The no-exception splitting theorem of Paper 5 was based on local accounting, in which each fragment carries its own container, and its conclusion was complete dispersal. The core of the present model, however, is that R² = Σ ν² sets a curvature-like scale, and local accounting on a flat background is not consistent with the model's own claim. This paper puts the curvature into the accounting and obtains three results. First, the inversion of the splitting theorem: when all fragments ride on a single shared container determined self-consistently by the total content S, minimizing the unfilled amount becomes equivalent to maximizing the total capacity, and an exhaustive check for S ≤ 25 shows that condensation (the single fragment) is strictly optimal. Local accounting (surface term s3/2, subadditive) → dispersal; shared accounting (capacity term s2, superadditive) → condensation — the divide is a single exponent, and physically it is the choice of whether the container = the curvature belongs to each fragment or is shared. In a one-parameter family mixing the two accountings, the condensation onset threshold behaves as w*(S) ≈ 3.40/√S → 0: the larger the system, the more a slight sharing of geometry suffices for condensation to dominate (a scaling of the same form as the Jeans criterion; a structural correspondence, not an identification). Under the exclusion rule (Paper 7), the competitor on the dispersal side changes from the all-1 partition to a Fermi-sea-type configuration, so the threshold drops further (0.6397 → 0.5872 at S = 9). Second, the internal expansion law: when the internal splitting cascade of a closed state space (container fixed, no exterior) is read through the ratios of an internal observer (who uses their own fragment as the measuring rod and carries the dual clock τ = 1/R'), the apparent scale factor satisfies the exact identity a⁴f = π²/2 (f is the occupancy fraction) and, in the leading term, follows a ∝ t1/2 (a universal law independent of the branching number; the fitted exponent in the central window of a deep ternary cascade is 0.507). f ∝ a⁻⁴ is the 4-dimensional dilution of a conserved number, and the appearance of a radiation-type dilution law in a system whose content is oscillation only is a consistency check. Expansion requires neither an exterior nor growth of the container: a system that subdivides inward looks expanding when measured with a measuring rod that itself shrinks. We give the numbers including the exclusion-rule correction of the endpoint (the cascade terminates at s' = 3, and the all-1 state is unreachable). Third, the area law and the reservoir: the phase-space area per state is a rigid body of exactly 2 (= 4δmin²), and the conservation of total area is the same law as the dual reading of energy conservation. On the other hand, imposing exact conservation of mode area freezes the cascade at stage 0. The exact conservation law that actually holds is "occupied area + gap = container volume," and splitting is the process of transferring occupied area into the gap. The vacuum-like state (the gap) is not a structureless void but a reservoir of area that makes evolution possible. Bilingual edition (Japanese and English): Markdown, LaTeX, and PDF for each language, plus three figures (PNG).
Noriaki Kihara (Sun,) studied this question.