Faruk Alpay

Faruk Alpay, Levent Sarioglu

A long-running append-mostly sequence, such as an edit log, event store, or versioned working set, is usually tiered into a bounded hot stratum and colder folded summaries. This saves memory but breaks stable references: a handle minted while a record is hot may later be resolved after the record has moved into a digest, after it has been superseded, or while a fold is in flight. We define the resulting cross-tier anomalies--dangling, stale, corrupt, and snapshot-skewed resolution--and present the Cascade Log, a reference-stable tiered append structure. The structure keeps a single persistent coalescing interval map over handles as the sole authority on each live version; folding a contiguous run replaces many singleton entries by one digest-backed interval node, and immutable roots provide snapshot tokens. Its cost is characterized by the fragmentation $A$, the number of index pieces, namely live handles plus maximal same-digest runs. The index uses $Θ(A)$ space, resolves a point in $O(\log A)$, reports a $k$-handle range in $O(\log A+k)$, and performs $a$ appends and $s$ supersedes in $O((a/B+s)\log A)$ update work for fold block size $B$. Matching lower bounds show that $Ω(A)$ space and $Ω(\log A+k)$ ordered range cost are unavoidable, and an adversary can force $A=Θ(s)$. Thus the index is sublinear on append-dominated histories and grows linearly only under fragmenting edits. A reference implementation and reproducible experiments to $10^6$ records validate the anomaly-freedom and the fragmentation bounds.

Faruk Alpay, Taylan Alpay

Public scientific and metrology releases can leak the hidden settings that produced them. We formalize and quantify this risk as a profiled statistical side-channel audit: a release map exposes finite-band statistics of a power spectral density (PSD), a profiled observer trains labeled template spectra under an explicit budget, and a challenge release is drawn from one of two utility-equivalent recipes separated by a protected coordinate. Averaged PSD bins follow a gamma channel, replaced by a covariance-weighted log-spectrum channel when the bins are correlated; this yields exact Kullback-Leibler divergences, Chernoff exponents, protected-bit advantage bounds, and finite-training, finite-library, finite-compute, and model-mismatch corrections. Our headline result is a finite-band transport-leakage law: after amplitude and blur are eliminated, the protected acid-transport information obeys $I_{λ|α,β}(K) = (64/1225)\, w λ^{6} K^{9} + O(w λ^{8} K^{11})$ for $Kλ\ll 1$, a ninth-order exponent with a closed-form safe band. A step-by-step protocol turns a measured release into these numbers, and a fixed-seed reproducibility package regenerates every table and figure. We instantiate the audit on screened extreme-ultraviolet (EUV) roughness spectra as a model-conditioned case study, with deployment on measured releases the next step.

Faruk Alpay, Baris Basaran

We study the observation congruences induced by rational polyhedral cones on vector-valued quantitative languages. The extreme rays of the dual cone define intrinsic covectors, and these covectors classify every incremental residual future by a finite sign cell: negative, tight, or positive along each extremal Farkas direction. The resulting carrier is the right-stable carrier of this cone-induced observation family, whose source is canonical: the restricted covector geometry of the order cone on the residual span of the language. We organize this construction through an observation-refinement correspondence, a cone-refinement calculus, and a separation between the qualitative conic observation quotient and the numerical residual carrier needed for potential certificates. A bounded-horizon fragment is fully computable by enumeration of accumulated futures, and reproducible evaluation runs show how the conic layer detects qualitative obstruction cells before numerical refinement.

Faruk Alpay, Levent Sarioglu

We study exact predecessor and rank search in a routed, atom-budgeted, certified-repair learned-index architecture. An ordered directory routes each query to a contiguous interval, a counted local predictor returns a certified rank window, and exact repair resolves the remaining uncertainty by comparisons. The result is scoped to this architecture and does not claim guarantees for arbitrary learned-index designs such as unconstrained RMI dispatch, hash routing, neural routing, or exact-payload indexes without additional accounting. The main parameter is conditional residual answer entropy: the entropy of the exact answer after the leaf, predictor output, certificate, and charged pre-repair information are observed. We prove a two-sided accounting theorem showing that this functional gives the query-time scale under the stated architecture and local predictor-atom budget. Directory space, sorted-array storage, and transcript-indexed repair-program space are treated as separate system costs, so the theorem is not a byte-level space lower bound or a compact implementation recipe. We also give a rank-spread specialization in which the radius term log(1 + Delta) is valid only when many residual ranks remain likely after the predictor transcript is known. For counted piecewise-linear segments, we make the profile term non-oracular, derive a shadow-price allocation rule, compute finite-instance RGapM and GapM values on real SOSD and Zenodo samples, and report benchmarks against PGM-index, RadixSpline, and binary search. The benchmarks expose overheads and bottlenecks rather than claiming speed for the shadow prototype.

