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Equation 1 · Part 3 · A Cap Set Needed No World to Be True In, Only a Checker to Accept It

superscript

Z3n\mathbb{Z}_3^n
superscript

What this part means

A raised number can be a power. When it is a label or bound, it selects a case or the upper limit of a sum; the formula’s structure distinguishes these uses.

Its job in the formula

A raised mark can be a power or an index. Its position and the surrounding notation determine which.

The passage around this formula

FunSearch’s evaluator for the cap set problem has none of this, at any remove. The discovered function — reproduced in full in DeepMind’s own supplementary material — is nine lines long, and every one of those lines is arithmetic on the candidate point’s own coordinates: a dot product of the vector with itself, reduced modulo three; two overlapping partial sums treated the same way; a count of how many coordinates equal a fixed reference coordinate, reduced modulo three again; and a weighted sum of those quantities that ranks the point for inclusion [ 2 ] . Nothing in that function, or in the closure check that scores its output, refers to anything outside Z3n\mathbb{Z}_3^n — no chemistry, no…

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Learn the underlying idea

An exponent tells how a base is used in multiplication. In x³, x is the base and 3 is the exponent: x³ = x × x × x.

Open the illustrated exponents: repeated multiplication and powers guide →

Sources cited in the surrounding passage

These citations provide research context; check each source for the exact claim it supports.