Test 20 — Lattice QCD Equation-of-State Transition — A Framework for a Theory of Everything

Test 20 — Lattice QCD Equation-of-State Transition Test

Across the QCD crossover, does the pressure \(p(T)\), energy density \(\varepsilon(T)\), entropy density \(s(T)\), trace anomaly \((\varepsilon-3p)/T^4\), and \(g_*(T)\) evolve the way first-principles lattice QCD says they must — and does the framework's "quark/gluon distinctions collapse into hadron distinctions" reading add anything beyond that lattice result, or only relabel it?

Agrees QCD crossover equation of state

What we're checking

Observable: The QCD equation of state across the crossover — pressure p(T), energy density ε(T), entropy s(T), the trace anomaly (ε−3p)/T⁴, and the relativistic species count g_*(T), anchored at the transition temperature T_c ≈ 156.5 MeV.
Standard cosmology: First-principles lattice QCD: continuum-extrapolated EoS curves (HotQCD — Bazavov et al. 2014; Wuppertal–Budapest — Borsanyi et al. 2014), with T_c = 156.5 ± 1.5 MeV (Bazavov et al. 2019).
Granularity (consistency reading): Free quark/gluon distinctions give way to confined-hadron distinctions at the crossover; the record-cost reading tracks the same lattice curve — high-T g_* ≈ 61.75 down to low-T g_* ≈ 17.25. It reads the curve, it does not re-derive it.
Measured: T_c ≈ 156.5 ± 1.5 MeV, cross-checked at 155 ± 9 MeV (Borsanyi et al. 2020); trace-anomaly peak (ε−3p)/T⁴ ≈ 3.5–4 near 200 MeV.
The epoch (context, not the compared quantity): ~20 microseconds after the Big Bang — when the crossover happens, not what is being matched.
Verdict: Agrees — consistent with the lattice curve end to end; a consistency reading against imported lattice data, not an independent derivation.

About twenty microseconds after the Big Bang the universe cooled through the single most violent phase change in its early life: the moment free quarks and gluons could no longer roam, and matter locked itself into protons and neutrons. Physicists have spent years mapping exactly how the pressure, energy, and entropy of that boiling plasma behave as it cools — not with a napkin sketch, but with continuum-extrapolated lattice-QCD supercomputer runs from two independent groups, HotQCD and Wuppertal–Budapest, that agree with each other.

Come at that same transition from a completely different direction — bookkeeping the cost of what the universe can still record, where a free quark is not a countable object but a proton is — and you trace the identical curve. The species count falls from about 61.75 degrees of freedom in the hot quark–gluon soup down to about 17.25 in the cooler hadron gas; the trace anomaly peaks in the same place; the crossover sits at the same T_c ≈ 156.5 MeV. Two roads that started nowhere near each other arrive at one shape.

Here is the honest part, stated up front. This is a consistency reading, not a fresh calculation. The lattice numbers are imported from HotQCD and Wuppertal–Budapest and the measured hadron spectrum — the framework consumes that curve rather than computing it from scratch. So the column below is labeled “Granularity (consistency reading),” not “derived.” The result is still worth stating plainly: a framework built for entirely different reasons has to live on the same equation of state that decades of lattice QCD nailed down, and it does — with no room left to nudge it.

What this test actually checked: two independent lattice-QCD collaborations (HotQCD and Wuppertal-Budapest) have measured how pressure, energy density, and entropy behave as the universe cools through the QCD crossover — the moment quarks and gluons stop roaming free and lock into protons, neutrons, and other hadrons. Their number for the crossover is \(T_c = 156.5\pm1.5\) MeV, around 20 microseconds after the Big Bang. This framework's reading of that same moment — that quark/gluon distinctions give way to confined-hadron distinctions with nothing lost or invented along the way — lines up with the lattice curve exactly, at both the high-temperature and low-temperature ends. That agreement is worth taking seriously: two very different ways of thinking about this transition land on the same physics. It is not proof that the framework's picture is right — the lattice numbers themselves are imported from HotQCD/Wuppertal-Budapest and the known hadron spectrum, not computed fresh here. What is open: this test only confirms consistency with an already-published result; it does not by itself derive the crossover temperature or the hadron spectrum from first principles.

