The Cross-Ratio: Quark–Meson Epoch

   
   Methodology for Computing the Cross-Ratio in the Quark–Meson Epoch

To explore the projective structure of the micro-cosmic universe, we treat quark–meson systems as the simplest “measurement environments” available in nature. Mesons—bound pairs of quarks and antiquarks—provide a clean, well-studied spectrum of energies, masses, and decay channels. These serve as our experimental anchors for constructing the four-point tuples needed to evaluate the cross-ratio.

The methodology proceeds in four steps:

1. Select a coherent family of mesons

We begin by choosing a set of mesons that belong to the same quark content or interaction channel—for example, the pseudoscalar or vector meson families. This ensures that all physical quantities arise from a single underlying interaction governed by QCD, minimizing external distortions such as electroweak mixing or hadronic broadening.

2. Identify four physically meaningful points

From the selected family, we extract four quantities that play the role of projective points. These can be:
    •    measured masses,
    •    energy level splittings,
    •    interaction strengths,
    •    or characteristic transition frequencies.

The ordering of these four points follows the same “observer-capability” philosophy used throughout the Theory of Measurements: we move from the least bound or least stable state to the most tightly bound or most stable. This ensures that the tuple reflects an increasing level of underlying geometric structure.

3. Normalize the quadruple using a projective map

As in Hime’s analytic formalism, the triple (A, B, D) is transformed to the canonical configuration:


via a Möbius transformation.  This step removes all unphysical scaling—mass units, energy conventions, decay-rate normalizations—leaving only the internal projective geometry of the quark–meson system.

4. Compute the projective invariant

With the quadruple in canonical form, the final step is to evaluate:


which captures the relative configuration of the four chosen states.
By construction, this χ is independent of:
    •    the mass scheme used,
    •    renormalization conventions,
    •    experimental calibration,
    •    or energy-scale choice.

It reflects only the projective relation among the four physical points—exactly the quantity predicted to remain stable across observers and across cosmic epochs.



     To compute the projective invariant for the quark–meson epoch, we rely exclusively on experimentally measured meson properties. All numerical values—masses, mass splittings, decay widths, and quantum numbers—are taken from the most authoritative and universally accepted source in particle physics:

Particle Data Group (PDG)
R.L. Workman et al., Particle Data Group, Progress of Theoretical and Experimental Physics 2022, 083C01 (2022),  https://pdg.lbl.gov.

Depending on the specific tuple chosen, the following Meson families provide clean, well-measured data:

    •    Light pseudoscalar mesons: π, K, η, η′
    •    Light vector mesons: ρ, ω, φ
    •    Charmonium states: J/ψ, ψ(2S), χc states
    •    Bottomonium states: Υ(1S), Υ(2S), Υ(3S), χb states

All of these have precisely measured masses with uncertainties typically at the level of keV to MeV, making them ideal for projective analysis.

Why is PDG data ideal for this calculation?

•    It is fully experimental (no theoretical bias).

•    Mass hierarchy within each family is clean and monotonic.
•    Errors are extremely small compared to the mass scales.
•    The families reflect pure QCD binding with minimal   electroweak contamination.
•    The data spans multiple quark compositions, allowing cross-checks of invariance.

These properties ensure that our four-point tuples reflect the true physical ordering of states, as required by the observer-capability methodology.



  To probe the quark–meson epoch with minimal distortion, we use heavy quarkonia—bound states of a heavy quark and its antiquark. These are ideal because they are:

    •    dominated by the strong interaction,
    •    relatively narrow and well isolated,
    •    and very precisely measured.

We choose one charmonium state and three bottomonium states:
    •    $J/\psi(1S): m_{J/\psi} = 3096.900 \,\text{MeV}$
    •    $\Upsilon(1S): m_{\Upsilon(1S)} = 9460.40 \,\text{MeV}$
    •    $\Upsilon(2S): m_{\Upsilon(2S)} = 10023.26 \,\text{MeV}$ 
    •    $\Upsilon(3S): m_{\Upsilon(3S)} = 10355.2 \,\text{MeV}$ 

1. We define the ordered quadruple

We order by increasing mass:

$A = m_{J/\psi},\quad B = m_{\Upsilon(1S)},\quad C = m_{\Upsilon(2S)},\quad D = m_{\Upsilon(3S)}$.

