Connecting gravity and strong interactions

Piotr Zenczykowski

REVIEW 4 major objections 4 minor 18 references

Connecting gravity and strong interactions

T0 review · 4 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read This paper claims that the MOND acceleration scale and hadronic mass-radius ratio are the same combination of constants, about 1 g/cm^2, suggesting space begins from hadronic 2D subspaces.

desk verdict The paper's central numerical coincidence is a tautology, so the gravity–strong force connection rests on nothing independent; it's a clear but unsupported speculative essay. read the letter →

arxiv 2412.00096 v1 pith:AW2EXWVB submitted 2024-11-27 physics.gen-ph

Piotr Zenczykowski This is my paper · ORCID

classification physics.gen-ph

keywords gravityMONDstronginteractionsemergentspacedimensionalanalysishadronicscalecosmologicalconstant

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper argues that the constants governing hadrons also fix the acceleration scale at which galaxies appear to leave Newtonian gravity. Its central evidence is equation (10): the MOND acceleration $a_M$ divided by Newton's constant $G$ is numerically the same, to within a factor of a few, as hadron mass $m_H$ divided by hadron radius squared $r_H^2$, both quantities being of order $1$ g/cm$^2$. The author takes this as a real physical signal rather than a coincidence, and concludes that the classical 3D space of Newtonian gravity is built out of physical 2D subspaces whose properties are set by strong-interaction scales. If the paper is right, the first signs of space's emergence should be looked for at hadronic distances near $10^{-12}$ cm, not at the Planck length.

What carries the argument

The load-bearing identity is equation (10), $a_M/G = m_H/r_H^2$, with $a_M$ the MOND acceleration scale ($\approx 1.2\times10^{-8}$ cm/s$^2$), $G$ Newton's constant, $m_H$ the hadronic mass scale ($\approx10^{-24}$ g), and $r_H$ the hadronic radius scale ($\approx10^{-12}$ cm). It is derived by equating the MOND transition radius $r_M=\sqrt{G/a_M}\,\sqrt{m}$ with the Compton wavelength $r_Q=h/(mc)$, and it connects the gravitational bound on acceleration to the quantum–relativistic bound on localization. The identity carries the interpretive load of the paper: because both sides have units of mass per area, it suggests a planar mass density shared by hadrons and by gravity, and it motivates reading MOND as gravity in two dimensions with 3D Newtonian gravity assembled from 2D pieces.

What would settle it

Use the electron-scattering proton charge radius, about $0.84\times10^{-13}$ cm, in place of $r_H=10^{-12}$ cm: then $m_p/r_p^2$ is roughly $2\times10^2$ g/cm$^2$, two orders of magnitude above $a_M/G\approx0.18$ g/cm$^2$, so the claimed equality disappears. A precise independent determination of $a_M$ from rotation-curve data that puts $a_M/G$ outside the $0.1$–$1$ g/cm$^2$ range would also refute the relation.

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Extended reading notes

Core claim

The author claims that experiment-supported dimensional analysis, using the constants $h$, $G$, and $\Lambda$ while leaving $c$ out of the mass–length construction, selects the hadronic scale $m_H\approx10^{-24}$ g and $r_H\approx10^{-12}$ cm as the physically meaningful quantization scale for space. The Planck and Wesson scales are then artifacts of extrapolating quantum theory beyond its tested range. When the MOND acceleration $a_M$ is used as the gravitational bound, the same scales are recovered through $m=((h/c)^2 a_M/G)^{1/3}$ and $r=(G/a_M \cdot h/c)^{1/3}$. Equation (10), $a_M/G=m_H/r_H^2$ to within an order of magnitude, equates a purely gravitational ratio with a purely hadronic ratio, and the author concludes that this equality is not a coincidence: MOND's two-dimensional behavior (a two-dimensional Gauss law) composes into the 3D Newtonian gravity and macroscopic space.

Load-bearing premise

The argument stands on the assumption that the order-of-magnitude equality between $a_M/G$ and $m_H/r_H^2$ is a real physical signal, not a consequence of choosing $r_H=10^{-12}$ cm and $m_H$ from the same constants that define $a_M/G$.

