From the Energy of a Quasilocal B⁴ Filling to an Effective Residual Sector on Compact S³

What if the homogeneous background behaviour usually attributed to a residual dark sector could be reconstructed directly from geometry? This work develops a quasilocal B⁴/S³ geometric-variational map in which the physical arena remains a closed FLRW spacetime M = I × S³_A, while the Riemannian four-ball B⁴_A acts only as an auxiliary filling attached to its compact metric representative. The construction follows the chain S³_A → B⁴_A filling → E₄(A, ξ) → (ε_D, p_D, Q_D). Its geometric core is remarkably simple: ∂B⁴_A = S³_A, V₃ = 2π²A³ and V₄ = (π²/2)A⁴, giving the exact ratio V₄/V₃ = A/4. This dimensional relation turns a filling energy E₄ into an effective homogeneous boundary component without introducing a physical fifth dimension. The direct reconstruction is governed by two compact relations: ε_D = E₄/V₃ and p_D = −dE₄/dV₃. If E₄ = ρ₄V₄, they become ε_D = (A/4)ρ₄ and p_D = −(A/3)[ρ₄ + (A/4)dρ₄/dA]. Pressure therefore appears as the variational response of the filling energy to a change of boundary volume rather than as an independently postulated quantity. The resulting local equation of state is w_D(A) = −1 − (1/3)d ln ε_D/d ln A = −4/3 − (1/3)d ln ρ₄/d ln A. Internal variables ξ generate the exchange channel Q_D, while linear superposition permits several independently reconstructed components. In particular, a two-branch filling yields the exact background decomposition ε_D(a) = ε_q + ε_w a⁻³, p_D = −ε_q, separating a vacuum-like slot from a pressureless dust-like slot. A dimensional extension gives w_d = (β − d − 1)/d for power-law fillings in d + 1 dimensions. The inverse result is equally important: ρ₄(A) = 4ε_D(A)/A. Consequently, every separately conserved homogeneous background fluid possesses a unique B⁴ representation within the adopted homothetic class. This is a precise background reconstruction theorem—not a claim that B⁴ reproduces arbitrary physics. The filling does not determine microscopic origin, particle content, perturbations, sound speed, anisotropic stress or complete spacetime dynamics. The work therefore distinguishes the topological carrier M_car ≃ S³, the homothetic metric representative (S³, g_A), and the observer’s causally accessible domain D_O(τ). It also separates background fluid from microscopic matter, kinematic admissibility from dynamical prediction, and auxiliary filling from physical bulk. The limits A → 0 and A → ∞ remain kinematic limits of the metric family unless a completed dynamics selects them as physical solutions. S⁴ compensation, Hopf structure, causal measure, radius protection, perturbations and the observational map are retained as clearly identified open branches of the wider research programme.