Capacity-Fixed Trace-Free Gravity and S3 Valuation Closure: A Unified Invariant Sieve with DESI DR2 BAO Diagnostics

This work develops a unified invariant sieve connecting local trace-free gravity, global horizon capacity, and finite valuation closure on compact S³/B⁴ geometry. Beginning with the trace-free field equation R_μν − ¼Rg_μν = 8πG(T_μν − ¼Tg_μν), stress–energy conservation and the Bianchi identity recover the Einstein system with an integration constant Λ_int. Rather than inserting a bare cosmological constant freely, the framework subjects this mode to the global capacity condition Λ_cap = 3π/(N_∂ℓ_P²), where N_∂ = Cap(H_∂; A_grav, Q_DM, P_val) is a logarithmic entropy-capacity eigenvalue—not dim H_∂. The manuscript carefully separates the integration, boundary-capacity, valuation, and cosmological ledgers, thereby turning their mutual compatibility into a set of exact residual tests, including R_Λ = Λ_obs − 3π/(N_∂ℓ_P²) and R_DM = ∇μJ_DM^μ. The microscopic derivation of N∂ is not assumed; instead, the theory identifies the precise boundary quantity that must ultimately be computed. For the S³/B⁴ sector, the valuation density obeys P_val(D)ρ_val = 0, with P_val(z) = (z − 1)z(z + 1)(z + 2)(z + 3), restricting the admissible scaling powers to {1, 0, −1, −2, −3}. After quotienting the vacuum, curvature, and dust directions, the sieve isolates a unique nonstandard boundary contribution ρ_X ∝ a⁻¹, corresponding to w = −2/3 and distinguished specifically in three spatial dimensions. The resulting closure is finite and falsifiable, with the nonlinear identity B₅ + 5B₄ + 5B₃ − 5B₂ − 6B₁ = 0 providing one of its exact signatures. An audit against the published DESI DR2 BAO Table 4 vector yields Ω_X = 0.146840, Ω_m = 0.293136, and χ² = 9.5299 for the positive a⁻¹ branch. This represents consistency-level improvement, not a detection: the positive a⁺¹ branch is driven toward Ω_P ≃ 0, while w₀wₐ remains preferred by BAO-only AIC. The DESI analysis therefore functions as a diagnostic sieve rather than a discovery claim—and, at the present level of testing, the invariant framework is not falsified.