Dark Energy as Cascade Dynamics: No-Phantom Constraint, DESI Retrodiction, and Sealed Predictions for DESI DR3

The Discrete Stochastic Cascade model on finite graphs is applied to the dark energy equation of state through a single physical identification: the cascade's global beta density maps to the dark energy density. Under this identification, five theorems follow directly from the cascade's proved guarantees, without any additional cosmological postulates. Most critically, the no-phantom theorem establishes that w(z) > −1 at all active steps, a constraint proved from the energy monotonicity supermartingale and enforced as a hard structural prior rather than fitted freely. This constraint is tested against the DESI Year 1 Baryon Acoustic Oscillation dataset (April 2024). A blind retrodiction protocol calibrating on three low-redshift points and predicting three withheld high-redshift points yields a cascade chi-squared of 5.50 on unseen data, compared to 5.54 for ΛCDM and 22.81 for CPL, which has one more free parameter and no phantom bound. A cross-release test training on DESI Year 1 and predicting DESI DR2 (March 2025) shows cascade performing at the ΛCDM level (chi-squared 4.45 vs 4.39), while CPL achieves a lower chi-squared of 2.17 only via a phantom crossing at z = 0.345 that violates the Null Energy Condition. An AIC comparison confirms ΛCDM is mildly preferred on current data; the cascade's contribution is a structural no-phantom prior, not a better fit. Six specific D_H/r_d predictions for DESI DR3 (expected 2027) are sealed in this document. The primary falsifiable claim is the absence of phantom crossing: if DR3 finds w < −1 at ≥3σ, the cascade application is definitively ruled out. All simulation scripts and the sealed prediction record are available at github.com/Shiv2071/Discrete-cascade-model.