The Four Planck Quantities Are Four Instances of One Rule: Mass Dimension Fixes the Power of 4pi, and l_p/t_p = c Holds in Every Unit System. Third edition.
This paper carried a table showing that under the Yamagishi patch G = 4pi, hbar = 1/(4pi) the four Planck quantities read l_p = c^(-3/2), t_p = c^(-5/2), m_p = sqrt(c)/(4pi), E_p = c^(5/2)/(4pi). THE TABLE ITSELF IS CORRECT AND IS UNCHANGED IN THIS EDITION. What the third edition adds is the REASON for that table, and the removal from it of two items that were never claims.
FIRST, TWO OF THE FIVE LISTED RESULTS ARE IDENTITIES. l_p/t_p = c and E_p t_p = hbar USE hbar G = 1 NOWHERE: in the ratio of l_p = sqrt(hbar G/c^3) and t_p = sqrt(hbar G/c^5) the factor hbar G CANCELS. Ten assignments of hbar and G were tested (the patch, SI units, hbar = G = c = 1, and six random assignments with hbar G ranging from 0.139 to 6.4e3). The worst relative error was 0.00e0 for l_p/t_p = c and 1.81e-16 for E_p t_p = hbar (rounding). THEY HOLD IN EVERY UNIT SYSTEM AND ADMIT NO WAY OF FAILING. Since only what can fail is worth checking numerically, the second edition’s line “all five items confirmed by numerical verification” PLACED TWO FACTS AND TWO IDENTITIES IN THE SAME COLUMN. An equation with no violable null hypothesis confirms nothing when verified (Papers 21, 16). The third edition counts them apart: FOUR PATCH-DEPENDENT FACTS AND TWO UNIT-SYSTEM-INDEPENDENT IDENTITIES.
THE ASYMMETRY IS ITSELF THE BOUNDARY LINE. Fixing c = 3.7 and multiplying G by 1.3, or hbar by 2, BREAKS ALL FOUR FORMS (l_p, t_p, m_p) while l_p/t_p = c SURVIVES UNTOUCHED. Under one and the same perturbation one side collapses entirely and the other does not move. THAT ASYMMETRY IS THE LINE DIVIDING A FACT FROM AN IDENTITY – and the line is seen by perturbing, not by arguing.
SECOND, THE 4PI IS DOING NO WORK. Paper 116 showed that setting G = a, hbar = 1/a for ANY a greater than zero makes hbar G = 1 hold identically, with the four quantities becoming l_p = c^(-3/2), t_p = c^(-5/2), m_p = sqrt(c)/a, E_p = c^(5/2)/a. NOTHING ANYWHERE REQUIRES a = 4pi. The second edition’s conclusion that “all four are expressible in powers of c and THE SOLID ANGLE 4pi alone” therefore reads accurately as “expressible in powers of c and OF THE PATCH CONSTANT a alone”. Writing 4pi is a CHOICE OF NAME, not a result (in Paper 95’s vocabulary, A CONVENTION), and no meaning as a solid angle follows from these four quantities. Moreover the 4pi in m_p is not independent: since hbar = 1/G, the relation hbar/G = 1/G^2 holds EXACTLY (verified here for G = 4pi, 2, 11.7 to within 1e-14), so m_p = sqrt(hbar c/G) = sqrt(c)/G. THE 4PI IN m_p IS G’S 4PI, SQUARED AND SQUARE-ROOTED BACK – not a second one coming from hbar (Paper 39). In these four quantities the 4pi has exactly one address.
THIRD, WHAT THEN EXPLAINS THE TABLE? MASS DIMENSION. Applying Paper 116’s theorem epsilon = mu (under the one-parameter patch, X = hbar^p G^q c^N = a^(-mu) c^N): l_p (mu = 0) and t_p (mu = 0) carry no 4pi, while m_p (mu = 1) and E_p (mu = 1) carry one. ALL FOUR AGREE. What decides whether a 4pi appears is NOT THE FLAVOUR of the words “geometric” and “material” but A NUMBER, the mass dimension. And because no quantity of mass dimension 2 or more occurs among the four, the power of 4pi never reaches 2. THIS PAPER HELD THE TABLE AND DID NOT HOLD ITS REASON. The reason lies in Paper 116, written later, and these four quantities are the smallest roll-call of that theorem.
