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2025
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| Online Access: | https://doi.org/10.57967/hf/7080 |
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| _version_ | 1866901258031333376 |
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| author | Zixi, Li |
| author_facet | Zixi, Li |
| contents | <p>We establish that reasoning incompleteness is not a logical deficiency but a geometric neces-<br>sity. Through three complementary proofs—the Euclidean Proof, the Manifold Proof, and<br>the Yonglin Proof —we demonstrate that any reasoning system operates on a prior-shaped<br>geometric manifold with structural collapse points.<br>The Euclidean Proof shows how axiomatic systems generate irreconcilable ontological and<br>cognitive priors through antinomy. The Manifold Proof establishes that all reasoning manifolds<br>contain singularities where inferential geodesics collapse back to prior anchors. The Yonglin<br>Proof formalizes the limit structure of reasoning as a reflexive transition from prior to meta-<br>prior (A → A∗ where A̸ = A∗), revealing object-level closure with meta-level rupture.<br>We validate these theoretical results through visualization experiments on ARC reasoning<br>tasks, showing that each task induces a distinct topological manifold structure—refuting the<br>notion of “universal pure reasoning.” Our conclusion: the historical separation of algebra and<br>geometry is not logically necessary but metaphysically mistaken</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_57967_hf_7080 |
| institution | Zenodo |
| language | |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | The Geometric Incompleteness of Reasoning Zixi, Li <p>We establish that reasoning incompleteness is not a logical deficiency but a geometric neces-<br>sity. Through three complementary proofs—the Euclidean Proof, the Manifold Proof, and<br>the Yonglin Proof —we demonstrate that any reasoning system operates on a prior-shaped<br>geometric manifold with structural collapse points.<br>The Euclidean Proof shows how axiomatic systems generate irreconcilable ontological and<br>cognitive priors through antinomy. The Manifold Proof establishes that all reasoning manifolds<br>contain singularities where inferential geodesics collapse back to prior anchors. The Yonglin<br>Proof formalizes the limit structure of reasoning as a reflexive transition from prior to meta-<br>prior (A → A∗ where A̸ = A∗), revealing object-level closure with meta-level rupture.<br>We validate these theoretical results through visualization experiments on ARC reasoning<br>tasks, showing that each task induces a distinct topological manifold structure—refuting the<br>notion of “universal pure reasoning.” Our conclusion: the historical separation of algebra and<br>geometry is not logically necessary but metaphysically mistaken</p> |
| title | The Geometric Incompleteness of Reasoning |
| url | https://doi.org/10.57967/hf/7080 |