Theorem: Principal renormalization theorem, unrooted

Finite-dimensional coherent contact freedom is characterized abstractly by all numerical-kernel relations; selected constraints are exhibited. Concretely: the raw contact assignments $(C_{\lambda \gamma })$ with $|\gamma |\le N_{\lambda }$ (finitely many slots) are subject to the linear coherence constraints of Proposition prop:Lcoh (edge interchange, wave-operator, and all other constant-coefficient numerical-kernel relations). The numbers $14321$ (raw contact slots), $321$ ($V_{\mathrm{K}}$ block), and $14000$ (complement) recorded by counterterm_map.py are raw counts of pairs $(\lambda ,\gamma )$, not dimensions of constrained spaces; no computed constrained dimension is claimed. Converse. Any assignment $(C_{\lambda \gamma })$ satisfying all numerical-kernel relations descends to a linear map on $\mathrm{span}\{{q^{\circ }}_{\lambda }\}$; adding the corresponding contact combination to ${\mathfrak{R}}_{\mathrm{L}}$ yields an admissible same-degree extension. The map

sends such coherent assignments to local collision distributions of the form (eq:L-deltaQ), with jet bounds as in Proposition prop:Ljet, and with the four interpretive categories of Section sec:l5D. The regression-preserving subspace is the further linear condition $\Delta {\mathfrak{R}}_{\mathrm{L}}{|}_{{\mathcal{V}}_{\mathrm{K}}}=0$. No root attachment, no scale variation, and no bulk-action matching is claimed.

Proof. Finiteness of the raw contact space: finitely many unordered labels (Lemma lem:L3comp: $255$) and finitely many multi-indices $|\gamma |\le N_{\lambda }\le 4$. Constraints: Proposition prop:Lcoh (selected wave-operator constraints are exhibited computationally; the abstract characterization uses all numerical-kernel relations). Converse: a coherent assignment annihilates every constant-coefficient relation among the ${q^{\circ }}_{\lambda }$, hence factors through the quotient span; adding the contact combination preserves the extension property off $0$ and the scaling-degree bound. The map is Lemma lem:Lamb. Jet bounds: Proposition prop:Ljet. Categories: Section sec:l5D. Regression-preserving: Definition def:Lreg.