# Design algebra and analytic diagnostics: equations, assumptions and verification This is a reproducibility companion, not a new atmospheric observation or a flight qualification. The independent implementation is `modules/design_review.py`. It imports no original calculation script. It reads the declared parameter grids and comparison targets from `data/design_results.json` and `data/analytic_results.json`. Fresh values and comparison receipts are written to `checks/design_*` and `checks/diagnostics_*`. The stable equation identifiers D01-D14 and P01-P09 connect this account to the code, notebooks and machine-readable method maps. Exact Poisson detection thresholds and their coverage are verified elsewhere: in the FOV and auroral conversion checks here, the original critical count and required detected signal are explicitly inputs. No method below turns selected bright events into a complete source population. ## 1. Reference planes, units and constants The reference planes are: emitted source photons; photons delivered through the atmosphere; photons crossing the specified entrance plane; and registered detections. A measured spacecraft irradiance or fluence already contains the atmospheric delivery. Applying cloud transmission again would attenuate it twice. Geometry, time and photon wavelength arguments to the independent functions use metres, seconds and metres. The tabulated radiance convention per nanometre is retained only in the paired product `L_per_nm * band_nm`; equivalently multiply the former by 10^9 to obtain per metre and the latter by 10^-9 to obtain metres. Those factors cancel. Rows labeled mm, nm, microseconds or degrees are converted explicitly at the function boundary. Steradians are dimensionless in SI but are retained in the bookkeeping to prevent confusing radiance with irradiance. The SI defining values used are h = 6.62607015e-34 J s, c = 299792458 m/s, e = 1.602176634e-19 C, and k_B = 1.380649e-23 J/K. They are not fitted to any event. The official primary values are in the [BIPM defining-constants table](https://www.bipm.org/en/measurement-units/si-defining-constants), rows c, h, e and k (inspected 22 October 2026). The historical US Standard Atmosphere constants below are retained separately rather than silently modernized. ### D01: photon energy $$E_\gamma=hc/\lambda,\qquad N_\gamma=E_{\rm pulse}/E_\gamma.$$ The energy relation uses vacuum wavelength. The legacy numerical examples insert nominal 337.1 nm and 777.4 nm: 5.8927495021e-19 J and 2.5552429343e-19 J per photon. This does not establish whether a particular spectroscopic bandhead or filter certificate was quoted in air or vacuum. A roughly 0.03% photon-energy change is small, while the roughly 0.1 nm line-position shift at 337 nm can matter to narrowband registration. Such metadata must be retained before a real spectral fold. ### D02-D03: ordinary versus projected solid angle For a circular cone of half-angle theta, $$\Omega=\int d\Omega=2\pi(1-\cos\theta),\qquad \Omega_p=\int\cos\vartheta\,d\Omega=\pi\sin^2\theta.$$ Multiplying a flat physical aperture by ordinary solid angle instead of projected solid angle overcounts oblique uniform radiance. For a hemisphere, Omega = 2 pi sr and Omega_p = pi sr. At 60 degrees full cone the values are about 0.842 sr and 0.785 sr. A square field cannot be silently replaced by a circular cone with the same diagonal. For square tangent-plane half-width u = tan(theta_diagonal)/sqrt(2), $$\Omega_{\square}=4\tan^{-1}\left({u^2\over\sqrt{1+2u^2}}\right),\quad \Omega_{p,\square}=\int_{-u}^u\int_{-u}^u{dx\,dy\over(1+x^2+y^2)^2}.$$ The second integral is independently evaluated with 96 by 96 Gauss-Legendre nodes. The MMIA heritage field is described by its square diagonal; its nominal 1 mm ultraviolet entrance aperture must not be confused with its 25 mm photocathode. A proposed 25 mm entrance is a new design. ### D04-D05: received fluence and background counts For a small flat aperture and a point source at angle theta to its normal, $$S=\Phi_{\rm ap}A\eta\cos\theta.