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Source. Published pp. 259–266, 272, and 285–286 (PDF).

Harper's vertex-isoperimetric theorem, external. In the Boolean cube, among sets of a given size the initial segment in simplicial order has the smallest closed Hamming neighborhood of every integer radius. The specific consequence used here is: if

$$|\mathcal A|>\sum_{j=0}^{a-1}\binom nj,$$

then its closed radius-$t$ neighborhood contains at least $\sum_{j=0}^{a-1+t}\binom nj$ vertices, with sums truncated at $n$. The source cites L. H. Harper, Optimal numberings and isoperimetric problems on graphs, J. Combinatorial Theory 1 (1966), 385–393. The original isoperimetric proof is not compiled in this source unit. Its exact role is in theorem_2_1.

Elementary analytic inputs expanded here. Binomial tails, uniform factorial estimates, product-measure concentration, and density-loss bookkeeping are proved in entropy_estimates and product_measure_separation. They are not additional unproved probability inputs.

Historical background, not dependencies of the joint-pattern proof. The introduction quotes Katona's intersection theorem, Erdős–Ko–Rado, Frankl–Wilson, Frankl–Füredi, packing asymptotics, and bounds for codes. Their original proofs and the quoted sharp numerical constants are not reconstructed here. Theorem 3.2 on p. 272 is an explicitly cited alternative from Frankl's 1976 paper, not an indispensable same-paper lemma: our weighted proof uses Theorem 3.1 instead. The older Ramsey assertions are linked to their canonical sources where used; no contemporary status is inferred from the 1987 introduction.

Geometry. Rotation averaging uses the normalized invariant probability measure on the orthogonal group, and its push-forward to uniform measure on a sphere. Existence and invariance of this measure are standard external measure-theoretic inputs. All subsequent averaging deductions are written out on the relevant result page.