Membership of the orientation mixture in the separated alternatives
→ averaging a fixed test’s errors over separated components. With
d comparison coordinates and opposite orientation vectors
b,-b, the laws (1+epsilon b)/d and
(1-epsilon b)/d average to the uniform IIA law
1/d; positive component separation need not survive mixing.
Some component has error at least the average. Each component is
delta-separated when
epsilon>=2 mu(sigma) delta, where mu(sigma)
is the cycle decomposition’s mean length. The same
4 n delta sufficient condition and testing conclusion
follow, with n the number of items.
Theorem 1 perturbation range
Theorem 1’s display → explicit range
2 mu(sigma) delta <= 1. The source defines
perturbations only for epsilon in [0,1]; its
proof chooses epsilon=2 mu(sigma) delta, where
mu(sigma) is mean cycle length and delta the
testing separation. This states the witness’s valid range; no
counterexample to a broader lower bound is supplied.
Le Cam coefficient
in the proof of Theorem 1
Coefficient 1/4 before total variation in the first
proof line → 1/2. Applying
TV <= (1/2)sqrt(chi^2) then gives the subsequent
1/4 coefficient used by the theorem.
Appendix
comparison-incidence bound
The rational mean-cycle-length branch without a denominator
condition → require d-4n+8 log_2(n)>0, where
d is the number of comparison-incidence edges and
n the number of items. The checked result states that case
condition explicitly while retaining the unconditional 2n
branch. The real-logarithmic simplification used for the printed
dispersion bound is checked for n ≥ 4.
All-even-subsets endpoint
The appendix corollary prints the range n ≥ 2. At
n = 2, the relevant comparison-incidence graph is not
Eulerian, so the stated construction does not apply. The checked theorem
begins at n = 3, proves the n = 3 case
directly, and proves all n ≥ 4 cases from the general
construction.