Abstract

We construct a controlled classical continuation of the relative Batalin–Vilkovisky structure of X-H-ST on the unchanged I-B/X-A-ST conical host. All analytical operations remain on the declared Minkowski completion, and the regulator width stays fixed and positive. The new interaction is the explicit compact vertex

supported inside the existing relative plateau. A finite coefficient specification of the reference Einstein–scalar action, including its fifth geometric-order contribution, gives a classical BV differential satisfying ${Q_{\lambda }}^2=0$ modulo ${\kappa }^4$. The interaction changes the scalar antifield equation by $\lambda {\kappa }^{3}a{\varphi }^3/6$, while leaving every coefficient through order two unchanged.

Two debugging results are essential. Exact recovery of the full free BV complex requires the centered background sector; for a general fixed off-shell background the inherited tadpole remains. Moreover, a complete finite instrument can report supplied numerical coefficients of a residual, but does not measure an arbitrary continuum or Grassmann-valued residual. We prove the appropriate finite-report theorem, including all selective outputs, failures, surviving systems, and untouched references, and prove its stability under finite adaptive composition and prescribed regular transport.

The chosen polynomial lift has an explicitly evaluated fourth-order master residual. Its new part is

with a nonzero real coefficient on compact ghost data. Thus finite-order classical closure is not silently promoted to an exact master identity. The positive reciprocity witness and the independent convolution gap remain unchanged in their specified diagnostic sectors; the three-tree plateau is retained once, not reconstructed or reinterpreted.

The contribution is a supported, order-controlled analytical–operational assembly, not a new general BV method. Its prospective usefulness is to make a genuine classical interaction compatible with explicit report interfaces while keeping the next obstruction visible. No interacting quantum BRST differential, characteristic quantum coefficient, loop-anomaly cancellation, ultraviolet completion, or spacetime emergence is established.