The 61-Decimal Limit of π as a Universal Resolution Scale: Geometry, Entropy, Fine-Tuning and the

We identify a universal geometric hierarchy RU/ℓP ∼ 10⁶¹, where RU is the radius of the observable universe and ℓP is the Planck length, and show that this single dimensionless ratio unifies five apparently independent results of modern cosmology through the master inequality 2RU · 10⁻ⁿ ≤ ℓP. The same scale governs the Bekenstein-Hawking entropy bound, the flatness fine-tuning condition, and the vacuum catastrophe discrepancy factor ∼ 10¹²². We propose two falsifiable predictions: a dynamic resolution parameter n(t) evolving with cosmic expansion, and a time-varying cosmological constant Λ(t) ∝ t⁻², testable with Euclid and LSST missions.