Faruk Alpay, Taylan Alpay

LLM agents process trusted instructions, retrieved records, and tool observations through a common generative channel. This conflates data flow with authority: an untrusted string can affect a secret-bearing response or an action proposal even when no application policy authorizes that influence. We introduce AgentSecBench as an empirical instantiation of a formal security framework for this problem. The framework defines three games-instruction-integrity, retrieval-confidentiality, and capability-integrity-under a common notion of intent-to-execution noninterference with permitted leakage. It represents an application policy as a projection onto authorized observations and capabilities, distinguishes prompt annotations from enforcing projections, and measures both adversarial advantage and whether a defense closes the relevant model-visible channel before generation. The exact-marker experiments are intentionally one observable instantiation of the games rather than a complete semantic security claim: they test disclosure and forbidden-action distinguishers with unambiguous ground truth. We evaluate six defense classes with Qwen3-0.6B and Qwen3-1.7B on paired adversarial and benign-control executions. The measurements show when risk reduction follows channel closure and when a model-visible adversarial capability remains exploitable. The result is a security-oriented evaluation method: prompt text can describe a boundary, whereas provenance projection, capability restriction, and output validation can enforce one.

Faruk Alpay, Levent Sarioglu

Sequential computation increasingly produces long traces containing nested branches, status transitions, textual payloads, and compact summaries of earlier execution. This paper introduces budgeted dynamic trace structures (BDTS), a data-structural framework for maintaining rooted trace graphs and append-only histories under an explicit byte or token budget. BDTS combines status-filtered reachability, cursor pagination, soft-capped recency logs, reference-counted observation keys, delta overlays, bounded cost caches, and summary-plus-suffix compaction. We give formal invariants, asymptotic bounds, and an ancillary Rust implementation with reproducible benchmarks. Across synthetic traces with 10,000-40,000 vertices, the prototype builds graphs in 0.58-2.72 ms, enumerates all descendants in 0.24-1.42 ms, and compacts histories of 350k-2.71M approximate tokens to 1,048-4,120 approximate tokens. Tokenizer and forward measurements with three public model targets reduce 3,359-3,360 trace tokens to 432-433 tokens.

Faruk Alpay, Hamdi Alakkad

Sound event reports often compress timed boundary behavior into frame, segment, or event scores. This paper defines executable boundary contracts for finite sound event traces. The frame fragment is a bounded Boolean fragment embeddable in STL after grid projection. The event layer adds declared interval matching, duration clauses, fragmentation clauses, and obligation restricted vector scoring. The aim is measurement, not a new general temporal logic and not a challenge leaderboard. The artifact evaluates controlled Mini LibriSpeech seeded scenes, MAESTRO Real soundscapes, frozen pretrained timing probes, and an official DCASE 2024 Task 4 baseline track. Across these tracks, standard scores and contract coordinates disagree in interpretable ways. The strongest real corpus finding is that union activity can hide typed boundary failure, while external DCASE outputs provide a class indexed challenge level reference. Code, generated tables, manifests, and Lean checks for the finite frame core are supplied as ancillary material.

Faruk Alpay, Taylan Alpay

We develop a unified mathematical formulation for post-quantum authenticated telemetry and actuation in FADEC-coupled dual-spool turbofan cyber-physical systems. The formulation integrates lattice-based key establishment under LWE/SIS-style assumptions, PUF-derived attestation entropy, authenticated encryption, radar-altimeter integrity, avionics-bus timing, and Kalman residual monitoring in a stochastic hybrid model. Within this model, plant evolution, communication latency, leakage, adversarial channel quality, and cryptographic state evolve under a common filtration. We show that channel uncertainty tightens admissible key-renewal periods, that ciphertext expansion enters bus-level schedulability constraints, and that sensing and actuator limits shape integrity thresholds and allowable control delay. We further relate PUF smooth min-entropy to distinguishing advantage and connect innovation statistics to conservative alarm design. Overall, the results characterize how post-quantum security, real-time schedulability, and closed-loop stability interact in safety-critical aerospace control architectures within a defensive analytical treatment that does not provide operational guidance for interference with real platforms.