1. Verdict

The number, both ways

Number we’re testing: QCD equation of state across the crossover — p(T), ε(T), s(T), trace anomaly (ε−3p)/T⁴, and g_*(T), anchored at T_c
Standard cosmology: Continuum-extrapolated lattice EoS curves (HotQCD, Bazavov et al. 2014; Wuppertal–Budapest, Borsanyi et al. 2014), with T_c = 156.5 ± 1.5 MeV (Bazavov et al. 2019)
This framework (granularity): Record-cost reading tracks the same lattice curve — g_* ≈ 61.75 (hot quark-gluon soup) down to g_* ≈ 17.25 (hadron gas); it reads the curve, it does not re-derive it
Measured: T_c ≈ 156.5 ± 1.5 MeV, cross-checked at 155 ± 9 MeV (Borsanyi et al. 2020); trace-anomaly peak (ε−3p)/T⁴ ≈ 3.5–4 near 200 MeV
Agreement: Consistent with the lattice curve end to end — both endpoints match known dof counts, trace-anomaly peak in the same place; two independent lattice collaborations agree within a few MeV in T_c and a few percent in p/T⁴ (consistency check — shared inputs)

Check the source → the calculation shown on this page (Data Used · Calculation Summary)

Agrees. Across the QCD crossover (\(T \approx 130\)–\(200\) MeV, roughly 20 microseconds after the Big Bang), the pressure, energy density, entropy, and trace anomaly this framework relies on move exactly the way first-principles lattice QCD says they must, and no extra hidden particles or degrees of freedom are needed to make the numbers fit. Two independent lattice groups (HotQCD and Wuppertal-Budapest) agree with each other, and the framework's reading of the crossover agrees with both.

2. Tested Claim

The precise granularity claim under test: as the universe cools through the QCD crossover, the thermodynamic functions \(p(T)\), \(\varepsilon(T)\), \(s(T)\), and the trace anomaly \(I(T) \equiv (\varepsilon-3p)/T^4\) must (a) interpolate smoothly and continuously between the weakly-interacting quark-gluon-plasma (QGP) value at high \(T\) and the hadron-resonance-gas (HRG) value at low \(T\) — consistent with a crossover, not a first-order phase transition, as established for physical quark masses; (b) show a trace-anomaly peak located at the pseudo-critical temperature \(T_c\); and (c) reduce, at both ends, to the same \(g_*\) counts already used in the framework's degrees-of-freedom ledger (Test 03: \(g_*\approx 205\)/4 ≈ effectively ~35–40 active dof in the QGP phase counted per lattice conventions below, collapsing to \(g_*\approx 17.25\) in the pion-dominated hadron gas just below \(T_c\)). The framework interprets the whole crossover as a single "granularity event": deconfined color-charge distinctions (quarks and gluons, individually thermally coupled and exchanging color) stop being stable, distinguishable record-carriers and are replaced by confined, color-singlet hadron distinctions — with no discontinuity in the underlying thermodynamic functions, matching the lattice-established fact that this is a crossover, not a sharp phase transition, for physical (non-zero, realistic) quark masses.

3. Data Used

Continuum-extrapolated lattice QCD equation of state, \(2+1\) flavor, physical quark masses: HotQCD Collaboration, Bazavov et al. (2014), "Equation of state in (2+1)-flavor QCD," Phys. Rev. D 90, 094503 (arXiv:1407.6387) — tabulates \(p/T^4\), \(\varepsilon/T^4\), \(s/T^4\), and the trace anomaly \((\varepsilon-3p)/T^4\) continuously from \(T\approx100\) MeV to \(T\approx 400\) MeV using the HISQ/tree action, continuum-extrapolated.
Independent cross-check, Wuppertal-Budapest Collaboration: Borsanyi et al. (2014), "Full result for the QCD equation of state with 2+1 flavors," Phys. Lett. B 730, 99–104 (arXiv:1309.5258) — an independent staggered-fermion lattice discretization reaching consistent results with HotQCD within quoted uncertainties, used here as the cross-validation the test's guardrails require before trusting a single-collaboration number.
Pseudo-critical (crossover) temperature: \(T_c = 156.5 \pm 1.5\) MeV, from the chiral susceptibility peak, HotQCD Collaboration, Bazavov et al. (2019), Phys. Rev. D 100, 094510 (arXiv:1907.07344); consistent with the Wuppertal-Budapest value \(T_c \approx 155\pm 9\) MeV (Borsanyi et al. 2020, Phys. Lett. B 795, 15–21, arXiv:2002.02821 — quark-mass extrapolated and stricter continuum limit). Both collaborations agree this is a smooth analytic crossover, not a genuine (first- or second-order) phase transition, for the physical up/down/strange quark masses realized in nature (established since Aoki et al. 2006, Nature 443, 675).
Trace anomaly peak: lattice data (Bazavov et al. 2014, Fig. 4) show \((\varepsilon-3p)/T^4\) rising from near zero in the hadron-gas phase, peaking at \(\approx 3.5\)–\(4\) near \(T\approx 200\) MeV (somewhat above \(T_c\), as expected for a crossover — the interaction-measure peak generically sits above the chiral-susceptibility-defined \(T_c\)), then falling toward the (small, asymptotically decreasing) perturbative-QCD trace anomaly at high \(T\).
Degrees-of-freedom endpoints: Husdal (2016), "On Effective Degrees of Freedom in the Early Universe," Galaxies 4(4), 78 (arXiv:1609.04979), Table 1 — \(g_* \approx 61.75\) just above \(T_c\) (quarks + gluons + leptons + photons active, using the up/down/strange-flavor-dominated count relevant at this temperature, distinct from the full high-\(T\) electroweak-era \(g_*=106.75\) used in Test 03) falling to \(g_*\approx 17.25\) just below \(T_c\) (pion-dominated hadron gas + leptons + photons), consistent with the lattice trace-anomaly and pressure curves integrating to the same asymptotic dof counts via \(p/T^4 \to (\pi^2/90)g_*\) at each end.
Big Bang Nucleosynthesis cross-check (downstream sensitivity): standard BBN light-element predictions (\(Y_p \approx 0.247\), D/H \(\approx 2.53\times10^{-5}\); Particle Data Group 2024 BBN review; Planck 2018 baryon density \(\Omega_b h^2 = 0.0224\pm0.0001\), A&A 641, A6) are computed using the standard \(g_*(T)\) history that already assumes the lattice-QCD crossover reported here — used only to confirm that adopting this equation of state does not create tension with an independently and separately tested downstream quantity (see Test 23).

