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 foreach odd composite n the interval-local Miller--Rabin and Euler--Jacobi spectra(n;A),(n;A), the strong-Lucas grid spectrum (n;A) overP∈[1,A], Q∈\±1\, and Baillie--PSW, Frobenius and extra-strong Lucasindicators. A CUDA backend (RTX~5090), verified bit-for-bit against a CPUreference, reaches N=10¹¹ at $A=1000$ in $68.8$ hours; it reproduces the priorrecord-holder $3215031751$, the OEIS base-2 strong-pseudoprime counts ($8607$ below10¹¹), and the empty Baillie--PSW survivor set below 10¹¹. The strong-Lucas-extremal composites are essentially disjoint from the MR-extremalones (all carry =0; at 10¹¹ the top-$100$ sets share one composite),and are dominated by the semiprime family $n=p(2p+3)$. We prove a local CRTcriterion for strong-Lucas pseudoprimality in terms of the rank of apparition (theLucas analogue of the corrected Miller--Rabin local-order criterion, independentlyverified computationally). It identifies $q=2p+3$ as the coefficient-$2$ affinefamily aligning the nonsplit $p+1$ rank with the split $q-1$ rank, sharpens in themixed character regime to an exact criterion (strong-Lucas v₂(ρₚ)=v₂(ρq)), and exhibits (n;A) as a small-height sum offixed-discriminant slices of Arnault's strong-Lucas parameter counts. The residue clustering n≡77100 among the strong-Lucas extremes is aselection effect of via the character (5n) and thep4 structure, not a prime-pair density artifact. We give the unconditionalmechanism and a conditional average-order dominance theorem, and isolate the exactanalytic obstruction, a Bateman--Horn and $2$-adic rank-equidistributioninput, to a ranked-tail asymptotic; asymptotic claims beyond the enumerated rangeare explicitly avoided.