What question did this study set out to answer?

This work investigates the relationship between color charge and hyperbolic phase rotation in quark structures within a geometric framework. The aim is to derive string tension values based on brane geometry.

June 8, 2026Open Access

Paper CCLXXVI: Color Charge as Hyperbolic Phase Rotation -- Fano Vertex Structure, Poincare Projection, and QCD String Tension from Brane Geometry

Key Points

This work investigates the relationship between color charge and hyperbolic phase rotation in quark structures within a geometric framework. The aim is to derive string tension values based on brane geometry.
Utilized the One-Octonion Brane-Bulk framework to analyze quark behavior across quaternionic sectors.
Conducted mathematical formalization and derived expressions for color charges and QCD string tension.
Explored the implications of non-associativity in the octonion structure for quark configurations.
Derives QCD string tension as sigma_QCD=M_rho^2/pi, within 3% of lattice value.
Identifies Nc=3 from Fano incidence and predicts Luscher coefficient as pi/12.
Demonstrates exact meson cancellation and anticolor orientation reversal with no residual tension.

Abstract

Within the One-Octonion Brane-Bulk (OOB) framework, quarks are anchored in the positive quaternionic sector H+ (on-brane) with their deep ends extending into the negative quaternionic sector H- (off-brane). We show that color charge is the hyperbolic phase rotation in H-, related to the familiar circular U (1) phase by the Gudermannian map phi=gd (psi). The three color charges correspond to the three vertices of a 3, 7 triangle in the Klein quartic tiling, with color neutrality enforced by the exact Z7 closure 1+2+4=0 (mod 7) on Fano quadratic residues. Anticolor is the orientation reversal psi->-psi on the same Gudermannian sheet, which yields exact meson cancellation with no residual tension. The QCD string tension is derived from the flat GGRT result corrected by the Poincare disk conformal factor lambda (0) =2 at the center of the 3, 7 fundamental domain, giving sigmaQCD=Mᵣho²/pi, within 3% of the lattice value. The Luscher coefficient pi/12 follows from two transverse H- modes. The number of colors Nc=3 follows from Fano incidence (the order-3 quadratic-residue subgroup of Z*₇ labeling the 3, 7 heptagon vertices). Non-associativity of the ambient octonion structure is flagged as load-bearing throughout (steps: color-boost independence, Z7 neutrality, SU (3) ->G2 embedding). Predictions: P1 sigmaQCD=Mᵣho²/pi at leading order (2% above lattice; breakable on the choice of Regge input -- the asymptotic-slope input is overcorrected by the full lambda (0) =2 and must be reconciled in follow-up) ; P2 Nc=3 from Z7 Fano closure; P3 Luscher pi/12 from D-2=2 transverse H- modes; P4 (falsifiable) anticolor is psi->-psi, not the Fano complement k->7-k. Clinical cross-link: the H+ anchor / H- extension and the auxetic nu=-1 brane describe the Pipeline Embolization Device (PED) braid (wall anchor vs lumen-extending crossings), governed by the same lambda (0) =2 hyperbolic correction. Builds on Papers I (S7 split / G2 / Fano), CCLXIX (Klein quartic systolic fbulk), and CCXLVI (GGRT rotating-rod Regge relation). Method: fundamental insights B. D. J. ; mathematical formalization with Claude (Anthropic) ; B. D. J. author of record. 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.

Read Full Paperexternally

Bookmark

View Full Paper

Cite This Study

Bharathi Jagadeesan (Sat,) studied this question.

synapsesocial.com/papers/6a265c69ad53cfb9357c59c7 https://doi.org/https://doi.org/10.5281/zenodo.20566854

Bookmark

View Full Paper