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REVIEW 4 major objections 3 minor 66 references

Effects of Quantum Spin-Connection Foam in the Solar System, Galaxies, and the Universe

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Pith's one-line read Quantum spin-connection foam is proposed as a single origin for MOND's acceleration scale and the cosmological constant.

desk verdict A new qMOND potential and a Milgrom-scale estimate are buried under a flawed derivation: the advertised MOND derivation fails at Eqs. (39)–(41). read the letter →

arxiv 2608.12404 v1 pith:YEWNM6PA submitted 2026-08-11 gr-qc astro-ph.GA

Igor V. Kanatchikov , Valery A. Kholodnyi This is my paper · ORCID
classification gr-qcastro-ph.GA MSC 83C4583F05 PACS 04.60.-m98.80.-k95.35.+d
keywords PrecanonicalquantumgravitySpinconnectionfoamMONDaccelerationscaleqCosmologicalconstantDarkmatterenergy
open problems Quantum Gravity
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 spin-connection foam—the picture of quantum gravity that emerges from precanonical quantization of general relativity, a covariant Hamiltonian quantization that treats space and time on equal footing—may already be visible in cosmic data. It claims that quantum fluctuations of the spin connection generate a fundamental acceleration scale $a_* = 8\pi G\hbar\kappa$, and that this same scale shows up as the acceleration scale of modified Newtonian dynamics (MOND) in galactic rotation curves, as a modification of classical gravitational dynamics at small accelerations, and, through $a_*^2$, as the cosmological constant. If the argument holds, the phenomena usually attributed to dark matter and dark energy would trace back to a single quantum-gravitational parameter. The paper also derives a specific MOND interpolating function and predicts small but potentially measurable deviations from classical gravity in the outer Solar System and in laboratory force sensors.

What carries the argument

The load-bearing object is the precanonical quantum-gravity wave function $\Psi(\omega,x)$ on the bundle of spin-connection coefficients over spacetime—precanonical quantization being a covariant Hamiltonian quantization that treats space and time on equal footing. In the nonrelativistic limit of the quantum version of empty flat spacetime, its ground state is a spherically symmetric, exponentially decaying profile $\Psi(\tilde\omega^i) \propto e^{-\omega/(8\pi G\hbar\kappa)}$ in the space of fluctuating spin-connection components; the width of this profile defines the invariant acceleration $a_* = 8\pi G\hbar\kappa$. The variance relation $\langle\tilde\omega^2\rangle = \bar a^2 = \frac12 a_*^2$ enters the squared geodesic equation and turns the classical inverse-square law into qMOND. The final step to MOND is carried by the mean-field redefinition $g = a - \bar a$ and by the derived interpolating function $\mu(x)$, which replaces MOND's usually ad hoc interpolation choice.

What would settle it

Substitute $g = a - \bar a$ into the paper's qMOND law $a=\sqrt{(GM/r^2)^2+\bar a^2}$. Squaring the claimed identity $GM/r^2 = g^2/(2\bar a)$ turns it into an equation for $g/\bar a$ that has only one isolated solution, so a direct algebraic check of equations (35), (40), and (41) decides whether the derivation of MOND is valid.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that a nonrelativistic test particle moving in the gravitational field of a point mass immersed in the quantum spin-connection foam obeys the modified law $a = \sqrt{(GM/r^2)^2 + \bar a^2}$, where $\bar a^2 = \langle \tilde\omega^2\rangle$ is the variance of the fluctuating spin-connection components. Re-expressing accelerations relative to the mean-field background $\bar a$ is then claimed to produce the deep-MOND relation $GM/r^2 = g^2/g_0$ with $g_0 = 2\bar a$, and comparing with the general MOND form yields the interpolating function $\mu(x) = \frac{1}{2x}(\sqrt{4x^2+1}-1)$. The scale $a_* = 8\pi G\hbar\kappa$ is estimated through the non-abelian gauge mass-gap relation $\kappa \sim (\Delta m)^3/(\hbar^4 g_s^2)$ to be $\sim 10^{-27}\,\mathrm{m}^{-1}$, numerically consistent with the MOND acceleration, and the relation $a_*^2/\lambda$ with $\lambda = 3$ gives a cosmological constant of order $10^{-52}\,\mathrm{m}^{-2}$.

