arXiv subjects

Faruk Alpay

Publications and source records attributed to Faruk Alpay.

At least 73 records · Page 4Linked to original sources

arXiv2026

The Geometry of Thought: Disclosing the Transformer as a Tropical Polynomial Circuit

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.

cs.LG↗

arXiv2026

Bare-Metal Tensor Virtualization: Overcoming the Memory Wall in Edge-AI Inference on ARM64

The deployment of Large Language Models (LLMs) on edge devices is fundamentally constrained by the "Memory Wall" the bottleneck where data movement latency outstrips arithmetic throughput. Standard inference runtimes often incur significant overhead through high-level abstractions, dynamic dispatch, and unaligned memory access patterns. In this work, we present a novel "Virtual Tensor Core" architecture implemented in software, optimized specifically for ARM64 microarchitectures (Apple Silicon). By bypassing standard library containers in favor of direct memory mapping (mmap) and implementing hand-tuned NEON SIMD kernels, we achieve a form of "Software-Defined Direct Memory Access (DMA)." Our proposed Tensor Virtualization Layout (TVL) guarantees 100% cache line utilization for weight matrices, while our zero-copy loader eliminates initialization latency. Experimental results on a 110M parameter model demonstrate a stable throughput of >60 tokens/second on M2 hardware. While proprietary hardware accelerators (e.g., Apple AMX) can achieve higher peak throughput, our architecture provides a fully open, portable, and deterministic reference implementation for studying the memory bottleneck on general-purpose ARM silicon, meeting the 200ms psycholinguistic latency threshold without opaque dependencies.

cs.CL↗

arXiv2026

Rate-Distortion Analysis of Compressed Query Delegation with Low-Rank Riemannian Updates

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.

cs.CL↗

arXiv2025

Truth-Aware Decoding: A Program-Logic Approach to Factual Language Generation

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.

cs.AI↗

arXiv2025

Calibrated Counterfactual Conformal Fairness ($C^3F$): Post-hoc, Shift-Aware Coverage Parity via Conformal Prediction and Counterfactual Regularization

We present Calibrated Counterfactual Conformal Fairness ($C^3F$), a post-hoc procedure that targets group-conditional coverage parity under covariate shift. $C^3F$ combines importance-weighted conformal calibration with a counterfactual regularizer based on path-specific effects in a structural causal model. The method estimates group-specific nonconformity quantiles using likelihood-ratio weights so that coverage degrades gracefully with the second moment of the weights. We derive finite-sample lower bounds on group-wise coverage and a bound on the equalized conditional coverage gap, and we show first-order control of a counterfactual coverage-parity surrogate via smooth threshold regularization. The approach is model-agnostic, label-efficient, and deployable without retraining. Empirical evaluations on standard classification benchmarks demonstrate improved group-conditional coverage and competitive efficiency relative to shift-aware and fairness-oriented conformal baselines. We discuss practical considerations, including partial availability of sensitive attributes and robustness to structural causal misspecification.

stat.ME↗

arXiv2025

A Riemannian Variational and Spectral Framework for High-Dimensional Sphere Packing: Barrier-Dynamics Reconciliation, Periodic Rigidity, and Discrete-Time Guarantees

We develop a unified framework that reconciles a barrier based geometric model of periodic sphere packings with a provably convergent discrete time dynamics. First, we introduce a C2 interior barrier U_nu that is compatible with a strict feasibility safeguard and has a Lipschitz gradient on the iterates domain. Second, we correct and formalize the discrete update and give explicit step size and damping rules. Third, we prove barrier to KKT consistency and state an interior variant clarifying the role of the quadratic term. Fourth, we show that strict prestress stability implies periodic infinitesimal rigidity of the contact framework. Fifth, we establish a Lyapunov energy descent principle, an energy nonexpansive feasibility projection (including a joint x and lattice basis B variant), and local linear convergence for the Spectral Projected Interior Trajectory (Spit) method. We also provide practical Hessian vector formulas to estimate smoothness and curvature, minimal schematic illustrations, and a short reproducibility stub. The emphasis is on rigorous assumptions and proofs; empirical evaluation is deferred.

math.OC↗

arXiv2025

Glass-Box Analysis for Computer Systems: Transparency Index, Shapley Attribution, and Markov Models of Branch Prediction

