Tension to Structure: Conditions for Emergent Geometry

We propose a minimal, testable theory for when geometry emerges from pregeometric tension in meaning—framing the transition as an observable event rather than a modeling assumption. We formalize the Span layer as the interference/condensation zone where folds, tension, and resonance accumulate into neighborhoods that prepare a spectrum; Span ends only when an onset triplet co-occurs: a rise in spectral gap (Δλ₂ ≥ ε), an intrinsic-dimension knee, and zigzag-persistence consolidation. On this basis we state six Emergence Conditions (E1–E6): local enabling (non-degenerate tension and directional fold-causality; resonance density; order-faithful composability; tension regularity), a global Cheeger floor for conductance, and the measurable onset triplet that licenses geometry. Together they yield a concise Geo-Bridge—P→Fold⁡σG→Spec(L,λ2,Ek)→Fisher(M,gV)\mathcal P \xrightarrow{\operatorname{Fold}_\sigma} G \xrightarrow{\text{Spec}} (L,\lambda_2,E_k) \xrightarrow{\text{Fisher}} (\mathcal M,g_V)—and delimit the post-onset window where the unique linear, local, reciprocal closure holds: B=σ∇gVSB=\sigma\nabla_{g_V}S. We corroborate the framework with Span mini-experiments: (I) resonance toggles advance onset and increase integrability, (II) Cheeger bottlenecks delay onset and destabilize topology, and (III) regularity breaks produce ill-conditioned Fisher charts despite spurious spectral hints. The work integrates an evidence-before-output gate (S⁵OS) and a reproducibility kit (seeds, thresholds, DOI’d artifacts), and clarifies minimal vs. sufficient conditions, nonlinear pre-onset dynamics, and cross-model/domain robustness for emergent discourse geometry.