The Power of 4pi Is the Mass Dimension: “Zero or One” Is a Property of the List, and What Fixes a = 4pi Is an Electromagnetic Unit Convention. Third edition.
This paper showed that under the Yamagishi patch G = 4pi, hbar = 1/(4pi) every Planck quantity reduces to c^N or c^N/(4pi), and called that branching the “geometric / material” dichotomy. THE DICHOTOMY HOLDS AND THE THIRD EDITION DOES NOT ALTER IT. What the third edition adds is the reason for it, and a correction to the range over which it may be asserted.
FIRST, EPSILON IS THE MASS DIMENSION. Paper 116 proved that under the one-parameter patch G = a, hbar = 1/a with arbitrary positive a, one has X = hbar^p G^q c^N = a^(-mu) c^N, that is EPSILON = MU with mu the mass dimension. This was verified here for all nine quantities: the Planck length c^(-3/2) and the Planck time c^(-5/2) have mu = 0 hence epsilon = 0; the Planck mass sqrt(c)/a, energy c^(5/2)/a, temperature c^(5/2)/a, force c^4/a, power c^5/a, density c^5/a and pressure c^7/a all have mu = 1 hence epsilon = 1. ALL NINE AGREE. The words “geometric” and “material” were another name for mu = 0 and mu = 1: what decides whether a 4pi appears is not whether the quantity feels geometric or material, but A NUMBER, the mass dimension. The second edition’s Theorem 3.1 (epsilon = 0 iff the G-dependence runs only through hbar G) is correct, since running only through hbar G and having mu = 0 are equivalent. But EPSILON = MU IS STRONGER: it states the matter as a property of the quantity itself rather than of a procedure, and it DERIVES the second edition’s dichotomy. This paper held a dichotomy and did not hold its name.
SECOND, AND THE MOST IMPORTANT CORRECTION HERE. The second edition’s Theorem 2.1 stated that “the power of 4pi is confined to zero or one, and the second power never appears”. THIS IS A PROPERTY OF THE LIST, NOT OF THE PATCH. The counterexample sits inside this paper’s own proof: writing the basic combinations used in that proof in one-parameter form gives hbar G = 1 (epsilon = 0), hbar G^2 = a (epsilon = -1), and hbar/G = a^(-2), that is EPSILON = 2 (verified here symbolically; at a = 4pi, hbar/G = 1/(16 pi^2)). The mass dimension of hbar/G is 2, and epsilon = mu gives epsilon = 2 directly. THE CORRECT STATEMENT IS THEREFORE “no quantity of mass dimension 2 or more occurs in this paper’s list”, not “4pi never reaches the second power”. The reason the second edition gave – the cancellation chain hbar G^2 = (hbar G) G – is correct for each item on that list but guarantees nothing for a general quantity. THIS IS NOT A RETRACTION BUT A STATEMENT OF RANGE: for the nine listed quantities the second edition’s conclusion stands, and what changed is the universal phrasing “never appears” (the discipline of Paper 17: saying “distinct roots” is only half, and the address must be written before the claim becomes checkable).
THIRD, THE ONLY THING FIXING a = 4pi IS ITEM (x), THE PLANCK CHARGE, AND THE 4pi THERE IS AN ELECTROMAGNETIC UNIT CONVENTION. Paper 116 showed the patch constant may be any positive a. What, then, inside this paper makes a = 4pi the choice? Item (x) and nothing else. With eps0 = c and hbar = 1/a: in SI (rationalised) units q_P = sqrt(4 pi eps0 hbar c) = 2 sqrt(pi) c / sqrt(a), a pure power of c precisely when a = 4pi. But in GAUSSIAN units the definition carries no 4pi and q_P = sqrt(hbar c) = sqrt(c)/sqrt(a), a pure power precisely when a = 1 (verified here symbolically). The 4pi inside the SI definition is placed there by hand, for the sake of rationalisation – a way of writing the units. SINCE WHICH VALUE LOOKS “FORCED” DEPENDS ON HOW ELECTROMAGNETISM IS WRITTEN, IT IS A CONVENTION AND NOT A FACT (Paper 95). Item (x) has a second problem as well: it rests on eps0 = c (the electromagnetic patch of Papers 13 and 14), and Paper 14’s third edition concluded that the switch itself survives while its two ends do not – eps0 = c being one of those ends. The third edition therefore HOLDS ITEM (x) IN ABEYANCE and removes it from the grounds of the dichotomy. That is why the table closes over nine quantities; the dichotomy stands without the charge.
