The same average can hide different arrangements.

Four points placed in half a box end up with two inside on average, whether each is placed on its own or all four are locked together. The average cannot tell the two apart; how often all four land inside, one time in 16 against one time in 2, can.

The light-quanta paper's §5 asks how likely it is that n points moving independently in a volume v₀ are all found, at a chosen moment, in a part v of it. For independent points the answer is (v/v₀)ⁿ, and through Boltzmann's principle the logarithm of that probability gives the entropy change (R/N) n ln(v/v₀), with the number of points as the coefficient. §6 then reads radiation's entropy in the same form and concludes that, where Wien's law holds, radiation behaves as if it consisted of independent quanta. The workbench tests the premise that makes the exponent n rather than 1. It keeps four points and a region of half the volume, f = 1/2, and compares two ideal candidates: points placed one by one, and all four locked to one shared position. Both give a mean of 2 points inside, and both give each point a one-half chance of being inside, so neither measurement distinguishes them. The chance that all four are inside does: 1/16 = 0.0625 for independent points and 1/2 for locked ones, which is fⁿ against f. So does the variance of the number inside, 1 against 4. A count record can be entered as how often 0, 1, 2, 3 or 4 points were inside. A single placement with 1, 2 or 3 inside is impossible for the locked candidate, and the workbench reports its likelihood as zero, not as a small number. With one point, or a region of none or all of the volume, the candidates agree on everything and no measurement separates them.

Put four points into a box one at a time, each with a one-half chance of landing in the left half. The chance that a particular point is inside is 1/2. The chance that all four are inside is 1/2 × 1/2 × 1/2 × 1/2 = 1/16, because the chances of independent events multiply. Counting every arrangement gives the whole distribution: 0, 1, 2, 3 or 4 points inside happen in 1, 4, 6, 4 and 1 of the 16 equally likely arrangements, so with chances 0.0625, 0.25, 0.375, 0.25 and 0.0625. The mean is (0 × 1 + 1 × 4 + 2 × 6 + 3 × 4 + 4 × 1)/16 = 32/16 = 2. The variance, the average squared distance from that mean, is (4 × 1 + 1 × 4 + 0 × 6 + 1 × 4 + 4 × 1)/16 = 16/16 = 1. Now lock the four points together, so that one placement decides all of them. Half the time all four are inside and half the time none are: the chances are 1/2 for 0 and 1/2 for 4, and zero for 1, 2 or 3. The mean is (0 + 4)/2 = 2, the same as before, and each point is still inside half the time. But the chance that all four are inside is 1/2, not 1/16, and the variance is ((0 − 2)² + (4 − 2)²)/2 = 4, not 1. In general, with n points and a fraction f of the volume, the all-inside chance is fⁿ for independent points and f for a locked group. §5 takes the logarithm: ln fⁿ = n ln f, while ln f = 1 × ln f. Multiplied by R/N, that logarithm is the entropy change, so the number of independent placements becomes the coefficient in front of ln(v/v₀). Four labels locked into one unit give a coefficient of 1, not 4. The two constructed records show how counts decide. The record 1, 4, 6, 4, 1, sixteen placements in exactly the independent proportions, contains counts of 1, 2 and 3, which the locked candidate cannot produce, so its likelihood under locking is zero. The record 8, 0, 0, 0, 8, all or nothing every time, is possible under both. Each all-or-none outcome has chance 1/16 for independent points and 1/2 for locked ones, a factor of 8, so sixteen of them are 8¹⁶ ≈ 2.8 × 10¹⁴ times more likely under locking, and the workbench prints the logarithm of the ratio, ln(L_independent/L_locked) = −33.27. It reports no prior, posterior or cutoff. Both records were constructed to illustrate the comparison and neither is a measurement. Last, the degenerate cases: with n = 1, fⁿ is just f, and with f = 0 or 1 both candidates are certain, so every measurement agrees.

The §5 argument is Boltzmann's principle, S = (R/N) ln W, applied to n independently moving points, and the paper's title calls the resulting view of light a heuristic one. The Brownian paper's §2 relies on the same independence: the configuration integral of n suspended particles grows as Vⁿ, which gives their osmotic pressure RTν/N. Neither paper considers a partly correlated alternative. The quanta are independent only where Wien's law holds: in 1909 Einstein showed that the energy fluctuations of radiation obeying Planck's law have a term like that of independent particles and a term like that of waves, and Bose's derivation of Planck's law in 1924 counted quanta in a way that does not treat them as independent. The two candidates here are ideal extremes built for the comparison, not historical models.