[4th edition (August 2026)] No conclusion has changed. Following Paper 149, the origin of the quarter-integer exponents has been rewritten from the determinant to the largest invariant factor. The absolute determinant gives an upper bound on what the denominator divides, not the denominator itself: the diagonal matrix with entries two and two has absolute determinant four yet yields halves only. The invariant factors of the dimension-exponent matrix of the kinematic viscosity and the dissipation rate are one and four and the cokernel is cyclic, so the conclusion of the 3rd edition stands unchanged. Only the way the reason is written has been corrected.
Non-dimensionalising the incompressible Navier–Stokes equations reduces the dimensionless parameters to the single Reynolds number Re = UL/ν: setting boundary conditions aside, the whole of the difficulty is packed into one number. Starting from that fact, this paper writes out the standing of the scales that appear in turbulence.
First, the Kolmogorov length is the output of a dimensional equation. In (L, T) one has [ν] = (2, −1) and [ε] = (2, −3), and requiring η = ν^a ε^b to be a length determines (a, b) = (3/4, −1/4) uniquely, giving η = (ν³/ε)^(1/4). The same procedure gives the Kolmogorov time (ν/ε)^(1/2) and velocity (νε)^(1/4).
Second, and this is the point: the dimension-exponent matrix of (ν, ε) has |det| = 4. For an integer matrix A one has A⁻¹ = adj(A)/det(A) with adj(A) integral, so every entry of the inverse is an integer divided by four — a quarter-integer. That the Kolmogorov exponents are without exception quarter-integers, and therefore that η carries a fourth root, is a consequence of that determinant. This is the same theorem, with a different determinant, as the one Paper 39 established for (ħ, G, c): |det| = 2, hence half-integers, hence the square roots that pervade Planck units. The answer to “why this particular radical?” is, in both cases, the determinant.
Third, η is precisely the place where a dimensionless number crosses one. With u_η = (νε)^(1/4) one finds Re_η = u_η η/ν = (ν⁴)^(1/4)/ν = 1 identically — an identity, not an approximation (verified numerically: indoor air with ν = 1.5×10⁻⁵ and ε = 10⁻³ gives η = 1.355 mm and Re_η = 1.000000; water in a pipe with ν = 10⁻⁶ and ε = 1 gives η = 0.032 mm and Re_η = 1.000000). Above η the inertial term dominates, below it viscosity governs. The module boundary of Paper 113 is here not an analogy but the definition itself. Note that ε is an input: the numbers are values for that choice of ε, not universal constants.
Fourth, η therefore has the same standing as the Planck length. Both are produced by the same procedure — extracting from given dimensionful quantities the unique combination with the dimensions of a length — and Paper 39 recorded that the Planck length is not a proven minimum length. The same holds here: η is a statistical, phenomenological scale for developed turbulence, not a rigorous lower bound on the size of structures in the Navier–Stokes equations. This distinction is the heart of the problem. If one could assert that no structure smaller than η arises, global regularity would follow — but dimensional analysis supplies no such theorem. Extracting the unique combination and proving a bound are different things.
Fifth, the exponent 3/4 is not a dimension. By the sorting of Papers 94 and 28 (3rd edition) — fractal dimension, topological dimension, scaling exponent — it is the solution of a dimensional equation, a coefficient settled by the dimensions of ν and ε alone; there is no such thing as “0.75 dimensions”. Fractal dimension does genuinely appear elsewhere in turbulence, in intermittency and multifractal descriptions of the dissipation field, but that is a different matter and is not entered into here.
Sixth, the Navier–Stokes equations know what dimension they are in. The vorticity obeys ∂_t ω + (u·∇)ω = (ω·∇)u + νΔω, and in two dimensions, with ω = (0,0,ω) and u independent of z, the stretching term (ω·∇)u = ω ∂_z u vanishes identically. A maximum principle then applies and global regularity is known; in three dimensions the term does not vanish and finite-time divergence of the vorticity is open (Beale–Kato–Majda 1984). By the measure of Paper 114 the equations are therefore on the dimension-selecting side — while the exponent 3/4, settled by the dimensions of ν and ε alone, is on the non-selecting side. Objects that select a dimension and objects that do not coexist within one paper.
Seventh, ν → 0 is a singular limit, which refines Paper 113. Formally it yields the Euler equations, but the coefficient of the highest derivative vanishes, so solutions do not converge uniformly near a boundary (boundary layers; Prandtl 1904): the layer thins as ~ν^(1/2) without disappearing, and gradients within it grow. This is not a smooth group contraction of the kind c → ∞ provides. Paper 113’s frame needs a second kind — a boundary that does not contract. “Boundaries always lie where a dimensionless number crosses one” holds; “boundaries are always smooth limits” does not.
Eighth, blow-up and the singularities of Ricci flow are alike in picture but the machinery does not transfer. Both are questions of singularity formation in a nonlinear evolution equation, with the structure “control the singularities and a global conclusion follows”. But Perelman’s resolution of the three-dimensional case rests on the discovery of a monotone functional (the W-entropy) and on continuation by surgery, and no monotone quantity capable of controlling blow-up is known for Navier–Stokes; the energy inequality (Leray 1934) exists but does not constrain the L^∞ norm. The resemblance is a rhyme, not a tool, and must not be read as “it worked for Ricci flow, so it should here”. The same applies to the cascade and the renormalisation group: RG methods have been applied to turbulence (Forster–Nelson–Stephen 1977 and later work), but there is no counterpart of what Paper 110 called gold — Friedan’s theorem, the equality of the sigma-model metric’s RG flow with Ricci flow.
Scope: a structural account of established mathematical theorems (Layer 2). Global regularity for the three-dimensional Navier–Stokes equations is a Clay Millennium Prize Problem and is unsolved; this paper claims neither its solution, nor a partial solution, nor the exclusion of blow-up. η is not treated as a lower bound on structure size, 3/4 is not conflated with a fractal dimension, intermittency corrections are not entered into, Ricci flow machinery is not transferred, and the cascade is not identified with RG flow. The Universe OS is a non-scientific Layer-3 lens.
(scope note) Positioning of this paper: This paper offers a structural interpretation of established physical laws and mathematical theorems. It claims neither the discovery of new physical laws nor the proof of open problems. The algebraic relations presented are transformations of known expressions; the aim is to propose a geometric reading of them. No claim contradicting established theory is made.
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). Dimensionless Numbers and Scales in the Navier–Stokes Equations [Paper 3]. Zenodo.
DOI: 10.5281/zenodo.21404030
ORCID: 0009-0006-8587-3450