FOURTH, 8 pi G/c^4 = 32 pi^2/c^4 IS A PRODUCT OF A FACT AND A CONVENTION. In general it reads 8 pi a/c^4, where the 8pi is A FACT (a number belonging to Einstein’s equation; twice Poisson’s 4pi) and the a is A CONVENTION (the patch constant; any positive value will do). Only on setting a = 4pi does one get 8pi x 4pi = 32 pi^2 = 2(4pi)^2 (verified here). But that is a fact multiplied by a convention, and in the resulting number no record survives of which pi came from where. This is the same shape Paper 43’s third edition exhibited, and it violates the discipline established by Paper 35’s third edition (the 4pi of Poisson’s equation is a fact and the 4pi of the G-patch a convention, and the two must not be gathered into one). THE CLAIM THAT THIS IS “THE SAME GEOMETRIC QUANTITY” AS IN PAPERS 32 AND 48 IS THEREFORE RETRACTED. Beyond the two coming out equal after substitution, this paper holds no ground for it.
FIFTH, THE GENERATION OF THE HIERARCHY IS CORRECT. Symbolic computation confirmed that F_P = E_P/l_P = c^4/a, P_power = E_P/t_P = c^5/a and P_Pl = E_P/l_P^3 = c^7/a all hold exactly. The second edition called this generation “evidence that the table is not a mere juxtaposition of numbers but carries structure”. The algebra is right, but the strength of the evidence is limited: E_P, l_P and t_P are all built from hbar, G and c, and the dimensions of force, power and pressure are fixed uniquely by products and quotients of them. THE GENERATION SUCCEEDS BECAUSE DIMENSIONAL ANALYSIS IS AT WORK, not because a hierarchy exists. The second edition itself drew this distinction (“since the exponent of c is fixed uniquely by dimensional analysis, the value of the exponent cannot itself be evidence for the hierarchy reading”). That self-limitation is correct and is kept unchanged.
THE SCOPE, RESTATED as ten rows. (1) Nine quantities close into c^N or c^N/a: a FACT (unchanged). (2) The rule deciding epsilon = 0 or 1: MASS DIMENSION (Paper 116). (3) The words “geometric / material”: ANOTHER NAME for mu = 0 / mu = 1. (4) That the power of 4pi never reaches 2: A PROPERTY OF THE LIST (hbar/G has epsilon = 2). (5) The value a = 4pi: a CONVENTION. (6) Item (x), the Planck charge: HELD IN ABEYANCE (Paper 14, 3rd ed.). (7) 32 pi^2 identical with Papers 32 and 48: RETRACTED. (8) Generation of the hierarchy: DIMENSIONAL ANALYSIS (correct, but not evidence). (9) That the exponent of c means a spatial dimension: NOT SHOWN (the self-limitation is kept). (10) c^5 as a 5D bulk and c^6 as a 6D phase space: RETRACTED (Papers 34, 42, 43). Of the ten rows, ONE is a fact proper to this paper – that the nine quantities close into two forms. This is not a shrinking but an addressing (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, dimensional analysis, symbolic computation). NO NEW THEOREM AND NO PREDICTION. What is not claimed: the exponent of c is not read as a spatial dimension (the second edition’s self-limitation is kept) / 4pi is not called a forced value (it depends on the electromagnetic units; Papers 95, 116) / “4pi never reaches the second power” is not asserted as a general proposition (hbar/G is a counterexample) / a factual pi and a conventional pi are not gathered into one (Papers 35, 43, 3rd eds.) / 32 pi^2 is not identified with the quantity in Papers 32 and 48 / results resting on eps0 = c are not treated as settled (Paper 14, 3rd ed.) / the generation of the hierarchy is not treated as evidence that the hierarchy is real / the cosmic-OS hierarchy reading is not placed in the body (Layer 3, quarantined; the c^5 and c^6 readings were retracted in Papers 34, 42, 43). The second edition’s correction of the fine-structure constant (alpha = e^2/c^2) is kept, but since it too rests on eps0 = c it is held in abeyance alongside item (x).
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 Power of $4\pi$ Is the Mass Dimension — “Zero or One” Is a Property of the List, and What Fixes $a=4\pi$ Is an Electromagnetic Unit Convention — [Paper 44] 3rd ed. [Paper 44]. Zenodo.
DOI: 10.5281/zenodo.21602231
ORCID: 0009-0006-8587-3450