Breaking the |τ| > 3 Barrier: The True Systole of the Gauss–Picard Modular Group PU(2,1;Zi)

We study the primitive loxodromic length spectrum of the Gauss–Picard modular group Γ =PU(2, 1; Zi) acting on complex hyperbolic 2–space H2C, and obtain two results. First, weenumerate the loxodromic conjugacy classes exactly up to a height bound: the characteristicpolynomial of g ∈ SU(2, 1) has the special form χg (λ) = λ3 − τ λ2 + ¯τ λ − 1 with τ = tr g,so the regular complex–conjugacy class is fixed by the trace alone—the loxodromic ones beingindexed by Gaussian integers τ with positive Goldman discriminant f (τ ) > 0—and the geometricinvariants (ℓ, φ) are read off the cubic. Enumerating integral J–orthonormal frames with thecorrected selector f (τ ) > 0 yields a systole ssys = 0.5915 attained at τ = 1 ± i, short–endtrace counts that saturate at 8/16/20 below spectral lengths 0.90/1.00/1.10, and shows thatthe discarded heuristic “|τ | > 3” misses 18 realized classes, the systole among them. Second,we prove an unconditional completeness statement: an elementary eigenvalue inequality boundsthe trace of any loxodromic of spectral length ≤ s, |τ | ≤ es + 1 + e−s, producing a finite explicitcandidate set; and a normalization–free entry bound |gij | ≤ |g33| together with an axis–to–corereduction certifies realizability. Because every candidate of spectral length ≤ 1.56 (= L∗) isrealized by an explicit Gaussian frame of height ≤ 13, the candidate set is exhausted and thetrace–spectrum below L∗ is complete unconditionally, comprising exactly 56 Gaussian–integertraces. The first unrealized candidate is τ = −4 (s = 1.567), whose certification is the precisefrontier of the R0–conditional extension. We fix conventions explicitly throughout: all reportedlengths are the normalization–free spectral length s = log |λmax|, equal to the Goldman (−1)translation length ℓ divided by two