The Proton’s Internal Pressure

The elastic ratio that governs the Aether applies inside a proton as well as outside it. It yields a single pressure with nothing left to adjust, and the experiment that tests it is already running.

MemoTOE 2.4 — The Proton’s Internal Pressure

AuthorBrett Murrell

Versionv0.9 — draft

DateSeptember 2026

SeriesTOE — Theory of Everything

Categorieslife-science

The preceding memo showed that a lattice of identical particles bound by linear central forces has a bulk-to-shear modulus ratio of exactly 5/3. That ratio is dimensionless and applies wherever the medium is compressed, including inside a proton. Combined with the measured proton mass and charge radius it yields the proton’s internal pressure directly. No parameter is fitted and the void density cancels out of the result. Burkert, Elouadrhiri and Girod first measured this pressure and reported it only as exceeding that of a neutron star; the gravitational form-factor programme now under way will test the figure directly.

Draft. This memo is a working draft, published for scrutiny rather than as a settled result. Figures, derivations and conclusions are open to revision, and the ledger at the foot records what is derived, what is measured, what is assumed and what remains unanswered. Where a number is carried from a secondary source and not verified against the original, the sources section says so.

The TOE series supersedes the earlier A Classical Aether Model and Governor Atom Model white papers one memo at a time. Where this memo and those papers disagree, this memo is the later position.

1.004 × 10³⁵ PaPredicted pressure at the centre of a proton

5 : 3The elastic ratio that fixes it, from lattice geometry

0Parameters fitted to obtain the figure

11%How far the disputed proton radius moves the prediction

Proven in this memo

The void density cancels out of the prediction entirely — it depends only on the proton’s own measured mass, its measured radius, and the speed of light.
Force-law exponents Γ ≥ 2, including the Lennard-Jones 1/r¹² wall, are excluded by the proton’s measured internal pressure.

The pressure figure itself is a prediction, not a proof. It follows from memo 1.3 given the assumptions in the ledger, and it stands or falls on measurement.

In plain terms

A proton is not a solid lump. It is a knot in the Aether — a place where the medium has been squeezed to enormous density. Squeeze any material and it pushes back, and how hard it pushes back is fixed by its stiffness.

The preceding memo showed that the Aether’s stiffness against squeezing is locked to its stiffness against twisting, at exactly five to three, by the geometry of how it is built. We already know the twisting stiffness, because that is what sets the speed of light. So the squeezing stiffness is known too — and that tells us how hard the Aether pushes back inside a proton.

The answer is a hundred thousand billion billion billion billion pascals. Nothing was adjusted to get it. And in 2018 an experiment measured that pressure for the first time, saying only that it was stronger than the inside of a neutron star. Better measurements are being made now.

1. What is inherited

Memo 1.3 established that an isotropic lattice with linear central forces between identical particles satisfies the Cauchy relation, and that this forces the ratio of its two elastic moduli:

K / G = 5 / 3(1)

Nothing in that derivation refers to a length, a density or an energy. It is a pure number, fixed by the symmetry of the lattice and the nature of the force between its particles. It therefore holds wherever the medium is found, at any density, including inside matter.

Memo 1.3 also established that the shear modulus is the density times the speed of light squared, because light is a transverse wave in the lattice:

G = ρ c²(2)

2. Applying it to the proton

A proton is a region where the Aether is compressed to the density of nuclear matter (memo 2.1). The bulk modulus governs how hard the medium pushes back against that compression, so the internal pressure is set by K. Combining (1) and (2):

P = K = (5/3) ρ c²(3)

Written in terms of the proton itself, the prediction is that the pressure inside a proton is five-thirds of its own rest-energy density. The inputs are its mass and its radius, both measured to high precision.

Input	Value	Source
Proton mass	1.67262192369 × 10⁻²⁷ kg	CODATA 2018
Proton charge radius	0.8414 fm	CODATA 2018
Proton volume	2.495 × 10⁻⁴⁵ m³	(4/3)πr³
Proton density	6.704 × 10¹⁷ kg/m³	mass / volume
Rest-energy density	6.025 × 10³⁴ J/m³	ρc²
Predicted pressure	1.004 × 10³⁵ Pa	(5/3) × the above

For scale, the centre of the Sun sits at about 2.3 × 10¹⁶ Pa. The prediction is some nineteen orders of magnitude above that, and comparable to the core of a neutron star.

The part that makes this worth staking on

Most of this framework rests on one shaky number — how dense the Aether is out in empty space. It is inferred rather than measured, and if it is wrong, a lot goes with it.

