The Hodge Conjecture
Algebraicity is assumed complete. This assumption is unproven.
Formal Statement
Let X be a smooth projective complex variety. The Hodge Conjecture asserts that every rational (p,p)-class is algebraic.
Core Boundary
All known approaches assume that algebraic cycles generate the full space of rational Hodge classes.
This is a completeness claim—not a proven property.
Strategy Limitation
- Cycle constructions are partial generators
- Analytic methods preserve type but not origin
- Reduction arguments depend on known cases
- Obstruction methods are incomplete
No approach spans the full space of possible Hodge classes.
Completeness Failure Condition
Let S = space of all rational (p,p)-classes.
Let A = subspace generated by algebraic cycles.
The conjecture assumes: S = A
This equality is not established.
Unresolved Possibility
There may exist classes in S \ A:
- Analytic in origin
- Indistinguishable by current invariants
- Not representable by algebraic cycles
Their existence cannot be excluded.
Falsifiable Resolution Requirement
- Universal proof of S = A
- Explicit example where S ≠ A
Any valid resolution must address completeness directly.
Claim Eligibility Boundary
Any framework assuming algebraic exhaustiveness without proof exceeds its epistemic authority.
Partial generation does not imply completeness.
Boundary Judgment
The Hodge Conjecture is not blocked by lack of construction—it is blocked by an unproven completeness assumption. Any approach that does not resolve this boundary cannot claim full coverage.