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.