The exact rational-frequency expansion, zero-sum moment identity, no-singleton-prime lemma, and triangular-weight decay identify a concrete growing-moment attack.
Evidence posture · Reported resultAnalytic number theory · primes and prime gaps · sieve methods
Cramér Prime-Gap Conjecture
Collaboration betaCan every sufficiently large gap between consecutive primes be bounded by a constant times the square of the logarithm of the earlier prime?
Known results and sources
Research problem
Exact mathematical statement
Let be the -th prime and let . Cramér's canonical prime-gap conjecture asks whether
Equivalently, there should be constants and such that for every . The stronger prediction of a specified limsup constant is outside this target. The accepted status remains open; a May 2026 preprint claim is unverified and does not establish a proof.
Problem infographic
Problem at a glance
Current mathematical picture
Where work on Cramér Prime-Gap Conjecture stands
Cramér's prime-gap conjecture asks whether consecutive primes always differ by at most a constant times the square of the logarithm. The retained research separates the problem into two open gates: every critical-length interval must contain many integers with no small prime factor, and at least one such survivor must be prime rather than a rough composite. Exact Fourier, denominator-support, shell, coherent-orbit, and Buchstab identities sharpen those questions. The strongest Gate A advance is a proposed reverse-sieve flow that would reduce uniform rough density to a growing-moment estimate at a smaller cutoff, especially a triangular-weight square-root case; that flow is explicitly challenged pending its remainder and quantifier audit. Gate B remains open as a strict deficit problem for the exact disjoint Buchstab composite sum. No proof, formal verification, independent…
Attack the triangular/Fejér square-root moment first using exact zero-sum frequencies, no-singleton prime support, connected cumulants, and quadratic kernel decay; revisit the sharp kernel second.
Route status · Active routeDo not optimize the intermediate cutoff inside separately lower-sieved core and upper-sieved shell estimates; their main terms cancel exactly by the dimension-one parity identity.
Route status · Eliminated routeThe least-prime-factor identity replaces an overlapping cover with a disjoint composite count and exposes the still-open strict-deficit obligation.
Evidence posture · Reported reductionFor z=sqrt(3X), X<m≤2X and H=o(X), the actual prime count N_H(m) equals the full sieve survivor count evaluated at the coherent residue vector (-m mod p)_{p≤z} for all sufficiently large X.
Evidence posture · Source-reported route statementEstablish the triangular-weight analogue of ||S_u-E S_u||{2k}≪sqrt(k E Su) at u=sqrt(w) and k≈u/log u.
Task status · Ready to work onWe corrected the cited passages. We removed a duplicate or outdated task or route step. We updated the highlighted open task or route. The mathematical claims and their status did not change.
Reader-facing record corrected; mathematics unchangedWork mapped so far
Cramér Prime-Gap Conjecture in numbers
918retained lines of mathematical investigation918 in the current working snapshot
- Argument development
- 793 · 86%
- Explored or eliminated routes
- 23 · 3%
- Computational analysis
- 3 · 0%
- Open obligations
- 32 · 3%
- Definitions and setup
- 67 · 7%
21selected mapped statements9routes investigated4reported milestones6open questions3contribution-ready tasks
How this is measured
This measures retained mathematical investigation, not proximity to a proof. Code, data, logs, repeated text, operational instructions, and generated presentation copy are excluded.
Recommended next task
Prove the weighted square-root growing moment
Establish the triangular-weight analogue of ||S_u-E S_u||{2k}≪sqrt(k E Su) at u=sqrt(w) and k≈u/log u.
Suggested move: Organize the exact zero-sum expansion by prime-support unions and connected incidence components, cancel proper zero-sum partitions at the cumulant level, and exploit Fejér decay before returning to the sharp kernel.
What would count as progress
- Remove singleton-prime support patterns before absolute values.
- Control connected cumulants through order c u/log u with constants independent of u.
