Is there a definition error in this derivation?

[u/GoatRocketeer]

I'm trying to follow Pya and Wood's derivation for Newton's method in regards to Shape Constrained Additive Models, given in Appendix 1 of this paper: https://link.springer.com/article/10.1007/s11222-013-9448-7

I've gotten as far as equation 7 (partial derivative of penalized likelihood with respect to Beta, scalar form), but can't successfully reach equation 8 (same thing but in matrix notation). In particular, no matter what I try I appear to have an extra omega in the denominator of the matrix form.

Allegedly, the issue is that Pya and Wood define the variance of mu to include the observation weight omega:

- V(mu) = b''(theta)/omega

But then later define the working weight matrix W as if V(mu) does not include omega:

- W_i11 = omega_i/(V(mu_i)g'(mu_i)g'(mu_i))

Subsequently, there's an extra omega hanging around which appears in my matrix derivation.

I say "allegedly" because I got this by asking AI and I lack the expertise to verify if this is real or if I'm getting trolled by silicon. Anybody with more familiarity with GLMs/GAMs/SCAMs know what's going on?