small-ratio-numerics

Small-Ratio Numerics

A synthesis of how communities that live in the small-ratio regime — nuclear Skyrme, baby-Skyrme, magnetic skyrmions, Einstein–Skyrme, boson stars, liquid-crystal defect energetics — handle a problem that defeats naive single-grid minimization. The recurring pattern: a strong short-distance scale sets the core and the bulk of the energy; a much weaker long-distance scale carries the physics you care about; the question (a soft Hessian eigenvalue, a tiny energy difference, a weak force) lives in the gap between them. Once $\varepsilon$ or $\lambda_{\min}/\lambda_2$ drops below ~1e-3, the weak signal is usually smaller than truncation error on any single grid, and the calculation reports spurious instability, wrong force signs, runaway frequencies, or "minima" that move with the box.

The field's response was not more precision — it was splitting the question into pieces that each have a well-conditioned answer. That split, and the techniques around it, is what this skill shares. It is advice, not a work plan; pick the pieces that fit the situation. (Origin: the methods note in issue #155 and the literature it cites.)

Eligibility before precision

Do not form a Hessian, generalized eigenproblem, fixed-$J$ functional, force, splitting, or radiation threshold until the mathematical object, ensemble, admissible representation, and observable have earned their upstream licenses. The background must satisfy the correct Euler–Lagrange equations in the claimed function space; record both its residual and a conditioning-aware forward-error estimate. Build the full second variation by differentiating the complete functional before restricting zero-valued fields or blocks. Declare the constrained tangent space, gauges and moduli, kinetic-metric rank and sign, asymptotic operator, and essential-spectrum threshold.

For a collective clock, first establish a globally defined compact action, fixed period and generator normalization, action invariance, vacuum fixation, conserved Noether charge, and nondegenerate Legendre map. Only then may “fix $J$” mean minimization on a physical fixed-charge manifold. A localized positive-norm eigenmode establishes MODE_EXISTS; it does not establish a cyclic phase, Noether charge, fixed-$J$ clock, or nonlinear periodic orbit.