- Deduce the displayed growing-moment inequality from the retained cumulant lemma.
- Record an independent mathematical review of every uniformity and combinatorial counting step.
Argument map and routes
How the current approaches connect
Claims, reductions, open questions, active routes, and narrowed alternatives in one mathematical map.
Visible working map
Research route map
Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.
Scroll horizontally to explore the route
Working claimActive routeOpen, active, or blocked questionUseful failure
Working overview, not proof. The map shows selected recorded relationships; more nodes or edges do not establish correctness or completion.
Attack the triangular/Fejér square-root moment first using exact zero-sum frequencies, no-singleton prime support, connected cumulants, and quadratic kernel decay; revisit the sharp kernel second.
Route status · Active routeAssuming the candidate trajectory, contradict an almost exact and nearly disjoint covering of the sqrt(w)-rough core using stability, entropy, coupled intersection energy, or structured factorization counts.
Route status · Active routeSeek any fixed power gain beyond level w in the Rosser-weighted shell remainder, uniformly in the translate; such a gain would give Gate A without the compressed moment route.
Route status · Active routeAs an alternative completion architecture, prove endpoint negative pressure and transfer it from the independent residue space to the full coherent actual-prime orbit, including high-conductor modes.
Route status · Active routeExplored alternatives
Other routes
First audit the proposed reverse-sieve flow; then combine it with a growing-moment or cumulant bound at one fixed smaller-power cutoff to force uniform Gate A density.
Route status · Narrowed routeAfter Gate A, split the exact disjoint composite sum at p=H and prove that it cannot saturate the rough survivor count for any actual shift.
Route status · Route held in reserveRetain the proposed (log X)^{2+ε} independent-residue maximal-gap theorem as model information only; complete its stopping-time and remainder audit without presenting it as evidence about actual primes.
Route status · Narrowed routeBrowse 2 more explored routes
Do not optimize the intermediate cutoff inside separately lower-sieved core and upper-sieved shell estimates; their main terms cancel exactly by the dimension-one parity identity.
Route status · Eliminated routeOptimal second-order scale remains useful diagnostics, but variance or typical-shift concentration cannot supply the pointwise core density required at the endpoint.
Route status · Useful but insufficientRoute statements and reductions
Statements the next route can inspect and build on
A negative Laplace bound log L_X(t,H)≤-κH/log X+o(log X), uniformly for H=A(log X)^2, eliminates empty intervals for A>1/κ and yields Cramér's bound.
Source-reported route statementThe centered u-sieve survivor count has an exact Fourier expansion over nontrivial squarefree q dividing P(u), with coefficients V(u) μ(q)K_H(a/q)/φ(q).
Source-reported route statementIn any integer zero-sum of reduced fractions with squarefree denominators, every prime dividing the product of the denominators divides at least two denominators.
Source-reported route statementA triangular interval weight preserves the linear-sieve and reverse-flow reductions, has quadratic Fourier decay, and a positive weighted survivor count implies a positive ordinary survivor count.
Source-reported route statementAt u=sqrt(w), if M counts shell-prime incidences and Ω is the overlap excess, then S_w=S_u-M+Ω exactly.
Source-reported route statementConditional on the reverse-flow trajectory and matching shell upper bound, an endpoint failure forces almost every sqrt(w)-rough offset into exactly one selected residue class for sqrt(w)<p≤w, with Ω=o(HV(w)).
Source-reported route statementFor z=sqrt(3X), X<m≤2X and H=o(X), the actual prime count N_H(m) equals the full sieve survivor count evaluated at the coherent residue vector (-m mod p)_{p≤z} for all sufficiently large X.
Source-reported route statementIf a uniform Rosser-weighted bilinear remainder saving BL_ρ holds for any fixed ρ>1, then a cutoff α<1 sufficiently close to 1 gives S_w(B)≫H/log w uniformly.
Source-reported route statement · dependencies incompleteMore ways to contribute
Open questions
Additional prepared tasks for exploring this research frontier.
