Diagram inventory and amplitudes

The inventory status is manually specified and partially checked. It accepts the o60 repairs: mixed D04 and D06, $V_{\mathrm{af},\varphi }$ at order ${\kappa }^1$, $m=0$, and leg balance. D04 freezes the kinetic structure $h\partial \varphi \partial \varphi$, with each vertex carrying one external and one internal scalar, so the count is four legs, two internal, two external, and mass zero. That placement is not all-internal. Global leg balance is not connectedness and is not one-particle irreducibility. Classical tree seeds D07, D08, and D09 are excluded from one-loop counterterm free slots. The Einstein–scalar $hh$ sector retains the D11 scalar loop and the D12 scalar tadpole. D11 is evaluated as the matter-loop subsector under the prescription above. D12 is the massless dimensional-regularization prescription, not a default zero inserted when the ledger is absent.

Renormalized amplitudes, at this milestone, are the kernel statements already recorded. Route A and Route B give the same prefactor $\Pi =(i{\kappa }^2/4)T$. Extension of the D11 kernel under the active $W_w$ reference is Constructed. The contact Ward identity is Constructed as kernel_two_point_Ward. Gate 3 overall remains Partial only because the spanning set is a candidate list. Bulk amplitudes are not Constructed. O.1 stays NOT_CLOSED.