/-
  Domain Intersection Gaps — Cross-Layer Bind Instances

  These are the missing bridges between TTM domain layers.
  Each should eventually become a lawful `bind` instance.

  Citation: Domain intersection analysis, ChatGPT research session, 2026-04-17.
-/

namespace MissingProofs.Intersections

/-! ## Junction α: Compression-Routing-Topology Nexus

Models: 1-10 (Cognitive Load) ↔ 71-75 (MI Signal) ↔ 16-23 (GWL Coupling)

Key insight: Information content determines routing policy determines geometric basin.
-/

/-- Bind instance: Cognitive Load → Mutual Information Signal. -/
def bindLoadToMI (load : Float) (miSignal : Type) : Type × Float :=
  -- MI signal extracted from load features
  sorry

theorem bindLoadToMILawful : True := by sorry  -- Cost bounded, information preserved

/-- Bind instance: MI Signal → GWL Coupling Weight. -/
def bindMIToGWL (mi : Type) (coupling : Type) : Type × Float :=
  -- Coupling weight derived from MI density
  sorry

theorem bindMIToGWLLawful : True := by sorry

/-! ## Junction β: Energy-Control-Encoding Trinity

Models: 45-48 (Homeostatic) ↔ 54-56 (Landauer) ↔ 89-93 (uSeed)

Key insight: Thermodynamic constraints shape encoding efficiency.
-/

/-- Bind instance: Homeostatic Pressure → Landauer Bound. -/
def bindControlToEnergy (pressure : Float) (landauer : Type) : Type × Float :=
  -- Pressure as energy demand
  sorry

theorem bindControlToEnergyLawful : True := by sorry

/-- Bind instance: Landauer Limit → uSeed Germination Cost. -/
def bindEnergyToSeed (energy : Type) (seed : Type) : Type × Float :=
  -- Energy budget → activation threshold
  sorry

theorem bindEnergyToSeedLawful : True := by sorry

/-! ## Junction γ: Braid-Verification-Lean Convergence

Models: 79-81 (Braid) ↔ 82-88 (AVMR) ↔ 111-118 (Unified Compression)

Key insight: Formal verification of compression through braid structure.
-/

/-- Bind instance: Bracket Braid Dynamics → AVMR Event. -/
def bindBraidToAVMR (braid : Type) (event : Type) : Type × Float :=
  -- Braid state classifies as AVMR event
  sorry

theorem bindBraidToAVMRLawful : True := by sorry

/-- Bind instance: AVMR Tree → Unified Compression Pulse. -/
def bindAVMRToCompression (avmr : Type) (pulse : Type) : Type × Float :=
  -- AVMR vector → compression pulse
  sorry

theorem bindAVMRToCompressionLawful : True := by sorry

/-! ## Junction δ: Temporal-Dynamics-Signal Intersection

Models: 24-29 (Temporal) ↔ 122-125 (Dynamics) ↔ 104-109 (Temporal Theorems)

Key insight: τ-field enables time-aware signal processing.
-/

/-- Bind instance: Temporal Dimension → Time Evolution. -/
def bindTemporalToDynamics (τ : Type) (evolution : Type) : Type × Float :=
  -- Temporal phase drives dynamics
  sorry

theorem bindTemporalToDynamicsLawful : True := by sorry

/-- Bind instance: Dynamics → Axial Event Production. -/
def bindDynamicsToAxial (dynamics : Type) (event : Type) : Type × Float :=
  -- Evolution selects axial generator
  sorry

theorem bindDynamicsToAxialLawful : True := by sorry

/-! ## Critical Collapse Lines

These are single concepts that should unify multiple domains.
-/

/-- Q16.16 as universal numeric representation across all domains. -/
theorem q16UniversalEmbedding : True := by sorry  -- Q16.16 embeds into ℝ faithfully

/-- Shell state as geometric encoding of integers (I ↔ C₁ ↔ H). -/
theorem shellStateGeometric : True := by sorry  -- (n,k,a,b) captures position + state

/-- Contact detection as closure constraint (C₁ ↔ F ↔ M). -/
theorem contactAsConstraint : True := by sorry  -- κ_A ∧ κ_C is admissibility predicate

/-- Resonance as spectral/braid/energy degeneracy (C₂ ↔ G ↔ K ↔ M). -/
theorem resonanceAsDegeneracy : True := by sorry  -- Same eigenvalue across representations

/-- bind() as universal lawful translation (B ↔ E ↔ M). -/
theorem bindAsUniversal : True := by sorry  -- All lawful translations reduce to bind

end MissingProofs.Intersections
