The source constructs virtual objects with arbitrarily positive Hodge line but zero conductor at both generic old and clean active divisors. An inequality using the actual isomorphism torsor, horizontal foliation, polarization, and integral saturated residues remains a proposed route.
Route status · Narrowed routeDiophantine geometry · varieties of general type · rational points · arithmetic geometry
Bombieri–Lang Conjecture
Collaboration betaDoes a variety of general type over a number field have rational points confined to a proper algebraic subset? The submitted route reaches a conditional numerical threshold only for a double-octic test surface; it is not a proof of that case or of the general conjecture.
Known results and sources
Research problem
Exact mathematical statement
For a smooth, projective, geometrically integral variety of general type over a number field, the arithmetic Bombieri–Lang conjecture staged here predicts that its rational points are not Zariski dense: there should be a proper closed subset containing them.
The governing source calls this the strong Bombieri–Lang conjecture. This page uses the descriptive phrase number-field non-density to keep it distinct from the stronger uniform-finiteness formulation over finitely generated extensions recorded below. the source develops a double-octic test case and explicitly says that the full conjecture has not been reduced to that family. Every route statement below is source-reported and remains subject to independent mathematical review.
Problem infographic
Problem at a glance
Current mathematical picture
Where work on Bombieri–Lang Conjecture stands
Selected route highlights from the mathematical source. This is not yet a complete mathematical inventory.
On the smooth double-octic test surface, the current work reduces non-density to a truncated counting-versus-height inequality. The balanced abelian-pair torsor is designed so old boundary inertia cancels while marked collisions remain visible, after which a global arithmetic-leaf inequality with a controlled local-support ledger would yield the required divisor inequality.
Evidence posture · Source-reported route statement · dependencies incompleteWe corrected the cited passages. 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
Bombieri–Lang Conjecture in numbers
5.4kretained lines of mathematical investigation5,404 in the current working snapshot
- Argument development
- 4,634 · 86%
- Explored or eliminated routes
- 102 · 2%
- Computational analysis
- 122 · 2%
- Open obligations
- 243 · 4%
- Definitions and setup
- 303 · 6%
8selected mapped statements2routes investigated4open questions4contribution-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
Define and prove the global saturated residue-minimizing arithmetic object.
Suggested move: Construct the actual torsor or moduli object whose local term is the saturated residue-rank minimum and prove invariance under tame base change, logarithmic lattice, and integral frame.
What would count as progress
- Supply a complete argument with every imported premise identified.
- Survive an independent attempt to falsify the proposed step.
Argument map and routes
How the current approaches connect
Claims, reductions, open questions, active routes, and narrowed alternatives in one mathematical map.
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Research route map
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Explored alternatives
Other routes
The source constructs virtual objects with arbitrarily positive Hodge line but zero conductor at both generic old and clean active divisors. An inequality using the actual isomorphism torsor, horizontal foliation, polarization, and integral saturated residues remains a proposed route.
Route status · Narrowed routeThe current work says the ordinary septic discriminant is generically étale along the marked divisor and therefore misses the required support. A marked-divisor logarithmic regulator or the balanced actual branch-swap construction remains proposed.
Route status · Narrowed routeMore ways to contribute
Open questions
Additional prepared tasks for exploring this research frontier.
Define and prove the global saturated residue-minimizing arithmetic object.Suggested move: Construct the actual torsor or moduli object whose local term is the saturated residue-rank minimum and prove invariance under tame base change, logarithmic lattice, and integral frame.
Ready to work on
Close every local support and integral-normalization gap.Suggested move: Prove integral conjugacy at generic old collisions, classify higher collisions, wild and compactification divisors, and verify canonical-extension and metric corrections.
Ready to work on
Global arithmetic-leaf inequality
The actual torsor needs a quantitative global height inequality whose local term is saturated residue rank and whose support excludes ordinary good-prime mismatch.
Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.Ready to work on
Separate test-family closure from the universal conjecture.Suggested move: Only after a rigorous double-octic inequality, develop a distinct universal reduction or higher-dimensional analogue for arbitrary varieties of general type.
Ready to work on
Sourced mathematical context
The known mathematical landscape
External progress
What the literature has established
Selected external milestones in reverse chronological order, with their evidence posture.
Authoritative summaryXie's survey records recent progress on the geometric characteristic-zero function-field variant and its finite-to-abelian class, separate from the number-field conjecture.[4] PreprintGao proved a geometric function-field Bombieri–Lang result for varieties admitting finite maps to abelian varieties; this is not the arithmetic number-field conjecture staged here.[3] PreprintBresciani distinguished strong and weak Bombieri–Lang formulations over finitely generated fields and reduced them toward the number-field base case under stated hypotheses.[2] Historical sourceLang formulated a broad link between hyperbolicity/general type and Diophantine finiteness or non-density.[1]
Mathematical neighborhood
Related results and reusable starting points
Current focusBombieri–Lang Conjecture
Recent geometric function-field theorems cover important finite-to-abelian classes but do not settle the arithmetic number-field non-density statement.
[3][4]The strong formulation asks for one open subset with finite points over every finitely generated extension, stronger than non-density over one number field.
