The Holographic Circlette: Part 24. Discrete Tunnelling, the Hartman Effect, and the Golden Saturation on the 4.8.8 Lattice

The Hartman eectthe saturation of quantum tunnelling group delay with bar- rier thicknesshas been observed experimentally in photonic, acoustic, and atomic systems but remains without a rst-principles geometric explanation. We derive the Hartman eect natively from the spectral anatomy of the C4 gauge bridge on the 4.8.8 Archimedean lattice, extending the continuous-discrete interface results of Part 21. Six theorems are established. The Chebyshev transfer matrix across iterated C4 bridges yields hyperbolic saturation in the stop band, providing the exact discrete origin of the tanh(κL) dependence observed by Longhi et al. in bre Bragg gratings. The C4 resolvent gives Winful's stored-energy cavity interpreta- tion a rigorous topological foundation. The Hybrid Form Factor factorisation man- dates a bipartite separation of tunnelling time into interface and bulk components, matching the universal separable structure reported in recent weak-measurement experiments. The saturated Hartman delay deep inside the bandgap evaluates to exactly 6 algorithmic clock ticks (in units of Λ−1 QCD), an exact integer arising from the cancellation of irrational golden-ratio eigenvalues on the path graph P4. This identies the low-energy tunnelling delay as the o-shell tail of the ρ(770) resonance, unifying quantum tunnelling with the vector meson sector. A parameter-free pre- diction is derived: group delay divergences at the silver ratio band edges δ±n S /4, with successive divergence energies in the dimensionless ratio δ2 S = 3 + 2√2 ≈5.83.