FOURTH, WHERE DO THE HALF-INTEGER EXPONENTS COME FROM? THE DETERMINANT OF THE DIMENSION MATRIX. Arranging the dimensions of (hbar, G, c) in (L, M, T) gives a matrix A with det A = 2 (verified here in exact rational arithmetic, and by numpy as 2.0). If X = hbar^p G^q c^N satisfies [X] = L^lambda M^mu T^tau, Cramer’s rule determines (p, q, N) uniquely as rationals with denominator det A. Solved exactly: l_p gives (1/2, 1/2, -3/2), t_p gives (1/2, 1/2, -5/2), m_p gives (1/2, -1/2, 1/2), E_p gives (1/2, -1/2, 5/2) – ALL TWELVE EXPONENTS HAVE REDUCED DENOMINATOR EXACTLY 2. Substituting the SI values of hbar, G, c and comparing with the textbook Planck quantities gave a maximum relative error of 1.52e-16. The second edition said the exponents are “fixed uniquely by dimensional analysis”; that is HALF THE STORY – uniqueness is given by det A being nonzero, and the denominator 2 is given by |det A| = 2. These are two different facts (Papers 38, 39).
FIFTH, THE SCOPE IS RESTATED AS SEVEN ROWS. (1) l_p = c^(-3/2) and t_p = c^(-5/2) are patch-dependent facts; (2) m_p = sqrt(c)/a and E_p = c^(5/2)/a are patch-dependent facts; (3) the value 4pi is a convention; (4) whether epsilon is 0 or 1 is a consequence of mass dimension (Paper 116); (5) the denominator 2 in the exponents is a consequence of the determinant (Papers 38, 39); (6) l_p/t_p = c is an identity; (7) E_p t_p = hbar is an identity. OF THE SEVEN ROWS, ONLY THE FIRST TWO CARRY A CLAIM PROPER TO THIS PAPER – namely the forms of the four quantities. THIS IS NOT A SHRINKING BUT AN ADDRESSING: only by writing which row belongs to which root does the table become checkable (Paper 17).
IMPORTANT (scope): a structural interpretation and self-audit of elementary established facts (Layer 2). The mathematics used is entirely known and elementary (the definition of the Planck units, the solution of a 3×3 linear system, dimensional analysis). From within this series it uses Paper 116’s theorem epsilon = mu, the |det| = 2 of Papers 38 and 39, and Paper 44’s form X = c^N/(4pi)^epsilon. NOT ONE NEW THEOREM IS PROVED. What is not claimed: an identity is not called a “check of self-consistency” (Papers 21, 16) / 4pi is not called a forced value (the closure holds for any a greater than zero; Paper 116) / 4pi is not read as a solid angle / the exponent of c is not read as a spatial dimension / the 4pi in m_p is not counted as an independent second one (Paper 39) / no new fact about the Planck system is asserted – it is a rewriting following from a choice of units (Paper 95) / the claim that c is the building material of dimension is neither supported nor denied (Layer 3, quarantined under a non-scientific label). Items retracted in the second edition (the correspondence with Paper 34’s c^5, and the phrase “an algebraic proof of (3+1)-dimensional structure”) are not reinstated.
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 Four Planck Quantities Are Four Instances of One Rule — 3rd ed.: in the Yamagishi Complete Planck Unit Unpacking Theory, the mass dimension fixes the power of $4\pi$, and $l_p/t_p=c$ holds in every unit system — [Paper 40]. Zenodo.
DOI: 10.5281/zenodo.21435739
ORCID: 0009-0006-8587-3450