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.

The dilution premise is discharged here: 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 premise (the deviation accumulates linearly in the scale factor), 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.

This paper also maps that premise exactly rather than merely tagging it. Read as a dilution channel count, with each recognition channel attenuating by φ⁻¹ per rung of cosmic scale, the accumulation exponent is the channel count, and within this integer power-law family the premise is proved to hold if and only if d = 1: the single-channel law is the unique member affine in the expansion fraction u = z/(1 + z), the zero-channel law fails the kernel's own early-time limit, and every d ≥ 2 fails a proved second-difference probe across u ∈ {0, 1/4, 1/2}, while the rung machinery itself propagates but does not determine the count. The affinity condition is algebraically the premise itself, so this classifies the premise's content and its rivals; it does not reduce its strength.

The open residue is the whole premise, exactly stated: that the aging charge dilutes through the single temporal channel of the forced 1 + 3 spacetime split. Until that identification is derived and the scale factor is constructed from the ledger, 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.

The kernel carries a normalized machine-checked contract: amplitude identity, accumulator normalization, the asymptotic early boundary, and a non-vacuity witness, with no interior shape-selection clause anywhere in the contract. A canonical accumulator packages the cosmic-history provenance as an explicit hypothesis, with a compiled spoof shown to fail it. The temporal identification is attacked with certified-provenance discharge machinery and reaches named inclusion: the full premise stack is bundled exactly, and three certified walls delimit it (no constructor-trace channel converter exists; intensivity of the physical aging ratio is reduced to one named faithfulness clause with a compiled two-cell falsifier; the rung machinery certifiably cannot supply that ratio package). A Friedmann-level likelihood instrument stands behind the registration: a frozen, hash-stamped gate and parameter ledger, DESI DR2 BAO and masked Pantheon+ as named datasets, an independently controlled and fully interval-certified scorer, and a certified global-minimum fit by interval branch-and-bound, in progress at this writing. Its verdict adjudicates the conditional registered shape; it does not change the conditional status of the shape itself.