An exhaustive strong-liar census for the Miller--Rabin primality test to N=10¹¹

We compute the strong-liar set $L(n)$ of the Miller--Rabin primalitytest for every odd composite n in [9, 10¹¹] and every witnessa in $[2, 1000]$, recording for each witness the total countΛₐ = #:\, a∈ L(n)\ and for each composite thesmall-height intersection σₙ = |L(n)∩[2,1000]|,the statistic relevant to fixed small-witness testing. Monier'sclassical formula is exact for $|L(n)|$ in the full unit group(Z/nZ)× but has no resolving power forσₙ: for our extreme composites $|L(n)|$ exceeds σₙby six orders of magnitude. The enumeration evaluates approximately4.58× 10¹³ 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=10⁶, 10⁸, 10⁹, 10¹⁰,10¹¹ (OEIS A055552) together with three furtherindependent checks. Three findings emerge. First, the worst fixedwitness in $[2,1000]$ at every scale is a=256=2⁸, and perfectpowers dominate the witness ranking; a local Chinese RemainderTheorem (CRT) criterion characterising when a=bᵏ 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 $10$ atN≥ 10¹⁰ being Carmichaels. Third, the record-holdern^ = 151· 751· 28351 with σn^ = 371remains globally maximal under extension from 10¹⁰ to10¹¹. We isolate the small-height extremal problemmaxn≤ N|L(n)∩[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.