Test 03 — Effective Relativistic Degrees of Freedom Ledger — A Framework for a Theory of Everything

Test 03 — Effective Relativistic Degrees of Freedom Ledger Test

Does the standard-cosmology ledger of active relativistic species, \(g_*(T)\) and \(g_{*s}(T)\), change in the right order — and by the right accounting — as the universe cools, on the granularity reading that this is a history of species becoming massive, confined, nonrelativistic, or decoupled?

Agrees Relativistic degrees-of-freedom ledger

What we're checking

Observable: The effective number of relativistic species, N_eff — the radiation "headcount" set as neutrinos stop sharing heat with the rest of the plasma.
Standard cosmology: N_eff = 3.044 (entropy bookkeeping across the e⁺e⁻ threshold, with the near-instant-decoupling correction).
Granularity (consistency reading): N_eff = 3.044 — the record-cost ledger steps its species count down at the same temperatures, by the same amounts, landing on the same number.
Measured: Planck 2018: N_eff = 2.99 ± 0.17 — both roads sit comfortably inside one standard deviation.
The epoch: Roughly 10⁻¹¹ to 10 seconds after the beginning — the era, not the quantity being compared.
Verdict: Agrees

Freeze the universe at a single second and count the light it can carry. Standard cosmology has a way to do this — it tracks every particle species still fast and hot enough to count as radiation, watches the electrons and positrons annihilate and dump their heat into the photons, and adds up a single number for the whole relativistic bath: N_eff = 3.044. That number has been measured. Planck's map of the oldest light pins it at 2.99 ± 0.17.

This framework never sets out to reproduce that figure. It keeps a different set of books entirely — a ledger of what the early universe can actually record, of when a species stops being a distinct, trackable thing and folds into the background. And yet, following its own accounting down through the same temperatures, it steps off the same cliffs at the same moments and arrives at the same total: 3.044. Two roads drawn for different reasons, meeting at one number that experiment already confirmed.

Here is the honest edge of it. The framework does not force 3.044 out of the geometry from scratch — it reads the same entropy ledger the standard picture reads, and confirms it lands in the same place. So call this what it is: a consistency check that passes cleanly, not a fresh independent prediction of the count. The agreement is real, the number is right, and the reason the two roads meet is that, at this depth, they are honestly describing the same physics.

What we actually checked: standard hot-Big-Bang cosmology keeps a running count of how many particle species are still "active" — hot enough, and still talking to everything else, to carry energy and entropy as radiation. That count, \(g_*(T)\), drops in specific steps as the universe cools: the top quark drops out, then quarks and gluons lock into protons, neutrons and pions at the QCD transition, then electrons and positrons annihilate and dump their energy into photons rather than neutrinos. We reran that whole ledger under our own reading — each step is a species losing its distinguishability as a separate, thermally coupled record-carrier — and it reproduces the same thresholds, the same photon-to-neutrino temperature ratio \((11/4)^{1/3}\approx1.401\), and the same effective neutrino count \(N_\text{eff}=3.044\) that the Standard Model predicts, matching the Planck-measured \(N_\text{eff}=2.99\pm0.17\) to within 1σ. Two independent ways of keeping the books — the textbook thermal-history calculation and our own record-keeping language for it — land on the same numbers. That agreement is worth taking seriously, but it isn't a proof of the framework: the particle content, the masses, and the QCD transition temperature are all measured or lattice-computed inputs we borrowed, not things we derived.

1. Verdict

The number, both ways

Number we’re testing: N_eff — the effective number of relativistic species (the g*(T)/g*s(T) ledger and the T_γ/T_ν ratio)
Standard cosmology: N_eff = 3.044 (entropy bookkeeping across the e⁺e⁻ threshold with the near-instant-decoupling correction); g* from 106.75 down to 3.36
This framework (granularity): N_eff = 3.044 — the record-cost ledger steps its species count down at the same temperatures, by the same amounts; same (11/4)^(1/3) ≈ 1.401 ratio
Measured: Planck 2018: N_eff = 2.99 ± 0.17 (TT,TE,EE+lowE+lensing)
Agreement: Both roads land on 3.044, sitting 'comfortably inside one standard deviation' of the measured 2.99 ± 0.17 (page: 'matching ... to within 1σ') (consistency check — shared inputs)

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

Agrees with existing models. Track the number of relativistic particle species from the electroweak era down through electron-positron annihilation, and our ledger and the standard cosmological ledger step down at the same temperatures, by the same amounts, with no missing or double-counted entropy. The two headline numbers that fall out of the bookkeeping — the photon/neutrino temperature ratio and the effective neutrino count \(N_\text{eff}=3.044\) — match the Planck-measured value. One honest caveat: this is a bookkeeping check, not a derivation of the Standard Model spectrum or the QCD transition temperature — those numbers are imported from particle physics and lattice QCD.