4. Calculation Summary

The check is a continuity-and-endpoint verification against a published, continuum-extrapolated lattice result, plus a cross-collaboration agreement check and a downstream-consistency check — not a fresh lattice computation.

Governing relations (standard finite-temperature QCD thermodynamics):

\[ \frac{p(T)}{T^4} = \int_0^T \frac{dT'}{T'}\,\frac{\varepsilon(T')-3p(T')}{T'^4}\ + \ \text{const.}, \qquad \varepsilon = T\frac{\partial p}{\partial T} - p, \qquad s = \frac{\varepsilon+p}{T}. \]

In the free-field, high-\(T\) (deconfined) limit, \(p \to \frac{\pi^2}{90}g_* T^4\) with the same \(g_*\) convention as Test 03's \(g_*(T)\) ledger. In the hadron-resonance-gas, low-\(T\) limit, the same relation holds with a much smaller \(g_*\) dominated by the lightest hadrons (pions).

Endpoint check (high-\(T\) side): lattice \(p/T^4\) at \(T\approx 350\)–400 MeV (Bazavov et al. 2014, Fig. 1) approaches but stays visibly below the ideal free-quark-gluon-gas Stefan-Boltzmann limit \(p_\text{SB}/T^4 = (\pi^2/90)\times 47.5 \approx 5.2\) (using \(N_f=3\) massless-quark + gluon dof), consistent with residual QCD interactions reducing the pressure below the free-gas value — the expected, well-established lattice behavior, not a discrepancy.

Endpoint check (low-\(T\) side): lattice \(p/T^4\), \(s/T^4\) below \(T\approx120\) MeV track the hadron-resonance-gas model built from the full measured hadron spectrum (Particle Data Group hadron listings) to good precision (Bazavov et al. 2014, §III; independently confirmed by Borsanyi et al. 2014), i.e., no unaccounted-for extra light degrees of freedom are needed to match the lattice pressure curve from the hadron side either.

Trace-anomaly peak location check: the lattice interaction measure \((\varepsilon-3p)/T^4\) peaks at \(T\approx 200\) MeV, above the chiral-susceptibility \(T_c \approx 156.5\) MeV — the expected ordering for a crossover (the two "transition" indicators need not coincide exactly, unlike a true phase transition where all observables would show a coincident singularity). This ordering is itself evidence *for* a crossover rather than a genuine phase transition, cross-confirmed by both collaborations (Aoki et al. 2006; Bazavov et al. 2014; Borsanyi et al. 2014).

Cross-collaboration agreement: HotQCD (HISQ action) and Wuppertal-Budapest (stappered action, independent lattice discretization) agree on \(p(T)\), \(\varepsilon(T)\), \(T_c\) within quoted systematic uncertainties (typically a few MeV in \(T_c\), a few percent in \(p/T^4\) at fixed \(T\)) after continuum extrapolation. Two independent discretizations converging on the same continuum answer is the standard lattice-QCD cross-validation the guardrails call for before treating a numerical value as trustworthy.

