Alpay Algebra II: Identity as Fixed-Point Emergence in Categorical Data

• URL: http://arxiv.org/abs/2505.17480v1
• Date: Fri, 23 May 2025 05:15:34 GMT
• Title: Alpay Algebra II: Identity as Fixed-Point Emergence in Categorical Data
• Abstract summary: I define identity as a fixed point that emerges through categorical recursion.<n>I prove the existence and uniqueness of such identity--fixed via ordinal-indexed iteration.<n>This paper positions identity as a mathematical structure that arises from within the logic of change itself computable, convergent, and categorically intrinsic.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: In this second installment of the Alpay Algebra framework, I formally define identity as a fixed point that emerges through categorical recursion. Building upon the transfinite operator $\varphi^\infty$, I characterize identity as the universal solution to a self-referential functorial equation over a small cartesian closed category. I prove the existence and uniqueness of such identity-fixed-points via ordinal-indexed iteration, and interpret their convergence through internal categorical limits. Functors, adjunctions, and morphisms are reconstructed as dynamic traces of evolving states governed by $\varphi$, reframing identity not as a static label but as a stabilized process. Through formal theorems and symbolic flows, I show how these fixed points encode symbolic memory, recursive coherence, and semantic invariance. This paper positions identity as a mathematical structure that arises from within the logic of change itself computable, convergent, and categorically intrinsic.

