# 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.