We formalize glass-box analysis for computer systems and introduce three principled tools. First, the Glass-Box Transparency Index (GTI) quantifies the fraction of performance variance explainable by internal features and comes equipped with bounds, invariances, cross-validated estimation, and bootstrap confidence intervals. Second, Explainable Throughput Decomposition (ETD) uses Shapley values to provide an efficiency-preserving attribution of throughput, together with non-asymptotic Monte Carlo error guarantees and convexity (Jensen) gap bounds. Third, we develop an exact Markov analytic framework for branch predictors, including a closed-form misprediction rate for a two-bit saturating counter under a two-state Markov branch process and its i.i.d. corollary. Additionally, we establish an identifiability theorem for recovering event rates from aggregated hardware counters and provide stability bounds under noise.

cs.PF↗

arXiv2025

On the Fixed Point Property in Reflexive Banach Spaces

Fixed point theory studies conditions under which nonexpansive maps on Banach spaces have fixed points. This paper examines the open question of whether every reflexive Banach space has the fixed point property. After surveying classical results, we propose a quantitative framework based on diametral l1 pressure and weighted selection functionals, which measure how much an orbit hull of a fixed point free nonexpansive map can collapse. We prove that if either invariant is uniformly positive, then the space must contain a copy of l1 and thus cannot be reflexive. We present finite dimensional certificates, positive and negative examples, and an x86-64 routine that computes mutual coherence and a lower bound for the pressure. The paper clarifies why existing approaches fail and outlines open problems and ethical considerations.

math.FA↗

arXiv2025

Quantization Errors, Human--AI Interaction, and Approximate Fixed Points in $L^1(μ)$

We develop a rigorous measure-theoretic framework for the analysis of fixed points of nonexpansive maps in the space $L^1(μ)$, with explicit consideration of quantization errors arising in fixed-point arithmetic. Our central result shows that every bounded, closed, convex subset of $L^1(μ)$ that is compact in the topology of local convergence in measure (a property we refer to as measure-compactness) enjoys the fixed point property for nonexpansive mappings. The proof relies on techniques from uniform integrability, convexity in measure, and normal structure theory, including an application of Kirk's theorem. We further analyze the effect of quantization by modeling fixed-point arithmetic as a perturbation of a nonexpansive map, establishing the existence of approximate fixed points under measure-compactness conditions. We also present counterexamples that illustrate the optimality of our assumptions. Beyond the theoretical development, we apply this framework to a human-in-the-loop co-editing system. By formulating the interaction between an AI-generated proposal, a human editor, and a quantizer as a composition of nonexpansive maps on a measure-compact set, we demonstrate the existence of a "stable consensus artefact". We prove that such a consensus state remains an approximate fixed point even under bounded quantization errors, and we provide a concrete example of a human-AI editing loop that fits this framework. Our results underscore the value of measure-theoretic compactness in the design and verification of reliable collaborative systems involving humans and artificial agents.

math.FA↗

arXiv2025

New Approaches to the Fixed Point Property in L^1 Spaces

This paper presents new approaches to the fixed point property for nonexpansive mappings in L^1 spaces. While it is well-known that L^1 fails the fixed point property in general, we provide a complete and self-contained proof that measure-compactness of a convex set is a sufficient condition. Our exposition makes all compactness and uniform integrability arguments explicit, offering a clear path from measure-theoretic compactness to weak compactness, normal structure, and ultimately fixed points via Kirk's theorem. Beyond this geometric approach, we contextualize this result within broader structural strategies for obtaining fixed points in L^1 and related spaces. We discuss the roles of ultraproducts, equivalent renormings that induce uniform convexity on l^1, and the fixed point property in non-reflexive spaces like Lorentz sequence spaces. This work unifies these perspectives, demonstrating that the obstruction to fixed points in L^1 is not the space itself but specific geometric or structural properties of its subsets. The results clarify the landscape of fixed point theory in non-reflexive Banach spaces.

math.FA↗

arXiv2025

XML Prompting as Grammar-Constrained Interaction: Fixed-Point Semantics, Convergence Guarantees, and Human-AI Protocols

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.

cs.PL↗

arXiv2025

Fixed-Point Theorems and the Ethics of Radical Transparency: A Logic-First Treatment