Related papers

• A Compositional Theory of Causally Masked Transformers [50.88598486616582]
  We develop a formalization that derives expressivity directly from the model's implemented dynamics.<n>Each attention head updates its own state independently within a layer.<n>A modified form of soft attention supports irreversible, checklist-like state.
  arXiv  Detail & Related papers  (2026-07-29T14:47:19Z)
• Geometric Factual Recall in Transformers [57.48371649045765]
  A common view casts internal weight matrices as associative memories over pairs of embeddings, requiring memorization counts that scale linearly with the number of facts.<n>We develop a theoretical and empirical account of an alternative, emphgeometric form of memorization in which learned embeddings encode structure directly.<n>In a controlled setting where a single-layer transformer must memorize random vectors from subjects to a shared attribute set, we prove that a logarithmic embedding dimension suffices.<n>We extend these results to the multi-hop setting, providing constructions with and without chain-of-thought that exhibit a provable
  arXiv  Detail & Related papers  (2026-05-12T17:22:22Z)
• Deterministic Zeroth-Order Mirror Descent via Vector Fields with A Posteriori Certification [45.85698554568285]
  We develop a deterministic zeroth-order mirror descent framework by replacing gradients with a general vector field.<n>Our analysis provides an evaluation template for last-rate inequality evaluations.<n>Together, these results expose a hidden geometric linking Bregman identities, deterministic certification, and robust conic geometry in zeroth-order mirror descent.
  arXiv  Detail & Related papers  (2026-01-31T10:05:05Z)
• Deep Delta Learning [91.75868893250662]
  We introduce Deep Delta Learning (DDL), a novel architecture that generalizes the standard residual connection.<n>We provide a spectral analysis of this operator, demonstrating that the gate $(mathbfX)$ enables dynamic between identity mapping, projection, and geometric reflection.<n>This unification empowers the network to explicitly control the spectrum of its layer-wise transition operator, enabling the modeling of complex, non-monotonic dynamics.
  arXiv  Detail & Related papers  (2026-01-01T18:11:38Z)
• Internalizing Tools as Morphisms in Graded Transformers [0.0]
  We introduce a graded formulation of internal symbolic computation for transformers.<n>The hidden space is endowed with a grading $V=bigoplus_gin GV_g$, and symbolic operations are realized as typed block maps.<n>A self-supervised emphgraded utility functional governs activation and yields sparse, interpretable behavior.
  arXiv  Detail & Related papers  (2025-11-21T23:27:53Z)
• On Counts and Densities of Homogeneous Bent Functions: An Evolutionary Approach [60.00535100780336]
  This paper examines the use of Evolutionary Algorithms (EAs) to evolve homogeneous bent Boolean functions.<n>We introduce the notion of density of homogeneous bent functions, facilitating the algorithmic design that results in finding quadratic and cubic bent functions in different numbers of variables.
  arXiv  Detail & Related papers  (2025-11-16T15:33:40Z)
• Beyond Inference Intervention: Identity-Decoupled Diffusion for Face Anonymization [55.29071072675132]
  Face anonymization aims to conceal identity information while preserving non-identity attributes.<n>We propose textbfIDsuperscript2Face, a training-centric anonymization framework.<n>We show that IDtextsuperscript2Face outperforms existing methods in visual quality, identity suppression, and utility preservation.
  arXiv  Detail & Related papers  (2025-10-28T09:28:12Z)
• Transfinite Fixed Points in Alpay Algebra as Ordinal Game Equilibria in Dependent Type Theory [0.0]
  This paper contributes to the Alpay Algebra by demonstrating that the stable outcome of a self referential process is identical to the unique equilibrium of an unbounded revision dialogue between a system and its environment.<n>By unifying concepts from fixed point theory, game semantics, ordinal analysis, and type theory, this research establishes a broadly accessible yet formally rigorous foundation for reasoning about infinite self referential systems.
  arXiv  Detail & Related papers  (2025-07-25T13:12:55Z)
• Alpay Algebra IV: Symbiotic Semantics and the Fixed-Point Convergence of Observer Embeddings [0.0]
  We present a theoretical framework in which a document and an AI model engage in a transfinite fixed-point interaction.<n>We prove that such convergence is mathematically sound, semantically invariant, and permanent.<n>This fixed point acts as an "empathetic embedding," wherein the AI internalizes not only the meaning of the content but also the author's intent.
  arXiv  Detail & Related papers  (2025-07-04T18:49:18Z)
• Fixed-Point Traps and Identity Emergence in Educational Feedback Systems [0.0]
  We prove that exam-driven educational systems obstruct identity emergence and block creative convergence.<n>Our model mathematically explains the creativity suppression, research stagnation, and structural entropy loss induced by timed exams and grade-based feedback.
  arXiv  Detail & Related papers  (2025-05-27T11:19:33Z)
• Alpay Algebra III: Observer-Coupled Collapse and the Temporal Drift of Identity [0.0]
  Third installment formalizes the observer-coupled phi-collapse process through transfinite categorical flows and curvature-driven identity operators.<n>System surpasses conventional identity modeling in explainable AI (XAI) by encoding internal transformation history into a symbolic fixed-point structure.<n>Results also offer a mathematically rigorous basis for future AI systems with stable self-referential behavior.
  arXiv  Detail & Related papers  (2025-05-26T10:20:12Z)
• Alpay Algebra: A Universal Structural Foundation [0.0]
  Alpay Algebra is introduced as a universal, category-theoretic framework.<n>It unifies classical algebraic structures with modern needs in symbolic recursion and explainable AI.
  arXiv  Detail & Related papers  (2025-05-21T10:18:49Z)
• Self-Attention as a Parametric Endofunctor: A Categorical Framework for Transformer Architectures [0.0]
  We develop a category-theoretic framework focusing on the linear components of self-attention.<n>We show that the query, key, and value maps naturally define a parametric 1-morphism in the 2-category $mathbfPara(Vect)$.<n> stacking multiple self-attention layers corresponds to constructing the free monad on this endofunctor.
  arXiv  Detail & Related papers  (2025-01-06T11:14:18Z)
• Understanding Matrix Function Normalizations in Covariance Pooling through the Lens of Riemannian Geometry [63.694184882697435]
  Global Covariance Pooling (GCP) has been demonstrated to improve the performance of Deep Neural Networks (DNNs) by exploiting second-order statistics of high-level representations.<n>This paper provides a comprehensive and unified understanding of the matrix logarithm and power from a Riemannian geometry perspective.
  arXiv  Detail & Related papers  (2024-07-15T07:11:44Z)
• Score-based Causal Representation Learning: Linear and General Transformations [31.786444957887472]
  The paper addresses both the identifiability and achievability aspects.<n>It designs a score-based class of algorithms that ensures both identifiability and achievability.<n>Results are validated via experiments on structured synthetic data and image data.
  arXiv  Detail & Related papers  (2024-02-01T18:40:03Z)
• Unsupervised Semantic Segmentation by Distilling Feature Correspondences [94.73675308961944]
  Unsupervised semantic segmentation aims to discover and localize semantically meaningful categories within image corpora without any form of annotation. We present STEGO, a novel framework that distills unsupervised features into high-quality discrete semantic labels. STEGO yields a significant improvement over the prior state of the art, on both the CocoStuff and Cityscapes challenges.
  arXiv  Detail & Related papers  (2022-03-16T06:08:47Z)
• Classification of equivariant quasi-local automorphisms on quantum chains [0.0]
  We classify automorphisms on quantum chains, allowing both spin and fermionic degrees of freedom, that are equivariant with respect to a local symmetry action of a finite symmetry group $G$. We find that the equivalence classes are uniquely labeled by an index taking values in $mathbbQ cup sqrt2 mathbbQ times mathrmHom(G, mathbbZ_2) times H2(G, U(1))$.
  arXiv  Detail & Related papers  (2021-06-03T21:41:05Z)
• Finite-Function-Encoding Quantum States [52.77024349608834]
  We introduce finite-function-encoding (FFE) states which encode arbitrary $d$-valued logic functions. We investigate some of their structural properties.
  arXiv  Detail & Related papers  (2020-12-01T13:53:23Z)
• Algebraic Neural Networks: Stability to Deformations [179.55535781816343]
  We study algebraic neural networks (AlgNNs) with commutative algebras. AlgNNs unify diverse architectures such as Euclidean convolutional neural networks, graph neural networks, and group neural networks.
  arXiv  Detail & Related papers  (2020-09-03T03:41:38Z)
• Infinite Feature Selection: A Graph-based Feature Filtering Approach [78.63188057505012]
  We propose a filtering feature selection framework that considers subsets of features as paths in a graph. Going to infinite allows to constrain the computational complexity of the selection process. We show that Inf-FS behaves better in almost any situation, that is, when the number of features to keep are fixed a priori.
  arXiv  Detail & Related papers  (2020-06-15T07:20:40Z)
• Assignment Flows for Data Labeling on Graphs: Convergence and Stability [69.68068088508505]
  This paper establishes conditions on the weight parameters that guarantee convergence of the continuous-time assignment flow to integral assignments (labelings) Several counter-examples illustrate that violating the conditions may entail unfavorable behavior of the assignment flow regarding contextual data classification.
  arXiv  Detail & Related papers  (2020-02-26T15:45:38Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.