The Redshift Kernel A Forced One-Parameter Dark-Energy Equation of State with a No-Phantom Bound

A companion paper proves that a self-similar recognition ledger admits a unique instance weighting: the continuum measure f(t) = φ⁻ᵗ, with φ the golden ratio, forced by composition and a single-step balance with no fitted scale. This paper evaluates that measure in one place, the cosmological redshift coordinate, and reads off a dark-energy equation of state whose functional form carries no fitted parameter. The construction needs one physical premise: cosmic scale is graded by φ-foldings and the aging dark-energy charge dilutes through the single temporal channel of the 1 + 3 split, one recognition step per rung. The continuum theorem then forces the kernel K(z) = φ^(−log_φ(1+z)) = 1/(1 + z) with no remaining freedom, so w(z) = −1 + δw₀/(1 + z) is exactly the Chevallier-Polarski-Linder form, but confined to the one-dimensional segment w_a = −(1 + w₀): an exact sum rule w₀ + w_a = −1. The kernel shape 1/(1 + z), the sum rule, and the no-phantom bound are base-independent: they follow from any exponential one-channel survival law and survive the complete failure of the golden-ratio measure theorem, since b^(−log_b(1+z)) = 1/(1 + z) for every base. Only the amplitude's numerical value and the saturation threshold φ⁸ ≈ 47 carry golden-ratio content. The amplitude's magnitude is forced by accumulated recognition cost; its sign is not, because the cost-blind measure is symmetric about w = −1, so the no-phantom bound w(z) ≥ −1 is stated as one explicit premise, the null energy condition ρ_DE(1 + w) ≥ 0, and read off as a falsifiable consequence rather than smuggled in. Under equilibrium occupancy (cited to the companion) the amplitude pinches to a horizon-independent near-point (w₀, w_a) ≈ (−0.882, −0.117) for any scale ratio above φ⁸ ≈ 47, carrying no new continuous degree of freedom (one named discrete closure η and one stated equilibrium premise aside), where the dark-energy density was at most 42% denser in the past, a finite, monotonic, CMB-safe departure from a constant Λ. The claim is sharp enough to die cleanly. I state its kill conditions and its present standing against DESI plainly: the current central values lie off the forced segment and, taken at face value, cross into the phantom sector, so the hypothesis is in tension now and will be decided by the model-independent reconstructions Roman and Euclid already have scheduled.