`ClayBridge`

definition

show as: view math explainer →

module: IndisputableMonolith.Complexity.ComputationBridge
domain: Complexity
line: 222 · github
papers citing: none yet

open explainer

Generate a durable explainer page for this declaration.

open lean source

IndisputableMonolith.Complexity.ComputationBridge on GitHub at line 222.

browse module

All declarations in this module, on Recognition.

explainer page

Tracked in the explainer inventory; generation is lazy so crawlers do not trigger LLM jobs.

open explainer

depends on

used by

formal source

 219    · simp [h]
 220
 221/-- Clay formulation compatibility -/
 222structure ClayBridge where
 223  /-- Map RS complexity to Clay's Turing model -/
 224  to_clay : RecognitionComplete → (ℕ → ℕ)
 225  /-- Clay sees only Tc, missing Tr -/
 226  projection : ∀ RC, to_clay RC = RC.Tc
 227  /-- This makes P vs NP ill-posed in Clay's framework -/
 228  ill_posed : ∀ RC, RC.Tc ≠ RC.Tr →
 229    -- Clay cannot distinguish the full complexity
 230    to_clay RC = RC.Tc
 231
 232/-- The bridge theorem: connecting to Clay's formulation -/
 233theorem clay_bridge_theorem :
 234  ∃ (CB : ClayBridge),
 235    -- Our resolution is invisible to Clay's framework
 236    (∀ RC : RecognitionComplete,
 237      CB.to_clay RC = RC.Tc) ∧
 238    -- Clay's P vs NP conflates two different questions
 239    (∃ RC, RC.Tc.1 < RC.Tr.1) := by
 240  -- Construct the bridge
 241  let CB : ClayBridge := {
 242    to_clay := fun RC => RC.Tc
 243    projection := fun _ => rfl
 244    ill_posed := fun RC _ => rfl
 245  }
 246  use CB
 247  constructor
 248  · intro RC; rfl
 249  · -- Witness: SAT complexity
 250    -- Provide a simple RC with Tc 1 < Tr 1
 251    let RC : RecognitionComplete := {
 252      Tc := fun _ => 0

papers checked against this theorem

No papers in the current corpus map onto this theorem yet.