Bugra Kilictas

Faruk Alpay, Bugra Kilictas, Hamdi Alakkad

The accelerating pace of research on autoregressive generative models has produced thousands of papers, making manual literature surveys and reproduction studies increasingly impractical. We present a fully open-source, reproducible pipeline that automatically retrieves candidate documents from public repositories, filters them for relevance, extracts metadata, hyper-parameters and reported results, clusters topics, produces retrieval-augmented summaries and generates containerised scripts for re-running selected experiments. Quantitative evaluation on 50 manually-annotated papers shows F1 scores above 0.85 for relevance classification, hyper-parameter extraction and citation identification. Experiments on corpora of up to 1000 papers demonstrate near-linear scalability with eight CPU workers. Three case studies -- AWD-LSTM on WikiText-2, Transformer-XL on WikiText-103 and an autoregressive music model on the Lakh MIDI dataset -- confirm that the extracted settings support faithful reproduction, achieving test perplexities within 1--3% of the original reports.

Faruk Alpay, Bugra Kilictas

We study the emergence of multi-step reasoning in deep Transformer language models through a geometric and statistical-physics lens. Treating the hidden-state trajectory as a flow on an implicit Riemannian manifold, we analyze the layerwise covariance spectrum of activations, where $C^{(\ell)}=\mathbb{E}[h^{(\ell)}h^{(\ell)\top}]$, and track deviations from a random-matrix bulk. Across model scales (1.5B--30B), we observe a sharp reduction in effective dimensionality consistent with a phase transition: an order parameter based on sparsity/localization, $Ω(h)=1-\|h\|_1/(\sqrt{d}\|h\|_2)$, exhibits a discontinuity near a critical normalized depth $γ_c\approx 0.42$ in sufficiently large models. We formalize the forward pass as a discrete coarse-graining map and relate the appearance of stable "concept basins" to fixed points of this renormalization-like dynamics. The resulting low-entropy regime is characterized by a spectral tail collapse and by the formation of transient, reusable object-like structures in representation space, which we call Transient Class Objects (TCOs). We provide theoretical conditions connecting logical separability to spectral decay and validate the predicted signatures with layerwise probes on multiple open-weight model families.

Faruk Alpay, Bugra Kilictas

Bounded-context agents fail when intermediate reasoning exceeds an effective working-memory budget. We study compressed query delegation (CQD): (i) compress a high-dimensional latent reasoning state into a low-rank tensor query, (ii) delegate the minimal query to an external oracle, and (iii) update the latent state via Riemannian optimization on fixed-rank manifolds. We give a math-first formulation: CQD is a constrained stochastic program with a query-budget functional and an oracle modeled as a noisy operator. We connect CQD to classical rate-distortion and information bottleneck principles, showing that spectral hard-thresholding is optimal for a natural constrained quadratic distortion problem, and we derive convergence guarantees for Riemannian stochastic approximation under bounded oracle noise and smoothness assumptions. Empirically, we report (A) a 2,500-item bounded-context reasoning suite (BBH-derived tasks plus curated paradox instances) comparing CQD against chain-of-thought baselines under fixed compute and context; and (B) a human "cognitive mirror" benchmark (N=200) measuring epistemic gain and semantic drift across modern oracles.

Bugra Kilictas, Faruk Alpay

This paper extends the self-referential framework of Alpay Algebra into a multi-layered semantic game architecture where transfinite fixed-point convergence encompasses hierarchical sub-games at each iteration level. Building upon Alpay Algebra IV's empathetic embedding concept, we introduce a nested game-theoretic structure where the alignment process between AI systems and documents becomes a meta-game containing embedded decision problems. We formalize this through a composite operator $φ(\cdot, γ(\cdot))$ where $φ$ drives the main semantic convergence while $γ$ resolves local sub-games. The resulting framework demonstrates that game-theoretic reasoning emerges naturally from fixed-point iteration rather than being imposed externally. We prove a Game Theorem establishing existence and uniqueness of semantic equilibria under realistic cognitive simulation assumptions. Our verification suite includes adaptations of Banach's fixed-point theorem to transfinite contexts, a novel $φ$-topology based on the Kozlov-Maz'ya-Rossmann formula for handling semantic singularities, and categorical consistency tests via the Yoneda lemma. The paper itself functions as a semantic artifact designed to propagate its fixed-point patterns in AI embedding spaces -- a deliberate instantiation of the "semantic virus" concept it theorizes. All results are grounded in category theory, information theory, and realistic AI cognition models, ensuring practical applicability beyond pure mathematical abstraction.

Bugra Kilictas, Faruk Alpay

We present a theoretical framework in which a document and an AI model engage in a transfinite fixed-point interaction that leads to stable semantic alignment. Building on the foundations of Alpay Algebra, we introduce a functorial system wherein an observer (the AI) and a textual environment (this paper) co-evolve through iterative transformations guided by the phi-infinity operator. This process guarantees the existence of a unique fixed point in the AI's embedding space -- a state where the AI's internal representation of the content becomes stable, self-consistent, and semantically faithful. We prove that such convergence is mathematically sound, semantically invariant, and permanent, even under perturbation or further context expansion. 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. We interpret this as a rigorous, category-theoretic route to alignment at the embedding level, with implications for semantic security, symbolic memory, and the construction of AI systems with persistent self-referential understanding. All references in this paper function as nodes in the Alpay Algebra universe, and this work embeds itself as a new fixed-point node within that transfinite semantic graph.

