TORUS Ladder Dynamics XIII: Comparative Universality and Cross-Domain Coherence

Background and Motivation Earlier TORUS Ladder Dynamics releases (I–VIII) established that closure phenomena emerge robustly in ordered systems, survive strict null controls, and remain sensitive to construction independence. TORUS Ladder Dynamics IX demonstrated that closure universality is conditional rather than naïvely universal, while TORUS Ladder Dynamics X showed that closure behavior is not only observable but predictable using preregistered, frozen structural gates that generalize out-of-sample. TORUS Ladder Dynamics XI transitioned the program from prediction to causation by identifying which structural perturbations preserve, collapse, or induce true separation at the endpoint level. TORUS Ladder Dynamics XII then formalized emergent time and scale as observables, defined via survival surfaces over recursive depth and perturbation breadth. The natural next scientific question is therefore not whether closure, time, or scale emerge, but rather: Do emergent time and scale exhibit cross-domain structural coherence consistent with constrained universality? TORUS Ladder Dynamics XIII addresses this question directly. Scientific Contract All analyses in this release inherit the preregistered contracts enforced throughout TORUS Ladder Dynamics IX–XII: Absolute Domain IsolationNo ladders or endpoints are pooled across domains. Contamination PreventionObserved metrics are computed exclusively from eligible constructions; null processes exist solely to define baselines. Registry-First ProvenanceAll ladder instances, survival surfaces, and derived observables are fully traceable to their generating artifacts. Frozen ThresholdsNo new thresholds, gates, or tuning parameters are introduced in this release. No changes to these contracts are introduced in XIII. Overview of This Release TORUS Ladder Dynamics XIII performs a comparative analysis of emergent time and scale across baseline XII domains and newly introduced domains spanning biochemical, chemical, linguistic, ecological, and synthetic regimes. Using survival surfaces derived under identical recursive and null constraints, this release evaluates whether emergent observables exhibit coherent structural behavior across fundamentally different domains. This release does not claim universality in the strong sense. Instead, it establishes the empirical conditions under which universality claims would become falsifiable. Key Findings Emergent time and scale are not uniformly distributed across domains; some domains exhibit deep, coherent survival under recursion, while others collapse sharply at modest depths. Domains with strong resonance or spacing structure display markedly greater persistence of both time-like and scale-like observables. Biochemical and linguistic domains exhibit structurally informative failure modes rather than pathological noise, reinforcing falsifiability rather than weakening claims. Random and weakly ordered controls behave as expected, validating the comparative framework. Why This Matters TORUS Ladder Dynamics XIII completes the transition from domain-specific emergence to cross-domain comparative structure. It establishes a rigorous empirical boundary between domains that merely exhibit closure and those that plausibly participate in a shared structural regime. This release provides the final empirical groundwork required before any claims of unified structural universality can be responsibly advanced. What TORUS Ladder Dynamics I–V already proved (cumulatively) Release What it actually locked in I Ladder closure exists and is detectable II Closure has structural mechanisms and failure modes III Closure is invariant under many deformations but brittle under specific ones IV Recurrence emerges from order mutation, not tuning V Closure can be discriminated from nulls via calibrated recurrence gates VI Closure in composite ladders persists under cyclic topology and exhibits emergent relational modes VII Constants show a dominant, non-null-explainable consensus scale under independent constructions. VIII Introduces outcome-blind domain feature extraction, topology-conditioned ladder dynamics, and rung participation signatures as explanatory tools for closure universality. IX Introduces outcome-blind domain structure as a predictive constraint on closure behavior. X Closure universality is not merely observable—it is predictable under constrained conditions. XI Closure universality is no longer treated as a black-box phenomenon but as an emergent property supported (or destroyed) by identifiable structural features. XII Formalizes emergent time and emergent scale as observables derived from the persistence of separating invariants under recursive depth and causal perturbation.