2. Tested Claim

The precise granularity claim under test: as the universe cools from the electroweak era (\(T \sim 160\text{ GeV}\)) through the QCD transition (\(T \sim 155\text{ MeV}\)) to \(e^+e^-\) annihilation (\(T \sim 0.5\text{ MeV}\)), the number of thermally active relativistic degrees of freedom \(g_*(T)\) and the entropy-weighted count \(g_{*s}(T)\) must (a) only ever change at physically motivated thresholds — mass thresholds, confinement, or decoupling — (b) never create or destroy degrees of freedom without a matching entropy transfer, and (c) leave the specific, already-observed relic signatures (photon-to-neutrino temperature ratio, \(N_\text{eff}\)) that standard cosmology requires. The framework interprets each such step-down in \(g_*\) as a distinct "granularity event": a class of field excitations losing thermal-coupling distinguishability and either freezing into a stable relic record or being absorbed into the entropy of the remaining bath.

3. Data Used

Standard Model relativistic species count at \(T \gtrsim 170\) GeV (full SM, above top-quark threshold): \(g_* = 106.75\). This is the textbook full-Standard-Model count (e.g. Kolb & Turner; Husdal 2016, "On Effective Degrees of Freedom in the Early Universe," Galaxies 4(4):78) — photon (2), gluons (16), \(W^\pm, Z\) (9, after electroweak symmetry breaking counting — see note below), 3 generations of charged leptons (Dirac, 4 each = 12) and neutrinos (2 each if Weyl = 6, or handled via the 7/8 fermionic weight), 6 quark flavors × 3 colors × 2 spin × 2 (particle/antiparticle, Dirac) with the 7/8 fermionic statistical weight, plus 1 real Higgs scalar. The standard tally (Husdal 2016, Table 1; Particle Data Group *Review of Particle Physics* 2024, "Big-Bang Nucleosynthesis" review, sec. on \(g_*\)) gives \(g_{*,\text{SM}} = 106.75\) for \(T\) above the top-quark mass (\(m_t \approx 172.5\) GeV, PDG 2024).
QCD confinement / hadronization transition: pseudo-critical temperature \(T_c \approx 156.5 \pm 1.5\) MeV, from lattice QCD (HotQCD Collaboration, Bazavov et al. 2019, Phys. Rev. D 100, 094510; consistent with Wuppertal-Budapest Collaboration, Borsanyi et al. 2020). Below \(T_c\), quarks and gluons (which carried \(2\times8=16\) gluon + \(6\times3\times2\times2\times\tfrac{7}{8}=63\) quark degrees of freedom while deconfined) are replaced by a hadron gas dominated by pions (3 light pion states), dropping \(g_*\) sharply.
Neutrino decoupling and \(e^+e^-\) annihilation: neutrinos decouple at \(T \sim 2\text{–}3\) MeV (well above electron mass \(m_e = 0.511\) MeV, PDG 2024), then \(e^+e^-\) annihilate into photons near \(T \sim 0.2\text{–}0.5\) MeV, heating the photon bath but not the already-decoupled neutrino bath. Standard result: photon temperature today exceeds neutrino temperature by the entropy-conservation factor \((11/4)^{1/3} \approx 1.401\), giving \(T_{\nu,0} \approx 1.95\) K against measured \(T_{\text{CMB},0} = 2.7255 \pm 0.0006\) K (Fixsen 2009; used as the standard reference value, e.g. Planck 2018 VI, A&A 641, A6).
Effective number of neutrino species: Standard Model prediction with instantaneous-decoupling correction, \(N_\text{eff} = 3.044\) (Bennett et al. 2021, Phys. Rev. D 104, 083524; de Salas & Pastor 2016 give 3.045 — later re-computation with QED corrections lowers to 3.043–3.044, adopted as 3.044 in Planck 2018-era literature). Measured value: Planck 2018 (A&A 641, A6) \(N_\text{eff} = 2.99 \pm 0.17\) (TT,TE,EE+lowE+lensing), fully consistent with the Standard-Model prediction at the ~1σ level. No new relativistic species are required or excluded by this test.
Full temperature-resolved \(g_*(T)\), \(g_{*s}(T)\) tables: Husdal, L. (2016), "On Effective Degrees of Freedom in the Early Universe," Galaxies 4(4), 78 (arXiv:1609.04979) — tabulates \(g_*\) from \(g_*=106.75\) (SM high-T) down to \(g_*=3.36\) (today, photons+neutrinos) through every SM threshold, cross-checked against lattice QCD equation-of-state data.

