Parts XXIX–XXXI mapped the linear constellations and the single nonlinear sequence n²+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 = Πq 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 n²+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 Qₐ (N) / ∫ dt/log (t²+a) tracks it across the family to a fraction of a percent. The degenerate a = −1, where n²−1 factors, has C → 0, the product correctly foreseeing the algebraic collapse. (ii) For Euler's n²+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 n²+n+A is A itself, and the richness climbs monotonically 1. 02, 1. 88, 3. 26, 4. 17, 6. 64. For n²+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
Ruqing Chen (Tue,) studied this question.