Appendix C: The two length scales ($D$ versus $a$)

The framework carries two microscopic lengths whose relation is not derived, and the appendix flags it as open. The angle scale of a few picometres sets the polarization geometry of light, while the much smaller cell size anchors the vacuum packing in SI units. Their ratio is about eight million, with no derivation connecting them.

The framework carries two microscopic lengths whose relation is not derived, and we flag it as open. The angle scale D=4.8526 pm (=2λ_(C,e)=4.852620477×10⁻¹² m, canonical) sets the polarization geometry of light (Chapter 2); the cell size a=6.33×10⁻¹⁹m is the SI anchor of the vacuum packing.

The framework carries two microscopic lengths whose relation is not derived, and we flag it as open. The angle scale D=4.8526 pm (=2λ_(C,e)=4.852620477×10⁻¹² m, canonical) sets the polarization geometry of light (Chapter 2); the cell size a=6.33×10⁻¹⁹m is the SI anchor of the vacuum packing. Their ratio is D/a≈7.7×10⁶, and no derivation connecting them is offered here. This matters directly for the vacuum-dispersion tension (Chapter 2): the quadratic dispersion scale is E_(QG)=√(2)hc/(πℓ), which is 115keV for ℓ=D but 882GeV for ℓ=a—an eight-order difference. Which length (if either) governs light propagation, and how the two are related, is one of the volume's open problems (Chapter 16) and is left to the physics volume. The same length D also fixes the propagation angle sinχ=λ/(mD) (physics volume §10.9.1), which classifies gamma as quasi-longitudinal and thereby reframes this very tension (§(angle-disp)): the transverse-branch dispersion scale above may be the wrong branch for gamma, so the D-versus-a ambiguity is entangled with the open question of which mode governs gamma propagation. The v2 datum of §(realmap) bears on this: the visible-band closure is a D-only, mass-free relation, which fixes the optical side of the ambiguity while leaving the gamma side open.