Count molecules by speed v = |v|. The velocity vector is an isotropic Gaussian per component with variance kT/m. Converting to speed multiplies by the spherical shell area 4πv²:
The v² is not decoration — it is the phase-space volume of the shell. Kill it and you have described one component, not the speed. That is exactly what window 6 does.
Natural units below use k = 1; the physics is unchanged.
These are the speeds behind temperature. Maxwell (1860) gave the velocity spread; the-boltzmann-distribution generalizes the weight e−E/kT over any energy, and the kinetic energy ½mv² is the molecular motion sitting underneath the-ideal-gas.
Downstream check (window 0/8): averaging ⅓(N/V)m⟨v²⟩ over this law returns PV = NkT exactly. The gas law falls out of the speeds.
Live re-derivation of the most-probable speed from the current engine, compared to the closed form vp=√(2kT/m). If window 6 tampers, the numeric peak drifts to 0 and this badge flips.
Re-checks on boot, on every tamper, and on restore.
Set temperature and molecular mass (natural units, k = 1).
Curve: f(v) from live closed form. Bars: 20 000 seeded samples (3 Gaussian components → speed). Markers: vp < ⟨v⟩ < vrms.
Mean kinetic energy ½m⟨v²⟩ = — (equipartition: (3/2)kT, three quadratic modes).
"It's a bell curve centered on the average speed." No. It is skewed: it starts at exactly 0 (the v² kills the origin), rises to a peak, then has a long fast tail. Peak, mean, and rms are three different numbers, and they never coincide.
"There's a slowest and fastest molecule." The support is [0,∞) — no upper cutoff. The tail is real: a tiny fraction always exceeds any bound, which is why evaporation and escape-to-space happen at all.
Most molecules move at the RMS speed.
→ Most sit near vp = √(2T/m); vrms is larger, pulled up by squaring the fast tail.
Mean speed = √(2kT/m).
→ That is vp. The mean is √(8kT/πm) ≈ 1.128 vp.
Heavier gas at same T is hotter/faster.
→ Same T ⇒ same &frac32;kT of KE; heavier ⇒ slower (all speeds ∝ √(T/m)).
The disclosed planted void. Drops the v² shell factor — the 1D form e−mv²/2T misused for a 3D speed. The peak collapses to v = 0 and the mean formula breaks. The witness (7) catches it live.