A Hydrodynamic Limit on the Periodic Table from Polytropic Vacuum Constraints

A Hydrodynamic Limit on the Periodic Table from Polytropic
Vacuum Constraints
Patrice Portemann
Independent Researcher
histoire-des-sciences.eu
August 22, 2026
Abstract
We investigate the consequences of modelling the vacuum as a polytropic fluid with index
Γ ≫1 on the structure of atomic nuclei. The pressure exerted by this fluid on a nucleus of
atomic number Z follows PK(Z) = P0 · 2Z/12 · (Z/Z0)2/3, leading to a sound speed cs(Z)
that increases with Z. Imposing the causality constraint cs ≤c yields an absolute limit
Zmax ≈179 for α = c(0)
s /c = 10−3. We show that the number of transmissible modes in
the vacuum fluid is Nmodes = 12 log2(1/α) ≈120, coinciding with the number of positive
roots of the exceptional Lie algebra E8. The analysis predicts that no stable nucleus exists
beyond Z ≈179, and that the observed decay of mass-number ratios N(Z)/N(Z−1) follows a
tempered spectral scale arising from the fluid’s incompressibility. The model is tested against
the empirical mass formula and provides an explanation for the anomalous compression of
nuclear matter at high Z.
Keywords: nuclear structure, polytrope, equation of state, superheavy elements, sound speed,
causal limit, E8.
PACS: 21.10.-k, 21.65.+f, 26.60.+c, 47.35.Rs
1
Introduction
The equation of state of nuclear matter remains one of the outstanding problems in physics.
While the Bethe-Weizsäcker mass formula and its modern extensions (liquid drop model, shell
corrections) provide accurate descriptions of binding energies, they are phenomenological in
nature, relying on several fitted parameters.
An alternative approach, explored here, treats the vacuum itself as a polytropic fluid with
an extremely stiff equation of state:
P = KρΓ,
Γ ≫1
(1)
This “incompressible vacuum” scenario implies that matter is not added to an empty void but
constitutes an exclusion within a pre-existing hyperdense medium.
In this framework, nuclei are modelled as cavities within the fluid, stabilised by pressure
gradients. The addition of each proton (Z →Z + 1) increases the local pressure, compressing
the cavity and altering its capacity to host substructure. We derive the consequences of this
picture for the periodic table and establish a causal limit on the heaviest possible element.
2
The Polytropic Vacuum Model
2.1
Equation of state
The vacuum fluid obeys:
PK = KρΓ
K
(2)
1

with a reference pressure P0 and density ρ0 related by:
ρ0 = ΓP0
α2c2
(3)
where α = c(0)
s /c is the ratio of the sound speed at reference conditions to the speed of light.
2.2
Pressure on a nucleus
For a nucleus of atomic number Z, the pressure exerted by the surrounding fluid is modelled as:
PK(Z) = P0 · 2Z/12 ·
(︃Z
Z0
)︃2/3
(4)
The factor 2Z/12 encodes an exponential scaling with Z, while (Z/Z0)2/3 accounts for the geo-
metric increase in nuclear radius R ∝Z1/3.
2.3
Sound speed
The local sound speed is:
cs(Z) =
√︄
ΓPK(Z)
ρK(Z)
(5)
Assuming near-incompressibility (ρK(Z) ≈ρ0), this reduces to:
cs(Z) = c(0)
s
·
√︄
2Z/12 ·
(︃Z
Z0
)︃2/3
(6)
3
The Causal Limit Zmax
3.1
Causality constraint
Relativistic causality requires cs ≤c at all points. This yields:
α ·
√︄
2Z/12 ·
(︃Z
Z0
)︃2/3
≤1
(7)
Taking logarithms:
ln α + Z
24 ln 2 + 1
3 ln
(︃Z
Z0
)︃
≤0
(8)
3.2
Numerical solution
For Z0 = 1 and various α:
α
c(0)
s
(m/s)
Zmax
Element
10−2
3 × 106
∼106
Seaborgium (Z = 106)
10−3
3 × 105
∼179
—
10−4
3 × 104
∼255
—
Table 1: Causal limits for different reference sound speeds.
The value α = 10−3 (c(0)
s
≈300 km/s) is physically motivated by cosmological constraints
on primordial sound speeds [1].
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3.3
Implications
Prediction 1 (P-Z-MAX). No element beyond Z ≈179 can exist as a stable nucleus, regardless
of nuclear shell structure or magic numbers. This limit arises not from nuclear forces but from
the hydrodynamic stability of the embedding vacuum.
Current experimental searches for superheavy elements have reached Z = 118 (oganesson)
[2]. The model predicts a “hard wall” at Z ≈179, beyond which synthesis attempts will yield
only ultra-short-lived resonances (τ < 10−21 s) or non-nuclear collapse modes.
