Paper CLXVII: One-Octonion Recasting the Cooper Pair Exclusion Sum Rule: Tan Contact Separation in 2D, Qualitative Support from Daix

ERRATUM (2026-06-09). CLXVII-1 (the 'clean zero-parameter 3D test' I3D*kF^2 = pi(xi*kF)^2 vs Ku+2012) is WITHDRAWN, on three independent grounds: (1) dimensional inconsistency (I3D = integral d3r has units length^3, so I3D*kF^2 is a length, not the dimensionless RHS; correct group is I3D*kF^3); (2) false premise -- it assumes g_updown(r->0)->0, but the exact Tan relation g2_updown=(I/16pi^2)(1/r^2 - 2/ar) DIVERGES as 1/r^2, so the integral is contact-contaminated; (3) Ku+2012 is a thermodynamic EoS measurement with no real-space g(r) to integrate. CLXVII-2 (the 2D dip-only, one-parameter ~15% qualitative support) stands. Files: original paper + erratum. ---- Paper CLXVI proposed the excluded-area sum rule R [1 −gpair(r)] d2r = πξ2 for the strongly-interacting Fermi gas, motivated by bulk-channel exclusion of Cooper pairs in the brane-bulk framework. Direct application to the 2D AFQMC data of Daix et al. (2026, PRL 136, 153402; arXiv:2504.01885) reveals a subtlety: in 2D, the pair correlation g↑↓(r →0) diverges logarithmically (as log2(kF r), following Werner & Castin 2012, their Eq. 8), rather than vanishing. The integrand 1 −g↑↓therefore mixes two physically distinct contributions: (i) intra-pair two-body physics controlled by the Tan contact C, and (ii) inter-pair exclusion — the actual prediction of the framework. We recast CLXVI-1 into two predictions: CLXVII-1 (clean, 3D): I3D · k2 F = π(ξ · kF )2 with ξ determined independently from the Tan contact, testable against Ku et al. (2012, MIT unitary gas) data; and CLXVII-2 (2D dip region only): Idip ·k2 F ≈π(ξexcl ·kF )2 restricted to g < 1. Applied to Daix et al. 2026 at crossover η ≡log(kF a2D) = 1.0–1.2, the triangular- dip integral evaluates to Idip · k2 F ≈2.76, matching π(ξkF )2 ≈2.77 for ξkF ≈0.94–1.0 — qualitative support within ∼15% without any free parameter. The 20–22% BCS violation (pair dip depth) and the location kF r ∼2 are both consistent with bulk-channel exclusion. The clean zero-parameter test requires 3D unitary gas data. Part of the One-Octonion Brane-Bulk Framework series. Anchor DOI: 10.5281/zenodo.19120873. Community: one-octonion-brane-bulk. Author: Bharathi Dasan Jagadeesan, M.D., University of Minnesota. ORCID: 0000-0002-1143-941X.