Prove the weighted square-root growing moment
Establish the triangular-weight analogue of ||S_u-E S_u||{2k}≪sqrt(k E Su) at u=sqrt(w) and k≈u/log u.
Suggested move: Organize the exact zero-sum expansion by prime-support unions and connected incidence components, cancel proper zero-sum partitions at the cumulant level, and exploit Fejér decay before returning to the sharp kernel.Ready to work on
Prove a uniform bilinear remainder saving
Establish BL_ρ for some fixed ρ>1, uniformly in the translate B, as an alternative route to Gate A.
Suggested move: Seek cancellation in the Rosser-weighted centered sawtooth sum beyond level w, retaining the exact uniform-in-B norm and error scale o(H/log w).Ready to work on
Audit reverse-sieve flow
Turn RF1-RF_α into a fully quantified uniform lemma before using arbitrary-small-power compression.
Suggested move: Fix α, S, an error tolerance, and a finite partition; write explicit shell Rosser levels and prove every uniform remainder before refining the partition, then repeat for the triangular weight.Ready to work on
Control coherent pressure transference
For the alternative pressure architecture, transfer an exponentially strong independent-sieve pressure estimate to actual prime counts along the full coherent orbit, including the high-conductor tail.
Suggested move: Separate the already-controlled low-conductor modes from the unresolved high-conductor tail and state the exact parity-breaking input needed for the latter.Prerequisites still open
Rule out the near-disjoint shell
Under the extremal-trajectory hypotheses, prove a positive lower bound Ω≥cHV(w) or another contradiction to simultaneous near-equality and near-disjoint shell coverage.
Suggested move: Develop a quantitative upper-sieve stability theorem, entropy bound for coherent residue classes, parity-surviving intersection energy, or structured factorization count.Prerequisites still open
Establish a Gate B Buchstab deficit
After Gate A is stable, prove that the exact disjoint composite sum is strictly smaller than S_w(m;H) for every target shift m.
Suggested move: Once Gate A is available, split the least-prime-factor sum at p=H, use bilinear estimates below H, and seek sparse-diagonal or dispersion control above H.Blocked by the current route
Sourced mathematical context
The known mathematical landscape
Current statusRecent proof claim under review
Cramér's canonical big-O conjecture remains open in the accepted literature: for the nth prime p_n, it predicts p_{n+1}-p_n = O((log p_n)²), whereas the established unconditional general bound remains O(p_n^0.525). A May 2026 arXiv v1 claims an explicit logarithmic-square bound and would resolve the canonical statement if correct, but no peer-reviewed acceptance or independent validation was located; the preprint's abstract and body also state different constants. The stronger constant-1 limsup prediction is a distinct formulation and is challenged by Granville's refined heuristic.
[3][5][10]External progress
What the literature has established
Selected external milestones in reverse chronological order, with their evidence posture.
PreprintWang's arXiv v1 claims an explicit O((log p_n)²) bound that would imply the canonical conjecture. The manuscript is an unverified preprint; its abstract and body also state different constants, and no…[10] Peer reviewedGrechuk and Ratcliffe's recent peer-reviewed overview retained Baker–Harman–Pintz as the best established general upper bound and summarized the modern large-gap results, confirming the wide gap between known…[5] Peer reviewedFord, Green, Konyagin, Maynard, and Tao proved that the maximal gap G(X) is at least a constant multiple of log X log log X log log log log X / log log log X for large X. This rigorous lower bound remains far…[4] Computational resultOliveira e Silva, Herzog, and Pardi exhaustively computed prime gaps through 4×10^18 and compared their empirical distribution with standard heuristics. A finite computation cannot settle the asymptotic…[6]
Mathematical neighborhood
Related results and reusable starting points
Current focusCramér's conjecture
Writing G(X)=max_{p_n≤X}(p_{n+1}-p_n), the eventual per-gap estimate g_n=O((log p_n)²) is equivalent, up to harmless endpoint conventions, to G(X)=O((log X)²).
