The Richness Spectrum of Quadratic Primes on the 6N Skeleton: Mask Rotation in n^2+a and the Heegner Corridor of n^2+n+A

Parts XXIX–XXXI mapped the linear constellations and the single nonlinear sequence n^2+1. Here we vary the quadratic and read the whole family through one principle: a quadratic generates primes densely exactly when its discriminant dodges the low primes — when no small q can divide its values. The Bateman–Horn richness is C = Π_q g(q) with local factor g(q) = (1 − ω(q)/q)/(1 − 1/q): where q cannot divide f (ω=0) it reads q/(q−1) > 1, a dodge; where it divides on two residues, (q−2)/(q−1) < 1, a suppression. We measure two families by a square-root sieve. (i) For n^2+a, the shift a rotates a geometric mask: a mod 6 decides which break-zones are open. We find a ≡ 1,4 feed both wings at 1:2; a ≡ 0,3 seal the left zone and feed only the right; a ≡ 2,5 seal the right and feed only the left. Independently of the mask, the richness C(a) swings widely with a — from 0.53 (a=5) to 1.97 (a=7) — and the measured count Q_a(N) / ∫ dt/log(t^2+a) tracks it across the family to a fraction of a percent. The degenerate a = −1, where n^2−1 factors, has C → 0, the product correctly foreseeing the algebraic collapse. (ii) For Euler's n^2+n+A, the discriminant is 1−4A, and the richness is governed by how long a corridor of small primes the polynomial threads without ever being divisible. The class-number-one (Heegner) values A = 3, 5, 11, 17, 41, with 4A−1 = 11, 19, 43, 67, 163, give the longest corridors: the first prime that can divide n^2+n+A is A itself, and the richness climbs monotonically 1.02, 1.88, 3.26, 4.17, 6.64. For n^2+n+41 we confirm ω(q) = 0 for every prime q ≤ 37 — it dodges all of them, first divisible at q = 41 — while a non-Heegner neighbour is divisible already at q = 3. The two families are one phenomenon: prime richness is the product of per-prime dodges, and the class-number-one discriminants are the champion dodgers, with −163 the largest. None of this is new as theory — Rabinowitsch, Heegner–Stark–Baker, and Bateman–Horn settled it — and we make no claim about the infinitude of any prime-rich polynomial. As throughout this series, this is a measurement, not a theorem; the Hardy–Littlewood heuristic is taken as input. Part XXXII of "Arithmetic Geodynamics on the 6N Skeleton." Code and measured data: https://github.com/Ruqing1963/6N-quadratic-spectrum