Hexadecimal games (GONC5 A3): exceptional values form cascades — ·b33b law G(n_{k+1})=n_k verified to 250000 with two new members, a self-stopping criterion, three Howse–Nowakowski Table-4 errata, and a 76-game negative sweep

Engine validation: two independent implementations (hexref.c O(N^3) reference; hexfast.c pair-XOR-bitset engine) agree on 48 game codes to n=500 and reproduce every published hexadecimal benchmark tested: the ·2048 21-value table; ·123456789 structure incl. G(2^k+6)=2^{k-1}; ·28=·29 (period 53, saltus 16); ·9c (36,16, preperiod 28); ·205200c sapp structure incl. G(40k+19)=6, G(40k+39)=14; ·f6 (604,43,32) on its initial range; and the ·7f census EXACTLY: 14 exceptional values for n≤100000, largest G(94156)=26614 (digit-for-digit match with GONC5 A3).

Structural obstruction (why ·f6-class games resist): Austin/Thm4 condition (2) requires max G on [0,e] < s while arithmetic-periodic values grow at slope s/p, so certifiability forces preperiod e ≲ period p, and the requirement is invariant under the (λp, λs) scaling. Any open game whose preperiod exceeds its period by more than the slope margin (e.g. ·f6, e=604 vs p=43) is beyond BOTH printed criteria at any computational depth; new criteria are needed for the long-preperiod class.

Negative sweep: all 76 open 2-digit family games of the GONC5 A3 list (·1x, ·2x, ·3x, ·4x, ·5x, ·6x, ·7x, ·9x, ·bx, ·dx, ·fx members as listed) were computed exactly to n=50000; NONE has an exact arithmetic-periodic tail with period p ≤ 4000 (per-game maxima and candidate scans in sweep_results.json). Additional bounded-value resisters: ·b6 has maxG=3891 at 1e5 with no period p ≤ 40000 (96% mismatch at every shift); ·f7 has maxG=1027 at 5e4 with no period p ≤ 20000. Structure censuses: ·9 P-positions exactly {0,2,3,4} to 1e5 (maxG 49662, 45841 distinct values); ·9d P-positions {0,3} to 1e5; ·e P-positions {0,1} to 3e4.

·b33b cascade law (new structure): the exception set E = {n : G(n) ≠ n} up to 250000 has exactly 34 members; sorted n_1<...<n_34, G(n_1)=G(5)=0 and G(n_{k+1}) = n_k for every k — the 'random' heap sizes of GONC5 A3 form a single chain. Two members are new beyond the published 1e5 horizon: G(159854)=96975 and G(217419)=159854; the first landed inside the prediction window (155k-185k) committed to the local git history (commit b8b3d25, b33b_proof.md) before the 250k extension run completed. Count discrepancy documented: 32 exceptions ≤ 1e5 vs GONC5's stated 27 (equal if the five below n=11 are excluded); dual-engine agreement on [0,3500].

Self-stopping criterion for ·b33b (proved, conditional): G(n) ≤ n always; G(n)=n iff all v<n are option values; cancellation splits (t,t,c) of the opposite-parity remainders n-1, n-4 realize every non-exceptional value; a two-high-bits construction (Lemma 4) realizes any remaining value whenever t=(m-v)/2 has two set bits above log2(max E), and a bit-transplant lemma reduces the degenerate-t cases to a finite table. Consequence: if computation to horizon N ≥ max(E) + 2^(floor(log2 max E)+3) + 8 shows no exceptions beyond max(E) (plus the finite table checks), then E is complete and ·b33b is arithmetic-periodic with period 1, saltus 1. At the current horizon the hypothesis fails (needed: 1266003; cascade alive), so ·b33b remains open; the criterion also bounds successive-exception ratios by ≈5, matching the observed 1.3-3.0.

·7f obeys the same cascade law relative to its n/2 trend: the outliers |G(n)-n/2|≥8 up to 250000 are exactly 16; each outlier value approximates half the previous outlier position within ±3 (exact in several links, e.g. G(192052)=75699=151398/2, G(39588)=12059=24118/2). The two outliers beyond the published 1e5 horizon are new: G(151398)=47081, G(192052)=75699.

Errata in Howse–Nowakowski, TCS 313 (2004), Table 4 (three rows contradict the paper's own rule definitions): ·338 has (preperiod, period, saltus) = (0,17,6), not (0,9,3) — hand-derivable: the options of heap 9 are {G(8)=2, G(7)=1, 1⊕1⊕1, 1⊕2⊕G(3)=3, 2⊕2⊕2=2} so G(9)=mex{1,2,3}=0 ≠ 3; ·3700f and ·3703f have (0,9,3) (matching their .3x00f siblings), not (0,6,3). All three remain arithmetic-periodic and are certified here by H-N's own Theorem 4 with the corrected parameters.

Criteria audit: implementations of Guy-Smith-style Theorem 3, Austin's Theorem 2, and H-N Theorem 4 exactly as printed (window formulas re-derived against the paper's own table, 8/8 rows) verify all 63 solved-game rows of H-N Tables 2-4 empirically with 0 failures (the three Table-4 errata rows under their corrected parameters) and certify 45 end-to-end (15 Thm3, 18 Austin, 12 Thm4). The other 18 (all Table 3, s in {1,2,4}) fail the hypotheses AS PRINTED: typically condition (4) requires nim-value 0 to recur at a positive/odd position (it never does in these games), or condition (2) is slope-marginal. The printed hypotheses are therefore stricter than those used in H-N's unpublished proofs (their ref [10]); no certification is claimed for those 18 games here.

·f6 empirical correction: the relation G(n+43)=G(n)+32 holds exactly on 604 ≤ n ≤ 1885 and FIRST FAILS at n=1886 (dip values ≈326-334 near n=1929; independently confirmed by the reference engine). Up to 50000 the sequence is only piecewise (43,32)-arithmetic-periodic: 22378 violations in 28 clusters (first [1886,2470], largest [34987,40333]), the difference relation resyncing between clusters. The clean 'apparent preperiod 604' picture does not survive n=1886.