What question did this study set out to answer?

The aim is to refine the excluded-area sum rule in a strongly-interacting Fermi gas framework, especially in 2D systems.

June 11, 2026Open Access

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

Key Points

The aim is to refine the excluded-area sum rule in a strongly-interacting Fermi gas framework, especially in 2D systems.
Recasting the previously proposed rules based on dimensional analysis and pair correlation behavior.
Applying predictions against 2D AFQMC data from Daix et al.
Direct evaluation of integrals with consideration for intra- and inter-pair dynamics.
CLXVII-1 predicts I3D · kF^2 matches π(ξ · kF)^2, testable against past data.
CLXVII-2 shows qualitative support within ∼15% for Idip · kF^2 and π(ξexcl · kF)^2.
Observed 20-22% BCS violation consistent with exclusion predictions.

Abstract

ERRATUM (2026-06-09). CLXVII-1 (the 'clean zero-parameter 3D test' I3D*kF² = pi (xi*kF) ² vs Ku+2012) is WITHDRAWN, on three independent grounds: (1) dimensional inconsistency (I3D = integral d3r has units length³, so I3D*kF² is a length, not the dimensionless RHS; correct group is I3D*kF³) ; (2) false premise -- it assumes gᵤpdown (r->0) ->0, but the exact Tan relation g2ᵤpdown= (I/16pi²) (1/r² - 2/ar) DIVERGES as 1/r², 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 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.

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