# The second moment of the rays

- 2026-08-28 [Refuted] The non-shift residual of the second moment converges as `R(n)/phi^(2n) -> C ~ 3` at rate `phi^2` - the normalised ratio reads 2.498, 3.045, 3.238, 3.182, 3.113, 2.983, 2.895 at `n = 8, 10, 12, 13, 14, 15, 16`, peaking at `n = 12` and falling by `0.9724` per level, and `R(n)/R(n-1)` sits at 2.573, 2.561, 2.509, 2.541, a rate near `2.54`, strictly below `phi^2 = 2.618`; what survives is `R(n) = O(phi^(2n))` on `n <= 16`, `lim R/phi^(2n)` undecided, `R = o(3^n)` with room `3/2.54` rather than `3/2.618`, and Conjecture Z untouched. Witness: lemma-b-pincer.
- 2026-08-31 [Refuted] The majorant `Sum_z M_n(z)(M_n(z)-1)` is a route to Conjecture W - it grows `2.907` a level at `n = 13` against `2.573` for `R` itself, because it drops the coprimality of `(s,t)` and so counts each collinear pair once per common divisor; the exact identity `R(n) = Sum_z P_n(z)` survives and the golden ceiling `M_n(z) <= F(n+1) - 1` survives, but the step from `P_n(z)` to `M_n(z)^2 - M_n(z)` does not. Witness: lab/py/gasket-witness-weights.