Light Quanta · Sections 1–2
Give every resonator its share,
and the total never stops growing.
§1 gives every linear resonator oscillation the same mean energy, whatever its frequency. Widen the range of resonator frequencies you allow, and see what that classical allocation actually predicts, and why the paper says it rules out any equilibrium between matter and radiation.
LQ-02 · Classical mode-energy allocation
Classical mode-energy allocation
Predict before you calculate
Widening the resonator range from 100 THz to 1000 THz changes the energy by:
Before removing the upper limit: will the total settle at a finite value?
| Mean energy per resonator oscillation | 2.070974e-20 J |
|---|---|
| Energy with resonators up to the cutoff | 6.439187e-3 J/m³ |
| Share of that energy above the probe frequency | 9.990000e-1 |
| Energy after widening the cutoff tenfold | 6.439187e+0 J/m³ |
| Ratio: tenfold-widened energy ÷ current energy | 1000.000 |
With resonators up to 1.000e+14 Hz at 1500 K, the classical allocation holds 6.439e-3 J per cubic meter; each tenfold widening multiplies it by 1000. Every resonator oscillation carries the same mean energy k_BT (source notation (R/N)T), which §1 notes is two-thirds of a free molecule's mean kinetic energy.
§2's Avogadro match and §4's later entropy argument (a separate instrument) use two disjoint limits of the same spectrum: the classical region, admitted at the 1% criterion for x ≤ 0.0200671, and the Wien region, admitted at x ≥ 4.605170. Meeting both arguments does not mean one "quantum regime" supplied both numbers.
The historical Avogadro readout, from Einstein's §2 constants
| N, from Einstein's printed §2 constants (historical) | 6.170486e+23 mol⁻¹ |
|---|---|
| N, as printed in §2 | 6.17e+23 mol⁻¹ |
| N_A, defined, 2019 SI (modern) | 6.02214076e+23 mol⁻¹ |
| Hydrogen-atom mass from the printed N (historical) | 1.62e-24 g |
| Hydrogen-atom mass, modern (labeled modern) | 1.673533e-24 g |
α is a suspected historical typographical correction (10⁻⁵⁷, not the witness's 10⁻⁵⁶); R = 8.31×10⁷ erg mol⁻¹ K⁻¹ and L = 3×10¹⁰ cm s⁻¹ are declared editorial inputs, not printed values, since §2 does not appear to print either. This readout is never mixed with a modern constant in the same number.
What this model leaves out
• The mechanism coupling matter and radiation beyond Planck's stated equilibrium condition.
• Cavity shape and walls.
• The approach to equilibrium.
• Any quantum hypothesis.
• Measured spectra (a separate instrument shows those).
Show the calculation owner: which real functions computed these numbers
src/physics/reference/radiation/classical.ts (classicalCutoffEnergyDensity, classicalTotalEnergy, meanResonatorEnergy), src/physics/reference/radiation/avogadro.ts (avogadroFromPlanckConstants), src/physics/reference/radiation/spectra.ts (regimeRelativeErrors), and src/experiments/lq02/session.ts (composing the snapshot; this component never recomputes any of these).
The law this instrument calculates
Every resonator oscillation, at every frequency, carries the same mean energy Ē = k_BT (source notation (R/N)T), two-thirds of a free molecule's mean kinetic energy, as §1 notes. Because that mean energy never falls off with frequency, the energy held by resonators up to a cutoff grows as the cube of the cutoff, without limit as the cutoff is removed.
Why the paper calls this a difficulty, not just an approximation
§1 draws two conclusions from this, not one: the classical allocation disagrees with the measured spectrum, and, independently, it rules out any equilibrium between matter and the radiation field at all, because the total grows without bound as the resonator range widens. This instrument's "remove the upper limit" action shows that second conclusion directly, as a typed refusal rather than a number that quietly becomes huge.
What §2 does with Planck's constants
§2 does not use Planck's radiation law as a hypothesis about light; it uses the two constants of Planck's fitted formula, in the classical limit, to solve for Avogadro's number. The historical readout below reproduces that calculation from Einstein's printed constants, labeled apart from the modern, defined value.