Pentad Hardy Feigenbaum

A directed fivefold rotation on the four-dimensional real sum-zero subspace producesa golden operator and a commuting complex structure. Its symmetric spectral readoutyields the golden pair, while its directed sectors carry angles 2π/5 and 4π/5. We provethat five rotations by φ−1turns leave the residual φ−5, equal to the maximum successprobability of the standard two-qubit Hardy paradox. We place this golden return onthe same pentad-derived carrier and recover further exact Hardy power relations on thecomplex seam conic. A Hardy-normalized unimodal family has golden critical point anda nondegenerate quadratic maximum, providing a dynamical setting for period-doublinganalysis. We then construct a five-sheet extension of the dyadic odometer and showthat its finite dyadic returns leave the oriented pentad residues 1, 2, 4, 3 modulo five. Ano-factor theorem proves that the phase is not a continuous five-cycle factor of the binaryodometer alone. An independent affine Cantor construction retains path order as thetranslation φ−2 − φ−5; its address coding carries dynamics topologically conjugate to thedyadic model. Its Hausdorff dimension differs from that of the quadratic Feigenbaumattractor, obstructing bi-Lipschitz identification. We construct explicit intertwiners fromthe pentad to a five-root Coxeter orbit and to the signed 120-line E8 graph, derive an exactorder-20 golden action modulo five, and identify the Cantor order residual as a binary-carrydisplacement. The 1,120 negative graph triangles give a sharp sum-of-squares identity andthe finite separation 30. An explicit Gaussian continuum model carries the bound throughexact block elimination and has vacuum mass gap √28m0 for an input scale m0 > 0. Aninteracting gauge-theory mass gap and an intrinsic Feigenbaum gap-geometry intertwinerremain open.