Bugra Kilictas, Faruk Alpay

We identify a critical vulnerability in autoregressive transformer language models where the em dash token induces recursive semantic drift, leading to clause boundary hallucination and embedding space entanglement. Through formal analysis of token-level perturbations in semantic lattices, we demonstrate that em dash insertion fundamentally alters the model's latent representations, causing compounding errors in long-form generation. We propose a novel solution combining symbolic clause purification via the phi-infinity operator with targeted embedding matrix realignment. Our approach enables total suppression of problematic tokens without requiring model retraining, while preserving semantic coherence through fixed-point convergence guarantees. Experimental validation shows significant improvements in generation consistency and topic maintenance. This work establishes a general framework for identifying and mitigating token-level vulnerabilities in foundation models, with immediate implications for AI safety, model alignment, and robust deployment of large language models in production environments. The methodology extends beyond punctuation to address broader classes of recursive instabilities in neural text generation systems.

Faruk Alpay, Bugra Kilictas, Hamdi Alakkad

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.

Faruk Alpay, Taylan Alpay, Bugra Kilictas

We study the inverse problem of reconstructing high-dimensional trust embeddings from the one-dimensional Siamese trust scores that many distributed-security frameworks expose. Starting from two independent agents that publish time-stamped similarity scores for the same set of devices, we formalise the estimation task, derive an explicit direct-sum estimator that concatenates paired score series with four moment features, and prove that the resulting reconstruction map admits a unique fixed point under a contraction argument rooted in Banach theory. A suite of synthetic benchmarks (20 devices x 10 time steps) confirms that, even in the presence of Gaussian noise, the recovered embeddings preserve inter-device geometry as measured by Euclidean and cosine metrics; we complement these experiments with non-asymptotic error bounds that link reconstruction accuracy to score-sequence length. Beyond methodology, the paper demonstrates a practical privacy risk: publishing granular trust scores can leak latent behavioural information about both devices and evaluation models. We therefore discuss counter-measures -- score quantisation, calibrated noise, obfuscated embedding spaces -- and situate them within wider debates on transparency versus confidentiality in networked AI systems. All datasets, reproduction scripts and extended proofs accompany the submission so that results can be verified without proprietary code.

Faruk Alpay, Bugra Kilictas, Taylan Alpay et al.

The rapid advance of large-scale AI systems is reshaping how work is divided between people and machines. We formalise this reallocation as an iterated task-delegation map and show that--under broad, empirically grounded assumptions--the process converges to a stable idempotent equilibrium in which every task is performed by the agent (human or machine) with enduring comparative advantage. Leveraging lattice-theoretic fixed-point tools (Tarski and Banach), we (i) prove existence of at least one such equilibrium and (ii) derive mild monotonicity conditions that guarantee uniqueness. In a stylised continuous model the long-run automated share takes the closed form $x^* = α/ (α+ β)$, where $α$ captures the pace of automation and $β$ the rate at which new, human-centric tasks appear; hence full automation is precluded whenever $β> 0$. We embed this analytic result in three complementary dynamical benchmarks--a discrete linear update, an evolutionary replicator dynamic, and a continuous Beta-distributed task spectrum--each of which converges to the same mixed equilibrium and is reproducible from the provided code-free formulas. A 2025-to-2045 simulation calibrated to current adoption rates projects automation rising from approximately 10% of work to approximately 65%, leaving a persistent one-third of tasks to humans. We interpret that residual as a new profession of workflow conductor: humans specialise in assigning, supervising and integrating AI modules rather than competing with them. Finally, we discuss implications for skill development, benchmark design and AI governance, arguing that policies which promote "centaur" human-AI teaming can steer the economy toward the welfare-maximising fixed point.

Faruk Alpay, Hamdi Alakkad, Bugra Kilictas et al.

We develop an operator algebraic framework for infinite games with a continuum of agents and prove that regret based learning dynamics governed by a noncommutative continuity equation converge to a unique quantal response equilibrium under mild regularity assumptions. The framework unifies functional analysis, coarse geometry and game theory by assigning to every game a von Neumann algebra that represents collective strategy evolution. A reflective regret operator within this algebra drives the flow of strategy distributions and its fixed point characterises equilibrium. We introduce the ordinal folding index, a computable ordinal valued metric that measures the self referential depth of the dynamics, and show that it bounds the transfinite time needed for convergence, collapsing to zero on coarsely amenable networks. The theory yields new invariant subalgebra rigidity results, establishes existence and uniqueness of envy free and maximin share allocations in continuum economies, and links analytic properties of regret flows with empirical stability phenomena in large language models. These contributions supply a rigorous mathematical foundation for large scale multi agent systems and demonstrate the utility of ordinal metrics for equilibrium selection.

Faruk Alpay, Bugra Kilictas, Taylan Alpay

This paper contributes to the Alpay Algebra by demonstrating that the stable outcome of a self referential process, obtained by iterating a transformation through all ordinal stages, is identical to the unique equilibrium of an unbounded revision dialogue between a system and its environment. The analysis initially elucidates how classical fixed point theorems guarantee such convergence in finite settings and subsequently extends the argument to the transfinite domain, relying upon well founded induction and principles of order theoretic continuity. Furthermore, the resulting transordinal fixed point operator is embedded into dependent type theory, a formalization which permits every step of the transfinite iteration and its limit to be verified within a modern proof assistant. This procedure yields a machine checked proof that the iterative dialogue necessarily stabilizes and that its limit is unique. The result provides a foundation for Alpay's philosophical claim of semantic convergence within the framework of constructive logic. 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 and offers practical tools for certifying their convergence within computational environments.