A spectral law for blackbody radiation without temperature or Planck’s constant

Planck introduced the constant h in 1900 to fit the blackbody spectrum but for years questioned whether it implied that energy is quantized or was merely a feature of its mathematical description. Here we show that his reservation was justified. The shape of the spectrum is determined by a single measured spectral rate, either its peak frequency ν max or its spectral width D , the rate at which the radiation loses phase correlation (its decoherence rate). Their ratio, D /ν max = 4.7368, is a dimensionless constant of the blackbody, internal to its spectrum. The shape follows from two relations among frequencies: the profile is a sum over integer harmonics of each frequency, as required for a recurrence rate, and each harmonic is weighted exponentially in its own frequency on the scale of D . The measured spectrum thus contains the structure needed to predict itself. A mathematical control without the recurrence constraint returns the Rayleigh–Jeans limit. On the COBE/FIRAS spectrum of the cosmic microwave background (CMB), the measured width alone predicts the shape with no fitted parameter, to a median absolute error of 0.005% of the maximum intensity, and the peak frequency ν max to 0.03%. A width–temperature calibration recovers the CMB temperature to 0.01%. Beyond spectral shape, a second spectral constant , governs the fourth-power relation between integrated intensity ( I ) and decoherence rate. Mapping the intensity–decoherence relation into temperature recovers the Stefan–Boltzmann constant 𝜎. The constants h , kB (Boltzmann’s constant) and 𝜎 appear only when the measured spectral relations are expressed in terms of temperature, energy, or absolute radiance, quantities assigned from outside the spectrum. The variable of the blackbody is therefore its coherence, not its temperature: for blackbody radiation, temperature is the decoherence rate expressed in kelvin. For other thermal radiation the two separate: structured thermal emitters at the same temperature emit with decoherence rates 7 to 150 times below the blackbody value. The discreteness the blackbody spectrum requires is the integer order of harmonics, a relation among frequencies, and not quantized energy.