Triadic Gauge Axioms — Methods & Notation for Size–Shape–Hodophase

**Scope (methods/notation card).** This note standardizes symbols, sectors, bundles, and reporting for the de-tripling (triadic) program: **Size–Shape–Hodophase**. It defines the gauge sectors \(G_s\) (size/time), \(G_{\mathrm{sh}}\) (shape frame × labels), and \(G_\phi\) (hodophase); the base \((S_m,g_m)\) (mass-weighted Kendall shape space); the triad connection/curvature; and the [DP] vs [H] regime tags (block-diagonal vs mixing). A single **abelian Kendall-loop** computation (unequal masses) anchors units and reporting, with the PT term tied to the Gaussian curvature \(K\) of \((S_m,g_m)\). **What’s included here.**- Axioms and notation; label-monodromy reporting \((\ell_\gamma,\theta_\phi)\).- Puncture vs blow-up domains and the **relative Stokes** boundary term.- A constructive, reproducible **recipe** for the horizontal-lift/history operator \(\mathcal K_T\) and the transport-fixed lift norm \(\widetilde C_{\mathcal K}(T)\) (triangulate \(\Sigma\), discrete Stokes for facewise fluxes, assemble \(\mathbf K\), take \(\|\mathbf K\|_2\)).- **Sensitivity box:** corrected *quadratic* scaling of bounds with \(\widetilde C_{\mathcal K}(T)\).- Minimal guidance for near-collision conditioning (use \(\tilde S_m\), refine locally).