isRecognitionConnected_empty
The empty set satisfies recognition connectivity for any recognizer, since no pairs of configurations exist to check. Researchers developing RG5 regularity in recognition geometry cite this as the base case for connectivity properties. The term proof reaches a contradiction from the assumption that a configuration belongs to the empty set.
claimFor any recognizer $r$, the empty set satisfies recognition connectivity: for all configurations $c_1, c_2$, if both lie in the empty set then they are indistinguishable under $r$.
background
Recognition connectivity is defined so that a set $S$ for recognizer $r$ is connected precisely when all its points are mutually indistinguishable. This notion appears in the module on recognition geometry, where it supports the RG5 axiom that resolution cells remain coherent rather than fragmenting into scattered points.
proof idea
The term proof introduces two configurations $c_1, c_2$ together with the membership hypothesis for $c_1$. It then applies the set-theoretic fact that the empty set contains no elements to obtain an immediate contradiction.
why it matters in Recognition Science
This result supplies the initial case for the connectivity_status summary in the same module. It fills the base step in the RG5 development, where recognition connectivity ensures that resolution cells support smooth geometric structure without pathological fragmentation. The forcing chain landmarks (T5 J-uniqueness through T8 for D=3) supply the ambient counting framework in which recognizers operate.
scope and limits
- Does not establish connectivity for nonempty sets.
- Does not prove local regularity (RG5) itself.
- Does not invoke phi-ladder mass formulas or the alpha band.
- Applies only within the recognition geometry setting.
formal statement (Lean)
48theorem isRecognitionConnected_empty (r : Recognizer C E) :
49 IsRecognitionConnected r ∅ := by
proof body
Term-mode proof.
50 intro c₁ c₂ h₁ _
51 exact absurd h₁ (Set.not_mem_empty c₁)
52
53/-- A singleton set is recognition-connected -/