Load-bearing premise

Everything rests on the assumption that the acceleration relative to the fluctuating background is the classical gravitational acceleration minus the mean-field fluctuation, $a-\bar a = GM/r^2$; the further algebraic rewrite that produces MOND does not follow from that assumption and holds only at one special acceleration.

Editorial extensions

If this is right

  • At galactic scales the modified law produces flat rotation curves over tens of kilolightyears, with the flat-region velocity tied to baryonic mass through the empirical baryonic mass–velocity relation, then a predicted upturn beyond roughly 100 kly.
  • In the Solar System the correction to the classical acceleration reaches about 1% near $3\times10^3$ au from the Sun, shifts the Earth-year period and the Sun–Earth equilibrium positions by $\sim10^{-9}\%$, and near a 1 kg mass at 10 cm yields an acceleration correction of $\sim10^{-12}\,\mathrm{m/s^2}$ that attonewton sensors could in principle detect.
  • The same acceleration scale fixes the cosmological constant through $a_*^2/\lambda$ with $\lambda=3$, giving $\Lambda\sim10^{-52}\,\mathrm{m^{-2}}$ without a separate dark-energy input.
  • The derived interpolating function is within about 12% of the simple interpolating function used in MOND phenomenology near $x\approx1.3$, so the predicted rotation curves stay close to standard MOND where MOND is already successful.
  • Because the modified potential deepens and lengthens gravitational wells, the paper expects faster matter clumping and a larger effective missing mass in clusters and large-scale structure than standard MOND or the standard cosmological model would give.

Reading between the lines

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

  • If one keeps the qMOND law but drops the disputed mean-field rewriting, the theory's distinctive observable signature becomes an upturn in rotation curves beyond $\sim100$ kly and a velocity minimum at $r_m=\sqrt{GM/\bar a}$; stacked galaxy-galaxy weak-lensing measurements at megaparsec scales can test this without resolving the derivation.
  • The same variance mechanism applied to two-point correlations rather than a single ground state would predict direction-dependent or stochastic corrections to planetary ephemerides, so existing spacecraft ranging data may already bound $\bar a$ more tightly than the outer-comet-cloud estimate.
  • The bridge between $\kappa$ and the non-abelian gauge mass gap turns the numerical estimate of $a_*$ into a sharp prediction: a lattice computation of the infrared gauge coupling and mass gap would confirm or rule out the claimed order of magnitude.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 3 minor

Summary. The paper claims that precanonical quantization of tetrad gravity produces a quantum spin-connection foam whose nonrelativistic mean-field effects modify Newtonian dynamics into Milgromian MOND, and that the same fundamental acceleration scale a_* = 8πGħκ also yields a realistic cosmological constant. The central derivation in Sec. 5 is intended to recover the deep-MOND law and a theoretically determined interpolating function from the qMOND law (35), while Sec. 6 estimates a_* from the QCD mass gap and discusses Solar System, galactic, and cosmological signatures. The abstract's claim that Milgromian MOND and its interpolating function are derived from precanonical quantum gravity is the load-bearing result of the manuscript.

Significance. If the derivations were correct, the paper would establish a striking connection between a specific quantum-gravity framework, the MOND acceleration scale, the cosmological constant, and galactic rotation-curve phenomenology, with falsifiable laboratory and Solar System predictions. The authors are transparent about many limitations, and the paper engages a concrete quantization scheme rather than a generic dimensional argument. However, the logical chain fails at the algebraic step connecting the qMOND law to Milgrom's law, and the proposed ground-state wave function in Sec. 4 does not satisfy the stated normalization. Because these are internal inconsistencies in the central derivation, the paper's main claims are unsupported in its present form.

