Overlap breaks the attempted context-free product rule
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One shared witness bit already defeats multiplicativity. With \(\tau_z(P)=P(0)+zP(1)\),
The product of the separate summaries is \(z\). Also \(w^2=w\) in the Boolean ring, so \(\tau_z(w^2)=z\), whereas \(\tau_z(w)^2=z^2\).
The field is \({\mathbb F}_{16}={\mathbb F}_2[\theta]/(\theta^4+\theta+1)\). The checker stores field elements as four-bit coefficient vectors: \(\theta\) is integer 2 and \(\theta^2\) is integer 4. These integers encode field elements; they are not ordinary integer multiplication. Exact field multiplication gives the reported mismatch.
The first row loses witness compatibility: \(w\) and \(1+w\) have no common accepting witness, yet their separate weighted summaries multiply to a nonzero value. The second row loses Boolean idempotence. This is a genuine counterexample to a context-free multiplicative trace compiler, not a floating-point effect or an implementation anomaly.
Exact GF(16) fixture imported from the source field routines. The general overlap failure is symbolic and does not depend on the selected field.