Definition: Parameterized contact-role realization

Fix a graph and a collision chart (eq:collisionchart). Choose the following data.

First, choose a finite-order differential operator

Its actual order and coefficient functions are parameters. The bound $|\alpha |\le 6$ in (eq:Cpartsource) does not bound $d_{\Gamma }$.

Second, list the fixed smooth weight generators already present for that graph, denoted

They are placeholders for inherited external weights, not freely chosen additional functions. Include the graphwise evaluations of the source-named functions $g,\partial g,{\sigma }_{\epsilon },{\sigma }_{\epsilon }^{\prime },{\sigma }_{\epsilon }^{\prime \prime }$ in this list. No formula for ${\sigma }_{\epsilon }$ is introduced.

For the polynomial realization convention of this patch, choose a finite set $F_{\Gamma }\subset {\mathbb{N}}^{{\ell }_{\Gamma }}$ and constant coefficients $a_{\Gamma ,\kappa }$, and define

This is an explicit polynomial family. The source does not state that this family equals its entire admissible weight class.

Third, let

be the spectator parameter. Its name records the contracted graph and retained slots. Equation (eq:Stemplate) does not determine the inherited spectator distribution.

For $|\alpha |\le 6$, define the local template value

Only choices satisfying the source's parity, ghost-number, field-degree, support, normalization, product, and pushforward requirements are candidates for membership in ${\mathfrak{C}}_{\mathrm{part}}$.