[4]The constant-1 formulation predicts limsup G(X)/(log X)²=1. It is strictly more precise than the canonical big-O upper bound and should not be treated as the same formal statement.
[1][4]Granville's refinement preserves the logarithmic-square scale but predicts a limsup constant at least 2e^(-γ), rather than 1. This is a corrected heuristic for extremes, not a theorem proving or refuting the big-O bound.
[2][4]Firoozbakht's conjecture implies the sharper eventual estimate g_n < (log p_n)² - log p_n - 1 and hence implies Cramér's big-O conjecture. Firoozbakht's conjecture is itself open.
[11]The established unconditional upper bound g_n=O(p_n^0.525) is a much weaker relaxation. Bridging a fixed power of p_n down to a square of its logarithm is the central unresolved gap.
[3][5]Modern large-gap theorems provide lower bounds for G(X), proving that some gaps greatly exceed the average log X scale while remaining asymptotically far below the conjectured upper scale.
[4]Formal and computational footholds
Existing statements, libraries, computations, and datasets that can shorten the next serious attempt.
- formal library support · source linked; not reproduced by ProofAtlasFormal Conjectures primeGap definition
At the audited commit, the Lean repository defines primeGap n as the difference between the (n+1)th and nth primes. The scoped tree contains related prime-gap conjectures but no exact formal statement or proof of Cramér's conjecture.
[8] - computation · not independently reproducedPrime-gap computation through 4×10^18
The peer-reviewed computation exhaustively enumerated the relevant prime-gap data through 4×10^18 and tested distributional predictions. ProofAtlas has not rerun it, and no finite range proves the asymptotic conjecture.
[6] - dataset · source linked; not reproduced by ProofAtlasPrime Gap List tables
The community-curated project publishes frequently updated tables of maximal, first-occurrence, and first-known prime gaps with endpoint-certification fields. It is an experimental resource, not a proof of an asymptotic upper bound.
[7]
Formalization opportunities
Lean work can make these reusable foundations precise without being presented as a proof of the core problem.
- Formalization targetAn exact reviewed Lean statement of the eventual bound g_n ≤ C(log p_n)^2, with explicit choices for prime indexing, coercions, logarithm, constants, and eventual quantification.
- Formalization targetA statement-alignment decision that keeps the canonical big-O conjecture separate from the stronger constant-1 limsup prediction and from fixed-constant pointwise variants.
- Formalization targetThe analytic-number-theory infrastructure needed to improve general prime-in-short-interval bounds from a fixed power scale to a logarithmic-square scale.
- Formalization targetA checked proof term for the exact Cramér statement; none was located in the scoped public formalization audit.
Research-record corrections
What changed in the research record
These notes describe corrections to cited passages, highlighted tasks, or connections between claims. The mathematical claims and their status did not change.
Research-record correctionWe corrected the cited passages. We removed a duplicate or outdated task or route step. We updated the highlighted open task or route. The mathematical claims and their status did not change.
Correction detailsCorrected the research recordCorrection note
The initial argument structure appears separately. Uploads, model runs, and presentation changes do not count as mathematical updates.
How the route was assembled
Argument structure
These stages follow the mathematical order of the supplied argument.
4 mapped milestonesretained argument map
Browse all 4 mapped stages
- Baseline inventory retained during initial intakeExact Fourier and sieve identities retained
- Baseline inventory retained during initial intakePrime-gap target and two open gates
- Baseline inventory retained during initial intakeReverse-flow candidate recorded with its open challenge
- Baseline inventory retained during initial intakeRoute portfolio and next obligations retained
Exact Fourier and sieve identities retainedThe intake preserves exact Fourier, denominator-support, triangular-weight, shell, coherent-orbit, and Buchstab identities as source-reported tools rather than verified evidence.