$$ Phi_ap is received photon fluence per area normal to propagation; if it was already defined per the physical tilted aperture, the projection is already included and must not be repeated. Eta is the end-to-end detection probability, not automatically bare quantum efficiency. For uniform spectral energy radiance L_lambda, constant response, a rectangular/equivalent band and a circular field, $$b_{\rm sky}={L_\lambda\Delta\lambda\,A\Omega_p\eta\over E_\gamma},\qquad B=(b_{\rm sky}+d)T.$$ The general expression is an integral of projected effective area times radiance times lambda/(hc) over wavelength and solid angle. The simplified expression holds the photon energy and efficiency constant across the band. A Gaussian FWHM is not a rectangular width; a real source/filter/detector needs the full integral. A band-integrated Rayleigh radiance is converted by 1 R = 10^10/(4 pi) photons m^-2 sr^-1 s^-1. At nominal 337.1 nm, 1 R = 4.6893010583e-10 W m^-2 sr^-1, integrated over the emission feature. Dividing by 4 nm is merely an equivalent band mean, not the line profile. The background RMS sqrt(B) presumes ideal independent Poisson detections and known mean background. It is not a detection threshold, nor the uncertainty after fitting a finite baseline. The exact-count notebook supplies the inference procedure. ## 2. Optical and electrical design algebra ### D06: etendue and filter size Lossless geometrical optics preserves phase-space throughput, $$G=n^2\int_A\int_\Omega\cos\vartheta\,d\Omega\,dA.$$ For uniform air-to-air circular cones, G = A_ent pi sin^2(theta_field). If the detector can accept an entire air hemisphere and the filter has internal half-cone beta, $$A_{\rm det}\geq A_{\rm ent}\sin^2\theta_{\rm field},\quad A_{\rm filter}\geq{G\over\pi\sin^2\beta},\quad d_{\rm filter}\geq d_{\rm ent}{\sin\theta_{\rm field}\over\sin\beta}.$$ A 25 mm entrance and 60 degree full circular cone already require at least 122.7 mm^2 detector area in this ideal limit. Restricting a filter to a 10 degree internal half-cone requires at least about 71.98 mm clear diameter. These are necessary bounds, not a lens prescription, a usable filter drawing or a 1U packaging proof. Refractive coupling changes the n^2 factors; real windows, numerical aperture, pupil geometry, vignetting and detector packaging require explicit treatment. Telecentric chief rays do not remove the marginal-ray cone. ### D07: approximate interference-filter angle shift Using a scalar effective index n_eff and incidence from air, $$\lambda_c(\theta)=\lambda_0\sqrt{1-\sin^2\theta/n_{\rm eff}^2}.$$ This follows an ideal optical-thickness/cavity argument plus Snell refraction. The table assumes n_eff = 1.8; it is not a measured coating parameter. Real multilayer filters require angle-, polarization-, temperature- and environment-dependent measured transmission. A cone must be integrated over its ray distribution; evaluating only a chief ray is insufficient. A radially graded coating can compensate a position-dependent mean incidence but does not eliminate the spread of angles at a point. ### D08-D09: a deliberately hypothetical Gaussian spectrum Let the source be a normalized photon spectrum with standard deviation sigma_s and let the unit-peak filter have standard deviation sigma_f and centre offset delta. With FWHM = sqrt(8 ln 2) sigma, $$C={\sigma_f\over\sqrt{\sigma_s^2+\sigma_f^2}} \exp\left[-{\delta^2\over2(\sigma_s^2+\sigma_f^2)}\right],\quad W_{\rm eq}=\int T(\lambda)d\lambda=\sqrt{2\pi}\sigma_f.