Latent Object Permanence: Topological Phase Transitions, Free-Energy Principles, and Renormalization Group Flows in Deep Transformer Manifolds

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.

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.

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.

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.

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.

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.

A Reproducible, Scalable Pipeline for Synthesizing Autoregressive Model Literature

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.

Alpay Algebra V: Multi-Layered Semantic Games and Transfinite Fixed-Point Simulation

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 $\phi(\cdot, \gamma(\cdot))$ where $\phi$ drives the main semantic convergence while $\gamma$ 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 $\phi$-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.

Recursive Semantic Anchoring in ISO 639:2023: A Structural Extension to ISO/TC 37 Frameworks

ISO 639:2023 unifies the ISO language-code family and introduces contextual metadata, but it lacks a machine-native mechanism for handling dialectal drift and creole mixtures. We propose a formalisation of recursive semantic anchoring, attaching to every language entity $\chi$ a family of fixed-point operators $\phi_{n,m}$ that model bounded semantic drift via the relation $\phi_{n,m}(\chi) = \chi \oplus \Delta(\chi)$, where $\Delta(\chi)$ is a drift vector in a latent semantic manifold. The base anchor $\phi_{0,0}$ recovers the canonical ISO 639:2023 identity, whereas $\phi_{99,9}$ marks the maximal drift state that triggers a deterministic fallback. Using category theory, we treat the operators $\phi_{n,m}$ as morphisms and drift vectors as arrows in a category $\mathrm{DriftLang}$. A functor $\Phi: \mathrm{DriftLang} \to \mathrm{AnchorLang}$ maps every drifted object to its unique anchor and proves convergence. We provide an RDF/Turtle schema (\texttt{BaseLanguage}, \texttt{DriftedLanguage}, \texttt{ResolvedAnchor}) and worked examples -- e.g., $\phi_{8,4}$ (Standard Mandarin) versus $\phi_{8,7}$ (a colloquial variant), and $\phi_{1,7}$ for Nigerian Pidgin anchored to English. Experiments with transformer models show higher accuracy in language identification and translation on noisy or code-switched input when the $\phi$-indices are used to guide fallback routing. The framework is compatible with ISO/TC 37 and provides an AI-tractable, drift-aware semantic layer for future standards.

Alpay Algebra III: Observer-Coupled Collapse and the Temporal Drift of Identity

This paper introduces a formal framework for modeling observer-dependent collapse dynamics and temporal identity drift within artificial and mathematical systems, grounded entirely in the symbolic foundations of Alpay Algebra. Building upon the fixed-point emergence structures developed in Alpay Algebra I and II, this third installment formalizes the observer-coupled {\phi}-collapse process through transfinite categorical flows and curvature-driven identity operators. We define a novel temporal drift mechanism as a recursive deformation of identity signatures under entangled observer influence, constructing categorical invariants that evolve across fold iterations. The proposed system surpasses conventional identity modeling in explainable AI (XAI) by encoding internal transformation history into a symbolic fixed-point structure, offering provable traceability and temporal coherence. Applications range from AI self-awareness architectures to formal logic systems where identity is not static but dynamically induced by observation. The theoretical results also offer a mathematically rigorous basis for future AI systems with stable self-referential behavior, positioning Alpay Algebra as a next-generation symbolic framework bridging category theory, identity logic, and observer dynamics.