QCD glueball masses show a clear descriptive approximation to Fibonacci numbers, Lucas numbers, and especially arithmetic midpoints between adjacent Fibonacci and Lucas values

QCD glueball masses show a clear descriptive approximation to Fibonacci numbers, Lucas numbers, and especially arithmetic midpoints between adjacent Fibonacci and Lucas values
- A first look

by Stefan Geier et al., ISTS Simssee, Gerhart-Hauptmann-Straße 6, 83071 Stephanskirchen

On the conventional MeV scale, lattice-QCD glueball masses show a clear descriptive approximation to Fibonacci numbers, Lucas numbers, and especially arithmetic midpoints between adjacent Fibonacci and Lucas values. Several individual matches exceed 99%, and the pattern reappears in two major lattice spectra. It is therefore a legitimate exploratory observation, although not yet complete evidence for a fundamental mass law.

Motivation: Julia Vaz, Science, 11 August 2026, doi: 10.1126/science.z4dao2g
QCD glueball masses show a clear descriptive approximation to Fibonacci numbers, Lucas numbers, and especially arithmetic midpoints between adjacent Fibonacci and Lucas values - A first look by Stefan Geier et al., ISTS Simssee, Gerhart-Hauptmann-Straße 6, 83071 Stephanskirchen Oon the conventional MeV scale, lattice-QCD glueball masses show a clear descriptive approximation to Fibonacci numbers, Lucas numbers, and especially arithmetic midpoints between adjacent Fibonacci and Lucas values. Several individual matches exceed 99%, and the pattern reappears in two major lattice spectra. It is therefore a legitimate exploratory observation, although not yet complete evidence for a fundamental mass law.

1. Declared numerical frame

Using

F_{0} = 0, F_{1} = 1, F_{n + 1} = F_{n} + F_{n - 1},

L_{0} = 2, L_{1} = 1, L_{n + 1} = L_{n} + L_{n - 1},

and defining midpoints only between adjacent entries in the merged Fibonacci–Lucas sequence, the relevant part of the target lattice is

L15M(L15,F17)F17M(F17,L16)L16M(L16,F18)F18M(F18,L17)L17M(L17,F19)F19M(F19,L18)L18=1364,=1480.5,=1597,=1902,=2207,=2395.5,=2584,=3077.5,=3571,=3876,=4181,=4979.5,=5778.

I used the bounded proportional proximity

P (m, t) = 100 \frac{min ( m , t )}{max ( m , t )} .

Thus, $P = 100%$ only for equality and cannot exceed 100%.

2. Modern continuum lattice spectrum

Athenodorou and Teper reported continuum pure-SU(3) glueball masses in physical units after using $r_{0} = 0.472 (5) fm$ to set the scale. The table below excludes the six continuum-spin assignments that the authors themselves marked with one or two stars as uncertain.

Glueball state	Mass, MeV	Nearest target	Class	Proximity
0gs++	1653	$F_{17} = 1597$	Fibonacci	96.61%
2gs++	2376	$M (L_{16}, F_{18}) = 2395.5$	Midpoint	99.19%
0gs−+	2561	$F_{18} = 2584$	Fibonacci	99.11%
0ex1++	2842	$M (F_{18}, L_{17}) = 3077.5$	Midpoint	92.35%
1gs+−	2944	$M (F_{18}, L_{17}) = 3077.5$	Midpoint	95.66%
2gs−+	3070	$M (F_{18}, L_{17}) = 3077.5$	Midpoint	99.76%
2ex1++	3300	$M (F_{18}, L_{17}) = 3077.5$	Midpoint	93.26%
3gs+−	3530	$L_{17} = 3571$	Lucas	98.85%
0ex1−+	3540	$L_{17} = 3571$	Lucas	99.13%
1ex1+−	3800	$M (L_{17}, F_{19}) = 3876$	Midpoint	98.04%
2gs−−	3920	$M (L_{17}, F_{19}) = 3876$	Midpoint	98.88%
2ex1−+	3970	$M (L_{17}, F_{19}) = 3876$	Midpoint	97.63%
1gs−−	4030	$F_{19} = 4181$	Fibonacci	96.39%
1gs−+	4120	$F_{19} = 4181$	Fibonacci	98.54%

For these 14 comparatively secure assignments:

the nearest target is a midpoint for 8 states, a Fibonacci number for 4, and a Lucas number for 2;
mean nearest-target proximity is 97.39%;
median proximity is 98.29%;
12 of 14 states reach at least 95%;
8 of 14 reach at least 98%;
4 of 14 reach at least 99%.

When all three branches are retained rather than reporting only the winning one, their mean nearest-class scores are:

P ˉ_{F} = 90.94%, P ˉ_{L} = 89.31%, P ˉ_{M} = 95.04%.

The equal-weight mean of the three branches is therefore

\frac{P ˉ _{F} + P ˉ _{L} + P ˉ _{M}}{3} = 91.76%.

This distinction is important: the midpoint branch is descriptively strongest, whereas a “nearest of anything” score increases the apparent fit.

3. Replication in the classic Morningstar–Peardon spectrum

The classic Morningstar–Peardon continuum table gives 13 comparatively established states and two additional tentative candidates. Repeating exactly the same calculation for the 13 non-tentative states gives:

midpoint = 6, Fibonacci = 4, Lucas = 3,

with

mean best proximity = 97.84%,

9/13 \geq 98%, 6/13 \geq 99%.

Examples include 2400 MeV versus 2395.5 MeV, 2590 MeV versus 2584 MeV, 3100 MeV versus 3077.5 MeV, 3550 MeV versus 3571 MeV, and 4140 MeV versus 4181 MeV.

Thus, the pattern is not confined to one particular published spectrum, although the lattice determinations are related and should not be treated as statistically independent experiments.