This prediction does not use it. The void density appears on both sides of the calculation and divides straight out. What is left is the proton’s own measured weight, its own measured size, and the speed of light — three of the best-known numbers in physics.

So the prediction is not hostage to the weakest assumption in the model. It tests one thing only: whether the medium really is held together by simple springs pulling straight between identical particles.

3. The void density cancels

This is the feature that makes the prediction worth staking the framework on. Rearranging (3):

K / G = P_proton / (ρ_proton c²)(4)

Only the proton’s own measured pressure, its own measured density, and the speed of light appear. The density of the void — the least certain quantity in the whole framework, and the one carried in from the neutron mass excess — is absent. It divides out of both sides.

So the prediction does not depend on the Aether’s absolute scale being right. If the void density is wrong by an order of magnitude, equation (3) is unaffected. What the prediction does depend on is the single structural claim that the lattice is held by linear central forces, which is exactly what should be under test.

ProofThe void density cancels

The prediction is P = (5/3)ρ_protonc², where ρ_proton is the proton’s mass divided by its volume. Both are measured directly. The void density ρ_void appears nowhere on the right-hand side.

The reason is that 5/3 is a ratio of two moduli, and both moduli scale with whatever density the medium happens to have. Forming the ratio removes it. Equation (4) makes this explicit: K/G = P_proton/(ρ_protonc²), with only proton quantities on the right.

Consequence: the least secure number in the framework — the Aether’s absolute density, inferred rather than measured — cannot affect this result. It could be wrong by orders of magnitude and the prediction would not move.

Chain: equation (3) → equation (4). Algebraic, with no physical assumption beyond memo 1.3.

4. It also discriminates between force laws

Equation (1) assumes a linear spring. A different force law gives a different exponent in the pressure–density relation: for a pair potential falling as r^−m the lattice pressure rises as ρ^Γ with Γ = 1 + m/3. Reaching 10³⁵ Pa at the compression inside a proton then requires a void bulk modulus that depends strongly on Γ:

Γ	Force law	Implied void bulk modulus
1.00	linear spring	1.35 × 10¹⁷ Pa	plausible
1.33	1/r² force	2.3 × 10¹¹ Pa	plausible
2.00	1/r⁴ force	0.37 Pa	excluded
4.67	1/r¹² wall	1.7 × 10⁻⁴⁸ Pa	excluded

ProofSteep force laws are excluded

For a pair potential falling as r^−m, lattice pressure rises as ρ^Γ with Γ = 1 + m/3. Requiring the pressure to reach 10³⁵ Pa at the proton’s compression then fixes what the void bulk modulus would have to be.

For Γ = 2 that value is 0.37 Pa — softer than air. For the Lennard-Jones wall at Γ = 4.67 it is 1.7 × 10⁻⁴⁸ Pa. Neither is a medium that could carry light at 3 × 10⁸ m/s, since c = √(G/ρ) requires a modulus of order 10¹⁶ Pa.

This is a proof by contradiction rather than a preference. Steep repulsive laws reach nuclear pressures at far lower compression, so they cannot also produce a medium stiff enough for light.

Chain: the measured proton pressure → the implied void modulus for each Γ → contradiction with c = √(G/ρ).

The Lennard-Jones 1/r¹² wall, widely used in molecular dynamics, would have reached the proton’s pressure at vastly lower compression. It is excluded here by forty-eight orders of magnitude. That exponent never had a physical justification in any case — it was chosen in 1924 because r⁻¹² is the square of r⁻⁶ and so was cheap to compute by hand.

The linear spring survives, and it is the form that equation (1) requires. The two results support each other.

5. Against measurement

The pressure distribution inside the proton was extracted from deeply virtual Compton scattering by Burkert, Elouadrhiri and Girod, published in Nature in 2018.^[1] They found a strong repulsive pressure near the centre, stated as exceeding the pressure inside a neutron star, with a binding pressure at greater radius.

That paper reports the distribution rather than a single central figure, and its measurability was challenged by Kumerički the following year.^[2] The programme has since advanced: Duran and colleagues reported gluonic gravitational form factors in 2023,^[3] and the field is reviewed in Reviews of Modern Physics.^[4]

The commonly quoted figure of order 10³⁵ Pa is consistent with the prediction. It is not a confirmation: the measurement is not yet precise enough to distinguish 1.0 × 10³⁵ from 1.5 × 10³⁵, and the two are different claims.

How to tell if this is wrong

There is no knob to turn. If the measured pressure at the centre of a proton comes in outside roughly 0.9 to 1.1 × 10³⁵ pascals, the idea that the Aether is a simple spring lattice is wrong, and everything built on it in this series goes with it.

