[4th edition (August 2026)] No conclusion has changed. Following Paper 149, the origin of the half-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 reduced Planck constant, the gravitational constant and the speed of light are one, one and two and the cokernel is cyclic, so the conclusion of the 3rd edition stands unchanged. Only the way the reason is written has been corrected.
Substituting the Yamagishi gravitational patch G = 4π and quantum patch ħ = 1/(4π) into the Planck length l_p = sqrt(ħG/c³) cancels ħG = 1 and gives l_p = c^(-3/2). The substitution is correct.
[What the 3rd edition adds] — this expression is put through the “counting of degrees of freedom” of Paper 38 (3rd edition).
First, the count. With base dimensions (M, L, T), the exponent vectors are ħ = (1, 2, −1), G = (−1, 3, −2), c = (0, 1, −1) and m = (1, 0, 0). Every one of the four ways of choosing three of them has a non-zero determinant — |det| = 2 for (ħ, G, c), which is Planck units, and |det| = 1 for the other three, among them the basis (G, m, c) of Paper 38 — so whichever three are chosen, a suitable choice of units assigns them arbitrary values simultaneously. But there are four quantities and only three base dimensions, so at most three may be assigned, and imposing a value on a fourth is not a convention but a claim about nature. Concretely: adopting m = 4π (Papers 18, 38), G = 4π (Paper 35) and ħ = 1/(4π) (this paper) exhausts the budget, after which the numerical value of c is no longer chosen but determined; adopting (ħ, G, c) instead is Planck units, whereupon the unit of mass is fixed to the Planck mass and m = 4π becomes a claim. Either is permissible; both together are not.
This paper assigns values to only two quantities, G and ħ, which is exactly why l_p remains a function of c. Spending the last degree on c = 1 gives l_p = 1 — the Planck length becomes the unit of length, which is not a new result but the definition of Planck units. That l_p = c^(-3/2) “still contains c” is not a defect but the display of a budget with one item remaining.
Second, where the exponent comes from. Writing l_p = ħ^a G^b c^d and matching dimensions gives a = b from the M component, d = −3a from the T component, and 2a = 1 from the L component, so (a, b, d) = (1/2, 1/2, −3/2) uniquely. The same procedure gives (1/2, 1/2, −5/2) for the Planck time, (1/2, −1/2, 1/2) for the Planck mass and (1/2, −1/2, 5/2) for the Planck energy. These are the unique solutions of a 3×3 system of linear equations: all that enters is the dimensions of ħ, G and c, and no equation of motion or orbit appears. The 3/2 of the Planck length is the closing of a dimensional ledger.
Third, why Planck units are full of square roots. Let A be the dimension-exponent matrix of (ħ, G, c); then |det A| = 2. For an integer matrix, A⁻¹ = adj(A)/det(A) and the entries of adj(A) are integers, so with |det| = 2 every entry of the inverse is an integer divided by two — a half-integer. That the exponents of the Planck length, time, mass and energy are without exception half-integers, and therefore that every Planck quantity is written with a square root, is a consequence of that determinant being 2. By contrast the basis (G, m, c) has |det| = 1, so its inverse has integer entries: the natural length is Gm/c², the natural time Gm/c³, the natural mass m and the natural energy mc² — not one square root appears. The impression that “Planck units are full of square roots” is not a property of nature but a property of the determinant of the chosen basis.
Fourth, the 4π inside ħ = 1/(4π) is not an independent occurrence. It was chosen so that ħG = 1; once G = 4π is fixed, wanting that cancellation leaves no option other than 1/(4π). This 4π therefore carries no information beyond the choice of G, has nothing to do with a forced 4π such as that of Poisson’s equation (Root 1 of Paper 35, 3rd edition), and must not be counted as “4π appearing in ħ as well”. The cancellation occurred because it was arranged to occur. Incidentally h = 2πħ = 1/2 here, and that 2π is Root B (periodicity of phase) of Paper 18 (3rd edition), of different origin again.
