Feigenbaum Residual Geometry

Feigenbaum Residual Geometry: Seam Derivations, Coupled Corrections and the Completed Gap Account investigates the Feigenbaum constants through the residual geometry of the golden-ratio/Seam construction developed in the accompanying MONAD and Elishia programme. Rather than treating the Feigenbaum constants solely as isolated numerical invariants of period-doubling universality, the paper asks what remains when golden recurrence geometry is folded, projected, and decomposed across the underlying pentadic structure. The resulting residuals are retained as first-class mathematical objects instead of being discarded as approximation error. Central to the analysis are coupled relations involving the Feigenbaum constants δF\delta_F and αF\alpha_F, the golden ratio Φ\Phi, conjugate recurrence structure, and the breakup of the pentadic torus. In particular, the ratios and residuals generated by structures such as δF/Φ\delta_F/\Phi and related golden-scaled quantities are examined as parts of a coupled geometric system rather than as independent numerical coincidences. The paper develops a residual account of the Feigenbaum sector: leading approximations are accompanied by the correction terms produced by the same generating operation. These carry-over terms expose information hidden by scalar comparison and allow apparently small discrepancies to be studied for shared structure, sign, scale, recurrence, and geometric origin. This treatment connects period-doubling universality with the broader Seam calculus, in which a transformation is represented not only by its projected output but also by the branch, fold, orientation, and residual information required to reconstruct how that output was produced. The Feigenbaum problem therefore becomes a particularly useful test case for the larger claim that residual structure can encode meaningful information about the geometry of a transformation. The manuscript separates exact identities and reproducible numerical relations from conjectural geometric and physical interpretations. Its purpose is not to replace the established theory of Feigenbaum universality, but to provide an additional structural description of its constants and residuals and to identify concrete mathematical questions generated by that description.