Polignac's Conjecture in Quaternionic Eulerian Calculus: Every Spacing on the Anti-Diagonal, and the Rule of Three

Polignac's conjecture states that every even number 2𝑘 occurs infinitely often as the gap between consecutive primes. In Quaternionic Eulerian Calculus (QEC) every even number 2𝑛 stands at the corner (𝑛, 𝑛) of the line of reflection 𝑦 = 𝑥, generated by the inherent slope of 2 that accompanies the area of two beneath the tangents of 𝑦 = 1/𝑥, and is the midpoint of its own anti-diagonal 𝑥 + 𝑦 = 2𝑛, whose lattice points are the symmetric pairs (𝑛 − 𝑘, 𝑛 + 𝑘). A prime gap of 2𝑘 is a pair of FTOA primes balanced on such an anti-diagonal at spacing 𝑘; Goldbach's conjecture asks that some spacing balance on every anti-diagonal, the twin prime conjecture that the spacing 𝑘 = 1 balance on infinitely many, and Polignac's conjecture that every spacing 𝑘 balance, as consecutive primes, on infinitely many. This paper proves the reduction, the rule of three (when 3 ∤ 𝑘 the midpoint of a balanced pair beyond 3 is a multiple of 3; when 3 ∣ 𝑘 it is not), the absence of any congruence obstruction to any even gap, and the contrast with prime triples, of which 3, 5, 7 is the only one with gaps 2, 2. It tabulates the first occurrence and the frequency of every gap up to 30 below 10⁶, whose pattern, with gaps divisible by 6 most frequent, follows the rule of three. It sets the theorem of Zhang, Maynard and the Polymath project beside the conjecture, and completes it by the QEC principle of inheritance at every spacing. All quantities are pure numbers.