4. Calculation Summary

The ledger is built as a monotone-in-temperature (but non-monotone-in-\(g_*\)) step function. Each step is checked against a threshold condition and an entropy-conservation identity rather than asserted qualitatively.

Governing relations (standard cosmology, e.g. Kolb & Turner Ch. 3):

\[ \rho_{\text{rad}} = \frac{\pi^2}{30}\,g_*(T)\,T^4, \qquad s = \frac{2\pi^2}{45}\,g_{*s}(T)\,T^3, \] with the standard fermion/boson weighting \[ g_*(T) = \sum_{\text{bosons } i} g_i \left(\frac{T_i}{T}\right)^4 + \frac{7}{8}\sum_{\text{fermions } j} g_j \left(\frac{T_j}{T}\right)^4 . \]

Threshold check — species becomes nonrelativistic when \(m_i \gtrsim 3\,T\) (standard Boltzmann-suppression rule of thumb). Applied to the top quark (\(m_t = 172.5\) GeV) this occurs at \(T \lesssim 57\) GeV, consistent with its removal from the active count used in tables below \(T\sim 80\)–\(170\) GeV. Applied to \(e^\pm\) (\(m_e = 0.511\) MeV) this occurs at \(T \lesssim 0.17\) MeV, consistent with the accepted \(e^+e^-\) annihilation window of \(T\sim 0.2\)–\(0.5\) MeV (annihilation is not instantaneous; the standard treatment integrates the Boltzmann equation, but the order-of-magnitude threshold check passes).

Confinement check: comparing the QCD confinement scale \(T_c \approx 156.5\) MeV (lattice, HotQCD 2019) against the temperatures at which quark/gluon degrees of freedom (79 dof) are replaced by a pion- dominated hadron gas (~17.25 dof just below \(T_c\), per Husdal 2016 Table 1) confirms the transition occurs in the correct regime — well above \(e^+e^-\) annihilation and well below electroweak symmetry breaking, so the ordering \(T_\text{EW} \gg T_\text{QCD} \gg T_{e^+e^-} \gg T_\text{rec}\) required by the ledger holds numerically: \(160\text{ GeV} \gg 0.156\text{ GeV} \gg 0.0005\text{ GeV} \gg 0.26\text{ eV}\) (recombination, Planck 2018 \(z_\text{rec}\approx1089\), for reference only — outside this test's window).

Entropy-conservation check (photon reheating from \(e^+e^-\) annihilation): before annihilation, photons and \(e^\pm\) share a common temperature with \(g_{*s} = 2 + \tfrac{7}{8}(4) = 4\); neutrinos have already decoupled at their own temperature with \(g_{*s,\nu} = \tfrac{7}{8}(6) = 5.25\) in a separate comoving entropy reservoir. Conserving entropy in the photon+\(e^\pm\) sector alone across annihilation, \(g_{*s,\text{before}}\,a_\text{before}^3 T_\text{before}^3 = g_{*s,\text{after}}\,a_\text{after}^3 T_\text{after}^3\), with \(g_{*s,\text{after}} = 2\) (photons only), yields \[ \frac{T_\gamma}{T_\nu} = \left(\frac{11}{4}\right)^{1/3} \approx 1.401, \] which is the standard textbook result and matches the observed \(T_{\text{CMB},0}=2.7255\) K against the predicted relic neutrino temperature \(T_{\nu,0} \approx 1.945\) K. This is the direct numerical fingerprint of the \(e^+e^-\) "granularity event" (the disappearance of \(e^\pm\) as active relativistic degrees of freedom) and it is not free — it is fixed once entropy conservation and the neutrino-decoupling timing are fixed. The test therefore has a genuine numerical trip-wire, not just a qualitative story.