Downstream-consistency check (rate vs. Hubble): using this equation of state (rather than a naive free-quark-gluon-gas or a naive free-hadron-gas approximation) in the Friedmann equation \(H^2 = \frac{8\pi G}{3}\varepsilon(T)\) changes the expansion rate at the percent-to-ten-percent level in the crossover window compared to the ideal-gas approximation, but this shift is already folded into the standard \(g_*(T)\) tables (Husdal 2016) used across Tests 03, 19, 22, and 23 — i.e., adopting the lattice EoS here does not introduce a *new* deviation relative to the values those tests already used; it is the same well-established input, not a new perturbation to propagate.

5. Granularity Interpretation

On the framework's reading, the QCD crossover is the epoch at which color-charge is no longer a stable, individually thermally-coupled distinction — quarks and gluons cease to be separately distinguishable record-carriers and are replaced by confined, color-singlet hadron distinctions (dominantly pions just below \(T_c\)). The lattice equation of state shows this transition as a smooth, continuous crossover with no latent heat and no discontinuity in \(p\), \(\varepsilon\), or \(s\) — which is exactly what the framework's validity rule would predict for a genuine granularity change that does not require inventing a new primitive object: the distinguishability of "quark" versus "hadron" changes character, but nothing is created or destroyed outside the accounting the trace anomaly and \(g_*\) ledger already track. The fossil record this test relies on is not a single relic abundance but the *entire subsequent thermal history* (BBN light-element abundances, \(N_\text{eff}\), and ultimately the expansion history recorded in the CMB) being consistent with having passed through exactly this equation of state — an indirect but pervasive record, not a directly observed relic of the QCD epoch itself (no direct astronomical observation probes \(T\sim150\) MeV directly; the crossover is only accessible via lattice QCD and heavy-ion collision experiments, e.g. RHIC/LHC, which recreate the deconfined phase but not the cosmological expansion conditions).

6. Gate Routing

This test informs the thermal plasma granularity gate. It directly supports and is supported by Test 03 (the \(g_*(T)\) ledger, which this test's equation of state must reduce to at both endpoints) and Test 19 (QCD Confinement / Hadronization, which addresses the confinement mechanism and hadron-spectrum side of the same transition qualitatively; this test supplies the quantitative thermodynamic function check). It also feeds forward into Test 22 (neutrino decoupling / \(N_\text{eff}\)) and Test 23 (BBN light elements) as a precision input to the expansion-rate history those tests depend on.

7. Failure Mode

Not applicable in the strict sense — the equation-of-state check passed against two independent lattice collaborations. For completeness, the specific failure modes this test screened for (per the test-suite document's guardrails), and confirmed absent:

No hidden extra degrees of freedom: the lattice pressure and trace-anomaly curves at both the high-\(T\) (QGP) and low-\(T\) (hadron-gas) endpoints are matched by the known Standard-Model quark/gluon content and the known hadron spectrum respectively — no additional color-sector or hidden dof is needed to fit the lattice data, so no unaccounted shift to the expansion rate is introduced.
No phase-transition mischaracterization: the data are treated as a crossover (continuous, no latent heat), consistent with the established result (Aoki et al. 2006) for physical quark masses — not mischaracterized as a first-order transition, which would require different bubble-nucleation and gravitational-wave-relic treatment (relevant instead to Test 15, Cosmic Strings and Topological Defects, for genuinely first-order scenarios in other sectors).
No single-collaboration reliance: HotQCD and Wuppertal-Budapest results are cross-checked against each other rather than a single lattice calculation being taken on faith.
No measured-value-as-derived confusion: \(T_c\), the full equation of state, and the hadron spectrum used to check the low-\(T\) endpoint are explicitly flagged as imported lattice/PDG inputs, not outputs of the framework's granularity doctrine.

8. Next Action

Data lookup / cross-check, not derivation: (a) carry this exact equation of state forward as the input used in Tests 22 and 23's expansion-rate calculations rather than re-deriving it there; (b) if the framework's technical program ever proposes additional colored or confined degrees of freedom (e.g., extra hidden-sector quarks), re-run this same lattice-comparison check to confirm the proposal does not shift \(p(T)\), \(\varepsilon(T)\), or \(g_*\) outside the bounds already fit by HotQCD/Wuppertal-Budapest; (c) no dead-end routing is needed for this test — it resolved as a consistency check against firmly established first-principles lattice data, not as an unresolved distinction/encoding question.

Bottom line

The lattice-QCD equation of state, its crossover character, its critical temperature, and its role in the expansion-rate history all belong to established QCD thermodynamics and the standard hot Big Bang model, which this framework inherits wholesale and agrees with. The framework's own contribution is a way of reading the crossover — quark/gluon distinctions giving way to confined-hadron distinctions — and that reading matches the lattice data checked here without needing anything extra.

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