From Folding to Reasoning: Spectral Bifurcations and Phase Transitions in Typed Semantic Graphs

We study Reasoning Phase Transitions (RPTs) in typed discourse graphs controlled by a fold parameter 𝛼(depth/resonance). Building on a field kernel with B = σ∇S, free energy 𝐹=𝑈−Θ𝐻sem, coherence 𝜅, dissipation Σ̇, and geometry-aware invariance metrics (ΔFold/IS/BS), we show that critical reasoning leaps occur at an 𝛼\*marked by a multi-signal coincidence: a peak in 𝑑𝜆2/𝑑𝛼, a sharp rise in the relational composition index 𝑑 RCI𝐿/𝑑𝛼≫0, and simultaneous improvements 𝑑 ΔFold/𝑑𝛼≪0, 𝑑Σ̇/𝑑𝛼≪0. We prove a necessity theorem: untyped densification can inflate 𝜆2without gains in RCI𝐿or ΔFold, hence typing (constraints & inverses) is necessary—not merely helpful—for reliable composition. We identify predictors—tension susceptibility 𝜒𝑆=∂∥∇𝑆∥/∂𝛼and the spectral ratio 𝜆3/𝜆2—and provide an online detector with peak search, multi-signal gating, and false-alarm control, integrated into a governance hook (“No leap without logs”). Experiments (E1–E4) link RPTs causally to jumps in invariance and task reliability across toy typed graphs, an ontology “mini-universe,” RAG pipelines, and LLM reasoning suites. Seeds are open and artifacts public.