\(N_\text{eff}\) cross-check: the same bookkeeping, carried through with the small non-instantaneous-decoupling correction (neutrinos are not perfectly decoupled when \(e^+e^-\) annihilation starts, so they absorb a small residual heating), predicts \(N_\text{eff} = 3.044\) (Bennett et al. 2021) against the Planck 2018 measured \(N_\text{eff} = 2.99 \pm 0.17\) — consistent within 1σ, with no room for an additional fully-thermalized relativistic species at this epoch (see also Test 22, Neutrino Decoupling and \(N_\text{eff}\) Test, for the dedicated treatment of this bound).

5. Granularity Interpretation

On the framework's reading, each downward step in \(g_*(T)\) is a point where a class of field excitations stops being a distinguishable, thermally coupled record-carrier and either (a) freezes into a new stable, recordable relic — the QCD transition freezes color charge into confined hadrons, and \(e^+e^-\) annihilation freezes the neutrino bath's temperature ratio into a permanent, in-principle-observable relic signature (the cosmic neutrino background, addressed separately in Test 41) — or (b) is absorbed as extra entropy into the remaining, still-coupled bath (the photon bath, which is reheated relative to neutrinos). Both outcomes are allowed under the framework's validity rule ("a distinction is valid in window \(W\) only if it can be coupled, encoded, stabilized, and/or observed at that window's resolution"): confinement produces a stabilized, later-observable record (hadron/baryon abundances, indirectly the \(\eta_B\) ratio); the photon-reheating step produces an encoded record in a global thermodynamic ratio that is, in principle, observable via \(N_\text{eff}\) and, if the cosmic neutrino background is ever directly detected, directly confirmable. No step in the standard ledger requires inventing a new distinguishable entity, and no step erases entropy that current cosmology says must persist. That is the entirety of what this test checks — it does not claim the framework derived the Standard Model spectrum, the value of \(T_c\), the electron mass, or the number of quark colors; all of those are imported, measured, or lattice-computed values that the ledger consistency check treats as given.

6. Gate Routing

This test feeds the particle appearance and disappearance ledger used elsewhere in the program. It is the bookkeeping backbone that the QCD confinement test (Test 19), the lattice-QCD equation-of-state test (Test 20), the neutrino-decoupling / \(N_\text{eff}\) test (Test 22), and the BBN light-element test (Test 23) each draw on for a consistent, non-double-counted degrees-of-freedom baseline. A failure here — an inconsistent or double-counted ledger — would propagate as a silent error into all four of those downstream tests, so this test's role is to certify the baseline they can trust, not to independently derive anything new.

7. Failure Mode

Not applicable in the strict sense — the ledger check passed. For completeness, the specific failure modes this test screened for (per the test suite's guardrails), and confirmed absent:

No premature appearance: no species in the ledger is placed above the temperature its mass or coupling structure would allow (checked via the \(m_i \gtrsim 3T\) threshold rule for the top quark and \(e^\pm\), and via the lattice \(T_c\) for confinement).
No field/particle conflation: gauge-redundant field components (e.g. the would-be Goldstone modes absorbed into \(W^\pm, Z\) longitudinal polarizations) are not double-counted as separate physical degrees of freedom in the standard tally used here; this is enforced by using the published Husdal (2016)/PDG-consistent counting rather than an independent re-derivation.
No un-sourced relic claim: the one relic abundance figure invoked (\(T_\gamma/T_\nu\), \(N_\text{eff}\)) is tied to a specific published calculation and a specific measured value with its uncertainty, not asserted qualitatively.
No measured-value-as-derived confusion: \(m_t\), \(m_e\), \(T_c\), and the SM particle content are explicitly flagged above as imported/measured/lattice-computed, not as outputs of the framework's granularity doctrine.

8. Next Action

Data lookup / cross-check, not derivation: (a) carry this exact \(g_*\)/\(g_{*s}\) baseline forward into Tests 19, 20, 22, and 23 rather than re-deriving it in each; (b) when the framework's dark-sector or hidden-sector proposals (if any) are evaluated (Tests 25–27), re-run this same threshold-and-entropy check to confirm no additional relativistic species is silently introduced without a matching \(N_\text{eff}\) or entropy accounting; (c) no further routing is needed for this test — it resolved as a bookkeeping consistency check against firmly established data, not as an open question.

Bottom line

Standard hot-Big-Bang cosmology and this framework's own record-keeping language for the same physics land on the same ledger: the entire \(g_*(T)\) history, its thresholds, and its entropy-conservation bookkeeping check out against the Standard Model of particle physics and lattice QCD. Our only addition is a way of describing each threshold — a species losing its distinguishability as a separate, thermally coupled record-carrier — and that description is consistent with, but does not itself add new numbers to, the physics checked here.

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