Ribbons and Braids: A Finite Constructor for Fermion Mass Exponents
Recognition Science, Recognition Physics Institute — Austin, Texas, USA
Summary
A finite, auditable constructor—Ribbons & Braids—regroups the Standard–Model mass anomalous dimension into a small motif dictionary that emits integers. At a single universal anchor \(\mu_\star\), the sector–global residue becomes the closed form \(\mathcal F(Z)=\ln(1+Z/\varphi)/\ln\varphi\). Equal–\(Z\) families are degenerate at the anchor and anchor mass ratios reduce to integer–\(\varphi\) powers. No per–species parameters are introduced.
Abstract
We present a finite, auditable constructor—Ribbons and Braids—that collapses the Standard–Model (SM) mass residue at a single, universal anchor \(\mu_\star\) to a closed form in one integer. From a reduced Dirac word \(W_i\) we extract integers \((L_i,\tau_g,\Delta_B)\) and a word–charge \(Z(W_i)\) computed from a small motif dictionary regrouping the SM anomalous–dimension insertions. At \(\mu_\star\) each motif contributes \(+1\) in a \(\varphi\)–normalized flow, yielding \(f_i(\mu_\star,m_i)=\ln(1+Z/\varphi)/\ln\varphi\). Equal–\(Z\) families are residue–degenerate, and when \(Z_i=Z_j\) the anchor mass ratios are exact, \(m_i/m_j|_{\mu_\star}=\varphi^{\,r_i-r_j}\) with \(r_k=L_k+\tau_g+\Delta_B\). The build uses only standard kernels (QCD 4L, QED 2L; fixed thresholds) and ships an executable audit verifying charged‑fermion equality at \(10^{-6}\) with no per‑species knobs.
Key Points
- Finite motif dictionary maps SM insertions to integer counts; species dependence becomes \(Z(W_i)\).
- At a universal anchor, residues equal \(\mathcal F(Z)=\ln(1+Z/\varphi)/\ln\varphi\).
- Equal–\(Z\) degeneracy and exact anchor ratios \(\varphi^{\Delta r}\); no per–species parameters.
- Audit artifacts and reproducible evaluation with QCD(4L)+QED(2L), fixed thresholds.