Numerically (in MeV):

$A = 3096.9,\quad B = 9460.4,\quad C = 10023.26,\quad D = 10355.2.$

2. Compute the raw cross-ratio

We use the standard projective cross-ratio

Plugging in the four masses gives

Crucially, this value lies in the preferred range $1 \le \chi_{\text{raw}} \le 2$, so we don’t need to change branch via permutations or $\chi \to 1/\chi$, etc. This is our canonicalized cross-ratio for this 4-tuple.

3. Normalize to the unit interval

To compare across very different systems (from quarks to     galaxies), we map $\chi_{\text{raw}}$ into the unit interval using

For $\chi_{\text{raw}} \approx 1.52$:

Mass uncertainties here are at the level of $10^{-2}–10^{-1}\,\text{MeV}$, far smaller than the GeV-level separations, so the uncertainty in $\chi_{\text{eff}}$ is tiny compared to the difference between 0.34 and, say, 0.2.

4. Interpretation

This heavy-quarkonia 4-tuple is:

    •    fully experimental,

    •    dominated by a single interaction (QCD),

    •    and very clean spectroscopically.

After projective canonicalization and normalization, it yields

We can repeat similar calculations for another 4-tuple as to obtain:



There are also exceptions.  For example, for light pseudoscalar mesons 4-tuple $(\pi, K, \eta, \eta')$:


We suspect that this difference is due to the fact, unlike other two examples, this quartet does not preserve monotonicity of the underlying ‘observer-capability ladder’ (π–K–η–η′).  Also there may be more than one interaction involved, whereas we are looking for a single interaction coupling the quartet when calculating the cross-ratio.
 

    Using experimentally measured masses from the Particle Data Group, we evaluated two independent heavy-quarkonia quartets—one built from $J/\psi$ together with $\Upsilon(1S), \Upsilon(2S), \Upsilon(3S)$, and another built from $\psi(2S)$ paired with the same three bottomonium states. After applying the projective canonicalization and normalization procedure, both quartets yielded nearly identical values of the effective cross-ratio, clustering tightly around $\chi_{\text{eff}} \approx 0.34$.

    This agreement is notable because the two quartets involve different internal structures and mass scales, yet produce the same projective signature. The result suggests that, within the quark–meson epoch, the underlying measurement geometry is already highly symmetric and remarkably stable—showing minimal sensitivity to the specific choice of heavy-quark states.

    In the following blogs, we will place this micro-cosmic value in context as we compare it to later cosmic epochs, eventually revealing the deeper pattern that motivates our “primordial stitch” interpretation.


     By examining the projective structure encoded in heavy-quarkonia spectra, we find that the effective cross-ratio remains strikingly stable, hinting that the earliest measurement geometry of the universe was already imprinted with a distinctive and enduring pattern.

    This symmetry, preserved across independent quartets drawn from different quark families, becomes our reference point for all later epochs. It is from here—at the micro-cosmic dawn—
this micro-cosmic value of $\chi \approx 0.34$ becomes our baseline—our primordial stitch in the Cosmic Quilt—against which all later epochs can be compared.



    With the micro-cosmic epoch established, we now move forward in the Cosmic Quilt to a very different arena: the era where atoms form and light begins to carry information across the universe.  At these lower energies, the strong force yields to the electromagnetic interaction, giving rise to a new set of projective structures encoded in the 21-cm hydrogen line (spin flip), optical atomic transitions, and hyperfine splittings.

    Although the physics is very different from that of heavy quarkonia, the same projective machinery applies: we can construct four-point tuples from atomic observables, and examine their cross-ratios to see whether the early geometric imprint we found—our micro-cosmic baseline of $\chi \approx 0.34$—continues into the atomic domain. The next blog explores this transition in detail, showing how the universe’s first spectral threads begin to weave themselves into the larger measurement geometry of the Cosmic Quilt.

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