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arxiv: 2605.18637 · v1 · pith:DSJ2NFUNnew · submitted 2026-05-18 · ⚛️ physics.gen-ph

Gravity, Fine-Structure Constant and Natural Units -- some Thoughts based on Dimensional Analysis --

Robi Banerjee This is my paper

Pith reviewed 2026-05-20 01:02 UTC · model grok-4.3

classification ⚛️ physics.gen-ph
keywords dimensional analysisfine-structure constantnatural unitsfundamental constantsgravityquantum mechanicsunit systems
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The pith

The form of the fine-structure constant follows directly from dimensional analysis with length, time, mass and the constants G, c, ħ, e.

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

The paper applies dimensional analysis using the three basic dimensions of length, time and mass together with the constants G, c, ħ and e to explore connections among fundamental constants. It establishes that this setup produces the observed form of the fine-structure constant as a necessary result of dimensional consistency. The analysis also draws attention to gravity as a field that remains distinct because it has not been successfully merged with quantum mechanics. Different systems of natural units are examined as outcomes of choosing which constants to include in the dimensional framework.

Core claim

Starting from Maxwell's dimensions L, T, M and the constants G for gravity, c for special relativity, ħ for quantum mechanics and e for electromagnetism, the dimensional equations lead directly to the form of the fine-structure constant. This outcome requires no extra physical assumptions beyond the count of independent dimensions and the included constants. The same framework underscores that gravity stands apart as the one interaction not yet combined with quantum mechanics.

What carries the argument

Dimensional analysis that connects the number of fundamental dimensions to the set of constants G, c, ħ and e, yielding the fine-structure constant as a direct consequence.

If this is right

  • The fine-structure constant is fixed by dimensional requirements rather than appearing as an independent input.
  • Gravity remains outside any quantized description within this dimensional counting.
  • Natural unit systems arise systematically from selecting subsets of the same constants and dimensions.
  • Dimensionless quantities such as the fine-structure constant can be understood through the balance between dimension count and constant count.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar dimensional counting might be applied to other constants by adding or removing elements from the set of dimensions or constants.
  • The separation of gravity could guide attempts to identify what additional structure would be needed to combine it with the other constants.
  • Unit choices derived this way might offer new ways to compare scales across different physical regimes.

Load-bearing premise

Dimensional analysis using only L, T, M together with G, c, ħ, and e is sufficient to derive the specific form of the fine-structure constant without further physical input or quantization assumptions.

What would settle it

A high-precision measurement of the fine-structure constant that deviates from the expression obtained purely from the dimensional equations in a manner not explained by the approximations of the analysis.

read the original abstract

Here we discuss direct links of the number of fundamental dimensions to the fundamental natural constants using simple arguments of dimensional analysis \corr{based on Maxwell's dimensions length (L), time (T) and mass (M) as well as the constants $G$, $c$, $\hbar$ and $e$}. We find that the \corr{form} of the fine-structure constant is a direct consequence of this connection. Additionally, our approach emphasises that gravity is a quite distinct area of physics which is not yet successfully quantised, i.e. not yet combined with quantum mechanics. We also discuss different unit systems based on dimensional analysis and natural constants.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript uses dimensional analysis based on the three fundamental dimensions (length L, time T, mass M) together with the constants G, c, ħ and e to explore connections among fundamental constants. It asserts that the specific form of the fine-structure constant follows directly as a consequence of this dimensional setup, notes that gravity remains unquantized and distinct from other interactions, and discusses implications for natural unit systems.

Significance. If the central derivation were valid and non-circular, the work would offer a compact dimensional perspective on the origin of α and the separation of gravitational physics from quantum mechanics, with potential relevance to discussions of natural units. The manuscript does not, however, supply machine-checked derivations, reproducible calculations, or falsifiable predictions beyond re-expressing known dimensionless combinations.

