Two-Scale Hierarchy on the Truncated Cubic Honeycomb: A Wilsonian Block-Spin Derivation of the Emergent QED Vacuum Screening Length a= λc/(2√2) from the Canonical C8 Spectral Eigenvalue

The Holographic Circlette framework operates on the canonical Truncated Cubic Honeycomb (TCH) substrate Z3 ⊗Q3 at the Planck scale ℓP ≈1.616×10−35 m. The hydrogen Lamb shift requires an effective lattice spacing a≈1.34 ×10−13 m to reproduce the experimental 1057 MHz 2S1/2–2P1/2 splitting. Naive identification of a as the bare substrate spacing would be catastrophically falsified by precision (g−2)/2 measurements down to sub-10−18 m. We resolve this apparent conflict by deriving a as the emergent Compton-scale correlation length of the dressed QED vacuum, generated by Wilsonian block-spin renormalisation through virtual electron–positron pair fluctuations of the canonical walk operator W= S·C. The leading-order closed-form identity is λc λc a= = M(C8 ) 0 2√2, where λc = ℏ/(mec) is the electron’s reduced Compton wavelength and M(C8 ) 0 = 2√2 is the canonical C8 cycle-graph standing-wave spectral eigenvalue that governs the static-defect nucleon mass M0 = 2√2 ΛQCD ≈939.04 MeV (ANCHOR §9.10; Part 11). Numerical evaluation yields a = 1.365 ×10−13 m at 1.89% agreement with the Lamb-derived value, with the residual gap attributed to NLO Watson-Integral G(0) ≈0.2527 dressing. A single substrate-level eigenvalue 2√2 therefore governs both the hadronic sector (static-defect nucleon mass) and the atomic sector (QED vacuum screening length), preserving the bare Planck-scale substrate and averting the precision-QED falsification threat. This note anchors the closure of ANCHOR §15 item 114 (Two-Scale Hierarchy Theorem) at Locked tier at leading order.