The Exact Solution for R(Dst) from the Classical Recursion Field

The previous paper in this series estimated the exponent n in the approximate logistic
solution for R(Dst) from the Hubble tension, and identified it as a mode-sum integral
requiring quantum field theory in curved spacetime to compute. This paper shows
that no quantum field theory is required. The recursion field in the Cohesion UFT
is a classical complex scalar field. Its equation of state w = (1 − 2R4
)/(1 + 2R4
) is
derived directly from the classical energy and pressure of the recursion Lagrangian in
the curved background. The resulting ODE dR/dDst = R(1 + 2R4
)/(6Dst) is separable
and integrates to the exact implicit solution:
R
(1 + 2R4)
1/4
= C · D
1/6
st .
This is a complete, closed-form result derived from the recursion Lagrangian and the
asymptote theorem alone. No quantum field theory, no mode-sum integral, and no
Bogoliubov transformation are required. The approximate logistic form used in the
previous paper has n = 1/6 exactly, derived rather than estimated. The asymptote
R0 → 0 as Dst → 0 confirms that the early universe had arbitrarily fast propagation —
inflation is the Dst → 0 limit of the exact solution. The finite observed R0 > 0 comes
from the fact that the real universe always has Dst > 0 (asymptote theorem). Open
Problem 1 advances to 90% complete.