BOST-CONNES-MARCOLLI SYSTEM FOR THE SIEGEL MODULAR VARIETY

The paper proves existence for 3<β≤4 and uniqueness in that interval via a precise BCM–measure correspondence, Dirichlet-series bounds, and Hecke-point equidistribution, culminating in Theorem 3.17 (uniqueness) and Proposition 3.10 (existence) , with the measure correspondence set up in Proposition 2.12 . The candidate solution reaches the correct conclusion but relies on unsubstantiated steps: (i) an unjustified disintegration claiming the PGSp+4(R)-marginal must be Haar and hence μ = m∞ ⊗ ν; (ii) a key uniqueness step that invokes Kazhdan’s property (T) to deduce a spectral gap for a finite set of normalized Hecke correspondences, without establishing that property (T) for Γ2=Sp4(Z) yields a spectral gap for these non-group averaging operators; and (iii) a heuristic “critical exponent equals 4 due to deg T(p) ~ p^3” that does not match the paper’s explicit Euler-product analysis leading to ζ(β)ζ(β−1)ζ(β−2)ζ(β−3)/ζ(2β−2) and the thresholds at β=3 and β=4 . Hence, while the conclusion matches the paper, the model’s proof is not correct as written.