An exhaustive strong-liar census for the Miller--Rabin primality test to $N=10^{11}$

We compute the strong-liar set $L(n)$ of the Miller--Rabin primalitytest for every odd composite $n$ in $[9, 10^{11}]$ and every witness$a$ in $[2, 1000]$, recording for each witness the total count$\Lambda_a = \#\{n:\, a\in L(n)\}$ and for each composite the\emph{small-height intersection} $\sigma_n = |L(n)\cap[2,1000]|$,the statistic relevant to fixed small-witness testing. Monier'sclassical formula is exact for $|L(n)|$ in the full unit group$(\mathbb{Z}/n\mathbb{Z})^{\times}$ but has no resolving power for$\sigma_n$: for our extreme composites $|L(n)|$ exceeds $\sigma_n$by six orders of magnitude. The enumeration evaluates approximately$4.58\times 10^{13}$ Miller--Rabin instances in $4.62$ wall-clockdays on a single $24$-core workstation, and is validated againstthe base-$2$ strong-pseudoprime decade counts$46, 488, 1282, 3291, 8607$ at $N=10^{6}, 10^{8}, 10^{9}, 10^{10},10^{11}$ (OEIS A055552) together with three furtherindependent checks. Three findings emerge. First, the worst fixedwitness in $[2,1000]$ at every scale is $a=256=2^8$, and perfectpowers dominate the witness ranking; a local Chinese RemainderTheorem (CRT) criterion characterising when $a=b^k$ is a strong liarsupplies the mechanism, though not a complete prediction of theranking. Second, the worst composites pivot fromPomerance--Selfridge--Wagstaff semiprimes $p(2p-1)$ to $3$-primeCarmichael numbers in the extreme tail, with $9$ of the top $10$ at$N\ge 10^{10}$ being Carmichaels. Third, the record-holder$n^\star = 151\cdot 751\cdot 28351$ with $\sigma_{n^\star} = 371$remains globally maximal under extension from $10^{10}$ to$10^{11}$. We isolate the small-height extremal problem$\max_{n\le N}|L(n)\cap[2,A]|$ as the central open question raisedby the data, and prove two structural mechanisms: the local CRTcriterion, and prime-square order halving under a Legendre-symbolnon-residue condition.