§Cosmic coherence
dark energy · bounded Z₂ channel · DESI DR2 comparison

Current epoch §status

scale factor a1.000
redshift z0.000
ρDE / ρ01.000
w(a)−1.000
C / C*0.00 %
B(C) / Bpeak0.000
approaching coherence peak

w(a) trajectory §w · DESI overlay

CRR DESI w0wa ΛCDM

Beauty trajectory B(C) §B

Falsifiability §F1 F2 F3

F1peak and w = −1 crossing at different z → CRR wrong
F2quintom A pattern (w > −1 past, w < −1 today) → CRR wrong
F3void vacuum CV ≠ 1 (bare) or Ω/2 (regulated) → CRR wrong
Z₂ Ω = 1/π · C* = π
cv—
target0.159
n0
CVZ₂ · C*SO(2) = —
SO(2) Ω = 1/2π · C* = 2π
cv—
target0.080
n0
CV · 2π·C* ratio = —
scale factor a 1.000
parameterisation
§CRR · dark energy explainer

The phantom crossing is the dark-energy density peak.

In CRR, these are the same event: a topological coincidence predicted with zero free parameters. DESI DR2 observes them at the same redshift.

The framework in one equation

Dark energy density along cosmic scale factor a traces the CRR beauty function for a bounded Z₂ channel:

ρDE(a) ∝ B(C(a)) = exp( C(a)/Ω ) · (C* − C(a))

Ω = 1/π and C* = π are topologically fixed for a Z₂ (bistable) system. They are not free parameters. The coherence variable C(a) accumulates monotonically with cosmic expansion. The beauty function peaks at C = C* − Ω = π − 1/π ≈ 2.823, which is one Ω before the rupture at C·Ω = 1.

What CRR predicts, structurally

Take the log of B and differentiate:

d(ln B) / dC = 1/Ω − 1/(C* − C)

This vanishes at C = C* − Ω. At that coherence value, ρDE is stationary in ln a (a density peak), and the continuity equation gives w = −1 exactly (a phantom crossing). The two events are structurally identical in CRR.

DESI DR2 observation

Abdul-Karim et al. (2025) and Lodha et al. (2025) fit DESI DR2 BAO + Planck CMB + supernovae to a w0waCDM model. The best fit places:

In w0waCDM these two features are independent fits. Their coincidence in DESI is surprising but not required. In the CRR beauty-function ansatz, their coincidence is structural.

What is load-bearing, what is phenomenological

load-bearing (no parameters)

Ω·C* = 1 · topological identity. Beauty peak at C* − Ω · analytic derivative. w = −1 at the peak · follows from continuity + analytic B. The peak/crossing coincidence · ansatz + algebra.

requires a choice

The C(a) accumulation rate · exp-sat vs power-law here. The architectural commitment that dark energy is a Z₂ CRR channel · a bet, not a derivation. Numerical match to DESI w0 · ~1.5σ with 1 parameter, exact with 2.

Falsification criteria

F1: if Euclid, DESI-III or Rubin LSST measure the ρDE peak at a redshift significantly different from the w = −1 crossing, the beauty-function ansatz is false.

F2: if the trajectory reverses to quintom A (w > −1 past, w < −1 today), CRR direction is wrong.

F3: cosmic-variance CV measurements in deep voids that fall significantly outside 1 (bare Z₂) or Ω/2 (regulated) contradict the structural identity CVZ₂ · C*SO(2) = 1.

What this sim renders

The cosmic view shows comoving galaxies distributed in a volume, rendered at physical positions scaled by a(t). The background luminance is modulated by ρDE(a)/ρDE,0. The w(a) plot shows the CRR trajectory against the DESI w0waCDM fit. The beauty trajectory shows B(C) with current C marked. Two live CV probes sample bounded Z₂ and SO(2) stochastic processes, confirming CV = 1/(2π) and CV = 1/(4π) respectively, and the structural identity CVZ₂ · C*SO(2) = 1.

References

Abdul-Karim M. et al., DESI DR2 Results II, arXiv:2503.14738 (2025).
Lodha K. et al., Extended Dark Energy analysis using DESI DR2 BAO, Phys. Rev. D 112, 083511 (2025).
Parr T., Pezzulo G., Friston K.J., Active Inference, MIT Press (2022).
Ito S., Dechant A., Stochastic time evolution, information geometry, and the Cramér-Rao bound, Phys. Rev. X 10, 021056 (2020).
Sabine A., CRR framework, www.temporalgrammar.ai (2026).

Pedagogical simulation. CRR used as computational engine; peer-reviewed FEP and FRW-cosmology vocabulary in the GUI. CRR claims coherence, not proof.