A parameter-free prediction for the present dark-energy equation of state, with the dilution law derived

The DESI second data release, combined with microwave-background and supernova data, prefers an evolving dark-energy equation of state at 2.8σ to 4.2σ, with present-day values w₀ between −0.84 and −0.67 depending on the supernova compilation. Nearly every theoretical response to that hint is a fitted model: given the data, a parameter is chosen. This paper registers the rarer move, made before the decisive data exist. A framework built on a unique recognition-cost functional, whose only distinguished scale is the golden ratio φ, yields without fitting any constant a single number for the present departure of the dark-energy equation of state from −1:

δw(0) = φ⁻⁴ J(φ) = φ⁻³/(3φ + 2) = (13√5 − 29)/4 ≈ 0.017221,

that is, w₀ = −1 + δw(0) ≈ −0.9828, with a proved band 0.015 < δw(0) < 0.018.

An earlier version of this paper registered the number as conditional on a named dilution premise. That premise is now dead: this paper proves that φ⁻ⁿ is the unique dimension-uniform multiplicative occupancy law closed under the framework's self-recognition closure, that the named rivals (φ⁻²ⁿ, (1 + nφ)⁻¹, J(φ)ⁿ, 2⁻ⁿ) each fail the closure, and that the premise cannot be replaced by ledger conservation or by cost minimality, two no-go theorems. What remains conditional is stated at each step: the cosmic-aging kernel (the model's identifying structure) and the closure instantiation every forced ratio in the framework shares. Everything else in the chain is machine-checked theorem.

What is frozen here is the framework's commitment, not a finished cosmology: the clean content is −0.985 < w₀ < −0.982, w(z) ≥ −1 at every redshift, and the thawing sign structure, plus two operational decision thresholds at |δw| = 0.005 and δw ≈ 0.03 marking where the mechanism would die.

Under the framework's accumulation hypothesis (linear in the scale factor; open), the registration gains a second number and a curve, all machine-checked: w_a = −δw(0) ∈ (−0.018, −0.015), the thawing line w₀ + w_a = −1 exactly, and w(z) = −1 + δw(0)/(1 + z) with landmarks (the departure at z = 1 is exactly half the present one), a strictly monotone thaw, and a density ratio bounded everywhere by exp(3 δw(0)) < 1.056.

Because the accumulation hypothesis is not yet derived, no Friedmann-level likelihood comparison with present data is possible; the paper reports where the numbers sit relative to the published DR2 fits without treating that placement as a test. Both current neighborhoods are excluded by the commitment: a convergence to w = −1 below precision 0.005, and the evolving-w best fits DESI currently reports, whose implied departure is ten to thirty-five times larger.

The registration is prospective, written after DR2 and ahead of DESI DR3 (expected 2027), Euclid wide-area cosmology, and the Roman Space Telescope. Should future data converge near w₀ = −0.983 with no phantom crossing, the commitment is on record.