This work presents the strict external-ceiling proof core of a conditional distinguishability calculus on a round S³ carrier. Its aim is to determine how many physically realizable distinctions can exist inside a finite causal region before an independently specified capacity bound becomes active. The construction begins with the exact causal fraction ΓR (T), combines it with the spectral count NP (ε) and the alphabet weight bP, and defines the physical capacity through the selector Σgrav (P) = min1, Bceil / bP ΓR (T) NP (ε). The zero-capacity boundary is treated separately, since the selector ratio is undefined when the raw capacity vanishes. The Bekenstein and holographic expressions enter only as external comparison ceilings: they constrain the available distinguishability but are not interpreted as a fundamental ontology of information. The framework also keeps the carrier curvature radius Rcar distinct from the causal-screen radius RT = cT, preventing a global geometric scale from being silently identified with the local causal domain. The decisive result emerges in the local carrier–Weyl regime, cT ≪ Rcar and Rcar/ε ≫ 1. Here the radius dependence of the causal fraction cancels the radius dependence of the spectral count, producing the curvature-free relation ΓR (T) NP (ε) ≈ (2CP/3π) (cT/ε) ³. The selector therefore assumes the universal cubic form Σ (u) = min (1, u³), with u = ε/ε*. When the active ceiling is holographic at RT = cT, the threshold becomes (ε) ³ = (2bP CP ln 2/3π²) cTℓP²*, and the associated mass–time law is * (m) ³T = ħ²/ (Gc) ·3π²/ (2bP CP ln 2) **. In the scalar binary sector, bP = 1 and C₀ = 1/3, yielding the exact dimensionless invariant J = 9π²/ (2 ln 2) ≈ 64. 0747. The same proof chain also fixes the critical screen energy Escr (R) = c⁴R/ (2G), the screen density ρscr (R) = c⁴/ (4π²GR²), and the exact same-scale ratio Ωscr = 2/ (3π) ≈ 0. 2122. These results form a compact theorem-level structure based solely on declared geometry, spectral conventions, causal localization, and external ceilings—without particle-data input, cosmological fitting, or phenomenological identification.
Batenin et al. (Fri,) studied this question.