Trace-Free Gravity and Horizon Capacity: Fibonacci Shell Closure and an Autonomous Edge-Hamiltonian Candidate for the Cosmological Constant

This work develops a horizon-capacity mechanism for the cosmological constant, shifting the problem from the magnitude of local vacuum energy to the selection of a discrete boundary sector. In trace-free gravity, constant vacuum-stress shifts disappear from the local gravitational source, while the contracted Bianchi identity restores Λ as a global integration mode. The central postulate identifies this mode with an integer horizon capacity q = A∂/(4ℓP²), giving the quantized relation Λ_q = 3π/(ℓP²q). The resulting question is therefore no longer “why is vacuum energy so small?”, but which horizon-capacity sector is dynamically selected? A Fibonacci capacity coordinate, n∂(q) = log_φ√(q/π), closes at the shell N_φ = 292, while the finite valuation polynomial P_val(z) = (z−1)z(z+1)(z+2)(z+3) fixes d_val = 5. Their combined valuation-compressed saddle is q₀ = πφ⁵⁸⁴/(1 + π/50) = 3.307251460713979 × 10¹²², corresponding to Λ₀ ≈ 1.0908980053 × 10⁻⁵² m⁻² for the stated Planck length. The key advance is to replace a numerical matching prescription with an explicit autonomous edge-selector candidate. The boundary construction combines a Fibonacci registry, common-kernel admissibility constraints, APS determinant phases, Harish–Chandra/stringy edge modes and a boundary-running transmutation scale. In a single declared scheme, the determinant calculation yields Ξ_edge = 0.99916928…, while the boundary beta sector gives b g∂²(ℓP⁻¹) = 0.5597545859987624455…, producing the same capacity scale through q_edge = Ξ_edge exp[16π²/(b g∂²)] = q₀. The corresponding edge free energy, Γ∂(q) = q[ln(q/q_edge) − 1] − log Z_edge(q), converts the real saddle into a physical discrete selection: q_phys = arg min Γ∂(q) over admissible positive integers, and finally Λ_phys = 3π/(ℓP²q_phys). If the scheme-fixed determinant derivative and convexity corrections remain below the finite Fibonacci-shell margin, the minimum stays inside the N_φ = 292 shell. Crucially, the construction is designed to be non-circular and falsifiable: Λ_obs, H₀, the observed horizon radius, q_obs, q₀ and N_φ are forbidden as defining inputs or renormalization conditions for the edge dynamics. The manuscript also identifies a significant obstruction—the naive combination of Fibonacci fusion, standard Hopf density and an O(1) APS defect selects a shell near N ≈ 9 rather than 292—showing that the full nonconformal/stringy determinant structure is essential rather than decorative. The proposed Hamiltonian is therefore presented as a concrete quantum-edge candidate, not as a derivation from an established UV-complete theory of quantum gravity. Its central claim is sharper: the observed cosmological scale may emerge as the integer-capacity minimum of a microscopic horizon partition function, turning the cosmological-constant problem into a testable problem of boundary spectrum, determinant structure and quantum-state counting.