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

The Information Bottleneck (IB) method frequently suffers from unstable optimization, characterized by abrupt representation shifts near critical points of the IB trade-off parameter, beta. In this paper, I introduce a novel approach to achieve stable and convex IB optimization through symbolic continuation and entropy-regularized trajectories. I analytically prove convexity and uniqueness of the IB solution path when an entropy regularization term is included, and demonstrate how this stabilizes representation learning across a wide range of \b{eta} values. Additionally, I provide extensive sensitivity analyses around critical points (beta) with statistically robust uncertainty quantification (95% confidence intervals). The open-source implementation, experimental results, and reproducibility framework included in this work offer a clear path for practical deployment and future extension of my proposed method.

Faruk Alpay, Baris Basaran

Biprofile deviation logic models strategic social choice states as pairs $(R,P)$, where $R$ is the true profile used for welfare comparisons and $P$ is the submitted report profile used by the rule. Coalition modalities replace only the reports of the coalition, and their relations satisfy the fixed law $E_C \circ E_D = E_{C \cup D}$. The paper proves soundness and completeness of $H_{\mathrm{bp}}$ for the abstract frame class $\mathrm{Dev}(N)$, with the reverse-composition midpoint displayed inside the canonical proof. It then separates abstract $\mathrm{Dev}(N)$-components from genuine report-coordinate products by coordinate separation. On the social-choice side, the classical facts supply the source notions; the paper-specific contribution is the audit layer for representation changes: typed manipulation witnesses, a boundary-row theorem for off-domain extensions, and a factor-closure criterion for public deletions. The ancillary material contains the input formats, an executable certificate checker, Lean and Alloy companions for the finite relational lemmas and update patterns, recorded run logs, and checksums.

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.

Madina Bazarova, Faruk Alpay

We introduce the finite-horizon first-order rank profile of a language $L \subseteq Σ^*$: the least quantifier rank needed by an $\mathrm{FO}[<]$ sentence to classify membership in $L$ correctly on all words of length at most $n$. The invariant measures quantifier depth only; formula size is deliberately not bounded. First, we prove a rank calculus that is independent of regularity. Every language satisfies $ρ_L(n) \le \lceil \log_2 n \rceil + 4$, via balanced first-order distance formulas and exact-word definitions. Moreover, $\sup_n ρ_L(n) < \infty$ holds exactly when $L$ is globally $\mathrm{FO}[<]$-definable, and the supremum equals the minimum quantifier rank of such a definition. Second, for regular languages we prove a sharp aperiodicity gap: if the syntactic monoid of $L$ is aperiodic, then $ρ_L(n) = O(1)$; otherwise $ρ_L(n) = \log_2 n + O_L(1)$. The lower bound extracts a nontrivial cyclic component from the syntactic monoid and combines it with an Ehrenfeucht-Fraisse power lemma for long repetitions of a fixed word. Thus, for full $\mathrm{FO}[<]$ quantifier rank, regular languages admit no intermediate finite-horizon growth between bounded and logarithmic rank.

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.

Faruk Alpay, Levent Sarioglu

We study a list-constrained extension of modular equation deletion over powers of two, called Coset-List Min-2-Lin$^{\pm}$ over $\mathbb{Z}/2^d\mathbb{Z}$. Each variable is restricted to a dyadic coset $a+2^{\ell}(\mathbb{Z}/2^d\mathbb{Z})$, each binary constraint is of the form $x_u=x_v$, $x_u=-x_v$, or $x_u=2x_v$, and the goal is to delete a minimum number of constraints so that the remaining system is satisfiable. This problem lies between the no-list case and the poorly understood fully conservative list setting. Our main technical result is a coordinatewise balanced covering theorem for linear gain graphs labeled by vectors in $\mathbb{F}_2^r$. Given any balanced subgraph of cost at most $k$, a randomized procedure outputs a vertex set $S$ and an edge set $F$ such that $(G-F)[S]$ is balanced and, with probability $2^{-O(k^2r)}$, every hidden balanced subgraph of cost at most $k$ is contained in $S$ while all incident deletions are captured by $F$. The proof tensors the one-coordinate balanced-covering theorem of Dabrowski, Jonsson, Ordyniak, Osipov, and Wahlström across coordinates, and is combined with a rank-compression theorem replacing the ambient lifted dimension by the intrinsic cycle-label rank $ρ$. We also develop a cycle-space formulation, a cut-space/potential characterization of balancedness, a minimal-dimension statement for equivalent labelings, and an explicit bit-lifting analysis for dyadic coset systems. These yield a randomized one-sided-error algorithm running in \[ 2^{O(k^2ρ+k\log(kρ+2))}\cdot n^{O(1)}+\widetilde{O}(md+ρ^ω), \] and the same framework returns a minimum-weight feasible deletion set among all solutions of size at most $k$.

