geometric-convergence-as-mathematical-attractor
OUT derived (depth 5)
Created 2026-08-25T03:10:25+00:00 · Reviewed 2026-08-25T04:02:18+00:00
The cross-model universality of feature geometry combined with its ontological status as a model-independent semantic object implies LLMs are converging to a shared mathematical attractor: the covariance/whitening geometry is the unique fixed point that any differentiable language model must instantiate, not an architectural artifact.
Justifications
SL — Universality (same structure across architectures) plus ontological status (it is a real object, not a tool) jointly entail the attractor interpretation; neither alone yields "unique fixed point of language modeling."
Antecedents (all must be IN):
- OUT superposition-geometry-explains-universality — The cross-model universality of feature geometry (SAE features more similar across architectures than within, Park orthogonality validated on both Gemma and LLaMA) is a consequence of superposition: the over-complete compositional basis is determined by the shared semantic grammar of language, making geometric structure an architectural invariant rather than a model-specific artifact.
- OUT geometry-ontological-status — The covariance/whitening geometry is not merely a convenient analytical tool but possesses ontological status as a genuine model-independent semantic structure, because three independent lines converge: it is the operational metric for editing and interpretation (depth-3), it is universal across architectures (depth-2), and it converges with externally-validated human-judgment metrics (depth-3).
Unless (any of these IN defeats this justification):
- IN geometric-convergence-as-mathematical-attractor-v2 — The cross-model universality of feature geometry, combined with its ontological status as a model-independent semantic structure, indicates that LLMs sharing over-complete superposition architectures converge toward a shared geometric regularity: the covariance/whitening geometry functions as a convergent structural attractor determined by the shared semantic grammar of language, rather than being merely a model-specific artifact.
Dependents
These beliefs depend on this one:
- OUT cross-model-convergence-conditional-on-artifact-control — The cross-model convergence of geometric structure (polytope geometry, feature universality, orthogonality) constitutes a genuine architectural property, but the orthogonality component specifically requires set-inclusion controls to distinguish genuine hierarchical semantics from combinatorial artifacts.
- OUT geometry-as-universal-semantic-currency — The covariance geometry is the single operational definition of "meaning" in LLMs, simultaneously determining what can be measured (evaluation via cosine/Spearman), what can be modified (rank-one editing via C⁻¹k*), what converges across architectures (SAE/Park universality), and what is hierarchically structured (feature neighborhoods as theorem instantiations)—making it a model-independent semantic currency rather than an architecture-specific artifact.