This paper establishes a formal framework, grounded in mathematical logic and order theory, to analyze the inherent limitations of radical transparency. We demonstrate that self-referential disclosure policies inevitably encounter fixed-point phenomena and diagonalization barriers, imposing fundamental trade-offs between openness and stability. Key results include: (i) an impossibility theorem showing no sufficiently expressive system can define a total, consistent transparency predicate for its own statements; (ii) a categorical fixed-point argument (Lawvere) for the inevitability of self-referential equilibria; (iii) order-theoretic design theorems (Knaster-Tarski) proving extremal fixed points exist and that the least fixed point minimizes a formal ethical risk functional; (iv) a construction for consistent partial transparency using Kripkean truth; (v) an analysis of self-endorsement hazards via Löb's Theorem; (vi) a recursion-theoretic exploitation theorem (Kleene) formalizing Goodhart's Law under full disclosure; (vii) an exploration of non-classical logics for circumventing classical paradoxes; and (viii) a modal $μ$-calculus formulation for safety invariants under iterative disclosure. Our analysis provides a mathematical foundation for transparency design, proving that optimal policies are necessarily partial and must balance accountability against strategic gaming and paradox. We conclude with equilibrium analysis and lattice-theoretic optimality conditions, offering a principled calculus for ethical disclosure in complex systems.

math.LO↗

arXiv2025

Manipulating Transformer-Based Models: Controllability, Steerability, and Robust Interventions

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.

cs.CL↗

arXiv2025

Nonlinear Continuum of States and Intuitionistic Flows in a Cognitive Space

We construct a rigorous mathematical framework for an abstract continuous evolution of an internal state, inspired by the intuitive notion of a flowing thought sequence. Using tools from topology, functional analysis, measure theory, and logic, we formalize an indefinitely proceeding sequence of states as a non-linear continuum with rich structure. In our development, a trajectory of states is modeled as a continuous mapping on a topological state space (a potentially infinite-dimensional Banach space) and is further conceptualized intuitionistically as a choice sequence not fixed in advance. We establish fundamental properties of these state flows, including existence and uniqueness of evolutions under certain continuity conditions (via a Banach fixed-point argument), non-measurability results (demonstrating the impossibility of assigning a classical measure to all subsets of the continuum of states), and logical semantic frameworks (defining a Tarskian truth definition for propositions about states). Throughout, we draw on ideas of Brouwer, Banach, Tarski, Poincaré, Hadamard, and others--blending intuitionistic perspective with classical analysis--to rigorously capture a continuous, ever-evolving process of an abstract cognitive state without resorting to category-theoretic notions.

math.LO↗

arXiv2025

Deterministic Coreset Construction via Adaptive Sensitivity Trimming

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.

stat.ML↗

arXiv2025

Escaping Saddle Points via Curvature-Calibrated Perturbations: A Complete Analysis with Explicit Constants and Empirical Validation

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.

cs.LG↗

arXiv2025

Calibrated Semantic Diffusion: A p-Laplacian Synthesis with Learnable Dissipation, Quantified Constants, and Graph-Aware Calibration

We develop a calibrated diffusion framework by synthesizing three established concepts: linear Laplacian smoothing, nonlinear graph p-Laplacian flows, and a learnable dissipation term derived from a strongly convex potential. This synthesis provides a general model for graph-based diffusion with controllable dynamics. Our key theoretical results include a quantified two-regime decay analysis for $p>2$, which provides stronger, p-dependent transient bounds not captured by standard ISS templates, and the first formalization of a "non-synonymy" impossibility principle, which proves that fixed-parameter models cannot meet universal performance targets across graphs with varying spectral properties. To address this, we propose a constructive calibration algorithm (SGPS) with formal guarantees for achieving target rates and mass. We derive explicit, closed-form lower bounds for the graph p-gap on canonical graphs a notable improvement over prior implicit estimates and provide sharp constants for discrete-time and stochastic stability, including a contextualized restatement of the necessary and sufficient Euler step-size and a strengthened analysis of the stochastic noise floor. Illustrative, small-scale empirical validations confirm the tightness of key theoretical bounds.

math.OC↗

arXiv2025

Reliability, Embeddedness, and Agency: A Utility-Driven Mathematical Framework for Agent-Centric AI Adoption

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.

cs.AI↗