# State of the Art — Axis #10: Prime Deserts (large prime gaps)

# What is known (checked against the literature, 2026-08-06)

Quantity	Known record	Source
Largest maximal prime gap	1854, after p = 101412319996363309069 (85th maximal gap, Robert Smith / Brian Kehrig code, 2026)	primerecords.dk, t5k.org GapsTable
Exhaustive verification bound for maximal gaps	≈ 2×10^19 (beyond 2^64)	t5k.org/notes/GapsTable.html
Largest known gap (any size, PRP endpoints)	16,045,848 near a 385,713-digit number (A. Höglund, 2024)	primerecords.dk/primegaps/megagap3.htm
Highest known merit g/ln(p)	41.93878… (gap 8350 after an 87-digit prime, Gapcoin network, 2017)	primegap-list-project
Highest Cramér–Shanks–Granville ratio g/ln²(p)	0.9206… (Nyman's gap 1132 after 1693182318746371, 1999)	MathWorld Prime Gaps
Theory (lower bound)	Ford–Green–Konyagin–Maynard–Tao 2016: max gap ≥ c·(ln x · lnln x · lnlnlnln x)/lnlnln x	published
Theory (upper bound)	Baker–Harman–Pintz: gap = O(x^0.525); RH gives O(√x ln x)	published
Heuristic	Cramér: limsup g/ln²p = 1; Granville correction: ≥ 2e^{-γ} ≈ 1.1229	published
Jumping champion	6 is the most common gap from ≈ 947 up to ≈ 1.7×10^35 (conjectured, verified in ranges); then 30	published (Odlyzko–Rubinstein–Wolf)

A new maximal gap — requires exhaustive sieving beyond 2×10^19: out of reach locally; not the goal.
A gap with merit > 41.94 — requires massive targeted search (Gapcoin-scale): out of reach in one cycle; a merit > 20 gap found by our own sieve pipeline would still only be "personal best" territory (thousands of merit-20+ gaps are catalogued). Any gap we find must be checked against the primegap-list-project tables before any claim.
Structural conjectures about gap statistics (residues of maximal-gap primes, corrections to Cramér in arithmetic progressions, gap-pair correlations) — realistic: findings are likely "(a) probably known", but a precise, massively tested statement with clean data is a legitimate output.
Certified desert constructions (covering systems): any construction is reproducible and certifiable, but primorial/covering constructions have low merit (~2–4) — value is the certificate pipeline, not records.

Maximal gap: must exceed 1854 AND be proven maximal (impossible without exhaustive sieve > 2×10^19) — or simply reproduce/verify the known table below our compute bound (verification, not record).
Merit record: > 41.93878.
CSG ratio record: > 0.92064.
First-occurrence gaps: the Nicely / primegap-list-project tables catalogue first known occurrences for every even gap; before claiming a "first occurrence", check those tables.