major comments (4)
  1. [Sec. 5, Eqs. (39) and (41)] The transition from the deep-qMOND expansion to Milgrom's law is algebraically incorrect. Equation (39) gives a ≈ \bar a + G^2M^2/(2\bar a r^4) for \bar a ≫ GM/r^2. Defining g := a - \bar a yields GM/r^2 ≈ sqrt(2\bar a g) = sqrt(g_0 g), whereas Eq. (41) asserts GM/r^2 = g^2/(2\bar a) = g^2/g_0. These expressions coincide only at the single acceleration value g = 2\bar a. Consequently Eq. (42), the deep-MOND relation, and the identification g_0 = 2\bar a in Eq. (43) do not follow from the preceding formalism.
  2. [Sec. 5, Eq. (44)] The claimed derivation of the interpolating function is also based on an algebraic error. Solving Eq. (35) for the Newtonian acceleration a_N = GM/r^2 in terms of g := a - \bar a gives a_N = sqrt(g^2 + 2\bar a g), not the expression in Eq. (44), a_N = sqrt(g^2 + \bar a^2) - \bar a. The latter would instead be the correct expression for g in terms of a_N. Equation (44) thus swaps the roles of g and a_N, and the interpolating function in Eq. (46), obtained by comparing Eq. (44) with Eq. (45), is not a consequence of the qMOND law.
  3. [Sec. 4, Eqs. (25)-(30)] The ground-state wave function (26) does not satisfy the normalization condition (27) as written. Substituting Ψ ∝ ω^{-1/2} exp[-ω/(8πGħκ)] into ∫ d^3ω Ψ^2 gives 4πGħκ (i.e., a_*/2) rather than 1. Similarly, the expectation values in Eqs. (29) and (30) do not follow from (26); for example, the integral defining <ω^2> yields a quantity proportional to (8πGħκ)^3, not (1/2)(8πGħκ)^2. Since \bar a is defined through Eq. (29), the qMOND law (35) is not based on the stated quantum ground state.
  4. [Sec. 6, rotation curves] The paper explicitly acknowledges that Eq. (52) predicts v(r) growing as r^{1/2} at large radii and thus contradicts observations of flat rotation curves at megaparsec scales, citing Ref. 63. The authors assert that the rescaling argument of Sec. 5 resolves this, but that argument is precisely the step invalidated by the algebraic errors in Eqs. (41) and (44). The contradiction with observed flat rotation curves is therefore not resolved within the manuscript, and the claimed phenomenological compatibility with MOND is not established. The numerical matching of a_* to a_0 and Λ is also admitted to hold only to within several orders of magnitude, with a_* fixed by the chosen hadronic mass gap via Eq. (47), so it does not provide an independent quantitative confirmation.
minor comments (3)
  1. [Eq. (16)] The notation 'the operator of e^{-6} = det(e^I_\mu)6' is confusing; the determinant operator and the exponent should be written out explicitly or defined in words.
  2. [Sec. 3, Eqs. (18)-(21)] The statement that Eq. (18) 'implies' the separate fiber and base equations (20) and (21) is abrupt; this appears to be a separation-of-variables ansatz for the modes, and it should be stated as such.
  3. [Sec. 6, Eq. (23)] The value λ = 3 for the Weyl-ordered operator in Eq. (14) is quoted without derivation or reference to the calculation; since the cosmological-constant estimate depends on it, the ordering calculation should be shown or cited in detail.

Circularity Check

2 steps flagged · score 6.0 of 10

Deep-MOND relation is asserted via Eq. (41) after an algebraically invalid rewrite of Eq. (39), and the numerical scale rests on the authors' own rough spectral estimate.

  1. self definitional [Sec. 5, Eqs. (40)-(43)]
    "In this reference system, the standard Newton’s law (for the absolute values) reads a−¯a= GM/r2 . (40) We can take this equation as a redefinition of the right hand side in terms of the left hand side, which is based on non-relativistic experiments at normal accelerations a≫¯a, and then try to extrapolate its validity (i.e. the definition of quantities which is based on the domain of classical Newtonian physics) to arbitrary accelerations."