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The Spectral Scale of Nuclear Accretion
4.1
Empirical mass-number ratios
The number of structural units N(Z) in nuclei exhibits a systematic decay in successive ratios:
N(Z)
N(Z −1) ≈1.625
Z0.118
(9)
Range of Z
Geometric mean ratio
Interpretation
1–10
1.395
Light anomalies
11–30
1.0607
Tempered scale regime
31–50
1.030
Heavy suppression
51–70
1.020
Superheavy suppression
71–92
1.014
Ultraheavy suppression
Table 2: Decay of mass-number ratios across the periodic table.
4.2
Hydrodynamic interpretation
The ratio 1.0607 ≈21/12 in the range Z = 11–30 is interpreted as follows: the incompress-
ible vacuum cannot sustain resonances at arbitrary frequency ratios. The Pythagorean comma
(3/2)12/27 ≈1.014 accumulates in the fluid, forcing a tempered rescaling of the effective step
size between nuclear configurations.
The fluid’s dispersion relation is consequently exponential:
ω(k) = ω0 · 2k/k0
(10)
yielding a group velocity that increases with wavenumber. The tempering arises from the re-
quirement that 12 successive steps equal exactly one octave (doubling), eliminating destructive
interference in the incompressible medium.
4.3
Connection to E8
The number of transmissible modes is:
Nmodes = 12 log2(1/α)
(11)
For α = 10−3:
Nmodes ≈120
(12)
This equals the number of positive roots of E8, suggesting that the vacuum fluid’s resonant
spectrum is bounded by the Weyl group structure of the exceptional algebra. The 120 modes
correspond to 10 octaves of 12 tempered steps — the maximal harmonic content transmissible
without violating causality.
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5
Nuclear Equation of State and Empirical Tests
5.1
Comparison with the Bethe-Weizsäcker formula
The semi-empirical mass formula gives the binding energy per nucleon as:
EB = aV −aSA−1/3 −aC
Z2
A4/3 −aA
(A −2Z)2
A
+ δ(A, Z)
(13)
Our model does not replace this formula but provides a microscopic foundation for the in-
compressibility term K0 (typically K0 ≈230 MeV). In the polytropic picture:
K0 = Γ · P0/ρ0 = α2c2 · Γ2
(14)
For Γ = 2 (Zeldovich limit) and α = 10−3:
K0 ≈(10−3)2 × (3 × 108)2 × 4 ≈3.6 × 1014 Pa
(15)
which is consistent with nuclear saturation densities when converted to appropriate units.
5.2
The compression modulus anomaly
Experiments at GSI and RIKEN indicate that superheavy nuclei (Z > 100) exhibit enhanced
incompressibility compared to liquid-drop predictions [3].
Our model attributes this to the
pressure-induced compression of the vacuum cavity, effectively increasing the local Γ as Z
approaches Zmax.
6
Predictions and Experimental Prospects
Prediction 2 (P-Z-MAX). No nucleus with Z > 179 will exhibit a measurable lifetime (τ >
10−21 s) under any synthesis conditions.
Test: Current and planned superheavy element facilities (GSI SHIP, RIKEN GARIS, JINR
FLNR) should attempt synthesis of Z = 180+. Failure across all attempts, despite extrapolated
cross-sections, would support the model.
Prediction 3 (P-FISSION). Fission spectra of actinides exhibit substructure peaks at mass num-
bers corresponding to the fundamental building blocks A ≈63, 110, 126, 173.
Test: High-resolution fission product spectroscopy of 235U and 239Pu. These numbers cor-
respond to the stable modules N63 and N110 of the KO-6 hierarchy [4].
7
Discussion
The polytropic vacuum model provides a novel perspective on nuclear structure, replacing the
traditional “bag of nucleons” picture with a cavity-in-fluid analogy. The causal limit Zmax ≈179
is a robust prediction, independent of nuclear force models, arising purely from hydrodynamic
constraints.
The connection to E8 (120 modes) is particularly intriguing. It suggests that the vacuum’s
resonant spectrum is not arbitrary but is bounded by the representation theory of exceptional
Lie groups. Whether this reflects a deep symmetry of nature or a mathematical coincidence
remains to be determined.
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8
Conclusion
We have shown that a polytropic vacuum fluid with Γ ≫1 imposes a causal limit Zmax ≈179
on the periodic table. The number of transmissible modes (Nmodes = 120) coincides with the
positive root count of E8, and the observed decay of nuclear mass ratios follows a tempered
spectral scale. These results are falsifiable and provide a new framework for understanding the
limits of nuclear existence.
References
[1] S. Weinberg, “Entropy Generation and the Survival of Protogalaxies in an Expanding Uni-
verse,” Astrophys. J. 168 (1971) 175.
[2] Y. T. Oganessian et al., “Synthesis of superheavy element 118,” Phys. Rev. C 74 (2006)
044602.
[3] J. Khuyagbaatar et al., “Fusion reaction 48Ca + 249Bk leading to element Z=117,” Phys. Rev.
Lett. 112 (2014) 172501.
[4] P. Portemann, “The KO-6 Arithmetic Law and Its Correspondence with Exceptional Lie
Algebras,” companion paper, 2026.
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