Editorial extensions

If this is right

The hadronic scale, not the Planck scale, would be the natural place where the classical concept of space starts to break down.
MOND's low-acceleration regime would be understood as two-dimensional gravity with a two-dimensional Gauss law, while ordinary Newtonian gravity is the three-dimensional composition of those 2D pieces.
The Planck and Wesson mass and length scales would be artifacts of applying quantum theory beyond its tested domain.
A precision measurement of hadron size near $10^{-12}$ cm would become a test of a gravitational relation, linking particle-physics experiments to galaxy dynamics.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The equality can be inverted to make a prediction that the paper does not: $a_M = G\,m_H/r_H^2$, which would fix the MOND scale from hadron parameters rather than from galaxy rotation curves.
If the two-dimensional reading is literal, the hadronic planar density of about $1$ g/cm$^2$ might be connected to galaxy disk surface densities, a comparison the paper leaves implicit.
The choice of $r_H=10^{-12}$ cm is discriminating: using the electron-scattering proton charge radius of about $0.84$ fm would break the equality, so the relevant hadronic radius would have to be something larger than the proton itself (for example a pion cloud or a two-pion system).

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

I'll cut to the chase: the central 'stunning' coincidence in Eq. (10) is an algebraic identity. Since m_H and r_H are defined by solving r_M = r_Q, the ratio m_H/r_H^2 equals aM/G exactly. Presenting that as an experimental match is misleading. The paper's only real numerical content is that the resulting mass is within an order of magnitude of the proton mass, but the radius is ~10^-12 cm, about an order of magnitude larger than the measured proton charge radius. If you plug in measured values, m_p/r_p^2 ~ 10^2 g/cm^2, three orders of magnitude off. So the abstract's claim about extracting information on the emergence of space has no quantitative basis.

What the paper does well: it's clearly written, and the dimensional analysis is straightforward and correctly executed. The observation that the (h,G,Λ) triple lands on hadronic scales is a well-known numerology, and the MOND-based variant is a nice way to see the same thing. The author is candid that the 2D-subspace speculation is a tendency, not a prediction. Anyone unfamiliar with the older papers [1,2] will get a readable summary.

The soft spots are the usual ones for this genre. The argument depends on choosing aM, which is fitted from galactic rotation curves, and then using derived quantities as if they were measured hadron properties. There is no mechanism, no new prediction, and no way to falsify the 2D-space claim. The novelty is low: this is a repackaging of prior work with the MOND route emphasized. Eq. (9) is also a loose numerical coincidence with a factor of ~8, not a precise relation.

Who is this for? A reader interested in speculative foundations of physics or in how dimensional analysis can be abused. It might make a lively reading-group discussion about circular reasoning. But as a research paper it doesn't carry evidential weight. I would not send it to a rigorous physics journal for full review; if it's submitted to a venue that explicitly publishes speculative essays, one referee could handle it in a few hours and write the identity in a paragraph. My advice: do not put it through a standard peer review process expecting substantive revision; the flaw is load-bearing and structural.

Referee Report

4 major / 4 minor

Summary. The paper argues from dimensional analysis that the relevant mass and distance scales for the interface between gravity, quantum theory, and the emergence of space are hadronic scales rather than Planck scales. It derives m_H ≈ 10^-24 g and r_H ≈ 10^-12 cm from the constants h, G, Λ, and/or the MOND acceleration scale aM, then presents Eq. (10), aM/G = m_H/r_H^2, as a 'stunning' coincidence linking gravitational and strong-interaction properties. The paper further speculates that Newtonian gravity and 3D space emerge from physical 2D subspaces. The central quantitative claim is Eq. (10), and the conclusion depends on its being a genuine numerical coincidence.

