Strong CP at the Bare Substrate on Z3 ⊗ Q3: Hermitian Code Mass Operators, the Mass-Phase Half, and Two Honest-Negative Results on Continuum Closure

The neutron-EDM–sensitive QCD vacuum angle is the basis-invariant combination¯ θ= θgauge + arg det(MuMd). The Standard Model provides no mechanism for¯ θ 10−10; the leading literature alternatives (Peccei–Quinn, Nelson–Barr) require either an as-yet unde- tected axion or postulated spontaneous CP breaking. We report a two-half closure of¯ θ at the bare substrate of the Holographic Circlette (Z3 ⊗ Q3) framework. (I) Gauge half. The substrate’s strong-channel adjacency is a real-symmetric graph Laplacian, which car- ries no continuous phase degree of freedom; θUV ≡ 0 identically (ANCHOR §15 item 93, Locked at bare). (II) Mass half. The code mass operators Hup, Hdown (and lepton analogues) are restrictions of the boundary walk operator W to the logical codebook. Her- miticity is therefore structural, not fitted. A Hermitian matrix has a real determinant; we verify explicitly that det Hdown ≈ 1.681 ∈ R. Hence arg det(MuMd) = 0 exactly (ANCHOR §15 item 143, Locked at bare). (III) Nelson–Barr realised automat- ically. The irreducible CKM phase φ ≈ 0.85π enters det Hdown only via the conjugate off-diagonal pair H12H21 = |H12|2eiφe−iφ and cancels. CP violation lives entirely in the mixing (sourcing the Jarlskog invariant J ≈ 4.3 × 10−5); the mass-generating basis is CP-real. (IV) Continuum closure: pinned residual. We characterise the bare-to- continuum lift via two parallel routes. The free-field domain-wall fermion’s 2D-boundary trace converges exponentially to the overlap operator (a script asserts ∥DLs ov− Dov∥ < 10−3 at Ls = 32, measured value 9.5 × 10−5). Substituting the genuine walk kernel honestly fails: the walk’s body-diagonal Wilson scalar vanishes at four even corners of the Bril- louin zone, leaving 4 species, not 1 the ultralocal-walk closure is ruled out. A second route via symmetric mass generation (CSS X-stabilizer completion of [8, 4, 4] to [[8, 0, 4]]) yields a unique gapped symmetry-respecting ground state at the cell level (assert: gap ≥ 1, measured 2.0; entanglement ≥ 1 bit, measured 2.0 bits) with anti-mirror operator ZχZW identified; the global dynamically-gauged closure remains open. A 2D strip cutoff scan as- serts the t = 4 mirror gap falls with cutoff (1.867 → 1.770 over n/q = 3 → 4, response 6.4% → 10.9%), resolving the saturate-vs-track-down fork toward track-down the second honest negative pinning the open frontier. (V) Perturbative backstop. Even ignoring substrate continuum matching, a¯ θ = 0 boundary condition under SM running generates only¯ θ∼ 10−16 via the Ellis–Gaillard / Khriplovich–Vainshtein 7-loop GIM-suppressed ef- fect, three orders below the experimental bound. Falsifier (Proposition tier, conditional on the open continuum step): dn ∼ 10−31 e· cm; any detection above 10−30 falsifies the discrete-substrate origin of QCD. All numerical claims in this paper are pinned by as- sertions in five public scripts (bulk_domainwall_overlap.py, walk_kernel_overlap.py, smg_construction.py, r2_smg_operator.py, smg_mirror_only.py) plus a cutoff scan (css_2d_strip_cutoff_scaling.py); exit-0 of each certifies the asserted bound. All scripts are publicly available at https://github.com/dgedge/strongCP. 2026-06-20 legacy canon revision: This is a canon-reconciled legacy version. CP/holonomy sector remains open; old strong-CP framing needs audit The paper retains its historical derivation trail but carries a 2026-06-20 canon revision note identifying current status and superseded claims. 2026-06-21 canon refresh: This version incorporates the 2026-06-21 ANCHOR/DRIFT/PTMS canon refresh and rebuilt local PDF.