    Eq. (40) is an input assumption, not a consequence of the qMOND law (35). Expanding (35) with g = a − a_bar gives g ≈ (GM/r^2)^2/(2a_bar), i.e. GM/r^2 ≈ sqrt(2 a_bar g) = sqrt(g0 g); Eq. (41) instead asserts GM/r^2 = g^2/(2a_bar), which coincides with the expansion only at the single value g = 2a_bar. Thus the deep-MOND relation (42) and the Milgromian scale (43) are not derived from the preceding formalism: (42) is the assumed mean-field Newton relation (40) with g0 defined as 2a_bar. The 'prediction' of MOND reduces by construction to the input assumption plus a renaming of the scale.

  2. self citation load bearing [Sec. 6, Eq. (47) and discussion following Eq. (49)]
    "The study of the spectrum of the DDW Hamiltonian of quantum Yang-Mills theory 34, that controls the mass spectrum of the theory, has shown that the order of magnitude of the mass gap ∆m is related to the scale of κ as follows: κ∼ (∆m)^3/(ℏ^4 g_s^2). This formula is based on a rather rough spectral estimate and it does not take into account a factor that depends on the dimension of the gauge group."

    The numerical bridge from precanonical quantum gravity to the Milgromian scale (a_*∼10^-27 m^-1, called 'consistent with the value of the Milgromian acceleration') is imported from the authors' own ref. [34], explicitly acknowledged as a rough spectral estimate with several orders of magnitude uncertainty. The claimed agreement with Λ≈10^-52 m^-2 likewise uses λ=3 from the authors' ref. [15]. These self-citations are load-bearing for the paper's quantitative claims, and they are not independently machine-checked, code-reproduced, or externally falsified within the paper, so under the review rules they do not provide independent support.

full rationale

The paper contains substantial non-circular work: Secs. 2–3 construct the precanonical spin-connection foam, derive the acceleration scale a_* = 8πGℏκ from Eq. (20), and compute the fluctuation moments (28)–(30). Eq. (35) follows from the assumed stochastic equation of motion (31) once <x_i ω_i/r^3>=0 (34). All of this is independent content. The circularity enters at the advertised 'derivation of Milgromian MOND' in Sec. 5. There Eq. (40) is asserted as a redefinition of Newton's law in the mean-field frame; but from the qMOND law (35) the small-g expansion is g ≈ (GM/r^2)^2/(2a_bar), so GM/r^2 ≈ sqrt(g0 g), whereas Eq. (41) states GM/r^2 = g^2/(2a_bar) = g^2/g0. These agree only at g = 2a_bar. Eq. (44), used to obtain the interpolating function, is likewise inconsistent with (35) and (40), so the interpolating function in (46) is not a consequence of the preceding calculation. The paper's own Sec. 6 admission that (52) gives v ∝ √r at large r, contradicting flat rotation curves 'described by MOND by design', confirms that the qMOND law was not the source of the MOND prediction. The deep-MOND law is instead the input mean-field relation (40) in new variables, with g0 defined as 2a_bar. The quantitative agreement of a_* with Milgrom's a0 and with Λ also leans on the authors' own rough spectral estimate (47) from ref. [34] and on λ=3 from ref. [15], both acknowledged as order-of-magnitude. Because the framework itself is not circular, but the central MOND derivation reduces at the critical step to an assumed equation plus definitions, the score is 6.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The central claim rests on a new quantization framework (precanonical quantization) and on several assumptions about how quantum fluctuations affect classical dynamics. The most fragile is the mean-field relation in Eq (40), which is not derived and is inconsistent with the qMOND law. The numerical prediction of a_star depends on a free parameter kappa and a chosen mass gap value.

