A structure theorem making the exponential metric an exact solution: the necessity of a ghost

This paper asks a dynamical question: under what action is the exponential(Yilmaz) metric an exact solution? It proves a structure theorem — underspatial conformal flatness and the condition that both sectors carry oppositepowers of one and the same function (A·B = 1), a static massless scalar sourceforces F to be exponential. The amplitude equation fixes the normalization ofthe scalar's kinetic term (|β_s| = 1/(8πG)), not the exponent; the exponent isfixed to 1 by the Newtonian limit. A second result is that G^r_r = −F'² ≤ 0holds independently of the form of F, so regular matter is excluded by thegeometry and the source must be a ghost scalar. Note on the series. This is a NEW, self-contained paper, not a revision of theauthor's earlier preprint "Gravity and Optics in a Double-Exponential Metric"(DOI 10.5281/zenodo.23571695). That earlier work established the metric as apostulate and computed its 1PN/2PN phenomenology; the present paper suppliesits dynamical origin. Sections 2 and 4 here restate the earlier results (withexplicit attribution) and do not claim them as new. Readers looking for thephenomenology should consult the earlier preprint; readers looking for theaction and the ghost argument should consult this one. This paper does not propose a new theory of gravity. Its claim is: thedynamical origin of the exponential (Yilmaz) metric can be written as astructure theorem, and this theorem forces a ghost scalar.