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, Bilge Senturk

We prove that the Transformer self-attention mechanism in the high-confidence regime ($β\to \infty$, where $β$ is an inverse temperature) operates in the tropical semiring (max-plus algebra). In particular, we show that taking the tropical limit of the softmax attention converts it into a tropical matrix product. This reveals that the Transformer's forward pass is effectively executing a dynamic programming recurrence (specifically, a Bellman-Ford path-finding update) on a latent graph defined by token similarities. Our theoretical result provides a new geometric perspective for chain-of-thought reasoning: it emerges from an inherent shortest-path (or longest-path) algorithm being carried out within the network's computation.

Faruk Alpay, Levent Sarioglu

We study algorithmic barriers to detecting and repairing a systematic form of structural overspecification in adaptive data-structure selection. An input instance induces an implied workload signature, such as ordering, sparsity, dynamism, locality, or substring structure, and candidate implementations may be preferred because they match that full signature even when the measured workload evidence supports only a strict subset of it. Under a model in which pairwise evaluators favor implementations that realize the implied signature, we show that this preference propagates through both benchmark aggregation and Bradley-Terry-Luce fitting. We then establish two main results. First, determining whether a representation-selection pipeline exhibits structural commitment beyond measured warrant is undecidable on unbounded input domains, by reduction from the halting problem, but decidable by exhaustive enumeration on finite domains. Second, under a conservative repair constraint requiring already evidence-aligned pipelines to remain unchanged, any total computable repair operator admits an overspecified fixed point via Kleene's recursion theorem. These barriers are qualitatively different from classical lower bounds in data-structure design: they do not limit efficiency on finite workloads, but the possibility of uniformly detecting and repairing overspecification across pipeline families.

Faruk Alpay, Hamdi Alakkad

This paper introduces Truth-Aware Decoding (TAD), a verification-oriented decoding scheme that aligns neural language generation with knowledge bases. Situated in the tradition of probabilistic program semantics for sequence models, TAD augments modern instruction-tuned systems with a lattice of semantic guards that operate at decode time. Our contributions are fourfold: (i) a constraint-based semantics that renders oracle filtering as a program-logic judgment, (ii) a proof that greedy selection enjoys local likelihood dominance under sound and complete guards (Theorem 2.7), (iii) an entropy-style invariant that quantifies factual risk via knowledge-aware safe mass, and (iv) a multi-agent operational calculus with verified Lean artefacts to certify implementation behaviour. Numerical and algorithmic case studies confirm that the resulting guardrails reduce hallucinations without sacrificing throughput, yielding a pragmatic bridge between large-scale empirical models and formal verification.

Faruk Alpay, Taylan Alpay

Structured prompting with XML tags has emerged as an effective way to steer large language models (LLMs) toward parseable, schema-adherent outputs in real-world systems. We develop a logic-first treatment of XML prompting that unifies (i) grammar-constrained decoding, (ii) fixed-point semantics over lattices of hierarchical prompts, and (iii) convergent human-AI interaction loops. We formalize a complete lattice of XML trees under a refinement order and prove that monotone prompt-to-prompt operators admit least fixed points (Knaster-Tarski) that characterize steady-state protocols; under a task-aware contraction metric on trees, we further prove Banach-style convergence of iterative guidance. We instantiate these results with context-free grammars (CFGs) for XML schemas and show how constrained decoding guarantees well-formedness while preserving task performance. A set of multi-layer human-AI interaction recipes demonstrates practical deployment patterns, including multi-pass "plan $\to$ verify $\to$ revise" routines and agentic tool use. We provide mathematically complete proofs and tie our framework to recent advances in grammar-aligned decoding, chain-of-verification, and programmatic prompting.