free parameters (5)
  • kappa (UV parameter of inverse spatial volume) = not directly fitted; chosen via Delta m ~ 0.1 GeV to yield a_star ~ 1e-27 m^-1
    Introduced in relation (1) on dimensional grounds. Its physical value is estimated from a rough spectral analysis of the Yang-Mills Hamiltonian, with admitted uncertainty of several orders of magnitude, and effectively selected to match the Milgromian scale.
  • Delta m (QCD mass gap) = 0.1 GeV
    Chosen from the expected range (100 MeV to a few GeV) to make a_star consistent with the Milgromian acceleration; this choice is a free parameter.
  • lambda (operator ordering constant in Eq 15) = 3
    Set to 3 for Weyl ordering of the spin connection operator in Eq (14); affects the relation between a_star and the cosmological constant.
  • g_s^2 (infrared QCD coupling) = 4 pi^2
    External input from nonperturbative QCD, used in Eq (47); not a free parameter of this paper but an assumed value.
  • a_bar (mean-field acceleration) = g0/2, with g0 from observations (0.12 nm/s^2)
    For Solar System and lab predictions, the paper explicitly sets a_bar from the empirical Milgromian acceleration g0 (Section 6, after Eq 49).
assumptions (6)
  • domain assumption Precanonical quantization framework and the Poisson-Gerstenhaber algebra are a valid quantization of gravity.
    The entire derivation relies on the precanonical quantization formalism developed by the first author (refs 6-14), which is not an established consensus framework.
  • domain assumption The correspondence condition <hat g_{mu nu}> = eta_{mu nu} in Eq (19) selects the quantum Minkowski wave function.
    Used to derive the massive fiber modes in Eq (20); it is a reasonable but unproven requirement.
  • domain assumption The ground state of Eq (25) is the Yukawa-like wave function in Eq (26).
    Called 'the simplest solution'; other solutions would give different moments and different modified dynamics.
  • domain assumption The cross-correlation <x^i tilde omega_i / r^3> = 0 in Eq (34).
    Needed to obtain the qMOND law in Eq (35); justified by symmetry, but the joint statistics of position and spin connection fluctuations are not derived.
  • ad hoc to paper The mean-field relation a - a_bar = GM/r^2 in Eq (40) defines how Newtonian acceleration is measured in the foam.
    This relation is asserted, not derived, and is inconsistent with Eq (39) except at one acceleration value. It is the key step that produces MOND.
  • ad hoc to paper The relation between kappa and the QCD mass gap in Eq (47).
    Based on a 'rough spectral estimate' from the authors' previous work (ref 34); it carries the numerical prediction.

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Cite this review

Pith. "Pith review of Effects of Quantum Spin-Connection Foam in the Solar System, Galaxies, and the Universe." pith.science (2026). https://pith.science/paper/YEWNM6PA

@misc{pith2026260812404,
  author       = {Pith},
  title        = {Pith review of: Effects of Quantum Spin-Connection Foam in the Solar System, Galaxies, and the Universe},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YEWNM6PA}},
  note         = {Machine review of arXiv:2608.12404}
}
abstract

We argue that effects of the quantum spin-connection foam, which describes quantum gravity according to the precanonical quantization of General Relativity, may already be observed in the form of the small cosmological constant and a modification of Newtonian dynamics at small accelerations, manifested in the flat rotation curves of galaxies. We obtain a modification of the Newtonian potential that takes into account the existence of a fundamental small acceleration scale, $a_* = 8\pi G\hbar\varkappa$, where $\varkappa$ is a parameter with the dimensions of inverse spatial volume that appears on dimensional grounds. The connection between $\varkappa$ and the hadronic scale of the mass gap in the pure Yang-Mills sector of the Standard Model leads to an estimated value of $a_*$ compatible with the Milgromian acceleration scale in MOND. The connection between $a_*^2$ and the cosmological constant leads to a realistic value of the latter. Milgromian MOND, together with a theoretically distinct interpolating function, is derived under the assumption that classical dynamics is modified by the mean-field acceleration calculated from the simplest solution of precanonical quantum gravity in the nonrelativistic approximation. We also indicate that the effects of Newtonian dynamics modified by the spin-connection foam may be observable in the Solar System and even in laboratory experiments.

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