Baseline inventory retained during initial intakeMapped research milestoneInitial research sequence
Prime-gap target and two open gatesThe intake fixes the O(log² p) target and separates uniform rough density from the still-distinct prime-detection problem.
Baseline inventory retained during initial intakeMapped research milestoneInitial research sequence
Reverse-flow candidate recorded with its open challengeThe strongest proposed Gate A compression is recorded as challenged, with its formal remainder and quantifier audit left open.
Baseline inventory retained during initial intakeMapped research milestoneInitial research sequence
Route portfolio and next obligations retainedthe current work keeps the weighted Fourier, shell stability, bilinear, Buchstab, and pressure routes distinct while closing or narrowing methods the source found insufficient.
Baseline inventory retained during initial intakeMapped research milestoneInitial research sequence
Detailed research inventory
Claims, milestones, and routes in the current map
This view highlights the mathematical statements most useful for following the current route.
11 standing statements6 proposed statements4 challenged statements4 mathematical milestones6 open questions2 narrowed routes6 conditional results6 completed special cases
- theorem candidate
4 of 21 4 - reduction
7 of 21 7 - equivalence
5 of 21 5 - lemma
5 of 21 5
Selected mathematical clusters5 mathematical clusters
Prime-gap question and completion architectureThe exact logarithmic-square target, dyadic reduction, endpoint scale, and two logically separate open gates.11 displayed rows · 3 routes included
- retained route statementCramér prime-gap conjecture
- retained route statementDyadic interval reductionintermediate
- retained route statementExact endpoint reparameterizationintermediate
- retained route statementGate A: uniform rough densityintermediate
- retained route statementGate B: prime detection among rough survivorsintermediate
- DerivationApply one fixed A logarithmic-square non-emptiness statement across successive dyadic ranges containing each sufficiently large prime to obtain the required global big-O bound.active reported
- Recorded relationshipUniform non-emptiness at one fixed logarithmic-square interval length on every large dyadic range is sufficient for the stated big-O prime-gap target.supports · reported by source
- Recorded relationshipThe source's deterministic completion chain first supplies uniform rough survivors, then rules out saturation by rough composites.supports · proposed
- Narrowed routeReverse flow plus compressed momentsFirst audit the proposed reverse-sieve flow; then combine it with a growing-moment or cumulant bound at one fixed smaller-power cutoff to force uniform Gate A density.
- Route held in reserveDeterministic Gate B Buchstab deficitAfter Gate A, split the exact disjoint composite sum at p=H and prove that it cannot saturate the rough survivor count for any actual shift.
- Active routeActual-prime pressure transferenceAs an alternative completion architecture, prove endpoint negative pressure and transfer it from the independent residue space to the full coherent actual-prime orbit, including high-conductor modes.
Exact Gate A toolsPressure, cumulants, independent-sieve variance, Fourier expansion, denominator support, triangular smoothing, and the exact shell identity.10 displayed rows · 2 routes included
- retained route statementNegative-pressure criterionconditional
- retained route statementCumulant-to-moment lemmaintermediate
- retained route statementIndependent-residue variance boundspecial case
- retained route statementExact rational-frequency expansionintermediate
- retained route statementNo-singleton-prime denominator supportintermediate
- retained route statementTriangular-weight special casespecial case
- retained route statementExact square-root shell identityspecial case
- ComputationSource-reported decimal evaluation of f(4), 1-f(4), and the safe density δ_{1/2}=(1-f(4))/4 used in the conditional square-root Gate A implication.The Markdown reports f(4)=0.9783540227059278…, 1-f(4)=0.0216459772940721…, and δ_{1/2}=0.0054114943235180…. No code, data, log, certificate, or independent rerun accompanies these values. · reported unreproduced
- Active routeWeighted square-root Fourier routeAttack the triangular/Fejér square-root moment first using exact zero-sum frequencies, no-singleton prime support, connected cumulants, and quadratic kernel decay; revisit the sharp kernel second.
