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

TL;DR

This paper introduces an operator-theoretic framework for temporal anchoring in embedding spaces, analyzing convergence, drift, and an internal computational architecture with rigorous proofs and applications to attention layers.

Contribution

It develops a novel mathematical framework for temporal anchoring in embeddings, including convergence theorems, robustness variants, and analysis of attention layer Lipschitz properties.

Findings

Proves a drift--projection convergence theorem with explicit bounds.

Establishes a finite-run equivalence for an internal computational architecture.

Shows softmax in attention layers is 1/2-Lipschitz in  norm.

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 $1/2$-Lipschitz in $\ell_2$ 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.