Boltzmann's phase-space cell as Fourier duality on the matter wave: a structural origin for ℏ

Boltzmann's combinatorial entropy S = kB ln W on continuous phase space requires a cell scale ω of action dimension to render the microstate count finite, and the placeholder is exposed at differing strengths by absolute entropies, the third law, the Gibbs paradox once paired with indistinguishability, and specific-heat freeze-out. Five quantitative regimes then converge on the same numerical value ω = h per canonical pair, namely the Sackur-Tetrode absolute entropies, the Planck spectrum derived from cell-counting by Bose 1924, the Fermi-Dirac and Sommerfeld free-electron metals, the Davisson-Germer matter-wave wavelength, and the harmonic-oscillator specific-heat threshold. This paper offers a structural reading of where that cell scale comes from, locating its origin in matter-wave Fourier content. On a matter wave whose wavelength satisfies the de Broglie relation λ = h/p, confirmed for electrons by Davisson and Germer 1927 and subsequently established for neutrons, atoms, fullerenes, and molecules beyond 25 kDa, the Heisenberg-Kennard variance bound of Kennard 1927 and Weyl 1928 reads σx σp ≥ ℏ/2, while the Weyl symplectic measure dq dp/h assigns one orthogonal state per phase-space cell of volume h per canonical pair. The two facts are distinct but consistent, both involve the same action constant h, and Boltzmann's cell scale ω = h emerges as the in-phase-space manifestation of matter's wave nature. Four converging experimental anchors then constrain the matter wave's mathematical form. Davisson and Germer 1927 measured the de Broglie wavelength directly through electron diffraction from a nickel crystal, while Stern and Gerlach 1922 observed splitting of silver atoms into two discrete magnetic-moment components along the field axis, with the angular-momentum interpretation, the universality across orientations, and the assignment of the value ±ℏ/2 supplied by Goudsmit and Uhlenbeck 1925 and the subsequent development of spin theory. Rauch et al. 1975 and Werner et al. 1975 then measured a 4π rotation period for neutron spin in interferometry experiments distinct from the gravitational interferometry of Colella, Overhauser, and Werner the same year, and Pauli's 1925 exclusion principle was reformulated by Heisenberg and Dirac in 1926 as antisymmetric exchange of multi-particle wave functions. Taken together these four anchors, some direct and some theory-mediated, constrain the candidate matter-wave object to be wave-valued, spinor-valued under spatial rotations with 4π period, and antisymmetric under particle exchange. With this much structural content empirically constrained, the matter wave is naturally read as physically real in the same sense the electromagnetic four-potential is naturally read as real after Aharonov and Bohm 1959. That reading is an interpretive position rather than an operational theorem, and an epistemic interpretation remains formally available, as does the parallel debate over Aμ realism. The standard pedagogical narrative dates ℏ's entry into physics to Planck's 1900 fit of the blackbody spectrum, whereas the structural reading offered here dates the placeholder for ℏ to Boltzmann's 1877 combinatorial entropy formula and treats the 1900-1928 quantum revolution as the period in which a value already presupposed by classical statistical mechanics was empirically pinned. The Dirac equation, curved-spacetime quantum mechanics, the configuration-space versus spacetime locus, the measurement problem, the Born rule, the spin-statistics theorem, and second quantisation lie outside the present scope.