decoy_q_zero

Module Wave C2 R0+R1 builds a dynamic structure-function bracket on two lattice sites. It closes the first two typed residuals of the QG Wave C2 gap-5 draft: R0 (decoy) shows that naively plugging a configuration-dependent inverse metric into the frozen Hamiltonian slot fails, and R1 shows that the honest Fréchet derivative cancels in the Hamiltonian–Hamiltonian bracket.

The decoy phase point is the explicit witness: unit configuration and unit momentum at site 0, zero at site 1, chosen so that both the gradient and $\partial g/\partial q$ are nonzero at site 0. Phase space on $n=2$ is a pair of maps $\mathbb{Z}/2\mathbb{Z}\to\mathbb{R}$ (configuration and momentum). The first projection is the configuration field $q$.

This lemma records the elementary evaluation $q(0)=1$ at that witness, which residual calculations need when comparing honest and frozen configuration partials.

One-line wrapper: simp unfolds the definition of the decoy phase point. The configuration component is the piecewise map that returns $1$ at site $0$ and $0$ elsewhere, so the goal reduces immediately.

Feeds the R0 decoy theorem TypedResidual_naive_dynamic_HamW_decoy_fails, which states that at this phase point, with the decoy lapse and site 0, the honest configuration partial of the dynamic Hamiltonian differs from the frozen partial evaluated at $w:=g(x)$ by an uncompensated $\partial g/\partial q$ term. Without the unit evaluation of $q(0)$, that residual comparison cannot fire.

In the broader Seven Gaps gravity program this is scaffolding for dynamic structure-function recovery on two sites. It does not flip gap5_constraint_recovery; continuum and HKT residuals remain open. The lemma is local bookkeeping inside the R0/R1 closure, not a forcing-chain step (T0–T8).