The Fermat-Type Dodge Ladder: Primes n^ (2ᵏ) +1 on the 6N Skeleton and the Non-Monotonic Richness

Part XXXII read prime richness as the product of per-prime "dodges" — primes forbidden to divide a sequence enhance it. We push that idea into higher degree along the Fermat-type family n^ (2ᵏ) +1: n²+1, n⁴+1, n⁸+1. The arithmetic is governed by a single cyclotomic fact: x^ (2ᵏ) ≡ −1 (mod q) is solvable exactly when q ≡ 1 (mod 2^ (k+1) ), with 2ᵏ roots. So n^ (2ᵏ) +1 can be divided only by q=2 and primes q ≡ 1 (mod 2^ (k+1) ), a vanishing fraction 2^ (−k) of all primes, while at those primes it is hit hard (ω = 2ᵏ). At n=2 this family is the Fermat numbers Fₖ = 2^ (2ᵏ) +1, whose early terms led Fermat to guess they were all prime until F₅ fell to 641: the same exponent that widens the dodge sets the depth of any hit. Two findings, measured by direct (Miller–Rabin / BPSW) primality testing to n ≤ 10⁶ (n⁴+1 up to 10²4). (i) The mod-6 wing structure is degree-invariant: every n^ (2ᵏ) +1 splits its primes right: left = 1: 2 exactly as n²+1 does, because n^ (2ᵏ) ≡ n² (mod 6) for all k ≥ 1; the exponent is invisible to the coarse lattice. (ii) The Bateman–Horn richness C = Πq g (q) is non-monotonic in k: as the degree doubles the dodge widens (more enhancing primes) but the hits deepen, and the two effects balance to a peak at n⁴+1. We measure C = 1. 376, 2. 676, 2. 075 for k = 1, 2, 3 — rising then falling — with the local product reproducing each to ≲1%, including a degree-eight Bateman–Horn check rarely attempted (where values reach 10⁴0 and direct testing is the only route). None of this is new as theory — the solvability of x^ (2ᵏ) ≡ −1 and its root count are classical cyclotomic facts, C and the Bateman–Horn asymptotic are standard, and the n²+1 constant is Landau–Shanks. What this paper adds is the 6N reading of the ladder, the degree-invariance of the wings, and a direct high-precision measurement of the richness, including the non-monotonic peak. The infinitude of primes n⁴+1 is, like Landau's problem, open; we make no claim about it. As throughout this series, this is a measurement, not a theorem; the Hardy–Littlewood heuristic is taken as input. Part XXXIII of "Arithmetic Geodynamics on the 6N Skeleton. " Code and measured data: https: //github. com/Ruqing1963/6N-dodge-ladder