# The novelty meter

- 2026-09-21 [Proved] Backward, elementarily: `S_f` vanishes for `y > 2`, so `F(s) zeta(s-1)/zeta(s) = (6/pi^2) F(2) 2^(s-2)/(s-2) + int_0^2 E_f(y) y^(s-3) dy`, and `E_f = O(y^(3/2 - eps))` continues the right side to `Re s > 1/2 + eps`; a zero `rho` there needs `F(rho) zeta(rho - 1) = 0`, impossible since `zeta(rho - 1) != 0` for `Re(rho - 1)` in `(-1/2, 0)` and `F(rho + k) != 0` for some `k`, else `f(u) u^(rho - 1)` is orthogonal to every polynomial on `[1, 2]` and `f = 0` by Weierstrass, with `u^k f(u)` again a bump; so the smoothed novelty rate for all bumps implies RH, and the meter is an equivalence as hard as RH, not a route. Witness: stack.md, The novelty meter.
- 2026-09-21 [Proved] The indicator's `1/2` is a sharp-cutoff artefact: `sum_(n <= x) phi(n) - 3x^2/pi^2` is `O(x log x)` by the Mobius route `sum_d mu(d) T(floor(x/d))` and `Omega(x)` because it jumps by `phi(p) = p - 1` at every prime, so the indicator meter sits at exponent `1` in `y`, `1/2` in `q`, unconditionally and sharply, above the `y^(3/2)` of the zeros, and says nothing about zeros; the jump between consecutive integers reads `phi(p) - 3(2p - 1)/pi^2`, `392073.4` at `p = 1000003` and `3920735.7` at `p = 10000019`. Witness: lab/py/smoothed-novelty `main`, SHARP CUTOFF.
- 2026-09-21 [Refuted] The smoothed novelty meter as a route to RH on this tree: on any finite range the smoothed error is a rendering of the first zeros, the bump's Mellin transform killing high zeros faster than any power, so a zero off the line at any height where RH is already checked is invisible here; the meter can neither certify nor refute the hypothesis. Witness: lab/py/smoothed-novelty `main`, the 138-zero agreement at `2.0e-06`.
- 2026-09-21 [Verified] Read at source, Verjovsky, Kodai Math. J. 17 (1994) 596-608: with `m_q(f) = q sum_n phi(n) f(q^(1/2) n)` and `m_0(f) = (6/pi^2) int_0^infty u f(u) du`, Theorem A gives `m_q(f) = m_0(f) + O(q^(1/2) log q)` for continuous compactly supported `f` on `(0, infinity)`; Theorem B part 1, RH iff `m_q(f) = m_0(f) + o(q^(3/4 - eps))` for every `f in C_c^r`, `2 <= r <= infinity`, and every `eps > 0`; part 2, for `alpha in (1/2, 3/4)` the error `o(q^(alpha - eps))` for all `f in C_c^2` iff `zeta` has no zero in `Re s > 2(1 - alpha)`; part 3, for the characteristic function of an interval `limsup q^-alpha abs(m_q(f) - m_0(f)) = infinity` for every `alpha > 1/2`; arXiv:1711.03593 Theorem 5.1 restates the smooth case in `y = q^(1/2)` as `o(y)` and `o(y^(3/2 - eps))` and refers its proof to the Kodai article. Witness: doi:10.2996/kmj/1138040054 p. 597 and arXiv:1711.03593 Appendix B.
- 2026-09-21 [Proved] The Mellin identity `sum_n phi(n) f(ny) = (1/(2 pi i)) int_(Re s = c) F(s) zeta(s-1)/zeta(s) y^-s ds` for `c > 2` and `f` a bump on `[1, 2]`, with main term `(6/pi^2) F(2) y^-2` from the pole of `zeta(s-1)` at `s = 2`; under RH and with `1/zeta(s) << abs(t)^delta` uniformly on `Re s >= 1/2 + eps`, the input the Gaussian Franel study reads at source at `K = Q`, the error `E_f(y) = y^2 S_f(y) - (6/pi^2) F(2)` is `O(y^(3/2 - eps))` for `f in C^2`. Witness: stack.md, The novelty meter, assembled from the Mellin transform; lab/py/smoothed-novelty `mellin_c2` prints `F(2) = 24/35` beside the closed form.
- 2026-09-21 [Verified] On `y = 2^-j`, `j` from 8 to 23.5 in steps of `1/16`, `phi` sieved to `3 * 10^7`, the slope in `q = y^2` of the per-octave root mean square of `E_f` reads `0.5023` for the indicator of `[1, 2]` (lower and upper eight octaves `0.5296` and `0.4883`), `0.7498` for the `C^2` bump `64 (u-1)^3 (2-u)^3` (windows `0.7553` and `0.7487`), `0.7471` for the `C^infinity` bump `exp(4 - 1/((u-1)(2-u)))` (windows `0.7474` and `0.7460`), window residuals `0.065` to `0.169` and full-range residuals `0.179`, `0.092`, `0.115`, no window pair differing by more than `0.05`; the sieve agrees with brute-force gcd counts to 2000 and with the Mobius route at `x = 10^6`, and the summation noise at `j = 23.5` is `5.6e-17` against `abs(E_f) = 5.4e-12`. Witness: lab/py/smoothed-novelty `slopes`, `totients`, `totient_sum_mobius`.
- 2026-09-21 [Proved] Under RH, for smooth `f`, `E_f(y) = sum_rho F(rho) (zeta(rho - 1)/zeta'(rho)) y^(2 - rho) + O(y^(3/2 + delta))`, the sum over nontrivial zeros as residues through the cited heights, written for simple zeros; each term has modulus `abs(c_rho) y^(2 - Re rho)`, so the exponent in `q` is `1 - beta/2` at a zero of real part `beta`, `3/4` on the line and smaller at the zero of a pair that sits right of it. Witness: stack.md, The novelty meter, by the contour shift to `Re s = 1/2 - delta`.
- 2026-09-21 [Verified] The residue sum over the first 138 zeros (height 300, `zeta(rho - 1)` and `zeta'(rho)` from PARI) reproduces the measured `E_f` of the `C^infinity` bump over the whole grid to a relative `2.0e-06` (one zero `0.50`, ten `2.7e-02`, thirty `2.0e-03`), coefficients `1.879e-01` at the first zero to `1.475e-07` at the 138th; for the `C^2` bump the coefficients fall like `gamma^-3.22` on the 138, its `F` being `O(abs(s)^-4)`, `1.386e-01` to `6.541e-06`, and the truncation stops at `2.3e-04`; `abs(E_f)/y^(3/2)` lies in `[1.3e-04, 0.558]` for the `C^infinity` bump against `2 sum abs(c_rho) = 0.755` over the same 138 zeros. Witness: lab/py/smoothed-novelty `main`, `zeros_from_pari`, `mellin_cinf`.
- 2026-09-21 [Verified] Cross-reference to the slopes row: the per-octave method reads "the samples with `j` in `[k, k+1)`, 16 of them except the 9 of the last octave `[23, 23.5]`, placed at `j = k + 1/2`"; the printed slopes are unchanged. Witness: lab/py/smoothed-novelty `slopes`, `octave_rms`.
- 2026-09-21 [Verified] Cross-reference to the 138-zero row: `gamma^-3.22` is now printed by the generator as the least-squares power of `abs(c_rho)` against `gamma` on the 138 zeros, `-3.22` for the `C^2` bump. Witness: lab/py/smoothed-novelty `main`, EXPLICIT FORMULA.