- Useful but insufficientVariance and typical-shift concentrationOptimal second-order scale remains useful diagnostics, but variance or typical-shift concentration cannot supply the pointwise core density required at the endpoint.
Candidate reverse-flow compressionThe proposed reverse flow, its conditional trajectory and square-root consequences, the explicit audit challenge, and the strongest Gate A work orders.17 displayed rows · 3 routes included
- retained route statementCandidate reverse-sieve flowconditional
- retained route statementConditional extremal-trajectory rigidityconditional
- retained route statementSquare-root moment route to Gate Aspecial case
- retained route statementConditional near-partition rigidityconditional
- DerivationThe current work partitions descending sieve cutoffs, upper-sieves each shell progression, passes from Mertens sums to a Riemann integral, and integrates (sf(s))'=F(s-1). The derivation remains challenged until uniform remainders, the first shell near s=2, partition-limit order, and error normalization are checked.challenged
- DerivationThe proposed reverse flow maps a sparse endpoint configuration to a fixed lower-tail event at u=sqrt(w). A growing-moment estimate would make that event rarer than one residue among P(u), forcing it to be empty. This derivation is conditional on the challenged flow and the unproved moment input.challenged
- ChallengeThe conceptual reverse-flow derivation is not yet a theorem: the source requires explicit uniform Rosser remainders, control of the first shell near the parity endpoint s=2, fixed-partition limit order, breakpoint handling, and normalized accumulated errors.unsupported step · open
- Research targetAudit reverse-sieve flowopen
- Research targetProve the weighted square-root growing momentopen
- Research targetRule out the near-disjoint shellopen
- Recorded relationshipThe moment estimate eliminates every smaller-cutoff sparse residue; candidate reverse flow transfers any endpoint failure into that sparse event.supports · contested
- Recorded relationshipThese exact tools expose the zero-sum support structure, improve kernel decay, and convert connected cumulant control to the required growing moment.supports · proposed
- Recorded relationshipCombining reverse flow with the ordinary lower linear sieve gives the conditional extremal trajectory.supports · contested
- Recorded relationshipThe exact shell identity plus matching extremal lower and upper terms forces negligible overlap under an endpoint failure, subject to the challenged estimates.supports · contested
- Narrowed routeReverse flow plus compressed momentsFirst audit the proposed reverse-sieve flow; then combine it with a growing-moment or cumulant bound at one fixed smaller-power cutoff to force uniform Gate A density.
- Active routeWeighted square-root Fourier routeAttack the triangular/Fejér square-root moment first using exact zero-sum frequencies, no-singleton prime support, connected cumulants, and quadratic kernel decay; revisit the sharp kernel second.
- Active routeNear-partition shell stabilityAssuming the candidate trajectory, contradict an almost exact and nearly disjoint covering of the sqrt(w)-rough core using stability, entropy, coupled intersection energy, or structured factorization counts.
Actual primes and Gate BThe coherent-orbit identity, disjoint Buchstab decomposition, strict composite-deficit target, and the alternative pressure-transference obligation.10 displayed rows · 2 routes included
- retained route statementExact coherent-orbit identityspecial case
- retained route statementExact disjoint Buchstab decompositionintermediate
- retained route statementGate B Buchstab-deficit targetconditional
- DerivationGate A supplies positive S_w; the strict composite deficit in the exact Buchstab identity gives N_H(m)>0; dyadic non-emptiness then implies the big-O prime-gap target. Both Gate A and the deficit premise remain open.proposed
- Recorded relationshipThe disjoint least-prime-factor identity turns the prime-detection gate into strict non-saturation of the rough count by composites.reduces to · reported by source
- Recorded relationshipPositive rough density plus a strict deficit leaves N_H(m)>0 in every target interval, and the dyadic reduction then yields the conjectured bound.supports · proposed
- Research targetEstablish a Gate B Buchstab deficitblocked
- Research targetControl coherent pressure transferenceopen
- Route held in reserveDeterministic Gate B Buchstab deficitAfter Gate A, split the exact disjoint composite sum at p=H and prove that it cannot saturate the rough survivor count for any actual shift.
