substrate-framework: Skill for Codex

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 elite move is to split scales into well-conditioned questions, then earn a trustworthy signal. Select applicable methods below and record their measured, bounded, or unresolved scope; this is a technical menu, not a fixed work plan. Origin: issue #155 and the cited methods literature. Apply AGENTS.md's reward-first supervision and language-editing contract.

Achievement — construct the object that precision will measure

Earn a production Hessian, generalized eigenproblem, fixed-$J$ functional, force, splitting, or radiation threshold by constructing its mathematical object, ensemble, admissible representation and observable first. The parent's analytic-closure receipt records the exact equations and complete variations; symmetries, constraints, gauges and zero modes; non-dimensional groups, dominant balances and scaling; available bounds, virial identities, limiting cases and asymptotic operators; the strongest conclusion already fixed without discretization; and one residual proposition that numerics must decide. “No convenient closed form” does not by itself open production numerics, while no proof that every unknown analytic route is impossible is required.