Significance. If the relation in Eq. (10) were an independent, numerically accurate comparison between gravitational and hadronic observables, it would constitute a remarkable hint about the connection between gravity and strong interactions. The paper is clearly written and the dimensional arithmetic is transparent. However, the coincidence is manufactured by the definitions of m_H and r_H, and the claimed agreement with measured hadronic properties fails when the measured proton mass and radius are used. The paper therefore offers no independent quantitative support for its speculative conclusion, despite the honest presentation of the unorthodox character of the idea.

major comments (4)

[§4, Eqs. (7)–(10)] The central quantitative claim, Eq. (10), is an identity rather than a numerical coincidence. Substituting m_H = ((h/c)^2 aM/G)^{1/3} and r_H = (h c G/aM)^{1/3} from Eqs. (7) and (8) into m_H/r_H^2 gives exactly aM/G. Thus Eq. (10) does not compare two independently measured ratios; it restates the definitions used to construct m_H and r_H. The 'stunning' agreement therefore carries no evidential weight for a gravity–strong-interaction connection.
[§4, Eq. (10) and §2] The claimed agreement with real hadronic properties disappears when measured values are used. The text sets r_H ≈ 10^-12 cm, but the measured proton charge radius is about 0.84 × 10^-13 cm. Using m_p ≈ 1.67 × 10^-24 g and r_p ≈ 0.84 × 10^-13 cm gives m_p/r_p^2 ≈ 2 × 10^2 g/cm^2, roughly three orders of magnitude larger than aM/G ≈ 0.18 g/cm^2. The 'hadronic radius' used in the paper is a derived quantity, not an experimentally measured hadronic size, so Eq. (10) is not an observationally supported relation.
[§2, choice #3] The 'nonrelativistic/hadronic scale' is not actually obtained from h, G, and Λ alone, as stated. The expression for m_H = (h^2/G × sqrt(Λ/3))^{1/3} has the correct mass dimension, but the length r_H = h/(m_H c) reintroduces c, which the text says is not used. No length scale can be formed from h, G, and Λ alone, so the claimed distinction between choice #3 and the Planck choice is not as clean as presented. This weakens the argument that the hadronic scale is selected independently of c.
[§4, Eq. (9)] The 'well-known coincidence' c^2√Λ ≈ 8.2 aM is used to connect the Λ-based derivation of the hadronic mass scale to the MOND-based derivation, but the relation is not derived or explained. If it is treated as an input, then the agreement between m_H from choice #3 and the MOND-based mass is not independent evidence; if it is intended to be a prediction of the proposed framework, no mechanism is provided. This is a load-bearing step in the argument and requires at least a concrete derivation or a quantitative test.

minor comments (4)

[§4, Eq. (10)] The notation '10^-12*2 g/cm^2' is ambiguous; it should read 10^{-24} g/cm^2 or the ratio m_H/r_H^2 should be written explicitly.
[Throughout] The term 'hadronic radius' is used without a clear operational definition; the proton charge radius, the pion Compton wavelength, and the strong-interaction cross-section radius differ by an order of magnitude. Please specify which observable is meant and why that particular definition is relevant.
[Abstract and §4] The phrase 'experiment-supported dimensional analysis' is misleading: the only experimental input is the MOND parameter aM, whose status is phenomenological, and the hadronic scales are not independently measured before being inserted into Eq. (10).
[References] Reference [2] lists both a journal citation and an arXiv identifier; please ensure the journal volume and page range are complete and consistent with the arXiv version.

Circularity Check

1 steps flagged · score 8.0 of 10

Eq. (10) is an identity: m_H and r_H are defined from aM/G, so m_H/r_H^2 = aM/G by construction.

self definitional [Section 4, Eqs. (7), (8), and (10)]
"Solving Eqs (2,6) for m by putting rM = rQ = r gives m = ((h/c)^2 aM/G)^{1/3} ... and radius r = (G/aM * h/c)^{1/3} = 10^{-12} cm ≈ rH. ... Formula (10) ... aM/G = mH/rH^2 ≈ 1 g/cm^2."