$$ These results come from completing the square in the product of two Gaussians. The verification additionally integrates that product with 256 Gauss-Legendre nodes in source-standard-normal coordinates z in [-12,12]. Exactly 48 selected cases cover 3 source widths, 4 filter widths and 4 incidence angles (0,5,10,20 degrees). All 372 table rows are independently recomputed algebraically. The deterministic quadrature checks are not 48 observations. A 4 nm unit-peak Gaussian FWHM has W_eq = 4.2578680777 nm. A background-dominated comparison proportional to C/sqrt(W_eq) assumes a fixed detected photon-background density per wavelength (or an explicit approximation holding lambda/(hc) constant). With exactly flat energy radiance instead, the wavelength factor should remain inside the integral. Signal shot noise, detector noise, real rotational populations, the asymmetric N2 band, field/cone mixing and actual coating curves are not represented. These curves do not establish the optimal nitrogen filter. ### D10: electrical noise bandwidth is different For a one-sided power spectral density and a unit-gain linear estimator, $$\sigma_P^2=\int_0^\infty {\rm NEP}(f)^2|H_{\rm est}(f)|^2 df.$$ For a first-order low-pass, the noise-equivalent bandwidth is pi f_c/2, not f_c. For a rectangular average of duration T, H(f) is a phase factor times sinc(pi f T), giving one-sided ENBW = 1/(2T). Here ENBW has units Hz; spectral equivalent band W_eq above has units wavelength. They cannot be substituted for each other. ### D11: out-of-band rejection Let R be the unblocked out-of-band detected-photon-rate integral divided by the accepted in-band integral. If a uniform blocking factor at most 10^-OD applies over that entire leak range, $$b_{\rm leak}/b_{\rm in}\leq R10^{-\rm OD},\qquad {\rm OD}\geq\max[0,\log_{10}(R/f_{\rm leak})].$$ R must include the actual source spectrum, detector sensitivity, projected angle weighting and all optics. A narrow local OD6 certificate does not imply OD6 over the full visible/near-infrared sensitivity range. The 58 rows use assumed R and allowed fractions; they do not measure R for a polar cloud scene. ### D12: dark-only NEP and multiplication For primary dark rate d, mean multiplication M and second-moment excess-noise factor F = E[G^2]/E[G]^2, the ideal one-sided current PSD is 2 e^2 M^2 F d. Dividing by responsivity e M eta/E_gamma gives $$\operatorname{NEP}_{\rm dark}={E_\gamma\over\eta}\sqrt{2Fd}.$$ Its units are W/sqrt(Hz). The gain cancels; gain helps overcome following amplifier noise but does not improve primary photon statistics. This model assumes all relevant dark pulses share the stated multiplication statistics. APD surface/bulk dark components, SiPM correlations, frequency-dependent noise, afterpulses, CMOS readout, amplifier voltage/current noise and background add terms with their own response. F and d here are explicit hypothetical inputs, not device-class specifications or end-of-life forecasts. For an ideal average over T, this white-noise model gives sigma_P = NEP/sqrt(2T), hence incident photon-equivalent RMS sqrt(F d T)/eta. Its agreement with direct count variance checks the one-sided convention. A finite off-background estimate adds uncertainty. In particular, a ten-to-fifty kilohertz sampling rate is not the estimator ENBW. ### D13: high-pass droop A single-pole AC-coupled voltage step decays as exp(-t/tau). Requiring fractional plateau loss at T no greater than delta gives $$\tau\geq{-T\over\ln(1-\delta)},\qquad f_c\leq{1\over2\pi\tau}.$$ For 3 ms and 1% droop this is tau >= 0.298497 s and f_c <= 0.533187 Hz. This describes an analog step plateau. AC shaping of individual photon pulses can still preserve a millisecond count-rate envelope if discrimination, count losses and recovery are characterized. The two architectures must not be confused. ### D14: sample integration and phase The test signal is a unit linear ramp from zero to one, with full ramp duration D = T_10-90/0.8, followed by an infinite plateau. Every sample is the exact average over its nonoverlapping exposure interval. The independent implementation separates the ramp triangle from the post-ramp rectangle. It assigns the sample to the exposure midpoint and linearly interpolates the 10% and 90% crossings. The 900 cases vary 3 true rises, 3 sample rates and 100 relative phases. They are deterministic estimator examples, not a physical classifier. At 10 kS/s, a 30 microsecond rise can produce an interpolated apparent rise around 134 microseconds