Three Distinct Roots of Aggregation — the shell theorem, Barnes-Hut and the fast multipole method drop terms of the same expansion for three different reasons.
That a distant crowd of sources may be replaced by a few numbers: this licence to aggregate is used in at least three places — Gauss’s law, Newton’s shell theorem, and the Barnes-Hut (1986) and fast multipole (Greengard and Rokhlin 1987) methods of N-body computation. Paper 51 traced the licence to a single origin, stating that what grants it is the exponent two and that four pi does not survive into the answer. This paper measures how far that attribution reaches. The conclusion is that all three stand on the same expansion while dropping its terms for three different reasons. NO NEW PHYSICAL LAW AND NO NEW MATHEMATICAL THEOREM IS CLAIMED.
The boundary of the category was drawn by Paper 51 itself. Its Section 1.1 separated culling from aggregation. Finitising the source set by a causal horizon, and the significant-figure threshold set by the Planck displacement, are culling and not aggregation; Yukawa-type exponential screening was withdrawn for gravity by that paper’s Theorem 2.1. The three counted here all lie on the aggregation side of that line, which is why the count closes at three.
First, exactly one thing is genuinely shared, and it is the addition theorem. The Legendre expansion of the Laplace kernel separates the source variable from the evaluation variable, and all three aggregations use that separation. That much is shared. The term with l equal to zero yields one over r, which is what gives the potential its exponent one and the force its exponent two: the exponent lives inside the expansion as its subject, not as its licence. The parting comes over what to do with the remaining terms, and there the answer splits three ways.
Second, the addresses of the three roots are these. Gauss’s law requires harmonicity alone and no symmetry at all — the interior may be arbitrarily lopsided — but what it yields is a single number, a flux, and not the field, and it has no notion of error, being an identity rather than an approximation. This is where the exponent d minus one lives. The shell theorem requires harmonicity and symmetry, and in return its error is exactly zero (verified here: for a uniform distribution on a sphere of radius one, the surface average at distance ten integrates exactly to the monopole value, with zero relative deviation to machine precision). Barnes-Hut and FMM require neither harmonicity nor symmetry, only geometric separation and smoothness of the kernel, and in return their error is never zero.
Third, Paper 51’s attribution therefore reaches the shell theorem but not FMM. The particles in one cell of an octree are not spherically symmetric, so the shell theorem’s second condition fails and the theorem does not apply. Placing two equal masses along the line of sight and computing the monopole error exactly, the error is the quadrupole itself, falling as the square of the size-to-distance ratio but never reaching zero (the ratio converges as 1.0101, 1.0025, 1.0006, 1.0002; computed here). The shell theorem has no truncation order; FMM has one. The presence or absence of that variable is the signature that the two roots are distinct.
Fourth, FMM’s real licence is low-rank separability, not the exponent two. When a source box and a target box are well separated, their interaction block is numerically low-rank. At a relative threshold of ten to the minus eight, the numerical ranks are 27 for the three-dimensional Laplace kernel, 29 for the two-dimensional logarithm, 37 for the Yukawa kernel, 105 for a Helmholtz kernel, 51 for the inverse fourth power, 85 for a Gaussian, and 29 for the inverse square root (computed here). Every one of them compresses — a non-harmonic kernel, kernels whose exponent is not two, and two dimensions alike. All that is required is smoothness of the kernel away from the diagonal. Moreover there exist methods that achieve the same linear complexity without expanding the kernel at all, using only its values on equivalent surfaces: kernel-independent FMM (Ying, Biros and Zorin 2004) and the black-box FMM (Fong and Darve 2009). Since the algorithm runs without knowing the functional form of the kernel, there is no room for the exponent two to be the licence. The theory of hierarchical matrices (Hackbusch 1999) says the same for general asymptotically smooth kernels.
Fifth, conversely, Paper 51’s other claim can be proved from the side of complexity. Scaling a kernel by a nonzero constant scales every singular value by the same constant, leaving their ratios unchanged, so the numerical rank at a relative threshold is unchanged and the complexity of a hierarchical low-rank method does not depend on any overall factor of the kernel. Indeed, scaling the Laplace kernel by one, by four pi, by one over four pi, and by a million leaves the numerical rank at 27 in every case (verified here). A mechanism entering a normalised quantity only as an overall factor is by definition unverifiable, and four pi stands in exactly that position with respect to FMM’s complexity. So the claim that four pi does not survive into the answer holds from complexity as well, independently of the geometric cancellation argument: this paper does not weaken that claim but gives it a second leg. What does appear in FMM’s cost is not four pi but the term count for a given truncation order together with the 27 and the 189 that come from the cubic lattice of the octree — numbers of a lattice, not of a sphere.
Sixth, dimension enters by a different root as well. In a d-dimensional hierarchical tree the child count is two to the d and the interaction list has length six to the d minus three to the d (189 in three dimensions, 1215 in four, 1673055 in eight). FMM’s dimensional signature is exponential in d, which is why FMM is a method for two and three dimensions, the curse of dimensionality. The corpus’s dimensional signature, the exponent d minus one, is linear. One word, dimension, with two distinct addresses.
Two corrections follow. Paper 51’s abstract described the shell theorem as the ground on which Barnes-Hut and the fast multipole method compress the cost of N-body computation; that line is too strong, and the attribution reaches the shell theorem and stops there. This is the shape the corpus has named repeatedly — a correction that did not reach the theorem (Papers 49, 50, 51, 55). And FMM is linear in N, while N log N is Barnes-Hut; Paper 51 wrote the two together. They differ in root and in complexity alike.
One further separation of addresses. The exponent two of Paper 01 is the exponent of a density profile, while the exponent two of the shell theorem is the exponent of a force, and they do not refer to the same distribution. Indeed, a density falling as the inverse square encloses a mass proportional to the radius, whose field goes as one over r and not as one over r squared. Both statements are correct; they point at different things. As Paper 17 demanded, to say distinct roots is only half the work — the address must be written before the claim can be checked.
IMPORTANT (scope): a judgement of attribution concerning established standard facts (Layer 2). Gauss’s law, the shell theorem, the addition theorem of the multipole expansion, Barnes-Hut, FMM, and the low-rank theory of hierarchical matrices and kernel-independent FMM are all established. No new algorithm whatsoever. What is not claimed: any new physical law (there is none) / any new mathematical theorem (there is none) / that FMM’s complexity follows from the exponent two (counterexamples are listed) / that four pi affects the complexity (being an overall factor, it cannot) / that the universe is an N-body simulator (as Paper 51 already stated, outside that premise the complexity problem does not arise at all, since fields respond locally). What remains: to share one machine is not to share one root.
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). Three Distinct Roots of Aggregation [Paper 124]. Zenodo.
DOI: 10.5281/zenodo.22057728
ORCID: 0009-0006-8587-3450