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What remains comparable when mathematical systems are described by fundamentally different equations?

Volume I of the Structural Atlas applies the same structural-reduction protocol to thirteen canonical systems, including:

• Schrödinger and wave equations
• Maxwell’s equations and general relativity
• Navier–Stokes, Burgers, and KdV equations
• Reaction–diffusion and Kuramoto systems
• The logistic map, Lorenz equations, and the 2D Ising model

The comparison does not ask whether these systems share the same equations. They clearly do not.

It asks whether recognizable organizational roles remain after the domain-specific mathematics is reduced into a common structural description.

Across the thirteen reductions, six recurring roles appeared:

• Stable regimes
• Boundary regions
• Coherence degradation
• Sensitivity amplification
• Transition behavior
• Attractor behavior

Their mechanisms and strength of expression varied substantially. The paper therefore treats them as candidate organizational invariants—not established universal laws.