Prime Gap Rational Distribution

Status: validated Domain: math Source: → wiki/research/rational-makeover-survey/ (§2)

What We Know

• Prime gaps are NOT Gaussian — they follow a rational mod-6 distribution
• Gap mod 6 distribution (first ~78,500 primes):

Gap mod 6	Count	Percentage
0 (divisible by 6)	32,821	41.8%
2	22,838	29.1%
4	22,837	29.1%
1, 3, 5	1	~0%

• The single gap ≡ 1 mod 6 is the gap from 2 to 3. All others are ≡ 0, 2, or 4 mod 6 — exactly the residues predicted by the 6k±1 scaffold
• The dominant gap size is 6 itself (17.3%)
• The {42%, 29%, 29%} split is well-approximated by {5/12, 7/24, 7/24} — simple rational fractions. No e, no π, no Gaussian needed.
• Near-exact symmetry: gap ≡ 2 and gap ≡ 4 counts differ by only 1 (22,838 vs 22,837)

Null Result: sopfr Does NOT Predict Individual Gaps

• Correlation(gap, avg_sopfr of composites between primes): r = 0.170 — weak
• Correlation(gap, sopfr(p+1)): r = 0.044 — negligible
• Correlation(gap, sopfr(p-1)): r = 0.040 — negligible
• Gap size is governed by sieve-theoretic effects (mod 6 structure, admissible tuples), not by factorisation depth of nearby numbers

What We Don’t Know

• Q-GG-01: Does the {5/12, 7/24, 7/24} rational model outperform Cramér’s Gaussian for predicting gap distributions?
• Q-GG-02: Within gap ≡ 0 mod 6 gaps, is there further structure by gap/6? (Are gap = 6, 12, 18, 24, 30 distributed geometrically?)
• Q-GG-03: Is the near-exact symmetry between gap ≡ 2 and gap ≡ 4 (22838 vs 22837) deep or coincidental?
• Does the mod-6 rational distribution hold at larger scales (10⁸, 10¹² primes)?

Relationships

• [[six-dimensional-scaffold]] — supports (strong): Gaps cluster on the 2×3 lattice — direct numerical evidence for the scaffold
• [[tusk-series]] — extends (moderate): Tusk Series’ 1/d amplitude decay is consistent with the mod-6 gap structure
• [[hydrogen-null-result]] — analogous-to (moderate): sopfr null result for gaps parallels the hydrogen null result — smooth/aggregate functions can’t predict individual discrete events
• [[square-root-problem]] — supports (moderate): The Gaussian “shadow” hides the rational 6k scaffold — another case of irrational formulations masking discrete structure
• [[prime-composite-duality]] — supports (moderate): The gap distribution IS the scaffold between consecutive primes — the composite “infrastructure”

Bridging Potential

• A rational gap distribution model could replace parts of probabilistic number theory where Gaussian assumptions fail
• Combined with [[six-dimensional-scaffold]], the mod-6 gap data provides one of the cleanest numerical demonstrations of the scaffold
• The sopfr null result bounds what aggregate arithmetic functions can predict about individual primes

Key Evidence

• 78,498 prime gaps computed and classified mod 6
• {42%, 29%, 29%} distribution — clean rational fractions
• sopfr correlation r < 0.05 for individual gap prediction
• Computation in wiki/research/rational-makeover-survey/investigate_all.py §2