On the Derivation of the Velado Constant: κ = √(2/7) as a Universal Threshold of Information-Geometric Phase Transitions

We establish that the detection-impact constant κ ≈ 0.534 appearing in the Velado Bound (D × I ≥ κ) admits two independent closed-form derivations converging to within 0.02%: an algebraic form κ = 4/7 − 1/27 arising from the universal phase transition threshold with cubic corrections, and a geometric form κ = √(2/7) emerging naturally from the 2-norm structure of the probability simplex. The appearance of 4/7 in both expressions connects this constant to the inverse susceptibility exponent (1/γ = 4/7) of the two-dimensional Ising model at criticality, to the Byzantine fault tolerance bound for k-hypergraphs, and to the empirically observed grokking threshold in neural network generalization. This dual derivation establishes the Velado constant as a fundamental constant of information geometry, governing the phase boundary between learnable and unlearnable, between consensus and chaos, between order and disorder.