This paper called the divergence of dm/dr = 4π r^(2−n) for n≠2 a "system crash theorem" and approached the fine-tuning problem from it. The second edition withdrew the claim that the anthropic principle had been "debugged" and separated the metaphors from the theorem. The 3rd edition settles the four points that remained.
First, only dimensionless numbers can be fine-tuned. c, ħ and G are dimensionful — that is, choices of units — and their numerical values depend entirely on the system of units, so the question "what if c had been a few percent different" cannot be distinguished from a redefinition of the units (Duff 2002; the 3rd edition of Paper 34 turned the same criticism on itself). Only dimensionless quantities can be tuned: the fine structure constant α=1/137.035999, the cosmological constant in Planck units Λℓ_p²~10⁻¹²², and so on. And the six numbers of Rees's Just Six Numbers (1999), cited in §1 of this paper, are all dimensionless: N~10³⁶ (ratio of electromagnetic to gravitational force), ε=0.00723 (nuclear binding efficiency H→He, computed here), Ω~1 (density parameter), λ~0.7 (contribution of the cosmological constant), Q~10⁻⁵ (amplitude of density fluctuations) and D=3 (spatial dimension). The fine-tuning debate was framed from the beginning as a discussion of dimensionless numbers. Yet §3 of the second edition ranged c, ħ, G, α, Λ in a single list, attaching to each a label such as "bandwidth", "resolution" or "compression ratio". The first three are choices of units and the latter two are (once made dimensionless) facts: things that must not be ranged together were ranged together — the same type of error as the 3rd edition of Paper 15 found in grouping m=4π (a choice of units) and ε₀=r (a function of the configuration) under the one word "Yamagishi patch". The format of a list concealed the difference of category.
Second, the real boundary of "crash" is n=3, not n=2. For ρ=ρ₀r^(−n) the mass within a sphere of radius R is m(R) = 4πρ₀∫₀^R r^(2−n)dr, which converges at the origin iff 2−n>−1, i.e. n<3. For n<3 it is m(R)=4πρ₀R^(3−n)/(3−n) and finite; at n=3 it diverges logarithmically (the real boundary); for n>3 it diverges at the origin. So for 2<n<3 the enclosed mass remains finite and nothing breaks. What the theorem of the second edition stated was the divergence of a rate, dm/dr, not the divergence of the mass itself, and a rate may diverge while its integral stays finite. The word "crash" was too strong for the divergence of a rate.
Third, real density profiles have n≠2 all the time. Uniform density is n=0, where dm/dr = 4πρ₀r² diverges as r→∞ — and that is the standard picture cosmology adopts on large scales (FLRW); nothing has broken. An NFW halo has n=1 inside (the cusp) and n=3 outside (where the mass diverges logarithmically and a cutoff is used), while the isothermal sphere is n=2. The second edition reserved judgement, writing that "what a divergent quantity means physically requires a separate dynamical argument, which this paper does not supply"; the 3rd edition supplies part of that argument — and it runs the other way. On the n<2 side the most basic case of all, n=0, actually holds, and the divergence of dm/dr there signifies no breakdown whatever. The label "n<2 is a macroscopic memory overflow" is therefore withdrawn, and the label "microscopic singularity freeze" on the n>2 side likewise — for 2<n<3 the enclosed mass is finite, and both the inner cusp and the outer slope of NFW haloes are structures actually observed.
Fourth, what then is n=2? At n=2 the quantity dm/dr = 4πρ₀ is a constant independent of r, giving the linear extension m(R)=4πρ₀R. This is correct and peculiar to n=2. But it is linearity, not stability, and which of the two words is used changes the strength of the claim completely. Moreover the two special values of n are distinct roots: n=2 is special because dm/dr does not depend on r, whose root is geometry (balance against the volume element r^(d−1)); n=3 is special because the enclosed mass ceases to be finite, whose root is analysis (convergence of an integral). The first and second editions spoke of the uniqueness of the former as though it carried the "boundary of breakdown" that belongs to the latter — in the vocabulary of Papers 86 and 100, two roots had been folded into one claim. In the vocabulary of Paper 113, the module boundary is the one at n=3, where the description changes and a cutoff becomes necessary. At n=2 nothing changes.
Fifth, n=2 is merely the case d=3 of n=d−1. In ℝ^d we have dm/dr = Vol(S^(d−1))·r^(d−1−n), so the condition of independence from r is n=d−1, giving n=1,2,3,4 for d=2,3,4,5. Paper 114 took a census of which structures select a dimension and which do not, classifying n=d−1 on the side that does not, and the 3rd edition of Paper 06 repeated the same judgement. The uniqueness of n=2 is therefore uniqueness given d=3. Section 4(3) of the second edition correctly wrote that "why d=3 lies outside this paper", but the 3rd edition writes it back beside the theorem — an assumption belongs next to the theorem, not in the scope section.
Finally, §3 of the second edition, the "universe OS system API specification", is removed from the body. The second edition placed it inside a definition environment, describing c as "bandwidth", ħ as "resolution" and G as "compression ratio", and noted at the head of the section that it was "an interpretation, not a theorem" — but the typesetting remained that of a definition. Paper 82 explicitly quarantined this kind of language (OS, rendering, bandwidth) in Layer 3, the non-scientific layer, and the 3rd edition follows that discipline in moving the section there. Noting something in a remark and showing it in the typesetting are different acts: the reader reads the name of the environment. For the same reason the 3rd edition of Paper 17 wrote that "naming is not deriving", a definition environment is used only for definitions. The theorem itself stands unchanged from the first edition: n=2 is the unique exponent for which dm/dr does not depend on r, giving the linear extension m(R)=4πρ₀R — but that is linearity, not stability, and it is uniqueness given d=3. All numbers were computed here (no external tables consulted). This paper claims neither the refutation nor the endorsement of the anthropic principle or the multiverse hypothesis.
(scope note) Scope of this paper: The algebraic content of this paper is the single theorem in Section 2. The System API correspondence in Section 3 is an interpretation, not a derivation. Sections 4 and 5 exist in order to limit the reach of that interpretation within the fine-tuning debate. Nothing here contradicts established physics.
Note on units (important): Substitutions such as m=4π are a choice of natural units. The numerical value of a dimensional quantity depends entirely on the unit system, and physics does not decide which choice is “correct.” What this paper shows is how the equations look in this unit system, not a new fact about nature.
A note on the aim of this work. This is a structural reinterpretation of well-established physics, not a proposal for a new physical law. Every relation shown is a substitution into, or a geometric re-reading of, equations that are already known and that remain unchanged. The aim is a modest one: to make a few recurring geometric structures — in particular the solid angle 4π and the exponent n = 2 — easier to see within a single picture. Where a particular unit convention is adopted (for example G = 4π), this is a choice of notation, not a claim about nature. Nothing here is intended to contradict existing results.
On the making of this work: The ideas and content of this work stem from the author's own considerations. Assistance from an AI (a large language model) was used for structuring, English translation, and checking the algebra. Any remaining errors or misinterpretations are solely the author's. Feedback and corrections are sincerely appreciated.
Reference
Yamagishi, Y. (2026). Only Dimensionless Numbers Can Be Fine-Tuned [Paper 10]. Zenodo.
DOI: 10.5281/zenodo.21404556
ORCID: 0009-0006-8587-3450