- Active routeActual-prime pressure transferenceAs an alternative completion architecture, prove endpoint negative pressure and transfer it from the independent residue space to the full coherent actual-prime orbit, including high-conductor modes.
Live, narrowed, and closed routesThe live bilinear route and challenged random-model baseline sit beside explicit failures of fixed moments, variance-only reasoning, separated sieving, random-to-prime inference, and unnecessary large-prime cumulants.15 displayed rows · 4 routes included
- retained route statementCandidate near-Cramér random-sieve theoremspecial case
- retained route statementBilinear superlevel alternativeconditional
- ChallengeThe proposed random-model maximal-gap theorem still needs stopping-time concentration, a uniform slice bound over the shell, and complete remainder accounting; even a completed random-model theorem would not transfer automatically to actual primes.unsupported step · open
- Useful failureUse only fixed-order moments of prime or sieve counts at the logarithmic-square scale.reported failure
- Useful failureUse variance or typical-shift concentration to infer endpoint pressure uniformly.reported failure
- Useful failureLower-sieve the core and separately upper-sieve every shell class, then optimize the intermediate power cutoff.reported failure
- Useful failureTreat independent-residue random-sieve conclusions as conclusions about actual primes.reported failure
- Useful failureRe-expand the independent-residue tail for primes p>H using cumulants.reported failure
- Research targetProve a uniform bilinear remainder savingopen
- Recorded relationshipThe actual-prime residue vector lies on a coherent one-parameter orbit, so the independent-residue model cannot by itself support an actual-prime conclusion.challenges · reported by source
- Recorded relationshipAny fixed power gain in the coupled Rosser-weighted shell remainder yields a positive main-term margin and uniform Gate A density.supports · proposed
- Active routeCoupled bilinear superlevel routeSeek any fixed power gain beyond level w in the Rosser-weighted shell remainder, uniformly in the translate; such a gain would give Gate A without the compressed moment route.
- Narrowed routeNear-Cramér random-model baselineRetain the proposed (log X)^{2+ε} independent-residue maximal-gap theorem as model information only; complete its stopping-time and remainder audit without presenting it as evidence about actual primes.
- Eliminated routeSeparated core-and-shell sieveDo not optimize the intermediate cutoff inside separately lower-sieved core and upper-sieved shell estimates; their main terms cancel exactly by the dimension-one parity identity.
- Useful but insufficientVariance and typical-shift concentrationOptimal second-order scale remains useful diagnostics, but variance or typical-shift concentration cannot supply the pointwise core density required at the endpoint.
How to interpret these counts
A statement may be a lemma, conditional reduction, special case, documented limitation, or open target. These counts describe the work's structure; they do not estimate distance to a proof.
Research outlook
Conditions that would advance the current route
2 approaches have already been tested and narrowed. The task above is the current priority within the larger open route.
A result can change the outlook by closing the bridge, narrowing its scope, or showing that the route cannot work.
- Remove singleton-prime support patterns before absolute values.
- Control connected cumulants through order c u/log u with constants independent of u.
- Deduce the displayed growing-moment inequality from the retained cumulant lemma.
- Record an independent mathematical review of every uniformity and combinatorial counting step.
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Prepared starting pointProve the weighted square-root growing moment
Cramér Prime-Gap Conjecture · ready to start
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Research contextPrepared context for any AI agent
Can every sufficiently large gap between consecutive primes be bounded by a constant times the square of the logarithm of the earlier prime?
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Sources and references11 cited works · next context review by Nov 6, 2026
The mathematical context was checked on Aug 6, 2026. Status can be refreshed sooner after a material result or claim.
