Franel one field up

• 2026-09-09 [Proved] Kluyver's identity in Z[i]: the Gaussian Ramanujan sum obeys c_d(lambda) = sum_{e | gcd(d, lambda)} mu_G(d/e) N(e) over ideal divisors, so the exponential sum of the Gaussian Farey set is S_N(lambda) = sum_{[e] | lambda, N(e) <= N} N(e) M_G(N/N(e)) with M_G the Gaussian Mertens function over associate classes; 2720 exact sums at norm bound 50 with 0 mismatches, 68 literal node sums agreeing to 1.281e-13, and the node set identified with the complex Farey set of the literature at T = 2 to 6 (4, 24, 64, 176, 320 points). Witness: lab/py/gaussian-franel check_theorem_1, check_sayous.
• 2026-09-09 [Proved] Franel's identity one field up: the Fourier L^2 discrepancy of the Gaussian Farey set on C/Z[i] is m^2 D_2(N)^2 = 4 zeta_K(2) sum_{[a],[b]} N(gcd(a,b))^2/(N(a) N(b)) M_G(N/N(a)) M_G(N/N(b)), a finite gcd-weighted quadratic form in Gaussian Mertens sums whose kernel is the Smith gcd matrix of the layer Gram; checked against the Fourier side truncated at N(lambda) <= 200000 with gaps 0.001883 and 0.010546 inside the printed tail bounds 0.226192 and 7.093392 at norm bounds 20 and 50; the same identity regenerates the classical Farey discrepancy with no Farey enumeration, C(Q) - 1 = 12 Phi(Q) sum delta_v^2 exactly at Q = 40 and S2 Q reading 0.5395, 0.5848, 0.6241, 0.6387, 0.6560, 0.6538, 0.6564 at Q = 125 to 8000, the Farey page's table digit for digit; the Gaussian global readout collapses like the rational one, sum_{N(a) <= x} M_G(x/N(a)) = 1 at every x to 2000. Witness: lab/py/gaussian-franel check_theorem_2, classical_exact, franel_form, readout.
• 2026-09-09 [Proved] F(N) = O(N^{1+eps}) for every eps > 0, equivalently D_2(N) = O(N^{-3/2+eps}), is equivalent to the Riemann hypothesis for zeta_{Q(i)}(s) = zeta(s) L(s, chi_-4): backward, the four units give M_G(N)^2 <= zeta_K(2) F(N) and partial summation makes 1/zeta_K analytic right of the critical line; forward, by divisor splitting from Littlewood's bound on M_G transcribed to zeta_K by Perron with the bounds of Hu, Kaneko, Martin and Schildkraut (Lemma 5.4 and Lemma 2.4), the biconditional being assembled here and written in neither source; the rational template is Huxley 1971 Lemma 10 and Huxley 2012 Theorem 1 over Q, the announced number-field part abandoned by its author's account, and the Gaussian identity not found in the sources read; an equivalence is exactly as hard as the hypothesis it names and the meter renders, it does not measure. Witness: lab/py/gaussian-franel main (the backward inequality alone, ratio at most 0.017 over norm bounds 100 to 64000), REFS.md.