What Does the Exponent of c Count? – The Exponent Formula of the Planck Reduction, epsilon as Mass Dimension, the Non-Uniqueness of 4pi, and the n = d-1 That Was Sitting in the Denominator.
Paper 44 showed that under the Yamagishi patch G = 4pi, hbar = 1/(4pi) every Planck quantity reduces to c^N or c^N/(4pi), and in the same breath imposed a self-limitation: it does not assert as a theorem that N means a spatial dimension. This paper converts that self-limitation FROM AN OPEN RHETORICAL STANCE INTO A CLOSED COMPUTATION – because “does not assert” is a POSTURE, not a VERDICT, and without a verdict the same claim returns in new clothes. FIRST: THE EXPONENT FORMULA. In d spatial dimensions, writing a quantity of dimension L^lambda M^mu T^tau as X = hbar^p G_d^q c^N, and setting n := d-1, one has q = (lambda + tau – mu)/n, p = mu + q, N = -(mu + tau) – 3q. The solution is UNIQUE, because the determinant of the linear system is 1 – d = -n. THAT THE PLANCK REDUCTION IS SOLVABLE AT ALL IS GOVERNED BY n = d-1. Evaluated at d = 3 the formula reproduces every row of Paper 44’s table. SECOND: THE “3” IN c^3 IS NOT A SPATIAL DIMENSION. It is the sum of the T^-1 carried by hbar (action) and the T^-2 carried by G (acceleration) – A WEIGHT IN TIME – and it contains no d at all. For every d one has l_P = (hbar G_d / c^3)^(1/n), matching the standard higher-dimensional form l_P^(D-2) = hbar G_D / c^3: THE 3 NEVER MOVES; only the index of the root moves. THE NAIVE READING IS THEREFORE FALSE: for Planck power N = 5 at d = 3, 4 at d = 4, 7/2 at d = 5, 16/5 at d = 6. 16/5 cannot be the number of a dimension. And N IS NOT INJECTIVE – Planck density and Planck power both have N = 5 – so N does not even function as a label. THIRD: WHERE THE DIMENSION ACTUALLY WAS. d enters N ONLY through 1/n: THE SPATIAL DIMENSION LIVES IN THE DENOMINATOR. And that n is the Gauss-law exponent – the n = 2 of Paper 01, the n = d-1 of Paper 114 – because the L^d of [G_d] comes from the r^(d-1) of F = G_d M m / r^(d-1). It was not where we went looking; IT WAS SITTING INSIDE THE SQUARE ROOT. (Brake: A RELATION, NOT A CAUSE – the chain passes through the definition [G_d] and so never leaves dimensional analysis. All that is claimed is an ADDRESS: the single place at which d touches N is 1/n.) FOURTH: epsilon IS THE MASS DIMENSION. Relaxing the patch to one parameter, G = a and hbar = 1/a for any a > 0, gives X = a^(-mu) c^N, that is epsilon_X = mu_X. Paper 44’s geometric/material dichotomy IS the statement mu = 0 versus mu = 1, and “the power of 4pi never reaches 2” IS “no quantity of mass dimension 2 or more appears in the list”. This DERIVES Paper 44’s Theorem 3.1 and names its root: hbar and G carry OPPOSITE M-weights, +1 and -1. FIFTH: 4pi IS NOT SELECTED. The closure holds FOR EVERY a > 0. The one item that fixes a is the Planck charge – and the 4pi there is the SI RATIONALISATION put in by hand, since in Gaussian units the same requirement selects a = 1 instead. A VALUE WHOSE FORCING IS A FUNCTION OF THE UNIT CONVENTION IS A CONVENTION, NOT A FACT (95): the reverse side of Paper 05 and a third instance of Paper 35. The invariant content of Paper 44 was hbar G = 1 (39), not 4pi; a is a coordinate on the family {(a, 1/a)}, not a determined value. SIXTH: SELF-AUDIT OF ITEM (x). In Gaussian units charge has mu = 1/2, hence epsilon = 1/2, which violates Paper 44’s epsilon in {0, 1}; in SI charge carries an independent base dimension and mu = 0, agreeing with Paper 44. There is no contradiction – but the scope of Paper 44’s Theorem 2.1 is “mass dimension 0 or 1 IN THE CHOSEN SYSTEM OF BASE DIMENSIONS”, a condition belonging to the design of the unit system, not to physics. Items (i)-(ix) hold unconditionally; only item (x) is convention-dependent. SEVENTH: THE KERNEL. The map (lambda, mu, tau) -> (N, epsilon) has a kernel: the single ray of dimension L^5 T^-3, which is hbar G itself. Two quantities collapse onto the same symbol exactly when their ratio is a power of hbar G. That – not a structural coincidence – is why density and power share the c^5 row. IMPORTANT (scope): THIS PAPER DOES NOT CROSS THE CIRCULARITY. The exponent formula is an identity INSIDE dimensional analysis and gives no independent route to the value of N; what it gives is CHECKABILITY. It requires X to be a monomial in (hbar, G_d, c): the c^6 of Hawking radiation is returned only for P M^2, which carries an external mass, so THERE IS NO NUMBERING AND NOTHING IS MISSING. It NEITHER SUPPORTS NOR REFUTES the author’s Layer-3 conviction that c is the building material of dimension – ONE ROAD TOWARDS IT IS CLOSED, that is all. 4pi IS NOT DEMOTED (its roles in Papers 01, 02, 13, 67 and 115 all stand) and PAPER 44 IS NOT RETRACTED (its Theorems 2.1 and 3.1 stand; what is added is the identification of the root and the statement of scope). Every piece of mathematics used is known and elementary (Buckingham; Bridgman; the standard higher-dimensional Planck length); no new theorem is proved, no prediction is made. No meaning is read into individual values of N (94, 101). Layer-3 lenses are quarantined with a non-scientific label.
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). What Does the Exponent of $c$ Count? — The Exponent Formula of the Planck Reduction, $\epsilon$ as Mass Dimension, the Non-Uniqueness of $4\pi$, and the $n=d-1$ That Was Sitting in the Denominator — [Paper 116]. Zenodo.
DOI: 10.5281/zenodo.22003794
ORCID: 0009-0006-8587-3450