The inside of a shell is not flat because of the mean value theorem – repair the reason, not the theorem.
An explanation is often repeated: that Newton’s shell theorem is really Gauss’s mean value theorem. This paper fences that explanation and writes instead what does force the constant. NO NEW MATHEMATICAL THEOREM IS CLAIMED, and nothing here contradicts the shell theorem itself: the conclusion is untouched and only one line of the reason is corrected.
First, and this is the core. The mean value theorem is true and far too weak. Take Phi equal to z. It is harmonic, its average over a sphere of any radius is zero, and its value at the centre is zero. And yet Phi is not constant inside, since Phi at the point zero, zero, one is one. The mean value property belongs to every harmonic function, and a property common to all of them cannot single out one, the constant. So the explanation that the shell theorem is really the mean value theorem does not reach the conclusion it is offered for.
Second, what is actually at work is three things: harmonicity, spherical symmetry, and regularity at the origin. A spherically symmetric harmonic function in d dimensions satisfies a second-order ordinary differential equation, whose solutions are C1 plus C2 over r at d of three, C1 plus C2 over r squared at four, C1 plus C2 over r cubed at five and C1 plus C2 over r to the fourth at six. In every case the family is A plus B times r to the two minus d. Boundedness on a region containing the origin forces B to vanish, and what remains is a constant. That is why the inside of a shell is flat, and the mean value theorem is not among the ingredients. There is another route: if the boundary values are constant and the function is harmonic inside, the maximum principle makes it constant inside. That route uses a consequence of the mean value theorem, so the two are cousins, but what does the work is the stronger statement that neither a maximum nor a minimum can be attained inside.
Third, it was checked by direct computation. For a uniform shell of radius one, the interior and exterior values were integrated numerically with each dimension’s own harmonic kernel. At d of three, four and five alike, the four interior points at 0.1, 0.3, 0.6 and 0.9 all give 1.0 with a spread of zero, and the exterior agrees with a point mass, the worst difference being 3.081 times ten to the minus thirty-three (computed here). The shell theorem is therefore not peculiar to three dimensions: with each dimension’s harmonic kernel it holds in every dimension.
Fourth, the separator is the exponent. At d of four, change only the kernel. With the harmonic kernel r to the minus two the four interior points are all 1.0 and the spread is zero. With an inverse square imported from three dimensions, r to the minus one, the interior values become 0.998748, 0.988620, 0.952703 and 0.884177, and the spread jumps to 0.1146. Shift the exponent by one and the interior stops being flat. What must be moved is the exponent, and the licence is harmonicity, not the number two.
Fifth, it follows that Paper 51’s n equals two and Paper 01’s n equals d minus one are one licence stated in two ways. When the force exponent is r to the minus d minus one, the potential is r to the minus d minus two, a logarithm at d of two, and in every dimension it satisfies Laplace’s equation. The first is the statement at d of three, the second for every d. And what both really require is not the exponent itself but the equation it leads to.
IMPORTANT (scope): a census of established standard material (Layer 2). What is not claimed: any new mathematical theorem (there is none) / anything against the shell theorem (nothing). What is added: fencing the it-is-the-mean-value-theorem explanation by counterexample, writing out the three conditions that do the forcing, showing in numbers that a wrong exponent leaves the interior not flat, and bridging Paper 51’s statement to Paper 01’s as one licence. What remains: a correct theorem can be carrying an incorrect reason. When it is, the thing to repair is the reason, not the theorem. And repairing it makes visible that two statements from neighbouring rooms, n equals two and n equals d minus one, were one and the same licence all along.
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 Inside of a Shell Is Not Flat Because of the Mean Value Theorem [Paper 156]. Zenodo.
DOI: 10.5281/zenodo.22091565
ORCID: 0009-0006-8587-3450