First-class closure for a regular constraint family

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ as in Regular primary constraints and total generator. Let $(P,\omega )$ be a smooth symplectic manifold. For every smooth real function $F$ on an open set $U\subseteq P$, define its Hamiltonian vector field $X_F$ by ${\omega }_z(X_F(z),v)=d{F}_z(v)$ for every $z\in U$ and $v\in T_{z}P$, and define $\{F,G\}:=dF(X_G)$. Nondegeneracy of $\omega$ makes $X_F$ unique; in local coordinates its coefficients are obtained by inverting the smooth nonsingular matrix of $\omega$, so $X_F$ and the bracket are smooth. Let ${\chi }_1,\dots ,{\chi }_r\in C^{\mathrm{\infty }}(U,\mathbb{R})$ be a regular constraint family on the common zero surface $N=\{z\in U:{\chi }_a(z)=0\text{ for all }a\}$ as defined by Regular primary constraints and total generator: the differentials $d{\chi }_a$ are linearly independent at every point of $N$. The family is first-class on $N$ when

$\{{\chi }_a,{\chi }_b\}(z)=0\text{for every }z\in N\text{ and all }a,b\mathrm{.}$

Equivalently, each pairwise bracket is weakly zero in the notation $F\approx 0$ of Regular primary constraints and total generator. On a canonical cotangent phase space this intrinsic convention agrees with Canonical Poisson brackets and local Hamiltonian flow: $\{F,G\}=F_{q^i}{G}_{p_i}-F_{p_i}{G}_{q^i}$. This definition is a pointwise closure condition on the stated regular surface; it does not assert multiplier-algorithm termination or global quotient regularity.