major comments (2)
  1. [Abstract] Abstract: The claim that 'the form of the fine-structure constant is a direct consequence of this connection' is not supported by any explicit derivation steps. Dimensional analysis with L, T, M and the listed constants can identify that e²/(ħ c) is dimensionless (in Gaussian units), but the conventional SI expression α = e²/(4π ε₀ ħ c) contains the factor 4π ε₀ whose origin lies in the definition of the Coulomb constant or the normalization of the electromagnetic Lagrangian, not in the dimensions of G, c, ħ and e alone.
  2. [Abstract] The manuscript provides no explicit combination or selection rule that isolates the fine-structure constant from other possible dimensionless ratios constructible from the same constants; the assertion therefore imports additional electromagnetic structure without stating the extra physical input required.
minor comments (2)
  1. The text would benefit from a dedicated section or appendix that lists the dimensional matrix and solves for the exponents leading to the claimed expression for α.
  2. Notation for the fine-structure constant and the constants (e.g., whether e is the elementary charge in Gaussian or SI units) should be clarified at first use to avoid ambiguity between unit systems.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thoughtful review and for identifying areas where the presentation of our dimensional analysis can be strengthened. We address each major comment below and outline the revisions we will make to improve clarity without altering the manuscript's core exploratory intent.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The claim that 'the form of the fine-structure constant is a direct consequence of this connection' is not supported by any explicit derivation steps. Dimensional analysis with L, T, M and the listed constants can identify that e²/(ħ c) is dimensionless (in Gaussian units), but the conventional SI expression α = e²/(4π ε₀ ħ c) contains the factor 4π ε₀ whose origin lies in the definition of the Coulomb constant or the normalization of the electromagnetic Lagrangian, not in the dimensions of G, c, ħ and e alone.

    Authors: We accept that the abstract states the claim concisely without showing the intermediate steps. The manuscript performs dimensional analysis using L, T, M together with G, c, ħ and e, demonstrating that the combination e²/(ħ c) is dimensionless when charge is assigned dimensions consistent with Gaussian units. We will revise the abstract and insert an explicit step-by-step derivation in the main text, together with a clear statement that the analysis adopts the Gaussian convention in which 4π ε₀ = 1. This addresses the distinction from the SI form. revision: yes

  2. Referee: [Abstract] The manuscript provides no explicit combination or selection rule that isolates the fine-structure constant from other possible dimensionless ratios constructible from the same constants; the assertion therefore imports additional electromagnetic structure without stating the extra physical input required.

    Authors: Dimensional analysis with the given constants yields several dimensionless ratios (e.g., e²/(ħ c), G m_p²/(ħ c)). The fine-structure constant is selected as the particular combination involving the elementary charge e because it quantifies the strength of the electromagnetic interaction in the theory. We will add a short explanatory paragraph that states this physical selection criterion explicitly, thereby making the additional electromagnetic input transparent rather than implicit. revision: yes

Circularity Check

1 steps flagged

Dimensional analysis of LTM + G,c,ħ,e restates the known dimensionless combination defining α rather than deriving its specific form

specific steps
  1. self definitional [Abstract]
    "We find that the form of the fine-structure constant is a direct consequence of this connection."

    The 'connection' is dimensional analysis performed with the constants G, c, ħ and e. Because α is defined precisely as the dimensionless ratio built from e, ħ and c, the identification of that ratio is tautological with the input rather than an independent derivation.

full rationale

The paper's central claim is that the form of the fine-structure constant follows directly from dimensional analysis in the LTM system using only G, c, ħ and e. However, the only dimensionless combination involving e is e²/(ħ c) (in Gaussian units) or the equivalent SI expression; this combination is the definition of α itself. No additional structure from the listed constants selects the numerical prefactor 4π or the precise normalization of the electromagnetic interaction. The derivation therefore reduces to re-expressing the input definition of α as a 'consequence' of the dimensional setup that already incorporates the constants appearing in that definition. The remainder of the paper (natural units, distinction of gravity) does not supply independent content that would break this reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The approach rests on the standard assumption that dimensional analysis with L, T, M suffices to link the listed constants; no new free parameters, axioms, or invented entities are identifiable from the abstract.

axioms (1)
  • domain assumption Dimensional analysis with Maxwell's L, T, M is sufficient to derive the form of the fine-structure constant from G, c, ħ, e
    Invoked as the basis for the central connection in the abstract.

pith-pipeline@v0.9.0 · 5629 in / 1188 out tokens · 66787 ms · 2026-05-20T01:02:23.087252+00:00 · methodology

discussion (0)

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Foundation/AlphaDerivationExplicit.lean alphaProvenanceCert echoes
    ?
    echoes

    ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

    we find that the form of the fine-structure constant is a direct consequence of this connection... xG=0... ℏc/e² ≈137.036 is a dimensionless number, namely the famous fine-structure constant

  • IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking echoes
    ?
    echoes

    ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

    The fact that G doesn’t contribute to the conversion of physical constants shows again that not only the gravitational constant G but the entire gravitational physics is special... gravity generates spacetime

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages · 1 internal anchor

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