The G and the c^5 of the Quadrupole Formula Cannot Be Separated: G/c^5 Is the Reciprocal of the Planck Power, and the Two 5s Have Different Origins. Third edition.
This paper carried an expression obtained by substituting G = 4pi into the gravitational-wave quadrupole formula P = (G/5c^5) avg[Qddd_ij Qddd_ij]. The third edition first corrects an arithmetic error contained in that rewriting, and then restates what the substitution does and does not reveal.
FIRST, THERE IS AN ERROR TO CORRECT. The second edition factored the coefficient as 4pi/(5c^5) = 4pi x 5/c^5, and also wrote P = 4pi (c^5/5)^-1 Qddd^2. BOTH ARE WRONG: (c^5/5)^-1 = 5/c^5, which is NOT 1/(5c^5), and THE TWO DIFFER BY A FACTOR OF 25 (2.594635e-41 against 1.037854e-42 with c at its SI value; verified here). The correct factorisation is 4pi/(5c^5) = 4pi x (1/5) x c^-5, a product of THREE factors; the quantity 5/c^5 does not occur. THE ERROR WAS REPEATED IN FOUR PLACES (twice in Section 2, once inside the proof, once in Section 5). However, the formula as written in the abstract, the scope statement and the conclusion was correct, so the conclusions are unaffected – but since the factorisation was wrong, whatever structure was read off it must be read again. The square on the right was also made explicit as the tensor contraction avg[Qddd_ij Qddd_ij].
SECOND, AND THE CORE OF THIS EDITION. G and c^5 cannot be separated inside this formula. What appears is the SINGLE COMBINATION G/c^5, which is nothing other than the reciprocal of the PLANCK POWER P_Pl = c^5/G = 3.628255e52 W. That is, G/(5c^5) = (1/5)(1/P_Pl), so P = (1/5) avg[Qddd_ij Qddd_ij] / P_Pl (verified here: G/(5c^5) = 5.512292e-54 and 1/(5 P_Pl) = 5.512292e-54, exactly equal). G NEVER APPEARS ALONE ANYWHERE. This weakens the second edition’s reading that source and geometry separate: they appeared to separate only because 4pi and c^-5 were written out separately AFTER the substitution, whereas in the original formula the two are one.
THIRD, THE SUBSTITUTION THEREFORE REVEALS NO STRUCTURE. Setting G = a makes the coefficient a/(5c^5), and the ratio to the true value is EXACTLY a/G (1.8828e11 for a = 4pi, 1.4983e10 for a = 1, 1.0937e11 for a = 7.3; verified here). That the ratio is proportional to a means the substitution rescales the whole formula uniformly; NO FACTOR OF 4PI EMERGED FROM WITHIN THE STRUCTURE. As Paper 116 showed, the patch constant may be any positive value, so choosing 4pi is A CONVENTION and not a fact (Paper 95); the reading that “a solid angle appears” does not follow from this formula. Setting G = 4pi is a choice of unit system, and in that system c does not carry its SI value either, so the numbers above are a test of “what happens if G alone is moved while c is held at its SI value”. What the test showed is one thing: however a is chosen, only the overall scale changes, and the bond between G and c^5 does not come undone.
FOURTH, THE TWO 5s HAVE DIFFERENT ORIGINS (the second edition’s verdict, now with numerical checks). The 5 of c^5 is fixed uniquely by DIMENSIONAL ANALYSIS: with [P] = M L^2 T^-3, [G] = L^3 M^-1 T^-2 and [Qddd]^2 = (M L^2 T^-3)^2, the length equation 3 + N + 4 = 2 and the time equation -2 – N – 6 = -3 both give N = -5 (the mass equation -1 + 2 = 1 fixes the power of G at 1 and is consistent). The 1/5, by contrast, arises from an ANGULAR AVERAGE over the sphere: avg[n_i n_j n_k n_l] = (1/15)(d_ij d_kl + d_ik d_jl + d_il d_jk), verified here by Monte Carlo over four million points on the unit sphere – avg[n1^4] = 0.199902 (theory 1/5 = 0.2), avg[n1^2 n2^2] = 0.066651 (theory 1/15), avg[n1^3 n2] = 0.000087 (theory 0) – with a maximum deviation of 9.8e-5 over the six components tested. NEITHER IS A COUNT OF SPATIAL DIMENSIONS.
FIFTH, THE CITATION TO PAPER 34 IS UPDATED. The second edition correctly recorded that the derivation in Paper 34 (obtaining c^5/5 from the integral of E^2 dc) had been retracted, and that verdict is kept. However, the second edition still cited Paper 34 under its RETRACTED TITLE – “5D Dimensional Leakage Theory: Unpacking (16 pi^2/5) c^5 and the Randall-Sundrum Bulk” – and what that title names is precisely what was retracted. Paper 34’s third edition has since completed a census of the c^5s, recording that the 5 and the 1/5 of the quadrupole formula are DISTINCT ROOTS and concluding that the c^5 of the Planck power, of the quadrupole formula and of the Planck time all fold into the SINGLE ROOT c^5/G. This third edition is the quadrupole instance of that census. The second edition’s verdict that “the G/(5c^5) of the quadrupole formula is determined independently by general relativity and needs no grounding in another paper” is kept unchanged; what this edition adds is only that the genuine coefficient HAS A NAME – one fifth of the reciprocal Planck power.
THE SCOPE, RESTATED as eight rows. (1) The quadrupole formula itself: an ESTABLISHED FACT (Einstein 1918; not altered here). (2) G/(5c^5) = 1/(5 P_Pl): an IDENTITY. (3) The expression with G = 4pi substituted: a CONVENTION. (4) That 4pi separates out of the structure: NOT SHOWN. (5) The 5 of c^5: a consequence of DIMENSIONAL ANALYSIS. (6) The 1/5: a consequence of an ANGULAR AVERAGE. (7) That 5 is a spatial dimension: NOT SHOWN. (8) New physics of gravitational waves: NONE. Of the eight rows, NOT ONE CARRIES A CLAIM PROPER TO THIS PAPER – one established formula, one identity, one convention, and the rest “not shown”. This is not a shrinking but an addressing (Paper 17).
IMPORTANT (scope): a structural interpretation and self-audit of established standard facts (Layer 2). The physics used is entirely standard (Einstein’s quadrupole formula, dimensional analysis, tensor averages over the sphere, the Planck power). NO NEW THEOREM AND NO PREDICTION. What is not claimed: the exponent 5 is not read as a count of spatial dimensions (dimensional analysis and an angular average; retracted in the second edition) / 4pi is not called a forced value (any patch constant will do; Papers 116, 95) / a substitution is not called a discovery (a rewriting following from a choice of units) / the reading that 4pi separates out on the source side is not asserted for this formula (G and c^5 are one) / Randall-Sundrum and extra dimensions are not touched (that reading was retracted and is not reinstated) / the quadrupole formula itself is not altered / the Planck power is not read as a physical upper bound or maximum luminosity (a separate discussion, not treated here) / Layer-3 lenses (the “cosmic OS” and the like) are kept out of the body (quarantined under 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). The $G$ and the $c^5$ of the Quadrupole Formula Cannot Be Separated — 3rd ed.: in the Yamagishi 5D Gravitational Wave Discharge Theory, $G/c^5$ is the reciprocal of the Planck power, and the two $5$s have different origins — [Paper 42]. Zenodo.
DOI: 10.5281/zenodo.21601988
ORCID: 0009-0006-8587-3450