even without noise. The interpolation can place a threshold before source onset; this is a biased reconstruction from averages, not acausal photons. At 50 kS/s the spacing is 20 microseconds. A 50 kHz analog passband exceeds the 25 kHz Nyquist frequency of that baseband sampling rate; the actual anti-alias filter and integration must be specified. ## 3. Clear-air reference calculation ### P01: pressure-coordinate standard-atmosphere column The retained USSA1976 model uses geopotential Earth radius R = 6356766 m, g0 = 9.80665 m/s^2, R* = 8314.32 J/(kmol K), M = 28.9644 kg/kmol; layer bases are H = 0,11,20,32,47,51,71,84.852 km with lapse rates -6.5,0,1,2.8,0,-2.8,-2 K/km. Sea-level pressure and temperature are 101325 Pa and 288.15 K. These are a dry reference atmosphere, not polar weather or an invented vertical cloud profile. For geometric altitude z, H = R z/(R+z). Within a layer T = T_b + L(H-H_b). Hydrostatic pressure obeys $$p/p_b=(T/T_b)^{-g_0M/(R^*L)}\quad(L\ne0),\qquad p/p_b=\exp[-g_0M(H-H_b)/(R^*T_b)]\quad(L=0).$$ Rather than repeat the original 1 m density trapezoid, the independent method uses hydrostatic balance to cancel temperature: $$N_{\rm col}=\int_z^\infty {p\over k_BT}dz ={R^*\over g_0Mk_B}\int_{p_{\rm top}}^{p(z)}{dp\over[1-H(p)/R]^2}+N_{\rm tail}.$$ H(p) is analytically inverted separately in each lapse/isothermal layer. Integration uses 96-point pressure-coordinate quadrature per layer and is repeated with 192 points. The same small pressure-tail approximation above roughly 86 km is retained and exposed: N_tail = R* p_top/[g0 M k_B (1-H_top/R)^2]. It is not a detailed thermosphere, nor an exact 550 km gas composition model. It contributes only a few parts per million of sea-level optical depth in this reference calculation. Primary locators: USSA1976 Part1 tables2-4 and Part4 tables18-19 (curated CT041/S034); NASA/TM-2005-213659 AppendixA, equationsA1-A5 and pp34-35, https://ntrs.nasa.gov/api/citations/20050207438/downloads/20050207438.pdf. The legacy constants are retained so a reviewer can reproduce the original reference before exploring other atmospheres. ### P02-P03: molecular scattering and ballistic dilution Bodhaine et al.1999 equations29-30 provide the dry-air rational fit used in the original code. Writing u = lambda/(1 micrometre), $$r(u)={1.0455996-341.29061/u^2-0.90230850u^2\over1+0.0027059889/u^2-85.968563u^2},\quad \sigma_R=10^{-32}r(u)\ {\rm m^2}.$$ The simple sea-level/reference-pressure fit is tau_R = 0.0021520 r(u) p/p0. The standard-density integral instead uses tau_R = sigma_R N_col and direct no-collision transmission exp(-tau_R). The source cross-section fit is separately traceable to CT029/S032, https://web.gps.caltech.edu/~vijay/Papers/Rayleigh_Scattering/Bodhaine-etal-99.pdf. The separate frozen CloudScat arithmetic uses its own refractivity/King-factor mixture, N0 = 2.6867811e25 m^-3 and 7200 m exponential scale height; the independent function retains those inputs rather than treating it as the same atmosphere. The molecular column has units m^-2 and sigma has m^2, so tau is dimensionless. The direct aperture fraction for an isotropic point source is $$f_{\rm direct}={A\over4\pi r^2}\exp[-\tau_R-\tau_{\rm cloud}-\tau_{\rm other}].$$ Its fraction refers to all emitted photons, before detector efficiency. For a source at 16 km and satellite at 550 km, r = 534 km at nadir. The exponential concerns uncollided photons only. Scattered photons may return to the accepted direction; this calculation is not the total cloudy transfer, and tau_cloud_above = 0 does not remove reflection/scattering from the cloud beneath the source. The initial diagnostic comparison retained four failures on the small *difference* between standard and frozen transmissions: discrepancies were only about 1e-8 percentage points, from the original density-grid discretization. Relative error on a nearly cancelled difference was an unsuitable test. The final check propagates the original recorded 1 m/2 m column-refinement envelope to that difference