Log-concavity and Kesten–Stigum tightness for planted 3-SAT broadcasts

We study tree reconstruction for planted random 3-SAT in which the planted assignment fixes the law of the number of true literals in a clause (Barthel et al.). The plantings without a local field form a family indexed by the probability p₀ ∈ [0, 1/4] that a clause has three true literals, from planted NAE-SAT to planted 3-XOR-SAT. On the Poisson hypertree that describes them locally we prove that the Kesten–Stigum (KS) bound is the reconstruction threshold for 0 ≤ p₀ ≤ 0.0321, with a computer-assisted step for p₀ > p_t ≈ 0.0283, and that it is not the threshold for p_tc < p₀ < 1/4, where p_tc = (7 − 3√5)/4 ≈ 0.0729; there reconstruction holds at T(p₀) < 1 times the KS density. For robust reconstruction (leaves seen through an arbitrary weak channel) KS is the threshold for every p₀ < 1/4. At the KS point robust reconstruction fails for p₀ < p_tc, with an explicit radius computer-assisted for p₀ > p_t, and holds for p₀ > p_tc. This is decided by the sign of the second-order coefficient c(p₀) = (2/3)(1 + 8p₀)(p₀p₂/p₁² − 1) of the χ²-capacity recursion, where p_j is the probability of a clause pattern with j false literals; c < 0 exactly when these weights are strictly log-concave. At second order this explains the numerical coincidence, found by Krzakala, Mézard and Zdeborová, of the tricritical point (where the predicted transition becomes discontinuous) with the power-law planting. The proofs use a gauge making the signed clause channel binary memoryless symmetric, Gu's less-noisy comparison with erasure channels, an explicit two-moment Gu–Pandey argument, and exact interval arithmetic. Finally, for every density above 0.9919 we show reconstruction below KS once p₀ is close enough to 1/4.