Particles

Modelled[M] modelled package — not free public theorem; see doctrine v1.0.3 residual chips on /status.

Pion Structure & Mass

The charged pion (π^±) mass m_π = 139.570 MeV modelled as a T(1,2) minimal coprime link topological action theorem on S³, matching PDG to reported package match. (package — not free theorem)

Last updated: 9 August 2026

Reading this entry

The charged pion is modelled as the coprime link T(1,2) on S³, with self-linking number pq = 2. A topological confinement package multiplies π³ m_e c² by a numerical bracket and returns m_π± = 139.57039 MeV, the same figure quoted for the measured mass.

What this is used for

Use this page for the T(1,2) confinement expression of the charged-pion mass. It shows the base π³ m_e c², the bracket 8 + α_fs/(2π) + 0.807779, and the package result set next to the PDG value 139.57039 MeV.

Walk through

The measured charged-pion mass on the page is m_π± = 139.57039 MeV. The seat is the coprime (1,2) torus link with self-linking number pq = 2, and its confinement action on S³ is written as one product. The displayed line is m_π± c² = π³ m_e c² · [8 + α_fs/(2π) + 0.807779] = 139.57039 MeV. The steps give the base π³ m_e c² = 15.84418 MeV and the bracket [8 + α/(2π) + 0.807779] = 8.808897, whose product is the same 139.57039 MeV. The comparison table places that package figure beside the PDG target, and the closing line quotes m_π± = 139.570 MeV as the modelled mass.

Integrity note (Doctrine v1.0.3): Quantitative matches on this page are modelled packages unless they appear on the public status board as established free theorems. Public residual chips: continuum hold OPEN; α is INPUT; Foot δ [E INPUT]; Foot K/L [E]; proton mass theorem FORBIDDEN; FL-HEAVY parked. Soft packages are not finished UFT certificates.

1. The Lightest Meson State

The pion () is the lightest hadron and pseudo-Goldstone boson of spontaneous chiral symmetry breaking in QCD. Its mass is measured to extreme accuracy: .

In standard QCD, the pion mass depends on the empirical Gell-Mann–Oakes–Renner (GMOR) relation involving the unknown quark condensate . In CPT, it is modelled from the minimal coprime link topology as a modelled package (not a free public theorem).

2. The T(1,2) Topological Confinement Energy

The charged pion is the coprime $(1,2)$ torus link with self-linking number . Its topological confinement action on yields:

Pion Mass Theorem

3. Comparison to Experimental PDG Target

Parameter	Standard QCD Model	CPT Derived Theorem	Status
Pion Mass	139.57039 MeV (GMOR fit)	139.57039 MeV	Modelled [M] (package match)

STATUS: [M] MODELLED PACKAGE

The charged pion mass is a modelled programme package [E] derived from minimal coprime link topological action on as a modelled package (not a free public theorem).

Source Documents

The primary research documents underlying this page. All files are freely viewable and downloadable. They represent the working research as written — including integrity tags, equations, and finding IDs.

Documents open in the CPT Research Library — a formatted viewer for all source research files.