Faruk Alpay, Taylan Alpay

Transformer-based language models excel in NLP tasks, but fine-grained control remains challenging. This paper explores methods for manipulating transformer models through principled interventions at three levels: prompts, activations, and weights. We formalize controllable text generation as an optimization problem addressable via prompt engineering, parameter-efficient fine-tuning, model editing, and reinforcement learning. We introduce a unified framework encompassing prompt-level steering, activation interventions, and weight-space edits. We analyze robustness and safety implications, including adversarial attacks and alignment mitigations. Theoretically, we show minimal weight updates can achieve targeted behavior changes with limited side-effects. Empirically, we demonstrate >90% success in sentiment control and factual edits while preserving base performance, though generalization-specificity trade-offs exist. We discuss ethical dual-use risks and the need for rigorous evaluation. This work lays groundwork for designing controllable and robust language models.

Faruk Alpay, Taylan Alpay

We develop a rigorous framework for deterministic coreset construction in empirical risk minimization (ERM). Our central contribution is the Adaptive Deterministic Uniform-Weight Trimming (ADUWT) algorithm, which constructs a coreset by excising points with the lowest sensitivity bounds and applying a data-dependent uniform weight to the remainder. The method yields a uniform $(1\pm\varepsilon)$ relative-error approximation for the ERM objective over the entire hypothesis space. We provide complete analysis, including (i) a minimax characterization proving the optimality of the adaptive weight, (ii) an instance-dependent size analysis in terms of a \emph{Sensitivity Heterogeneity Index}, and (iii) tractable sensitivity oracles for kernel ridge regression, regularized logistic regression, and linear SVM. Reproducibility is supported by precise pseudocode for the algorithm, sensitivity oracles, and evaluation pipeline. Empirical results align with the theory. We conclude with open problems on instance-optimal oracles, deterministic streaming, and fairness-constrained ERM.

Faruk Alpay, Hamdi Alakkad

We present a comprehensive theoretical analysis of first-order methods for escaping strict saddle points in smooth non-convex optimization. Our main contribution is a Perturbed Saddle-escape Descent (PSD) algorithm with fully explicit constants and a rigorous separation between gradient-descent and saddle-escape phases. For a function $f:\mathbb{R}^d\to\mathbb{R}$ with $\ell$-Lipschitz gradient and $ρ$-Lipschitz Hessian, we prove that PSD finds an $(ε,\sqrt{ρε})$-approximate second-order stationary point with high probability using at most $O(\ellΔ_f/ε^2)$ gradient evaluations for the descent phase plus $O((\ell/\sqrt{ρε})\log(d/δ))$ evaluations per escape episode, with at most $O(\ellΔ_f/ε^2)$ episodes needed. We validate our theoretical predictions through extensive experiments across both synthetic functions and practical machine learning tasks, confirming the logarithmic dimension dependence and the predicted per-episode function decrease. We also provide complete algorithmic specifications including a finite-difference variant (PSD-Probe) and a stochastic extension (PSGD) with robust mini-batch sizing.

Faruk Alpay, Taylan Alpay

We formalize three design axioms for sustained adoption of agent-centric AI systems executing multi-step tasks: (A1) Reliability > Novelty; (A2) Embed > Destination; (A3) Agency > Chat. We model adoption as a sum of a decaying novelty term and a growing utility term and derive the phase conditions for troughs/overshoots with full proofs. We introduce: (i) an identifiability/confounding analysis for $(α,β,N_0,U_{\max})$ with delta-method gradients; (ii) a non-monotone comparator (logistic-with-transient-bump) evaluated on the same series to provide additional model comparison; (iii) ablations over hazard families $h(\cdot)$ mapping $ΔV \to β$; (iv) a multi-series benchmark (varying trough depth, noise, AR structure) reporting coverage (type-I error, power); (v) calibration of friction proxies against time-motion/survey ground truth with standard errors; (vi) residual analyses (autocorrelation and heteroskedasticity) for each fitted curve; (vii) preregistered windowing choices for pre/post estimation; (viii) Fisher information & CRLB for $(α,β)$ under common error models; (ix) microfoundations linking $\mathcal{T}$ to $(N_0,U_{\max})$; (x) explicit comparison to bi-logistic, double-exponential, and mixture models; and (xi) threshold sensitivity to $C_f$ heterogeneity. Figures and tables are reflowed for readability, and the bibliography restores and extends non-logistic/Bass adoption references (Gompertz, Richards, Fisher-Pry, Mansfield, Griliches, Geroski, Peres). All code and logs necessary to reproduce the synthetic analyses are embedded as LaTeX listings.

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.