchy.linkage to force the synthetic vectors into a strict topological hierarchy (Murtagh, 2012).

   from 3.3 Algorithms for

What would falsify it

1. This links abstract projective mathematics directly to observable, empirical AI feature structures.

   from 2.6 Sparse

2. Stability The mathematical proof of cross-ratio stability fundamentally refutes the assertion that neural networks are merely complex statistical associators.

   from 5.1 Interpreting Cross-Ratio

Where the claim stops

1. the Gap Between Statistical Approximation and Formal Invariants in Large Language Models Author: Rowan Brad Quni-Gudzinas Contact: rowan.quni@outlook.com ORCID: 0009-0002-4317-5604 ISNI: 0000000526456062 DOI: https://papers.qnfo.org/papers/projective-geometric-frameworks-for-semantic-structures/ Date: 2026-04-14 Version: 1.0 Abstract : Current critical literature often dismisses Large Language Models (LLMs) as statistical parrots operating via mere stochastic approximation, yet their capacity for cross-lingual zero-shot inference suggests the internalization of deep, invariant topological rules.

   from Addressing

2. Context and Motivation: The Limits of Stochastic Approximation Statistical approximation models are fundamentally insufficient for capturing the deep, generative structure of semantic meaning in artificial intelligence.

   from 1.1

3. This assertion challenges the dominant associational paradigm, wherein critics argue that statistical models fail to capture the deep structure of meaning due to their reliance on surface-level frequency distributions (Huang, 2023).

   from 1.1

Where to read

1. Addressing251 words · 1 min
2. 1.01,077 words · 5 min
3. 2.0799 words · 3 min
4. 3.0612 words · 3 min
5. 4.0530 words · 2 min
6. 5.0408 words · 2 min
7. 6.0 Implications:399 words · 2 min
8. 7.0 Conclusion and Future Work292 words · 1 min
9. Appendices323 words · 1 min