High-Dimensional Relational Manifestation: A Geometric Framework Hypothesizing the Basis of Capability Emergence and Dialogue Simulation in Large Language Models

Language is intrinsically a structured network of relational constraints rather than isolated symbolic tokens. In this work, we propose the High-Dimensional Relational Manifestation Hypothesis (HDMH) as a theoretical framework addressing the fundamental mechanics of Large Language Models (LLMs). Our core thesis is that high-dimensional representation may facilitate the manifestation of latent relational patterns, providing a geometric explanatory hypothesis for capability emergence and a structural perspective on multi-turn dialogue simulation. Starting from thought experiments on degenerate representations (d in {1, 2}), we show that within linear superposition representations, representational dimension defines a geometric capacity limit: parameter expansion enlarges function complexity within existing dimensions, but within a single forward step cannot manufacture independent relational axes beyond the vector space dimension. When multi-relational graphs are compressed into low dimensions, cross-talk severely constrains relational structure. Conversely, high-dimensional spaces provide abundant near-orthogonal directions via concentration of measure, hypothesized to facilitate the decoupling and joint manifestation of latent relational patterns. Within an idealized spherical-superposition model under matched-filter readouts, we formally establish:1. A Provable Sufficient Bound (Theorem 1): For m >= 2 superimposed relational signals in R^d, reliable joint recovery with failure tolerance delta is mathematically guaranteed whenever d >= 2(m-1)log(m/delta).2. An Asymptotic Scaling Conjecture (Conjecture 1): The median transition dimension for non-destructive relational manifestation is conjectured to scale asymptotically as d_50(m) = Theta(m log m).3. Preliminary Probit Numerical Benchmark: Using maximum-likelihood probit generalized linear models (GLM) and Fieller confidence sets on preliminary simulation batches (m in {8, 16, 32, 64}), we observe working thresholds shifting monotonically (rho_50: 0.902 -> 1.014 -> 1.163 -> 1.250), qualitatively aligning with the extreme-value sub-leading logarithmic drift O(log log m / log m) derived from an analytical independent-Gaussian surrogate. Finally, we distinguish four conceptual and functional tiers: domain-level relations, internal representations, generated token sequences, and external judgments, while distinguishing domain constraints from textual corpus statistics within the first tier. We define the computational ontology of an LLM as a self-contained parametric functional mapping. Emergence is conceptualized as a threshold-like manifestation transition wherein multi-relational dependency graphs achieve joint non-destructive retrieval across active channels. In sparse-evidence regimes, residual superposition cross-talk is hypothesized to potentially contribute to representational drift and factual errors. Within the model's parametric mapping, the forward pass computes next-token probability distributions that do not themselves provide externally verified judgments of factual truth or formal correctness; factual and ungrounded statements are generated via the same autoregressive decoding interface, with epistemic distinction arising when decoded sequences are evaluated against external verification criteria.