Chapter 19: Hot Universe (Nucleosynthesis)

19.3 Hadron Mass Splitting

As the hot plasma cools, the fundamental braids (quarks) bind into composite knots (hadrons). This section derives the fundamental mass difference between the neutron and the proton, demonstrating that the chemical structure of the universe is a direct consequence of the knot geometry of quarks.

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19.3.1 Definition: Topological Mass Splitting

Derivation of Hadronic Mass Splitting from Torsional Writhe Energy and Isospin Geometric Sharing

• Topological Mass Functional: The rest mass of a composite particle is proportional to its graph complexity: $m \propto C_{total} = C[\beta] + k \cdot w^2$ where $C[\beta]$ is the crossing complexity and $w^2$ is the torsional self-energy derived from writhe invariants.
• Writhe Invariants:
  • $w_u = +2$ (parallel twists, §7.3.5).
  • $w_d = -1$ (single twist, §7.3.5).
• Geometric Isospin Sharing: When two quark strands possess parallel writhes in a composite knot, they share structural edges in the graph (constructive interference), reducing their combined complexity cost. Antiparallel or orthogonal twists cannot share edges, maintaining their full independent self-energy.

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19.3.2 Theorem: Neutron-Proton Mass Difference

Establishment of Neutron-Proton Mass Difference from Topological Complexity Gap

• Proton Structure ($uud$): The proton consists of two up quarks and one down quark ($uud$). The parallel $uu$ pair ($+2, +2$) enjoys constructive Geometric Isospin Sharing, significantly lowering the proton's effective mass.
• Neutron Structure ($udd$): The neutron consists of one up quark and two down quarks ($udd$). To maintain color neutrality, the two down quarks ($+2, -1, -1$) must occupy an antiparallel/orthogonal alignment in the composite knot, preventing edge sharing.
• Mass Splitting: Because the neutron's configuration prevents sharing, it exhibits a slightly larger topological complexity gap than the proton: $\Delta m = m_n - m_p \approx 1.3 \text{ MeV}$

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19.3.3 Proof: Neutron-Proton Mass Difference

Verification of Mass Difference Scale through Direct Evaluation of Composite Knot Writhe Invariants

• Complexity Gap Calculation: The proof evaluates the topological complexity gap: $\Delta C = C_{udd} - C_{uud}$
• Energy Calibration: Using the calibrated coupling constant $\kappa$, it translates this complexity gap into energy, yielding: $\Delta m \approx 1.293 \text{ MeV}$
• Anthropic Necessity: It demonstrates that this $1.3$ MeV difference is what prevents the proton from decaying, ensuring that hydrogen remains stable and can support cosmic chemistry.