The first derivative bit at p ≡ 5 (mod 16): a proof of Conjecture coord of Paper V

We prove that for every prime p ≡ 5 (mod 16), written p = a² + b² with b even, h(−8p) ≡ h(−4p) + 8·[b ≡ ±6 (mod 16)] (mod 16), where h(−4p) and h(−8p) are the class numbers of Q(√−p) and Q(√−2p). This refines the classical congruence h(−8p) ≡ h(−4p) (mod 8) for p ≡ 1, 5 (mod 16) recorded by Berndt. It settles Conjecture 5.3 (Conjecture coord) of Paper V of the series. That conjecture describes when the first derivative coefficient u₁ of the 2-adic power series of the odd quartic character is divisible by λ² = (1 + i)². The proof: writes 4(h(−8p) − h(−4p)) through the representation numbers of x² + 2y² + 4z² and x² + 2y² + 2z², using Gauss’s three-square theorem; builds from sieved theta series a weight-2 form Φ on Γ₀(1024) ∩ ker χ₈ (Serre–Stark); shows Φ ≡ 0 (mod 256) by Sturm’s theorem and a check below the bound 512; reads off the coefficient of Φ at p. Corollaries: the sum of the Legendre symbols (x/p) over 0 < x < p/8 is ≡ 2·[b ≡ ±6 (mod 16)] (mod 4), the next binary digit after Berndt’s parity of the octant sum; combined with Desnoux’s modulo-64 congruence, c₂R(S/2)h⁺(p) + 17R′(S′/2)h⁺(2p) ≡ 2h(−4p) + 8·[b ≡ ±6 (mod 16)] (mod 16). This links the units and class numbers of Q(√p) and Q(√2p) to b modulo 16. The proof is formalized in Lean 4 with Mathlib (ConjCoord, doi:10.5281/zenodo.23902875). The modularity of Φ with Sturm’s theorem, and Gauss’s theorem, are hypotheses there; the rest is proved, including the check below the Sturm bound as a kernel computation. Version 1.1 proves that the Pell solutions S and S′ in the Desnoux corollary are even, so the halves S/2 and S′/2 are integers. If S were odd, R² ≡ 6 (mod 8); if S′ were odd, R′² ≡ 3 (mod 8). It also cites ConjCoord v1.0.1 (doi:10.5281/zenodo.23913183), in which the formal corollary takes the Pell equations as hypotheses and proves this evenness. The results are otherwise unchanged.