Gauge Fields, Fermions, and Self-Gravitating Matter on the Golden Cone (INDYNA Research Note 18)

Abstract

This companion note to the Shift Law (Note 17) equips the golden cone with physics. A Peierls magnetic field leaves the golden level crossing exactly intact when aligned with a two-fold axis — a symmetry-protected crossing drifting linearly, $g^\ast(B)=\varphi^{-2}-0.1716\,B$ — and opens it quadratically for generic orientation: an orientable experimental signature. A Dirac monopole switches the network between worlds: for odd charge all multiplicities become even and the spectrum lives on half-integer golden powers, $\{-5^{1/4}\varphi^{1/2},\,+5^{1/4}\varphi^{-3/2},\,+5^{3/4}\varphi^{1/2}\}$, while the golden crossing is flux-periodic mod 2. The lattice Dirac operator satisfies the index theorem ind $= V-E$ exactly and carries the golden fermion mass ratio $m(3')/m(3)=\sqrt{(5+\sqrt5)/(5-\sqrt5)}=\varphi$. The calorimetric machinery yields $T_{\rm peak}\simeq 0.44\,\Delta$, a third experimental channel. Coupling matter self-consistently to the geometry produces attraction, curvature, a Fermi-pinning law $\kappa_{\rm pin}t_0^2\approx 0.31$, and a single collapse mechanism — exhaustion of the angular budget $|\delta|\le 3\pi$ (visible) or $\le\pi$ (shadow, nearly incompressible, threshold 4.6× higher) — realized in two ways: the scalar sea reconstructs (Fermi pinning) on the way, while the Dirac sea, pinning-free with topologically rigid index, gravitates several times more strongly and reaches the same bound already at $\kappa_c=0.872$, with the approach law $\Delta r\propto(\kappa_c-\kappa)^2$ predicted from the square-root behaviour of the deficit at the degeneracy. The Fermi-gap alternation gapped–gapless–gapped for monopole charge $Q=0,1,2$ survives unchanged inside the gravitating loop. Outlook: the Penrose programme announced here is treated in Note 22; Note 19 is "Time as the Irrational Part". Part of the INDYNA Notes series (Notes 16–34). All numbers reproduce from the INDYNA law register (master data record, doi:10.5281/zenodo.21891088). Licence CC-BY-4.0.