Cosmology

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

Hadronic Hagedorn Freeze-Out Temperature

The deconfinement phase transition temperature Tc = 156.38 MeV modelled as a 5D harmonic sphere theorem (kB Tc = π⁵ me c²), matching HotQCD lattice simulations to 0.08% accuracy. (package — not free theorem)

Last updated: 9 August 2026

Reading this entry

The Hagedorn page places a geometric freeze-out temperature for the quark-hadron transition as a modelled package [M], where the torus-knot partition function on S³ meets π⁵ times the electron rest energy. The print is k_B T_c = 156.38 MeV, set beside the HotQCD lattice value.

What this is used for

Early-universe and heavy-ion comparisons use this temperature as an analytical print next to the lattice window 156.5 ± 1.5 MeV, and 1.81 trillion K is the same package. Baryogenesis and the spectrum skeleton can cite T_c as this modelled deconfinement scale.

Walk through

The page takes the Hagedorn window T_c ≈ 155–160 MeV as the quark-hadron condensation, and quotes HotQCD at T_c = 156.5 ± 1.5 MeV. The partition function is Z(T) = Σ_{(p,q)} d(p,q) exp(−E/k_B T) over coprime torus knots on S³, and deconfinement is placed when that factor meets the 5D multiplier π⁵ on the electron mass. The displayed evaluation is k_B T_c = π⁵ · m_e c² = (306.0197) × (0.5110 MeV) = 156.38 MeV. Button arithmetic with π⁵ ≈ 306.019685 and m_e c² = 0.510999 MeV reaches 156.3758 MeV, or 1.81 trillion K, scored at 0.08 percent from 156.5 MeV, and the table lists that print beside 156.5 ± 1.5 MeV and 1.82 trillion K inside the modelled package [M].

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. Quarks-to-Hadrons Phase Transition

In early universe cosmology and heavy-ion collisions, the Hagedorn temperature marks the fundamental phase transition where free quarks and gluons condense into hadrons (protons, neutrons, pions).

While standard physics relies on extensive supercomputer grid simulations (HotQCD Collaboration finds ), CPT derives as an exact analytical geometric theorem.

2. The $\pi^5 m_e$ Deconfinement Theorem

The partition function over all coprime torus knots on deconfines when thermal fluctuations match the 5D harmonic sphere volume multiplier over the electron mass unit :

Hagedorn Freeze-Out Temperature Theorem

3. Comparison to Supercomputer Lattice QCD

Parameter	Lattice QCD Simulations	CPT Derived Value	Status
Deconfinement Energy $k_B T_c$	156.5 ± 1.5 MeV (Grid fit)	156.38 MeV	Modelled [M] (package)
Freeze-Out Temp (Kelvin)	1.82 Trillion K	1.81 Trillion K	Modelled [M]

STATUS: [M] MODELLED PACKAGE

The Hadronic Hagedorn freeze-out temperature is a modelled programme package [E] derived from 5D harmonic sphere geometry () 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.