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 composite the size$\sigma_n = |L(n)\cap[2,1000]|$ and for each witness the total count$\Lambda_a = \#\{n:\, a\in L(n)\}$. Where Monier's classical formulacontrols $|L(n)|$ inside the full unit group$(\mathbb{Z}/n\mathbb{Z})^{\times}$, the quantity $\sigma_n$ measuresinstead the intersection $L(n)\cap[2,A]$ of the strong-liar set withthe initial interval of small heights $[2, A]$ --- a\emph{small-height intersection} statistic that is the principalobject of study of this paper. The enumeration evaluates approximately$4.58\times 10^{13}$ Miller--Rabin instances in $4.62$ wall-clock dayson a single $24$-core workstation. The base-$2$ strong-pseudoprimecounts $\pi_{\spsp_2}(10^k)$ for $k\in\{6,8,9,10,11\}$ equal$46, 488, 1282, 3291, 8607$, matching OEIS sequence \texttt{A001262}.Three empirical findings emerge from the data. First, the worst fixedwitness in $[2,1000]$ at every scale is $a=256=2^8$, and perfectpowers dominate the witness ranking; we give a local Chinese RemainderTheorem (CRT) criterion characterising when $a=b^k$ is a strong liarin terms of the orders of $b$ modulo the prime-power factors of $n$,and use it (as a mechanism, not a complete explanation) to motivatethe empirical champion. Second, the worst composites pivot fromPomerance--Selfridge--Wagstaff semiprimes $p(2p-1)$ at $N=10^6$ to$3$-prime Carmichael numbers at the extreme tail, with $9$ of the top$10$ at $N\ge 10^{10}$ being Carmichaels; the data are consistentwith --- but do not establish --- an asymptotic transition. 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 show that Monier's bound on $|L(n)|$ is toocoarse to constrain $\sigma_{n}$ at this resolution (six orders ofmagnitude too large to be useful), and isolate the small-heightextremal problem\[\max_{n\le N}\,|L(n)\cap[2,A]|\]as the central open question raised by the data. As a partialstructural observation, we show that prime-square factors innon-perfect-power witnesses can produce order-halving under aLegendre-symbol non-residue condition on at least one prime factorof $n$.