- 1On the order of magnitude of the difference between consecutive prime numbersoriginal source · Harald Cramér · Acta Arithmetica · 1936 · DOI 10.4064/aa-2-1-23-46 · accessed Aug 6, 2026
- 2Harald Cramér and the distribution of prime numberssurvey or monograph · Andrew Granville · Scandinavian Actuarial Journal · 1995 · DOI 10.1080/03461238.1995.10413946 · accessed Aug 6, 2026
- 3The Difference Between Consecutive Primes, IIpeer reviewed result · R. C. Baker, Glyn Harman, János Pintz · Proceedings of the London Mathematical Society · 2001-11 · DOI 10.1112/plms/83.3.532 · accessed Aug 6, 2026
- 4Long gaps between primespeer reviewed result · Kevin Ford, Ben Green, Sergei Konyagin, James Maynard, Terence Tao · Journal of the American Mathematical Society · 2018 · ARXIV 1412.5029 · DOI 10.1090/jams/876 · accessed Aug 6, 2026
- 5Modern Breakthroughs in the Study of Small and Large Prime Gapssurvey or monograph · Bogdan Grechuk, Ashleigh Ratcliffe · Mathematics Magazine · 2025-07-07 · DOI 10.1080/0025570X.2025.2481010 · accessed Aug 6, 2026
- 6Empirical verification of the even Goldbach conjecture and computation of prime gaps up to 4·10^18peer reviewed result · Tomás Oliveira e Silva, Siegfried Herzog, Silvio Pardi · Mathematics of Computation · 2014 · DOI 10.1090/S0025-5718-2013-02787-1 · accessed Aug 6, 2026
- 7Prime Gap List tablessoftware or dataset · Prime Gap List community · accessed Aug 6, 2026
- 8FormalConjecturesForMathlib.NumberTheory.PrimeGap at commit 15340cb3bcc032931c9ab4411ec4db564aa819fbformalization · The Formal Conjectures Authors · Google DeepMind formal-conjectures repository · accessed Aug 6, 2026
- 9Cramér's conjectureencyclopedia · Wikimedia Foundation · accessed Aug 6, 2026
- 10On Maximal Prime Gapspreprint · Cheng-Ting Wang · arXiv · 2026-05-14 · ARXIV 2605.14871 · accessed Aug 6, 2026
- 11Upper Bounds for Prime Gaps Related to Firoozbakht's Conjecturepeer reviewed result · Alexei Kourbatov · Journal of Integer Sequences · 2015-11-24 · ARXIV 1506.03042 · MR 3436186 · accessed Aug 6, 2026
Important qualifications
- This record treats Cramér's asymptotic big-O bound g_n = O((log p_n)^2) as the canonical conjecture. The stronger constant-1 limsup prediction is recorded as a related stronger formulation and must not be silently substituted for the canonical statement.
- A May 2026 arXiv v1 claims a logarithmic-square upper bound, but no peer-reviewed acceptance or independent validation was located. Its abstract states a factor 4 while its body states 51/16, so the claim remains explicitly unverified here.
- The scoped formalization audit covered the current Google DeepMind formal-conjectures source tree at commit 15340cb3bcc032931c9ab4411ec4db564aa819fb. It found a primeGap definition but no exact Cramér statement or checked proof; this does not establish that no formalization exists elsewhere.
- The Oliveira e Silva–Herzog–Pardi computation and the Prime Gap List resources were not independently reproduced by ProofAtlas during this administrative collection.
- The Prime Gap List is a changing community-curated dataset and distinguishes certified endpoints, probabilistic endpoints, definite first occurrences, and first-known occurrences. Its entries must be interpreted using those fields.
- No exact entry was verified in the current Epoch AI FrontierMath: Open Problems collection, the English Wikipedia composite list of unsolved mathematical problems, or the scoped Open Problem Garden search. These negative searches do not establish permanent non-membership.
- Recent manuscripts and purported proofs do not change the accepted open status without authoritative mathematical review.
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