Research focus. This work develops a conditional selector calculus for physical distinguishability on a compact round S³ carrier. Its purpose is to determine how many states can become operationally distinguishable inside a finite causal region before an independently imposed capacity ceiling is reached. Information is therefore not introduced as a fundamental substance or as an automatic property of geometry. It appears only as a quantitative measure of realizable physical distinguishability. For each spectral sector P, the construction combines the causal fraction ΓR (T), the available spectral mode count NP (ε), and the information weight bP in the finite selector Σgrav (P) = min1, Bceil / bP ΓR (T) NP (ε). The Bekenstein and holographic bounds are used strictly as external comparison ceilings, not as assumptions about the microscopic nature of information. Core result. In the local Weyl regime, where the causal domain is much smaller than the curvature radius of S³, the global radius disappears from the leading causal–spectral capacity. The decisive cancellation gives ΓR (T) NP (ε) ≈ (2CP/3π) (cT/ε) ³. The resulting threshold is therefore controlled locally by causal duration, resolution scale, and the spectral coefficient CP, rather than by the total size of the compact carrier. Matching this capacity to the holographic ceiling produces the cubic relation (εP) ³ = (2bP CP ln 2/3π²) cTℓP²*, together with the dimensionless invariant *JP = (mP) ³TGc/ℏ² = 3π²/ (2bP CP ln 2) **. In the scalar binary sector, bP = 1 and CP = 1/3, which yields J = 9π²/ (2 ln 2) ≈ 64. 0747. Physical significance. A second exact component establishes the normalization of a critical spherical screen. Starting from Escr (R) = c⁴R/ (2G), the framework derives the universal fraction Ωscr = 2/ (3π) ≈ 0. 2122066 and the entropy relation Escr/TR = kB A/ (4ℓP²). The area normalization is thus recovered without identifying the screen count with an independent microscopic entropy ontology. The study isolates a compact proof core consisting of a finite distinguishability selector, a curvature-independent local cubic threshold, and an exact screen-normalization algebra. Comparisons with hadronic or cosmological scales remain diagnostic consequences of the formalism rather than premises of the derivation.
Batenin et al. (Fri,) studied this question.