Temporal Anchoring in Deepening Embedding Spaces: Event-Indexed Projections, Drift, Convergence, and an Internal Computational Architecture
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| Format: | Preprint |
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2025
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| _version_ | 1866916896179224576 |
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| author | Alpay, Faruk Kilictas, Bugra Alakkad, Hamdi |
| author_facet | Alpay, Faruk Kilictas, Bugra Alakkad, Hamdi |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_09693 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Temporal Anchoring in Deepening Embedding Spaces: Event-Indexed Projections, Drift, Convergence, and an Internal Computational Architecture Alpay, Faruk Kilictas, Bugra Alakkad, Hamdi Machine Learning Functional Analysis Optimization and Control 47H09, 47H10, 90C25, 65K10, 68T07 F.1.1; G.1.6; I.2.6; G.1.2 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. |
| title | Temporal Anchoring in Deepening Embedding Spaces: Event-Indexed Projections, Drift, Convergence, and an Internal Computational Architecture |
| topic | Machine Learning Functional Analysis Optimization and Control 47H09, 47H10, 90C25, 65K10, 68T07 F.1.1; G.1.6; I.2.6; G.1.2 |
| url | https://arxiv.org/abs/2508.09693 |