Local controls and fixed interactions

Set $H^F=U_F^{-1}HU_F$, using the block half-density action, and similarly for $G,FG$. Covariance (3.7) transfers the statistical evolution identity for $H$ to $H^F$. To see precisely which current is transferred, a real half-density coordinate conjugation of $-\Delta/2$ has the form

Its current is $a^{\alpha\beta}\operatorname{Im}(\overline\Psi\partial_\beta\Psi)$. Explicitly, if $v_F^\Psi$ denotes the conjugated velocity, then

The Jacobian half-density factor is real and hence contributes no imaginary phase gradient. Changing variables in the continuity equation transfers the assigned-density identity with this same velocity. Only the metric in the transformed particle block changes.

The operator difference

therefore has zero kinetic part. All one-body terms cancel. All pair terms except $W_{ij}$ cancel as well, leaving exactly multiplication by (4.1). The same mixed difference of the four currents is zero, block by block. Subtracting the four transferred statistical identities gives $D\Pcal_\Psi[-iM_{F,G}\Psi]=0$.

Take $F=F_s^u$, $G=F_t^v$, divide by $st$, and let $s,t\to0$. The limit is (4.2). The mixed differences and their derivatives have common compact support in the two active blocks; the other coordinates are spectators only in this multiplier estimate. Taylor's formula gives convergence on every Schwartz tangent vector. Thus (3.2) applies.

Some component $c=\partial_{i,a}\partial_{j,b}W_{ij}$ is nonzero on a product of sufficiently small balls $B_i\times B_j$. Taking $u=f e_a,v=g e_b$ in (4.2) gives $f(q_i)g(q_j)c(q_i,q_j)\in\Acal$. For $h$ compactly supported inside that product, $h/c$ is smooth there. Finite sums of separated compact functions approximate it in $C^\infty$, with common compact support. For example, extend it to a box, approximate its smooth periodic extension by Fourier partial sums, and multiply by fixed cutoffs in the two variables. Multiplication by $c$ and (3.2) give $h\in\Acal$.

Annihilator identities are invariant under pullback by the one-body coordinate transports: differentiate (3.7) along a phase variation. Independent compact diffeomorphisms can move any sufficiently small pair of balls to this nonvanishing product. Pullback therefore gives all compact multipliers in every such product cell. A finite partition of unity on the support of a target function proves the result globally.

A finite partition reduces the question to a small box with room to spare in a larger box. Write its coordinates as $(x_1,x')$, let $H(x')=\int f(t,x')\,dt$, and choose a unit-integral bump $\alpha(x_1)$ in a spare interval disjoint from the $x_1$-projection of $\supp f$. Then

is compactly supported. Choose a compact function $\beta(x_1)$ whose derivative is one on the projection of $\supp f$, zero on $\supp\alpha$, and has its compensating integral in another spare interval. Let $\chi(x')$ be one near the transverse support of $a$ and put $b=\beta\chi$. Transverse derivative products vanish, whereas $\partial_1a\,\partial_1b=(f-\alpha H)\beta'=f$. All supports fit in the larger box. Summation proves the lemma.

Induct along a spanning tree. One-body and edge multipliers are available. Suppose all compact multipliers on a connected collection $S$ of particle blocks are available, and attach a new particle $j$ by an edge at $i\in S$. For compact one-body factors $f_k$, apply lemma 4.3 to the desired factor on block $i$, writing $f_i=\sum_\ell\nabla a_\ell\cdot\nabla b_\ell$. Then

Only the shared block contributes. The two inputs are available by induction and edge saturation, so lemma 3.1 gives the desired separated product after summation. Separated compact functions are $C^\infty$-dense, with common blockwise compact supports, in compact functions on these blocks, by the Fourier-cutoff argument in lemma 4.2. This is convergence after multiplication by each fixed Schwartz state, including spectator variables. Thus (3.2) completes the induction.

Use proposition 4.4, proposition 3.3 in dimension $3N\ge2$. For $N=1$, one-body controls already give full scalar saturation. lemma A.1 below supplies normalized nowhere-zero approximations; A4 and (2.4) then extend the equality of measures to every state. Undoing mass scaling gives (1.1).