Fifth, the Planck length is not a minimum length. The claim that it is a lower bound on length is not part of established physics; the Planck length is the scale at which quantum-gravitational effects are expected to stop being negligible, and its provenance is the dimensional analysis above. Frameworks incorporating a minimum length have been proposed but none is established. What can be said is that the Planck scale is a module boundary in the sense of Paper 113 — the place where the dimensionless ratio E/E_p crosses one — and that Paper 35 (3rd edition) said the same from the other side: in natural units [G] = M⁻², and a coupling of negative mass dimension makes divergences worsen at higher orders (pure gravity diverges at two loops; Goroff–Sagnotti 1986). The Planck length is therefore not a pixel that cannot be subdivided but the boundary beyond which the present description stops working.
Finally, the second edition’s rejection of the identification with Kepler’s third law was correct, and the 3rd edition gives that distinction a name in the vocabulary of Paper 28 (3rd edition). Kepler’s 3/2 is a scaling exponent of dynamics, forced by the inverse-square law that Gauss’s flux imposes in three dimensions; it concerns a variable (the semi-major axis) and is testable — from Mercury to Saturn, T/a^(3/2) lies between 0.999 and 1.001. This paper’s 3/2 is the solution of a dimensional equation, and the c it acts on is a constant, so it is not a scaling law and there is no data against which to test it. All they share is the form “the square root of a cube”; an agreement of form is not an identity of phenomenon.
[On the planetary table] The Earth row equals 1 by definition: one astronomical unit is the Earth’s semi-major axis and one year is its orbital period, so the entry T/a^(3/2) = 1/1 = 1 holds exactly and by convention and cannot possibly fail. That row is a convention rather than a fact, and the independent confirmations the table supplies number five, not six — the discipline of separating fact from convention (Papers 95 and 35, 3rd edition) applied to this series’ own table. As for the residuals: the entries are computed from three or four significant figures, and since the relative sensitivity is −(3/2)·δa/a, rounding in the last printed digit of a alone shifts the ratio by ~1.9×10⁻³ for Mercury and ~1.0×10⁻³ for Venus, so the wobble among the inner planets lies inside the band created by the printing precision. That Jupiter and Saturn both fall below 1 is not explained by rounding (Jupiter’s rounding sensitivity is only ~1.4×10⁻⁴); the exact two-body problem gives T/a^(3/2) = (1+m/M)^(-1/2), and the mass ratios predict ≈0.9995 for Jupiter and ≈0.99986 for Saturn, agreeing in direction with the table in both cases. This is recorded as a remark only, not as a verification: the input precision is comparable to the effect and the mutual perturbations of Jupiter and Saturn are large.
Scope: a structural interpretation of established physical law (Layer 2). No new physical law and no prediction is claimed, no new fact about the Planck length is asserted, quantum gravity is not solved, and the “minimum UI pixel size” remains a name rather than a derivation. The main results ħG = 1 and l_p = c^(-3/2) are unchanged. 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 law. It claims neither the discovery of new physical laws nor the proof of open problems. The algebraic relations presented are substitutions into known expressions; the aim is to propose a geometric reading of them. No claim contradicting established theory is made.
Note on units (important): The substitutions m=4π, G=4π and the like employed here constitute a choice of natural units. G and ħ are dimensionful quantities whose numerical values depend entirely on the unit system. Setting G=4π is possible in exactly the sense that G=1 (geometrised units) or G=6.674×10⁻¹¹ (SI) is possible; physics does not determine which is “correct.”
What this paper exhibits is therefore how the equations look once this unit system is adopted, not a new fact about nature. Moreover the relations obtained hold in natural units, and conversion factors are required to carry them into SI. Whatever value this paper has lies in making geometric structure visible — rendering the provenance of π and the role of solid angle easier to read off the equations.
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). The Yamagishi Planck Length Unpacking Theory ($l_p=c^{-3/2}$) [Paper 39]. Zenodo.
DOI: 10.5281/zenodo.21430490
ORCID: 0009-0006-8587-3450