siddhartha-gadgil/leanaide/generic-arguments

Skill: Generic Arguments & "Almost All" Reasoning

Pre-condition: Do not apply this skill unless the problem involves a property claimed to hold for "almost all" objects, "generic" configurations, or requires proving a global result by analyzing a representative point in a dense/thick subset.

Goal: Leverage topological, measure-theoretic, or algebraic notions of "genericity" to simplify proofs or establish results on dense subsets.

Instructions:

1. Identify the Notion of Genericity

Select the appropriate framework based on the problem context:

• Topological/Baire Category: Property holds on a residual set (countable intersection of dense open sets). Use for complete metric spaces.
• Measure-Theoretic: Property holds "almost everywhere" (complement is a null set). Use for integration or probability.
• Algebraic Geometry: Reasoning via the generic point $\eta$ of an irreducible scheme or a Zariski-dense open set.
• General Topology: Property holds on a dense open subset.

2. The Generic Strategy

1. Define the Space: Identify the space $X$ (e.g., a manifold, a scheme, a function space $C^k$).
2. Select the "Generic" Representative:

• Algebraic: Work in the function field $k(X)$.
• Analytic: Pick a point $x$ avoiding a specific countable union of "bad" sets (e.g., resonances, singular values).
• Measure: Assume the property holds for $x$ outside a set of measure zero.

1. Prove at the Generic Level: Establish the result for this representative.
2. Spread/Extend:

• Use continuity to extend from a dense set to the whole space.
• Use compactness or Baire Category Theorem to show the intersection of generic properties is still generic.
• Use Fubini's Theorem for slice-wise genericity in measure theory.

3. Handle the Exceptional Set

Characterize the "bad" set $E$ where the property fails.

• Is $E$ meager (1st category)?
• Is $E$ a null set?
• Is $E$ a proper closed subvariety?