# Review contract and photon reference planes

## What can be verified

The package exposes equations, parameter values, units, data origins, implementation locations, residuals and tolerances. Original r4 inputs are read-only references. Changed assumptions define a new conditional scenario.

The evidence levels are independent mathematical checks, reconciliation of the same simulation, numerical convergence, and experimental validation. The last two may remain incomplete while the first two pass. This is consequential for the tabulated-ice point-nadir estimator.

## Photon reference planes

For an isotropic point source and a small planar nadir aperture:

$$K_{\rm vac}=\frac{1}{4\pi R_0^2},\qquad f_{\rm ap}=A K,\qquad N_{\rm PE}=\eta A\Phi_{\rm ap}.\tag{V1}$$

Here K is delivered photons per square metre per emitted photon. It is not an escape probability. The directional gain K/K_vac can exceed one. The transport chapter uses K_0 for its attenuated, unscattered direct term; K_vac always means the vacuum reference here. The aperture-interception fraction is a different quantity. When aperture-plane fluence has already been measured, applying atmospheric attenuation again counts that loss twice.

For a uniform extended radiance field and a circular cone in air:

$$G=A\int_\Omega\cos\theta\,d\Omega=A\pi\sin^2\theta_{\max},\qquad \Omega=2\pi(1-\cos\theta_{\max}).\tag{V2}$$

Using A times geometrical solid angle for a wide filled cone omits the cosine projection. A nonuniform angular optical efficiency requires its own integral.

A band radiance in rayleigh and an energy spectral radiance are different inputs:

$$1\ {\rm R}=\frac{10^{10}}{4\pi}\ {\rm photons\ m^{-2}\ sr^{-1}\ s^{-1}},\qquad E_\gamma=\frac{hc}{\lambda}.\tag{V3}$$

Dividing a band-integrated auroral brightness by 4 nm gives an average spectral comparison only under that declared convention. Its photon-count contribution is integrated once; it must not also be multiplied by the filter bandwidth.

## Numerical acceptance and uncertainty

Algebra checks state absolute and relative tolerances. Quadrature checks compare separately formulated integrals with closed forms. Monte Carlo checks disclose seeds, batch counts and standard errors. A native-array identity does not validate the distribution that generated those arrays.

Adjacent response bins share photon histories. Their error bars must not be added as independent variances. Finite-batch standard errors can miss rare large contributions. Small estimated errors do not remove the demonstrated GHM convergence gap.

## Conditions outside the model

The source is assumed isotropic. The new cylinder is homogeneous and conservative with vacuum outside. The molecular column is a separate direct-beam calculation. Scalar transport does not model polarization, event-specific geometry or an observed ice-habit distribution.

Ideal Poisson waveforms omit analog correlation, afterpulsing, particle bursts and operational event-selection effects. There is no empirical Kp/lunar narrowband radiance distribution, universal minimum thermal 777/337 ratio, selected-device orbit-degradation model or target fluence/exposure population in this package. These are missing inputs, not numerical zeros.