English

Temporal Anchoring in Deepening Embedding Spaces: Event-Indexed Projections, Drift, Convergence, and an Internal Computational Architecture

Machine Learning 2025-08-14 v1 Functional Analysis Optimization and Control Machine Learning

Abstract

We develop an operator-theoretic framework for temporal anchoring in embedding spaces, modeled as drift maps interleaved with event-indexed blocks culminating in affine projections. We provide complete proofs for a variable-block contraction lemma (products of Lipschitz factors), a drift--projection convergence theorem with explicit uniform-gap envelopes, and ontological convergence under nested affine anchors with a robustness variant. We formalize an internal Manuscript Computer (MC) whose computations are defined purely by these operators and prove a rigorous finite-run equivalence theorem (with perturbation bounds). For attention layers, we give a self-contained proof that softmax is 1/21/2-Lipschitz in 2\ell_2 and derive sufficient layer-contraction conditions (orthogonal/non-orthogonal heads). All floats are placed exactly where written; the manuscript uses only in-paper pseudocode and appendix figures.

Keywords

Cite

@article{arxiv.2508.09693,
  title  = {Temporal Anchoring in Deepening Embedding Spaces: Event-Indexed Projections, Drift, Convergence, and an Internal Computational Architecture},
  author = {Faruk Alpay and Bugra Kilictas and Hamdi Alakkad},
  journal= {arXiv preprint arXiv:2508.09693},
  year   = {2025}
}

Comments

16 pages, 2 figures, 2 tables