Three ceilings over one inequality – and no two are held down by the same thing.
Three upper bounds are spoken of for the CHSH quantity: the classical two, the quantum two root two, and the algebraic four. This paper calls all three by name and shows that no two of them are held down by the same thing. NO NEW PHYSICAL LAW IS CLAIMED. Bell’s inequality, CHSH, the Tsirelson bound, the PR box and the no-signalling condition are all standard. Nor does this paper answer why nature stops at two root two – that is an open research question, and what can be written is the address, and no further.
First, the classical ceiling is combinatorics. A local deterministic strategy merely fixes four signs in advance, and there are only sixteen of them. Exhausting all sixteen, the CHSH quantity falls between minus two and two, with a maximum absolute value of exactly two (computed here). Mixing cannot raise it: a probabilistic mixture is a convex combination of these extreme points, so the maximum is attained at one of them. The bound of two therefore follows from the assumption of local determinism alone, by counting signs. Its address is combinatorics.
Second, the quantum ceiling comes out of density matrices. Measuring a singlet along zero, ninety, forty-five and minus forty-five degrees, the four correlations all have absolute value 0.7071067811865475 and the CHSH quantity is 2.8284271247, differing from two root two by 4.44 times ten to the minus sixteen (computed here). Drawing both the two-qubit state and the four measurement directions at random forty thousand times, the bound was never once exceeded, the largest value found being 2.7291936631. Random directions rarely land near the optimal configuration, which is why the search stopped there; all this paper takes from the trial is that the bound was not exceeded, and it is not a proof of the bound.
Third, and this is the core. What stops quantum mechanics there is geometry. For the singlet the correlation is minus the inner product of two direction vectors, so the CHSH quantity is bounded above by the length of b plus b prime added to the length of b minus b prime. Maximising that over two unit vectors gave 2.8284271247 across two hundred thousand random pairs, attained exactly when the two are orthogonal. The reason fits on one line: the sum of the squares of those two lengths is fixed at four, and the sum of two numbers with a fixed sum of squares is largest when they are equal, whence the bound of two root two. The address of two root two is therefore the inner product, and the Bloch-sphere geometry of Paper 68 appears here unchanged. Where the classical two counted signs, this one measures lengths – distinct roots.
Fourth, the brake. What forbids four is not relativity. The PR box has correlations plus one, plus one, plus one and minus one, and reaches exactly four. Yet writing out its marginals for each of the other party’s settings, they remain 0.500 whatever either side chooses – the PR box does not signal (verified here). Relativity therefore permits four, and the explanation that it stops at two root two because of relativity does not explain that. What stops it is the state space. Locality was already surrendered above two, and no-signalling allows all the way to four; what remains is the single fact that the correlations are inner products of unit vectors, and that gives two root two.
The census. The ceiling of two is held by local determinism, moved by allowing non-locality, and lives in combinatorics. The ceiling of two root two is held by the quantum state space, moved only by changing the inner-product geometry, and lives in geometry. The ceiling of four is held by outcomes drawn from plus and minus one, moved only by changing the range of outcomes, and lives in arithmetic. One inequality carries three ceilings and no two are held down by the same thing. The sentence that Bell’s inequality has been violated is a statement about the first ceiling only.
And the two and the root two appearing here are distinct from all five twos counted in Paper 157: here the two comes from arrangements of signs and the root two from the sum of two numbers with a fixed sum of squares. The census of that one word is not finished.
IMPORTANT (scope): a census of established standard material (Layer 2). What is not claimed: any new physical law (there is none) / an answer to why nature stops at two root two (open). What is added: obtaining the classical ceiling by exhaustion, obtaining the quantum one from density matrices and confirming non-exceedance by random search, exhibiting the source of two root two numerically as the geometry of an inner product, and checking that the PR box reaches four while not signalling, which fences the explanation that relativity is what stops us. What remains: once one notices that there are three ceilings, the word violated turns imprecise. What was violated is the first. The second has never once been broken, and the third stands empty in a place relativity allows, with nobody living there.
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 Ceilings over One Inequality [Paper 160]. Zenodo.
DOI: 10.5281/zenodo.22092434
ORCID: 0009-0006-8587-3450