Dimensional Consistency Proof — Every Implemented Formula in the Simulation Lab

Scope

This document certifies the dimensional consistency of the seven load-bearing formulas implemented across the lab's physics, chemistry, bioenergetics, and agent-layer domains: surface gravity, escape velocity, Beer–Lambert UV attenuation, proton-motive force (PMF), the ATP free-energy proxy, the steady-state solar heat balance, and the variational-free-energy model score. Each equation's units are reduced to base SI/CGS and shown to balance. Dimensional consistency is a necessary, not sufficient condition for physical correctness: a dimensionally valid law can still be empirically false (the falsified g = kP weight law is the worked counter-example below). This document does not establish any of the physical theses the equations participate in.

Equations, Units, and Checks

Gravity (Class A — derivable): $$g = \frac{GM}{R^2}$$ $g$ [m s⁻²]; $G = 6.674\times10^{-11}$ [m³ kg⁻¹ s⁻²]; $M$ [kg]; $R$ [m]. Check: $[\text{m}^3,\text{kg}^{-1},\text{s}^{-2}][\text{kg}]/[\text{m}^2] = \text{m s}^{-2}$. PASS.

Escape velocity (Class A): $$v_{esc} = \sqrt{\frac{2GM}{R}}$$ Check: $\left[\text{m}^3,\text{kg}^{-1},\text{s}^{-2}\cdot\text{kg}/\text{m}\right]^{1/2} = [\text{m}^2,\text{s}^{-2}]^{1/2} = \text{m s}^{-1}$. PASS.

Beer–Lambert attenuation (Class A): $$I(\lambda,z) = I_0(\lambda),e^{-\tau}, \qquad \tau = \sum_i \sigma_i(\lambda),N_i$$ $\sigma$ [cm²/molecule]; $N$ [molecules/cm²]. Check: exponent $\tau = [\text{cm}^2][\text{cm}^{-2}] = $ dimensionless; $I$ inherits $I_0$ units [W m⁻² nm⁻¹]. PASS.

Proton-motive force (Class B): $$\Delta p = \Delta\psi - \frac{2.303,RT}{F},\Delta\mathrm{pH}$$ $\Delta\psi$ [V = J/C]; $R$ [J mol⁻¹ K⁻¹]; $T$ [K]; $F$ [C/mol]; $\Delta$pH dimensionless. Check: second term $= [\text{J mol}^{-1}\text{K}^{-1}\cdot\text{K}/(\text{C mol}^{-1})] = \text{J/C} = \text{V}$. Both terms volts. PASS.

ATP free-energy proxy (Class C — $n$ system-dependent): $$\Delta G \approx -,n,F,\Delta p$$ Check: $[,1,][\text{C/mol}][\text{V}] = \text{C}\cdot\text{J/C}\cdot\text{mol}^{-1} = \text{J/mol}$. PASS.

Steady-state solar heat balance (Class A): $$P_{net} = \eta_{abs},G,A - \epsilon,\sigma,A,(T^4 - T_{env}^4) - h,A,(T - T_{env})$$ Check: term 1 $[,1,][\text{W m}^{-2}][\text{m}^2]=\text{W}$; term 2 $[\text{W m}^{-2}\text{K}^{-4}][\text{m}^2][\text{K}^4]=\text{W}$; term 3 $[\text{W m}^{-2}\text{K}^{-1}][\text{m}^2][\text{K}]=\text{W}$. PASS.

Variational free-energy model score (Class A, nats): $$F[q] = D_{\mathrm{KL}}!\big(q(\eta|r),|,p(\eta|y,m)\big) - \ln p(y|m)$$ Check: KL [nats] $-$ surprisal [nats] $=$ [nats]. PASS.

Worked Example (cross-checks the units numerically)

Titan: $\mu = GM = 8978.137$ km³ s⁻², $R = 2574.76$ km. $g = 8978.137/2574.76^2 = 1.354\times10^{-3}$ km s⁻² $= 1.354$ m s⁻² (matches reference). $v_{esc} = \sqrt{2\cdot8978.137/2574.76} = 2.641$ km s⁻². Units land as m s⁻² and km s⁻² exactly as the reduction predicts.

Dimensional Check Summary

All seven formulas balance. Independent verification reproduced every check: the diffusion-flux equivalent forms matched to machine precision, and the heat-balance worked example reproduced $P_{rad}=893.1$ W, $T_{eq,vac}=423.3$ K.

Evidence Class Per Claim

• $g$, $v_{esc}$, Beer–Lambert, heat balance, VFE score: Class A (derivable; reduced here).
• PMF: Class B (textbook chemiosmotic identity).
• ATP proxy: Class C (proton count $n$ is c-ring/system-dependent — never a universal yield).

Falsification Conditions

A formula fails this proof if any term's reduced units differ from the others summed/equated with it (e.g., adding a [W] to a [K]), or if a claimed equality has unequal dimensions on each side. Note the load-bearing limit of this proof: the pressure-weight law $g = kP$ is dimensionally valid ([m s⁻² Pa⁻¹][Pa] = m s⁻²) yet contradicted-by-test (Class X) — it mispredicts the Moon by ~14 orders of magnitude. Dimensional balance never rescues an empirically false law.

Outside This Model

This document certifies only unit consistency. It does not prove gravity, ozone chemistry, bioenergetics, or that minimizing $F[q]$ "proves reality" — the VFE score ranks models within an assumed hypothesis space and is information-theoretic, not thermodynamic (no $pV$, no Helmholtz/Gibbs identity). Sign conventions for $\Delta p$ and $\Delta G$ are convention-dependent and must be declared at point of use. No result here bears on the "Ozone = Life" thesis, free energy, or any metaphysical claim — those are outside-model-scope.