The main wobble is not in the theory but in the proton’s size, which is itself disputed — the two best measurements of the proton’s radius disagree, and that moves the prediction by about eleven per cent. So the experiment has to beat eleven per cent to settle it.

6. How this is falsified

The prediction is a single number with no free parameters, so it fails cleanly.

A measured central pressure outside roughly 0.9 to 1.1 × 10³⁵ Pa falsifies the linear central-force lattice. There is no parameter available to absorb the difference.
The proton radius is the main sensitivity. Using the muonic-hydrogen value of 0.8409 fm gives 1.01 × 10³⁵ Pa; the older electron-scattering value of 0.8751 fm gives 8.93 × 10³⁴. The proton radius puzzle therefore moves the prediction by about 11 per cent, which sets the precision the measurement must beat to be decisive.
The prediction is for the pressure at the centre, where the medium is compressed. It says nothing about the binding pressure at larger radius, and should not be compared against a volume-averaged figure.

If the number holds, the case for the framework is not that it accommodates a measurement but that it determined one in advance from a structural argument about the medium.

7. Sources

Reference codes read source.work.passage and resolve on the Master Source Register, which carries every source used across this series.

V. D. Burkert, L. Elouadrhiri & F. X. Girod. The pressure distribution inside the proton, Nature, 557 (2018) 396–399. 80
K. Kumerički. Measurability of pressure inside the proton, Nature, 570 (2019) E1. 81
B. Duran et al. Determining the gluonic gravitational form factors of the proton, Nature, 615 (2023) 813–816. 82
V. D. Burkert, L. Elouadrhiri, F. X. Girod, C. Lorcé, P. Schweitzer & P. E. Shanahan. Colloquium: Gravitational form factors of the proton, Reviews of Modern Physics, 95 (2023) 041002. 83
E. Tiesinga, P. J. Mohr, D. B. Newell & B. N. Taylor. CODATA recommended values of the fundamental physical constants: 2018, Reviews of Modern Physics, 93 (2021) 025010. Source of the proton mass and charge radius. 70
R. Pohl et al. The size of the proton, Nature, 466 (2010) 213–216. The muonic-hydrogen radius used in the sensitivity check in section 6. 84

Volume and page details should be checked against the originals before formal publication.

Ledger — memo 2.4

Derived

P_proton = (5/3) ρ_proton c² = 1.004 × 10³⁵ Pa
That the void density cancels from the prediction entirely (section 3)
That force-law exponents of 2.00 and above are excluded by the proton’s measured pressure (section 4)

Measured

Proton mass, 1.67262192369 × 10⁻²⁷ kg (CODATA 2018)
Proton charge radius, 0.8414 fm (CODATA 2018); 0.8409 fm muonic, 0.8751 fm older electron scattering
Speed of light (defined)
Proton internal pressure, reported as exceeding neutron-star pressure (Burkert et al. 2018)

Assumed

K/G = 5/3 from memo 1.3, which rests on linear central forces between identical particles and on isotropy. This is the whole of the prediction’s content; if it fails, the number fails with it.
That the proton’s charge radius is also the radius of the compressed region. The proton’s mass radius is measured smaller than its charge radius — around 0.55 fm from J/ψ photoproduction. Using a smaller radius would raise the predicted pressure. This memo uses the charge radius, and that choice is not derived.
That the medium inside a proton is the same lattice, with the same force law, as the medium outside it.

Open

The measurement is not yet precise enough to be decisive. It is consistent with the prediction but does not confirm it
The proton radius puzzle moves the prediction by about 11 per cent, which the measurement must beat
The micropolar caveat carried over from memo 1.3: light strictly travels at √((μ + κ)/ρ), and equation (2) identifies G with μ alone. This holds only if κ is small compared with μ, which has not been computed
Whether to use the charge radius or the mass radius (see Assumed) is unresolved and changes the number

Prior art

Burkert, Elouadrhiri & Girod, 2018 — the first extraction of the pressure distribution inside the proton, and the measurement this prediction is aimed at
Kumerički, 2019 — the challenge to that measurement’s interpretation, cited here because the prediction should not rest on a contested figure without saying so
The prediction itself is not known to have been made elsewhere

Cite this memo

Murrell, B. (2026). The Proton’s Internal Pressure. Moral’s Mark — TOE Memo Series, Part 2 memo 4, v0.9 draft (September 2026). https://moralsmark.com/memo-toe-2-4-proton-pressure