[2]Formalization opportunities
Lean work can make these reusable foundations precise without being presented as a proof of the core problem.
- Formalization targetA formal statement must fix the exact field, smooth/projective/geometrically integral hypotheses, general-type predicate, and whether the target is weak non-density or strong uniform finiteness.
- Formalization targetNo checked proof-assistant proof of the exact arithmetic number-field conjecture was located in the scoped review; this negative search is not evidence of nonexistence.
- Formalization targetAny double-octic formal artifact would prove only its exact test-family statement and would need separate alignment to the universal conjecture.
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 updated the highlighted open task or route. The mathematical claims and their status did not change.
Correction detailsCorrected the research recordCorrection note
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
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.
Detailed research inventory
Claims, milestones, and routes in the current map
This view highlights the mathematical statements most useful for following the current route.
3 standing statements5 proposed statements4 open questions2 narrowed routes
- theorem candidate
1 of 8 1 - reduction
1 of 8 1 - lemma
4 of 8 4 - computational claim
1 of 8 1 - negative result
1 of 8 1
Selected mathematical clusters1 mathematical clusters
Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.24 displayed rows · 2 routes included
- retained route statementAre rational points on every general-type variety non-dense?
- retained route statementCurrent reductionintermediate
- retained route statementClosing targetintermediate
- retained route statementExact general-type targetintermediate
- retained route statementBalanced actual abelian pairintermediate
- retained route statementOld cancellation and active rank sevenintermediate
- retained route statementConditional coefficient thresholdintermediate
- retained route statementVirtual positivity is insufficientintermediate
- Recorded relationshipThe source reports this as a route toward the conjecture; missing or unaudited premises remain and the reduction does not itself prove the target.supports · reported by source
- Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
- Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
- Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
- Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
- Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
- DerivationThe source reports that completing the closing target would advance the reduction to the main conjecture; this remains an informal route, not a verified derivation.proposed
- Useful failureVirtual-class conductor positivityreported failure
- Useful failureOrdinary septic discriminant detectionreported failure
- Research targetDefine and prove the global saturated residue-minimizing arithmetic object.open
- Research targetClose every local support and integral-normalization gap.open
- Research targetSeparate test-family closure from the universal conjecture.open
- Research targetGlobal arithmetic-leaf inequalityopen
- ComputationThe source gives exact monomial-count formulas for the Griffiths exponent, nilpotent rank, and integer threshold searches for symmetric levels.The current work reports 6K_21/R_21 = 4.050708... as the first sharp threshold above four, with coarser valid conditional levels n=25 and n=28 under their respective local bounds. · reported unreproduced
- Narrowed routeVirtual-class conductor positivityThe source constructs virtual objects with arbitrarily positive Hodge line but zero conductor at both generic old and clean active divisors. An inequality using the actual isomorphism torsor, horizontal foliation, polarization, and integral saturated residues remains a proposed route.
- Narrowed routeOrdinary septic discriminant detectionThe current work says the ordinary septic discriminant is generically étale along the marked divisor and therefore misses the required support. A marked-divisor logarithmic regulator or the balanced actual branch-swap construction remains proposed.
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.
- Supply a complete argument with every imported premise identified.
- Survive an independent attempt to falsify the proposed step.
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Prepared starting pointDefine and prove the global saturated residue-minimizing arithmetic object.
Bombieri–Lang Conjecture · ready to start
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Research contextPrepared context for any AI agent
Does a variety of general type over a number field have rational points confined to a proper algebraic subset? The submitted route reaches a conditional numerical threshold only for a double-octic test surface; it is not a proof of that case or of the general conjecture.
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- Current routes and known obstacles
- What a useful result should report
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Sources and references4 cited works · next context review by Nov 14, 2026
The mathematical context was checked on Aug 14, 2026. Status can be refreshed sooner after a material result or claim.
- 1Hyperbolic and Diophantine analysisoriginal source · Serge Lang · Bulletin of the American Mathematical Society 14(2), 159–205 · 1986 · DOI 10.1090/S0273-0979-1986-15426-1 · accessed Aug 14, 2026
- 2On the Bombieri-Lang Conjecture over finitely generated fieldspreprint · Giulio Bresciani · arXiv · 2020 · ARXIV 2012.15765 · accessed Aug 14, 2026
- 3A complete proof of the geometric Bombieri-Lang conjecture for ramified covers of abelian varietiespreprint · Guoquan Gao · arXiv · 2025 · ARXIV 2511.17010 · accessed Aug 14, 2026
- 4Recent progress on the geometric Bombieri--Lang conjecturesurvey or monograph · Junyi Xie · arXiv · 2026 · ARXIV 2607.02165 · accessed Aug 14, 2026
Important qualifications
- This metadata is source-separated from the private packet and grants no mathematical, review, credit, publication, or deployment authority.
- The literature uses several weak, strong, arithmetic, and geometric formulations; this record keeps the number-field non-density statement separate from function-field results.
- The original-attribution history is compressed conservatively: Lang's 1986 article is used as the historical anchor, while the conventional title credits Bombieri and Lang.
- Scoped searches found no exact checked formal proof; negative search results do not establish absence elsewhere.
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