Theorem: Compact balanced-forest composition

Let a finite forest have vertex set $V$ and edge set $E$. Give every edge a balanced bidistribution and pull it back to $M^V$ by its endpoint projection. Multiply by a compact smooth joint weight. Then:

1. the finite edge product is defined;

2. its pushforward over any set of internal vertices is defined by proper support;

3. for $n\ge 1$ external slots, the output wavefront is contained in

   $$Z_n=\left\{(x_1,p_1;\dots ;x_n,p_n):{\sum _{a=1}}^n{p}_a=0\right\}\setminus 0.$$

Finite differentiations and smooth multiplications preserve this bound. It excludes an all-future or all-past nonzero causal tuple, and therefore implies the corresponding microcausal kernel condition. With no external slots, the compact pushforward is a scalar.

Proof. Endpoint projections are submersions, so their pullbacks exist. A forbidden product cancellation would assign nonzero edge covectors, with opposite signs at their endpoints, whose sum at every vertex is zero.

The participating edges form a subforest. A leaf of that subforest has one participating incident edge, so its vertex equation forces that edge covector to vanish. Removing the edge and repeating forces every participating covector to vanish, a contradiction. The product criterion therefore applies.

The compact smooth weight makes the product compactly supported. Projection is then proper on its support. For a pushforward wavefront point, the internal covectors are zero. Summing the vertex covectors cancels each edge pair, leaving zero total external covector. This proves eq:Zn. Differentiation and smooth multiplication do not enlarge that bound.

Finally, future and past causal cones are pointed. A tuple contained entirely in either cone with zero sum must have every covector zero, which is excluded.