Eqs. (7) and (8) define mH and rH by requiring the MOND transition radius rM = sqrt(G/aM)*sqrt(m) to equal the Compton wavelength h/(mc). Under that constraint, mH/rH^2 = [((h/c)^2 aM/G)^{1/3}] / [((G/aM)(h/c))^{2/3}] = aM/G identically. Thus Eq. (10) is not a comparison of two independently measured quantities; it is an algebraic rearrangement of the input aM/G. The claimed 'stunning' equality is forced by definition. The only remaining empirical content is the rough proximity of mH to the proton mass; using the measured proton radius (~0.84e-13 cm) instead of the derived rH=10^-12 cm gives m_p/r_p^2 ~ 2e2 g/cm^2, which is orders of magnitude away from aM/G = 0.18 g/cm^2.

full rationale

The dimensional calculation producing mH from aM/G, h, and c is internally self-contained, and the numerical coincidence between the derived mass scale and the proton mass is a genuine empirical observation. However, the paper's central quantitative claim, Eq. (10), equates aM/G with mH/rH^2, where mH and rH were themselves chosen by solving the equation rM = rQ. Substituting Eqs. (7) and (8) into mH/rH^2 yields aM/G exactly, so Eq. (10) is a tautology rather than a measured cross-domain connection. The self-citations [1] and [2] motivate the program but do not carry the load of this calculation; they are not the source of circularity. Because the main 'prediction' reduces by construction to its own input, the circularity score is high, though some independent numerical coincidences remain in the paper.

Assumptions & free parameters 3 free parameters · 5 assumptions · 1 invented entities

The central chain runs: pick aM (fitted) and the (h, G, Lambda) triple (selected for matching hadron data), derive m_H and r_H from aM, then celebrate that m_H/r_H^2 equals aM/G. The only external input is the loose resemblance of the derived scale to real proton properties, and that resemblance worsens if the measured proton radius is used.

free parameters (3)

aM (MOND acceleration scale) = 1.2 x 10^-8 cm/s^2
Fitted to galaxy rotation curves and used in Eqs (7), (8), (10) to generate the hadronic mass and radius. The paper's 'prediction' of the hadron scale is this parameter expressed in different units.
Hadronic radius r_H used in Eq (10) = 10^-12 cm
The paper uses its dimensionally derived radius in the m_H/r_H^2 comparison instead of the measured proton radius of about 0.84 x 10^-13 cm. This choice changes the ratio from ~0.18 g/cm^2 to ~200 g/cm^2, and is essential for the claimed coincidence.
Choice of constant triple (h, G, Lambda) = N/A
The hadronic triple is selected among the four constants because it reproduces proton data; Planck and Wesson triples are dismissed as untestable. This is a post hoc selection, not a derivation.

assumptions (5)

domain assumption MOND's formula a = sqrt(G m aM)/r describes real galactic dynamics without dark matter.
Invoked in Eq (2); MOND is a contested paradigm, and the paper does not defend it beyond citing supporters.
ad hoc to paper aM is a fundamental constant, not a phenomenological fit parameter.
Needed for the dimensional analysis generating hadronic scales; no independent justification is offered.
ad hoc to paper The relation c^2 sqrt(Lambda) is approximately 8.2 aM is physically meaningful.
Used in Eq (9) to connect the hadronic scale to MOND; the paper does not explain why this numerical near-equality should be more than coincidence.
ad hoc to paper Lambda is a relevant fundamental constant at hadronic scales.
Choice 3 uses h, G, and Lambda without c to obtain the hadronic scale; no physical reason links the cosmological constant to 10^-12 cm.
domain assumption Dimensional analysis with a hand-chosen constant set is a valid guide to physical scale.
The entire Section 2 relies on this heuristic.

invented entities (1)

Physical 2D subspaces composing 3D space
purpose: Explains the 'planar mass density' coincidence and claims low-acceleration gravity is 2D while Newtonian 3D gravity is built from these subspaces.
No mechanism or prediction is given; it is a speculation inferred from the dimensional-analysis coincidence.

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Cite this review

Pith. "Pith review of Connecting gravity and strong interactions." pith.science (2026). https://pith.science/paper/AW2EXWVB

@misc{pith2026241200096,
  author       = {Pith},
  title        = {Pith review of: Connecting gravity and strong interactions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AW2EXWVB}},
  note         = {Machine review of arXiv:2412.00096}
}

read the original abstract

We use experiment-supported dimensional analysis to further bolster our arguments that crucial information on the emergence and/or nature of space could be extracted from the combination of the properties of gravitational and strong interactions.

Discussion (0). Continue with ORCID to comment.

Reference graph

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