---

The complete source chain uses the following external results. Their exact interfaces and applications are recorded here; their proofs are outside this source unit.

1. Frankl--Wilson forbidden-intersection bound. The exact form is Theorem 2: when $k$ is a prime power and $n=4k$, a family of $n/2$-subsets of $[n]$ with no distinct pair intersecting in $n/4$ elements has size at most $2\binom{n-1}{n/4-1}$. Kahn and Kalai state and attribute this result on physical PDF p. 2 (journal p. 61). The original 1981 proof has not been recursively checked here.

2. Stirling's formula. As $n\to\infty$ through positive integers,

$$n!=\sqrt{2\pi n}\,(n/e)^n(1+o(1)).$$

The proof uses only its fixed-density binomial consequence

$$\log\binom n{\alpha n}=nH(\alpha)+O(\log n), \qquad H(\alpha)=-\alpha\log\alpha-(1-\alpha)\log(1-\alpha),$$

for $\alpha=1/2$ and $1/4$ along multiples of four.

3. Prime number theorem. The form used is the consequence that if $p(x)$ is the largest prime at most $x$, then $p(x)/x\to1$ as $x\to\infty$. Equivalently, for every $\varepsilon>0$, every sufficiently large $x$ has a prime in $[(1-\varepsilon)x,x]$. Kahn and Kalai explicitly invoke the prime number theorem in the last sentence of Section 2 on physical PDF p. 2.

The incidence-vector distance calculation, the affine-dimension bound, the binomial simplification, monotonicity under Euclidean embedding, and the conversion from covers of a finite configuration to partitions are proved directly in this source chain and require no further imported theorem.