A Local Rank Criterion for Strong Lucas Pseudoprimes and the Lucas-PSW Semiprime Family

We compute a multi-test census of small-height pseudoprimality liars, recording for
each odd composite $n$ the interval-local Miller--Rabin and Euler--Jacobi spectra
$\sigmamr(n;A),\sigmaej(n;A)$, the strong-Lucas grid spectrum $\sigmasl(n;A)$ over
$P\in[1,A]$, $Q\in\{\pm1\}$, and Baillie--PSW, Frobenius and extra-strong Lucas
indicators. A CUDA backend (RTX~5090), verified bit-for-bit against a CPU
reference, reaches $N=10^{11}$ at $A=1000$ in $68.8$ hours; it reproduces the prior
record-holder $3215031751$, the OEIS base-2 strong-pseudoprime counts ($8607$ below
$10^{11}$), and the empty Baillie--PSW survivor set below $10^{11}$.

The strong-Lucas-extremal composites are essentially disjoint from the MR-extremal
ones (all carry $\sigmamr=0$; at $10^{11}$ the top-$100$ sets share one composite),
and are dominated by the semiprime family $n=p(2p+3)$. We prove a local CRT
criterion for strong-Lucas pseudoprimality in terms of the rank of apparition (the
Lucas analogue of the corrected Miller--Rabin local-order criterion, independently
verified computationally). It identifies $q=2p+3$ as the coefficient-$2$ affine
family aligning the nonsplit $p+1$ rank with the split $q-1$ rank, sharpens in the
mixed character regime to an exact criterion (strong-Lucas $\iff v_2(\rho_p)=
v_2(\rho_q)$), and exhibits $\sigmasl(n;A)$ as a small-height sum of
fixed-discriminant slices of Arnault's strong-Lucas parameter counts.

The residue clustering $n\equiv77\pmod{100}$ among the strong-Lucas extremes is a
selection effect of $\sigmasl$ via the character $\left(\tfrac5n\right)$ and the
$p\bmod4$ structure, not a prime-pair density artifact. We give the unconditional
mechanism and a conditional average-order dominance theorem, and isolate the exact
analytic obstruction,  a Bateman--Horn and $2$-adic rank-equidistribution
input,  to a ranked-tail asymptotic; asymptotic claims beyond the enumerated range
are explicitly avoided.