Abstract

We evaluate one external-source benchmark on four-dimensional Minkowski space. The source is classical, and the quantum field is a free real scalar of fixed mass $m>0$, with its minimally coupled canonical stress tensor Wick ordered in the Minkowski vacuum. A two-particle calculation gives the exact spin-two and spin-zero spectral densities. A once-subtracted dispersion prescription for the reduced form factors defines a real, causal, tempered response distribution of scaling degree at most eight. We classify the complete parity-even, Lorentz-covariant, pair-symmetric, doubly transverse contact freedom in that extension class: one quadratic and two quartic polynomial tensors. A zero-momentum condition fixes all three coefficients uniquely. Both linear Ward identities hold distributionally.

The four closed-time-path kernels are constructed consistently at quadratic source order. Smooth preparation is incorporated by multiplying the external source by a fixed time ramp. The resulting response is finite on arbitrary compact joint tests. We calculate its transformation residual, prove plateau-supported intertwining, and construct a late, contact-independent detecting test for an unrestricted preparation-layer transformation.

Spectral detection is not a measurement instrument, a classical record, or a quantum-return decoder. Documentary comparisons retain their individual models and versions and are not premises of the calculation. No relational scalar two-point coefficient, inherited characteristic graph, graviton–ghost inventory, interacting BRST/BV identity, bulk restoration, or ultraviolet completion is established.