This paper formally integrated E²=(4πc²)²+(4πvc)² with respect to the speed of light c and, on the strength of the resulting (16π²/5)c⁵, claimed a "complete proof" of the Randall–Sundrum model and a "5D geometric leakage theorem". The second edition withdrew both in full, leaving only an observation about the exponent structure c⁵. The 3rd edition turns that observation into a census.
First, the exponent 5 of the quadrupole formula is forced. In P = (G/5c⁵)·Q⃛ᵢⱼQ⃛ᵢⱼ, counting dimensions gives [P]=ML²T⁻³ and [Q]=ML², so [Q⃛]=ML²T⁻³ and [Q⃛²]=M²L⁴T⁻⁶, while [G]=M⁻¹L³T⁻²; hence [G·Q⃛²]/[P] = (ML⁷T⁻⁸)/(ML²T⁻³) = L⁵T⁻⁵ = (L/T)⁵. The moment one decides to write the radiated power in terms of G and Q⃛, no exponent other than 5 is possible. There is no room for choice — it is a "forced" number in the classification of the 3rd edition of Paper 38.
Second, the 5 and the 1/5 inside that same formula are distinct roots. The exponent 5 comes from dimensional analysis, but the coefficient 1/5 arises from averaging the radiation over the full solid angle. The fourth-order angular average of a unit vector is ⟨nᵢnⱼnₖnₗ⟩ = (1/15)(δᵢⱼδₖₗ+δᵢₖδⱼₗ+δᵢₗδⱼₖ), verified here by quadrature over the sphere: ⟨n₁⁴⟩=0.199999 (exactly 1/5), ⟨n₁²n₂²⟩=0.066666 (exactly 1/15), ⟨n₁³n₂⟩=0.000000. Contracted through the transverse-traceless projection, this average yields the 1/5. The 1/5 therefore comes from an integral over the sphere — geometry of the same family as 4π. The two stand side by side in one expression but come from different places; this is the same work as Paper 48's separation of the three factors of 2 in Gauss–Bonnet and the 3rd edition of Paper 17 giving each of three twos an address, and the 5 and the 1/5 of G/(5c⁵) must not be read as "the same 5".
Third, this paper's own 5 and 1/5 come from one and the same operation. The (16π²/5)c⁵ it produced comes from ∫c⁴dc = c⁵/5, where the rise of the exponent by one and the appearance of the coefficient 1/5 are two faces of the same antiderivative — one root. The quadrupole formula has two roots; this paper has one. That the numbers "5 and 1/5" look aligned is precisely because the breakdowns differ, and looking aligned is exactly the reason to inspect the breakdown.
Fourth, four of the c⁵s in this series share one root. We have [c⁵/G] = (L/T)⁵/(M⁻¹L³T⁻²) = ML²T⁻³ = power, with the value P_Pl = c⁵/G = 3.6283×10⁵² W (computed here). What deserves notice is that this is the one Planck quantity containing no ħ: length, time and mass all require ħ, but power alone closes with G and c. And: the Planck power (Paper 44) is P_Pl = c⁵/G itself; the quadrupole formula (Paper 42) is P = Q⃛²/(5P_Pl), the same c⁵/G in reciprocal; the Planck time (Paper 40) is t_p = √(ħG/c⁵) = √(ħ/P_Pl), the same c⁵/G with ħ on top; and the causal-sphere entropy of Paper 65 is S = A/(4ℓ_p²) with A=4π(ct)² and ℓ_p²=ħG/c³, giving S = πc⁵t²/(ħG) = π(P_Pl/ħ)t², which with ħ=G=t=1 yields S=πc⁵. Numerical check: √(ħ/P_Pl)=5.3912×10⁻⁴⁴ s agrees with t_p to machine precision. These four are therefore not "different quantities sharing a number"; they are one quantity. This series usually splits one number into distinct roots; here, conversely, four fold into one.
Fifth, exactly one c⁵ does not belong to that family — this paper's. The (16π²/5)c⁵ contains no G, and its dimension [c⁵]=L⁵T⁻⁵ is neither a power, nor a time, nor an entropy. The four belong to the family of the physical quantity c⁵/G; this paper's is an artefact of calculus. Two different things stand under one notation "c⁵". The second edition wrote that it "shares the same exponent structure"; the 3rd edition withdraws that — it does not share it. The exponents agree because one was forced by dimensional analysis and the other was raised by one through integration; they did not become 5 for the same reason. Nor, as the second edition already held, is there any ground for reading the exponents 5 and 3 as the spacetime dimensions 5 and 3 — an exponent and a count of dimensions belong to different categories (the same distinction drawn by the 3rd editions of Papers 28 and 06).
Finally, a concrete weight. The peak power radiated at the merger of GW150914 is reported as about 3.6×10⁴⁹ W. Computing the ratio here, P_GW/P_Pl = 3.6×10⁴⁹/3.6283×10⁵² = 9.92×10⁻⁴ ≈ 1/1000: an event in the universe reached one thousandth of the Planck power. And what matters is that this is a dimensionless ratio — it is the same in every system of units (3rd edition of Paper 38: dimensionless ratios are the physics). The c⁵ itself changes with the units; this 10⁻³ does not. That this paper was drawn to c⁵ in the first place was not the mistake; the mistake was trying to manufacture one. The c⁵ was already there, carrying G with it, standing as the scale of power. All numbers were computed here (no external tables consulted). This paper contributes neither to the confirmation nor to the refutation of the RS model, and claims no new physical law or prediction.
(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). A Census of the $c^5$s [Paper 34]. Zenodo.
DOI: 10.5281/zenodo.21430340
ORCID: 0009-0006-8587-3450