The Planck Invariant as Metrical Foundation of the 4D Yang-Mills Vacuum: From the Photoelectric Effect to the Lund Zebra

Abstract We propose a geometric interpretation of Planck's constant ℏ as the Gromov widththe inmum of symplectic areas of closed orbitsin the phase space of Yang Mills gauge theory. In four spacetime dimensions (d = 4) this identication is unique: only in d = 4 is the YangMills action dimensionless, the Sobolev embedding W 1,2 ,→ L4 critical, and ℏ a genuine geometric scalar of the orbit space. Using Gromov's non-squeezing theorem, the Donaldson complexication of the gauge-orbit space, and the DunfordSchwartz spectral projection formula, we derive the lower bound ∆ ≥ ℏ/Rvac for the YangMills mass gap and identify ∆ as the rst non-zero pole of the Hamiltonian resolvent. We establish a semantic series of four physical phenomenaEinstein's photoelectric eect (1905), the DuaneHunt bremsstrahlung limit (1915), the Casimir eect (1948), and the predicted Lund Zebraas successive manifestations of the same geometric principle: ℏ as a hard spectral boundary set by the symplectic orbit-closure condition. As a falsiable experimental consequence we predict a discrete oscillatory pattern in the soft region of the Lund plane, kt ∈ [ΛQCD, 3ΛQCD] ≈ [200, 600] MeV, with logarithmic period T = ln(∆2/∆1) and exponential suppression outside this infrared strip. We provide a complete experimental protocol for detection at ALICE and ATLAS (LHC), including the optimal analysis scale µ∗ ≈ 540 MeV, the CambridgeAachen jet algorithm with Fourier analysis of the ln kt distribution, and a quantitative signal-to-noise formula. A schematic of the predicted Lund-plane distribution is given in Figure 1.