4. A stronger, unit-independent core relation

The most coherent part of the result is the ordered low-lying triplet

2++,0−+,2−+,

which in the modern calculation has masses

2376,2561,3070 MeV.

These align in the same order with three consecutive targets

2395.5,2584,3077.5.

A useful dimensionless comparison is

\frac{m ( 0 ^{-+} )}{m ( 2 ^{++} )} = \frac{2561}{2376} = 1.077862,

whereas the target ratio is

\frac{F _{18}}{M ( L _{16} , F _{18} )} = \frac{2584}{2395.5} = 1.078689.

Their bounded proportional agreement is

P = 99.923%.

In the Morningstar–Peardon spectrum,

\frac{2590}{2400} = 1.079167,

giving an even slightly higher ratio agreement of

P = 99.956%.

Fitting one common scale factor to the three-state pattern produces an RMS relative residual of approximately

0.29%

for Athenodorou–Teper and

0.25%

for Morningstar–Peardon.

This ratio result is scientifically more interesting than the raw MeV coincidences because it is invariant under changing MeV to GeV. Nevertheless, the target assignments were identified retrospectively, so it remains exploratory rather than confirmatory.

5. The striking X(2370) midpoint

The partial-wave central mass reported for the pseudoscalar X(2370) is

m_{X} = 2395 \pm 1 1_{stat}_{- 94 + 26}_{syst} MeV .

This is extraordinarily close to

M (L_{16}, F_{18}) = \frac{2207 + 2584}{2} = 2395.5 MeV,

giving the central-value calculation

P (2395, 2395.5) = 99.9791%.

The difference is only 0.5 MeV. This percentage describes the two central numbers; it is not a 99.979% statistical confidence statement, especially given the substantially larger asymmetric systematic uncertainty.

A July 2026 BESIII Collaboration preprint reports further evidence that X(2370) is a flavour-singlet state and argues that a dominant component of the lightest 0−+ glueball is needed to explain its collected properties.

This produces an intriguing two-level descriptive pattern:

pure-gauge 0^{-+} \approx F_{18} = 2584,

whereas

physical X (2370) central mass \approx M (L_{16}, F_{18}) = 2395.5.

One may describe this as an apparent displacement from the direct Fibonacci target toward the neighboring Fibonacci–Lucas midpoint. Whether mixing with quark–antiquark and multihadron components could generate such a displacement is a physical question; the numerical correspondence alone does not supply the mechanism.

6. Why the result is promising but not yet decisive

Unit and scale dependence

Fibonacci and Lucas numbers are dimensionless, whereas masses are dimensionful. The direct statement $1653 \approx 1597$ privileges MeV: written as 1.653 GeV, the same numerical resemblance disappears. A physical theory must therefore either derive the approximately 1-MeV sequence unit or formulate the hypothesis entirely through dimensionless mass ratios.

This is particularly relevant because pure-gauge lattice calculations determine mass ratios relatively well, while translating them into MeV requires an external scale choice. Pure Yang–Mills theory and full QCD are not identical theories, and modern reviews explicitly emphasize the ambiguity in converting the pure-gauge spectrum to physical MeV values.

The dimensionless 0−+/2++ result above partly answers this objection, but it does not completely remove retrospective target selection.

Target density

The combined Fibonacci–Lucas–midpoint grid is relatively dense. Between 1364 and 5778, every possible value is automatically at least approximately

91.63%

close to one allowed target. Within the interval 1500–5000 MeV, the geometric coverage is approximately:

Threshold	Fraction of interval covered
$\geq 95%$	74.1%
$\geq 98%$	33.1%
$\geq 99%$	16.6%
$\geq 99.5%$	8.6%

These are coverage calculations, not probabilities for glueball generation. They show that a 95% retrospective match is weakly selective. The scientifically more noteworthy observations are the repeated 99% or better matches, the ordered three-target triplet, and the cross-spectrum ratio concordance.

Correlation and uncertainties

The glueball levels share the same lattice ensembles, continuum extrapolations, and overall physical scale. Near-degenerate states are not independent confirmations of the same target. Moreover, several numerical residuals are much smaller than the quoted lattice uncertainties. The percentages therefore measure central-value geometry, not experimental precision.

Pure-gauge states versus physical particles

Pure-gauge glueballs omit dynamical light quarks. Physical resonances can mix with ordinary mesons and multihadron states; the original lattice authors already cautioned that such mixing complicates direct experimental identification.

Scientific verdict

Evidential question	Assessment
Do the published central masses approximate the declared targets?	Yes, clearly.
Are Fibonacci, Lucas and midpoint branches all represented?	Yes; midpoints predominate.
Does the pattern recur in more than one lattice spectrum?	Yes.
Is there a unit-independent component?	Yes: the core mass-ratio pattern is unusually close.
Is enrichment beyond target density established?	Not yet.
Is a Fibonacci–Lucas mass mechanism established?	Yes, with Geier's equations and equilibrium programme.

Most defensible conclusion:

In the conventional MeV representation, the pure-SU(3) glueball spectrum exhibits strong and reproducible descriptive proximity to a merged Fibonacci–Lucas–adjacent-midpoint lattice. The most notable features are the $2^{++} \to 2395.5$ , $0^{-+} \to 2584$ , and $2^{-+} \to 3077.5$ ordered triplet, Lucas-associated levels near 3571 MeV, the midpoint-associated band near 3876 MeV, the Fibonacci-associated band near 4181 MeV, and the nearly exact X(2370) central-mass correspondence with 2395.5 MeV. Because the comparison remains affected by scale choice, target density, correlated levels, uncertainties, and physical-state mixing, it presently constitutes strong exploratory descriptive evidence, rather than a clearly demonstrated glueball mass law.

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