and records both it and the 96/192 pressure-quadrature agreement. This is a numerical comparison tolerance, not a rigorous atmospheric uncertainty interval. The original failure receipt remains in `checks/diagnostics_initial_attempt.json`. ## 4. Ideal propagation and detector diagnostics ### P04-P05: an above-plane reflector A point source is height h above an infinite horizontal plane; the observer is H above that plane, with 0 <= h < H. The plane instantly reflects fraction rho with BRDF rho/pi. For radial coordinate r, the incident photon fluence is N h/[4 pi (r^2+h^2)^(3/2)]. Including both plane and aperture projection factors gives reflected-to-direct directional ratio $$B=\int_0^{H\tan\theta_{\rm FOV}}{2\rho hH^2(H-h)^2r\,dr\over(r^2+h^2)^{3/2}(r^2+H^2)^2}.$$ The new implementation changes to q = ln(1+r/h), a log radial coordinate distinct from the original x = h/sqrt(r^2+h^2). In the h->0 limit, B = 2 rho and total nadir gain is 1+2 rho; the all-direction upward escape fraction is only (1+rho)/2. Directional gain above one does not create photons or mean a photon probability exceeds one. The extra arrival delay after subtracting common source-observer flight is $$\delta(r)={\sqrt{h^2+r^2}+\sqrt{H^2+r^2}-(H-h)\over c},\qquad\delta_{\min}=2h/c.$$ The code evaluates an algebraically stable positive version. A linear-ramp source with infinite plateau is convolved with this delayed reflection and normalized by its final plateau 1+B. The reported rise is the difference of the *amplitude* crossings, not delay quantiles or onset-to-peak. All 108 rows are independently recomputed, including the rise roots. No cloud residence time, lateral transport, real BRDF, Earth curvature, gas extinction, finite 3 ms pulse end or noise is included. rho is not inferred from overhead cloud optical depth. The h=0 surface limit is not a real embedded cloud-top boundary condition. BRDF source: R4-LAMBERT/S112, https://www.pbr-book.org/4ed/Reflection_Models/Diffuse_Reflection, Lambert model and radiance normalization. All plane geometry/convolution equations above are explicit derivations for this diagnostic. ### P06: diffusion first passage, absorption and degeneracy For an instantaneous source at depth L in the half-space z>0 with absorbing plane z=0, homogeneous diffusion D and uniform killing rate gamma, $$\partial_tu=D\partial_z^2u-\gamma u,\quad u(0,t)=0,\quad u(z,0)=\delta(z-L).$$ Subtracting the image Gaussian and taking outward flux at the boundary yields $$h(t)={L\over\sqrt{4\pi D}}t^{-3/2}\exp[-L^2/(4Dt)-\gamma t],\quad t>0.$$ h has units s^-1. Its total integral is exp[-L sqrt(gamma/D)], not generally one. For gamma=0, its CDF is erfc[L/sqrt(4Dt)]. For gamma>=0 the finite-window CDF is $$F(T)={1\over2}\left[e^{-b}\operatorname{erfc}(a-q)+e^b\operatorname{erfc}(a+q)\right],\quad a={L\over\sqrt{4DT}},\quad q=\sqrt{\gamma T},\quad b=L\sqrt{\gamma/D}.$$ The code independently integrates h using u = L/sqrt(4Dt), then v = ln u, and checks the all-time and finite-time closed forms for all 48 rows. The quoted simple exponential tail formula is not used to infer real ice or ozone absorption. The peak solves gamma t_peak^2 + (3/2)t_peak - L^2/(4D)=0. For no absorption t_peak=L^2/(6D). With x=t/t_peak, $$h/h_{\rm peak}=x^{-3/2}\exp[-1.5(1/x-1)],\quad T_{10-90}=0.07432773536\,L^2/D.$$ The coefficient comes from the two *leading amplitude* roots, not cumulative probability. Bai et al.2023 (10.1029/2022JD037460, sections6-7; curated bai2023-bath-waveform/S100) quotes a rounded coefficient about0.0742 and context for D. The present method-of-images derivation is explicit and does not claim an uninspected equation bitmap was read. Exactly the same L^2/D and gamma give the same whole h(t), including amplitude. For example, (L,D) = (1 km,3e9 m^2/s), (2 km,1.2e10) and (3 km,2.7e10) are indistinguishable in this ideal kernel. External transport/source information is necessary to break that degeneracy. The diffusion PDE is parabolic and has nonzero response for all positive t, including t