Causal–Spectral Distinguishability Selection on a Conditional Round S³ Carrier

This work develops a causal–spectral selector on a conditional round S³ carrier. Its central point is the separation of four layers that are often confused: carrier distinguishability, spectral mode capacity, causal accessibility, and physically admissible code capacity. On a round carrier S³Rcurv, the causally accessible fraction is exact: ηR (T) = (τ − 1/2 sin (2τ) ) / π, with τ = cT / Rcurv for 0 ≤ T ≤ tcoh, and ηR (T) = 1 after the coherence time. For homogeneous shell-complete spectral projectors, causal spectral accessibility reduces to this same volume fraction, so the raw causal capacity becomes Bcausal^ (P) (ε, T) = bP ηR (T) NP (ε), where bP = log₂ qP converts finite alphabet states into bits. The key construction is the dimensionless gravitational selector ηgrav^ (P) = Bₚhys^ (P) / Bcausal^ (P). Physical admissibility is imposed by gauge constraints, gravitational dressing, energy bounds, entropy bounds, and stability conditions. Under the safe ceiling estimate, Bₚhys^ (P) ≤ min (Bcausal^ (P), BBek, Bₕolo), with BBek = 2π ED rD / (ℏ c ln 2) and Bₕolo = AD / (4 ℓP² ln 2). Hence the selector is a controlled contraction, 0 ≤ ηgrav^ (P) ≤ 1. In the min-law diagnostic model this becomes the compact normal form ηgrav^ (P) = min (1, Bceil / (bP ηR (T) NP (ε) ) ). The spectral side supplies explicit computable capacities. For scalar modes on round S³Rcurv, N₀ (ε) = M₀ (M₀ + 1) (2M₀ + 1) / 6, with M₀ = √ (1 + (Rcurv/ε) ²) and Weyl scaling N₀ (ε) ∼ (Rcurv/ε) ³ / 3. In the early-time, fine-resolution regime, ηR (T) ∼ 2τ³ / (3π) and NP (ε) ∼ CP (Rcurv/ε) ³, so the curvature radius cancels from the joint asymptotic law: DP ∼ (2 bP CP / 3π) (cT/ε) ³. This gives a clean self-similar rule: in the filtered scalar Weyl regime, halving ε reduces the selector by the cubic factor ηgrav (ε/2) / ηgrav (ε) ≈ 1/8. The manuscript is deliberately conservative in status. It does not claim that physical space has measured S³ topology, that arbitrary objects are three-spheres, that Hopf fibers are automatically material rings, or that the selector alone derives classicality, quantum gravity, or the Standard Model. The round-S³ formulas may be used as a screen transform and comparison language: for any positive length r one may compute round-screen quantities such as V₃ (r) = 2π²r³, R₃ (r) = 6/r², and tcoh = πr/c. But a topology claim requires an independent topology theorem or a measured physical field invariant. The durable contribution is therefore the exact causal–spectral integration, the selector ratio, the threshold structure, and the finite-alphabet discipline that turns spectral multiplicity into physically bounded code capacity.