Black-hole microstate counting in an asymmetric Z5 CHL model outside M24: positivity, blind growth tests and exact Rademacher expansions

We study the dyon counting functions of a four-dimensional N=4 string model whose symmetry lies outside the Mathieu group M_24: the CHL model built from the quantum Z_5 symmetry Q (Frame shape 1^-1 5^5, Conway class 5C) of the K3 model that Gaberdiel and Volpato obtain as an asymmetric Z_5 orbifold of Volpato's A_4-torus model, together with its non-cyclic Z_5 × Z_5 twisted-twining lifts. Reading these Siegel-type functions as (twined) quarter-BPS indices, normalised to leading coefficient 1, we run the black-hole consistency tests available at accessible charges, on exact coefficients and under pre-registration. Positivity: the single-centred index is strictly positive at all 237,378 charges tested for this model and for the geometric 5A control, without hair removal; this is large-scale evidence for the positivity conjecture beyond M_24, not a proof. Blind growth tests: on 29,047 fresh charges for the twined function and 19,749 fresh charge classes for the twining function, all predicted before the coefficients were computed, a coherent sum over the dominant zeros, with no fitted continuous parameter, gives the sign of every coefficient and its logarithm to about 0.02; this extends the coherent-saddle method of Banerjee-Jatkar-Sen and Chattopadhyaya-David to twined and twining lifts outside M_24 and, in such a model, makes Sen's discrete-information picture testable; the test confirms it through O(1). The first subleading term, 5/(π√Δ), is Banerjee-Jatkar-Sen's two-loop term rescaled by n_2 = 5 (an application of their result); in a later pre-registered test on fresh charges, with an independent recomputation, the residue series predicts the mean of ln|d| to 10^-5 (per-charge scatter 7·10^-4 for the twined function). Exact formulas: at every Fourier-Jacobi index m ≡ 2, 3 (mod 5) the coefficients are honest (not mock) Jacobi forms with no wall-crossing and no twisted polar part, and the single-centred index is given exactly by an I_{3/2} Rademacher series whose only inputs are twining-cusp polar coefficients (a theorem given the companion paper); procedurally blind tests reproduced 840 fresh coefficients at m = 2, 3 and 2,430 at the six honest m from 7 to 18, to at least 27.8 and 56 digits; this gives exact single-centred indices against which macroscopic entropy formulas can be compared. Half-BPS structure: we prove that the theta series of the rank-20 coinvariant lattice of the 5C Conway element is η^24 times an explicit combination of the half-BPS functions, and we confirm, on all tested charges, the mock-modular capture of single-centred indices by the finite parts of Fourier-Jacobi coefficients (Bhand-Sen-Singh, after Dabholkar-Murthy-Zagier) for this model and its twined lift. Walls, S-duality, hair and entropy: the non-cyclic lift has exactly the cyclic walls, with jumps that factorise into twined half-BPS indices; the indices are invariant under the expected S-duality shifts and under the Levi image of Γ_0(5); hair-removed positivity holds in both duality frames, unlike the Z_2 toroidal analogue; and the exact coefficients carry the parameter-free Fricke asymmetry predicted by the R^2 coupling, as do those of all 17 unbalanced Frame shapes (six of them in a disclosed, non-blind re-run after a coding error was corrected). All of this is low-charge consistency evidence for the dyon-index reading, not a proof of it; the absolute normalisation of the twisted index is stated as an open question.