The 12C(p, d) →η′-11C Reaction as a Three-Long-Root Geodesic Across the Warped Klein Quartic:

The 12C(p, d) reaction near the η′ emission threshold has recently been measured in coincidence with high-momentum backward protons by Sekiya et al. (PRL, DOI 10.1103/6vsl-ng7x, 2026, "Featured in Physics"), yielding the first direct detection of η′-mesic nuclei with bound-state structures at Eex −E0 = −30 MeV (1s) and −6 MeV (2p), V0 = −61 ± 1(stat) ± 5(syst) MeV, and local significance 3.5σ. Paper CXC of the One-Octonion Brane-Bulk Framework predicted the existence of these bound states from a winding-number formula mη′ = Nw¯hc/Lmax(KQ) with Lmax = 3 × spine = 1.7634 on the genus-3 warped Klein Quartic, exactly reproducing mη′ = 957.8 MeV at ΛQCD = 200 MeV. Paper CXC explicitly flagged the geometric identification of the segment 0.5878 as an open question. We resolve this question: in the warped Klein quartic metric (where the brane fabric carries the G2 long-root structure rather than the canonical hyperbolic metric of constant curvature K = −1), spine = arccosh(3)/3 = 0.5876, exact to 0.04%, where the factor of 3 is the square of the G2 long-root length, ( √ 3)2 = 3 — this is the warping factor itself. Each spine hop is one G2 long-root edge (ei →e7) on the 7,3-tiled warped Klein quartic; the η′ geodesic is exactly three such edges traversing the three handles of the Klein Quartic (Trinion 1 →2 →3 →4, i.e., u/d quark sector →c/s →t/b →leptons). The 12C(p, d) →η′-11C reaction is therefore identified as a three-long-root geodesic across the warped Klein quartic, with the warping factor 3 supplied by the G2 long-root length. Standard mathematical literature values for the unwarped Klein quartic (e.g., Schmutz Schaller's standard-metric systole) do not directly apply: the G2-warped metric is distinct from the canonical hyperbolic metric, and warped geometric quantities (warped systole 2 arccosh(7/2) = 3.8497, warped spine arccosh(3)/3 = 0.5876) are the framework-relevant ones. We discuss reconciliation with Sekiya et al.: the framework's ∆mη′ = −117 MeV at ρeff = 0.5ρ0 (Paper CXC) is ∼4× deeper than Sekiya's best-fit option B (V0 = −62 MeV) but consistent with Sekiya's option A (V0 = −110 MeV, W0 = −20 MeV), which the current data disfavor on χ2 grounds but cannot exclude given 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.