Formal diagram coefficient-to-observable adoption

Assume Countable Choice, ${\mathrm{A}\mathrm{C}}_{\omega }$ The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), as required by the cited mathematical phase-space map.

For a specifically named finite tree calculation, adopt its selected formal diagram coefficient as a leading perturbative coefficient with a stated phenomenological reading. For QED, use the vertex and free-line conventions of the current draft supplier Explicit formal QED vertex/free-line convention. A Dirac species of natural charge $q_n$ has vertex $i{q}_n{\gamma }^a$; the electron has $q_n=-e_n$ and vertex $-i{e}_n{\gamma }^a$. The rational fermion line is $i\frac{\mathrm{p̸}+\mu }{h(p,p)-{\mu }^2},$ for $\mu >0$ and off-shell $p$ with $h(p,p)-{\mu }^2\mathrm{\ne }0$. In Feynman gauge the auxiliary photon line is $-i\frac{h_{ab}}{h(k,k)+i0}=i\frac{{\eta }_{ab}}{h(k,k)+i0}\mathrm{.}$ For a finite matrix kernel the coefficientwise inverse in Explicit formal QED vertex/free-line convention is part of the formal convention and asserts no convergence. If a time-ordered distribution is used, use the actual free carrier specified there; that identification is conditional on The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$). The displayed pole notation does not define loop products. In the pair-production channel below the selected muon also has model charge $-e$, and only the photon line is internal.

For declared incoming spin probabilities, sum over declared final spin labels, and a specified formal amplitude $\mathcal{A}$, adopt the cross-section coefficient map of Complete two-body phase space and coefficient flux map. Its scattering inputs are positive-shell energy vectors $p_i=(E_i,{\mathbf{p}}_i)$ with $E_i=\sqrt{\mathrm{\mid }{\mathbf{p}}_i{\mathrm{\mid }}^2+{\mathsf{M}}_{i,\mathrm{i}\mathrm{n}}^2}>0$, ${\mathsf{M}}_{i,\mathrm{i}\mathrm{n}}\ge 0$, and common future-timelike total $P=p_1+p_2=k_1+k_2$. The outgoing $k_j$ are on their specified positive shells. Require $s=h(P,P)>({\mathsf{M}}_1+{\mathsf{M}}_2{)}^2$ for the outgoing rest energies and positive incoming flux $F>0$, precisely the supplier's admissible scattering domain. In the center-of-momentum frame it uses the invariant incoming flux $F$, the two-body measure $d{\mathrm{\Phi }}_2$, and the final-state factor $S_f=2$ for identical particles or $S_f=1$ for distinguishable particles, integrated over the full labeled sphere. When this coefficient is read phenomenologically, the initial preparation, channel, threshold, and approximation are part of the model statement. No universal error estimate for omitted terms is implied.

For the scalar examples, separately adopt the constant formal amplitudes $\mathcal{A}=-\lambda$ for the quartic contact channel and $\mathcal{G}=G_E$ for the cubic parent decay, with the units and external normalization stated at each use. The scalar scattering inputs lie on their stated positive shells with a common total and the same strict threshold and positive-flux conditions. For decay, the parent is $P_0=({\mathsf{M}}_0,\mathbf{0})$, ${\mathsf{M}}_0>{\mathsf{M}}_1+{\mathsf{M}}_2$, and its two positive-shell daughters sum to $P_0$. Their coefficient map is the one in Complete two-body phase space and coefficient flux map. If a rate $\mathrm{\Gamma }$ is reported as an energy width, the convention is explicitly ${\mathrm{\Gamma }}_E:=\mathrm{\hslash }\mathrm{\Gamma }$; it does not supply a survival law.

The readings above do not identify a formal coefficient with an exact continuum cross section, probability, decay rate, or measured estimate. In particular they do not construct an interacting continuum $S$-matrix or LSZ limit, establish convergence of a perturbation series, bound loop or soft-radiation effects, specify beam or detector cuts, resolve threshold resummation, or reproduce the statistical analysis of an experiment. A comparison with observations would require the actual apparatus, sampling, calibration, uncertainty, and radiative-inclusiveness conditions. The reported electron magnetic-moment measurements, including the Fan et al. report listed in the source references, remain source-reported empirical results and are not premises of this adoption.