Acyclic CIVP–Index Locking, Molecular Corner Statistics, Entropy–Flux Structure, and Capacity Selection in Trace-Free de Sitter Gravity

This work develops a type-safe statistical and operator-algebraic framework for sector selection in trace-free de Sitter gravity, built around intersecting null boundaries and the quantum characteristic initial-value problem (CIVP). The construction begins with a positive atomic null-area measure whose elementary constituents — embadons — define an unordered molecular configuration and a bosonic permutation quotient. Double-null CIVP gluing preserves the common atomic measure, while an independent holomorphic carrier K_q = H⁰(CP¹, O(q−1)) provides an exact collision-safe locking criterion: the residual evaluation complex is acyclic precisely when N_emb = q_ind. Importantly, molecular support number and geometric area are kept distinct; identifying them requires the additional unit-weight condition γᵢ = 1. This separation prevents several formally similar but physically inequivalent quantities — area capacity, molecular count, carrier dimension, Berry degree, and Jones index — from being identified by notation alone. A second layer isolates the genuinely finite-index sector. Berry–Kirillov geometry produces a line bundle of degree N−1 with analytic index N, while Capelli–Harish–Chandra data provide an independent rank-sensitive reconstruction mechanism. The analysis proves that neither finite matrix rank nor the continuous supertranslation semigroup can simply be interpreted as a Jones index. However, if a correctly typed CIVP defect satisfies the strict Pimsner–Popa window 1/3 < λ < 1/2*, the small-index classification rigidly forces Ind(X) = φ², selecting the A₄/Fibonacci standard-invariant branch under the stated hypotheses. This Fibonacci structure is therefore a consequence of a finite-index constraint, not a postulated macroscopic particle count. At the thermodynamic level, the admissible bridge is correspondingly additive: ΔS_gen = log Ind(X). A multiplicative area ladder is incompatible with this entropy typing; the natural multiplicative object is instead the state count Ω = e^(S_gen), for which Ωₙ₊₁/Ωₙ = Ind(X). Sector selection itself is supplied by a single reduced ultraviolet measure, with Z_q^UV = I_q^UV/Γ(q+1). Its scheme-independent shape is encoded by the discrete curvature κ_q^UV = Δ² log I_q^UV, which is the complete invariant under affine reweightings I_q^UV → C e^(aq) I_q^UV. Strict discrete convexity together with one adjacent sign crossing yields a unique integer sector q*, while explicit bounds determine whether that selection survives topology-dependent Casimir deformations. Under five independent physical certificates — existence of the finite-index defect, the Pimsner–Popa window, unit molecular weights, acyclic evaluation locking, and a stable UV selector — the framework closes to q_can = N_emb = q_ind, with the selected capacity mapped kinematically to de Sitter geometry through R_q = ℓ_P√(q/π) and Λ_q = 3π/(qℓ_P²). The central result is therefore not a fitted prediction of the observed cosmological constant, but a sharply falsifiable conditional architecture specifying exactly which mathematical locks are rigorous and which physical identifications still require independent derivation.