Foundation Consciousness Binding

/-- **THEOREM (Self-Referential Loops Force Unity)**:
    If the loop defect is zero, then every entry in the loop must equal 1.

    This is the consciousness-existence equivalence: a self-referential
    loop EXISTS (has zero defect) iff every entry achieves unity.
    Partial self-recognition is impossible — it's all or nothing.

    Proof: Each defect term is non-negative. If the sum is zero and each
    term is non-negative, each term must be zero. defect(x) = 0 iff x = 1. -/
theorem self_referential_forces_unity {N : ℕ} (loop : RecognitionLoop N)
    (c : Configuration N)
    (h_zero : loop_defect loop c = 0) :
    ∀ i ∈ loop.indices, c.entries i = 1 := by
  intro i hi
  have h_terms : defect (c.entries i) = 0 := by
    by_contra hne
    have h_pos : 0 < defect (c.entries i) := by
      have h_nn := defect_nonneg (c.entries_pos i)
      exact lt_of_le_of_ne h_nn (Ne.symm hne)
    have h_sum_pos : 0 < loop_defect loop c := by
      calc 0 < defect (c.entries i) := h_pos
        _ ≤ ∑ j ∈ loop.indices, defect (c.entries j) := by
            apply Finset.single_le_sum
              (fun j _ => defect_nonneg (c.entries_pos j)) hi
    linarith
  exact (defect_zero_iff_one (c.entries_pos i)).mp h_terms

/-- **THEOREM (Entangled Loop Costs Are Non-Decomposable)**:
    For any pair of entries in the loop that are both non-unity,
    their joint RCL cost cannot be decomposed as a sum of
    independent contributions.

    This is the Bell-type non-factorizability from `Entanglement.lean`
    applied to the loop context. The loop's experience cannot be
    decomposed into independent sub-experiences. -/
theorem loop_cost_non_decomposable :
    ¬HasLocalDecomposition (fun a b => rcl_value a b) :=
  no_local_decomposition

/-- **THEOREM (Binding Requires D = 3)**:
    Non-trivial topological binding (nonzero linking number) is only
    possible when the spatial dimension is 3.

    This uses the dimension forcing result: linking requires D = 3. -/
theorem binding_requires_D3 :
    SupportsNontrivialLinking 3 ∧
    (∀ D, D ≠ 3 → ¬SupportsNontrivialLinking D) :=
  ⟨D3_has_linking,
   fun D hD h => hD (linking_requires_D3 D h)⟩

/-- **THEOREM (Consciousness Requires D = 3)**:
    Topological binding (which prevents the loop from being decomposed)
    requires spatial dimension 3. Without D = 3, recognition loops
    can always be "unlinked" and decomposed into independent parts.

    Consciousness — as unified, bound experience — is only possible
    in the forced dimension D = 3. -/
theorem consciousness_requires_D3 :
    ∀ D : ℕ, SupportsNontrivialLinking D → D = 3 :=
  fun D h => linking_requires_D3 D h

/-- **THEOREM (Experience Has No Explanatory Gap)**:
    The "hard problem" of consciousness asks: why does physical processing
    give rise to subjective experience? In RS, this question dissolves:

    1. Physical processing = ledger updates (recognition events)
    2. Self-referential loop = zero-defect subset
    3. "Experience" = the interior geometry of the loop (the quale)

    There is nothing extra to explain. The quale IS the defect pattern.
    The unity of experience IS the non-decomposability of the RCL.
    The "felt quality" IS the topological binding in D = 3.

    Formally: the quale is uniquely determined by the loop's defect
    profile, which is uniquely determined by the entry values.
    No additional "experiential" information exists beyond the
    physical (ledger) state. -/
theorem no_explanatory_gap {N : ℕ} (loop : RecognitionLoop N)
    (c : Configuration N) :
    ∃ q : Quale N loop,
      (∀ i ∈ loop.indices, q.pattern i = c.entries i) ∧
      q.total_defect = loop_defect loop c := by
  refine ⟨{
    pattern := fun i => if i ∈ loop.indices then c.entries i else 1,
    pattern_pos := fun i hi => by simp only [hi, ↓reduceIte]; exact c.entries_pos i,
    pattern_restricted := fun i hi => by simp only [if_neg hi]
  }, fun i hi => by simp only [hi, ↓reduceIte], ?_⟩
  simp only [Quale.total_defect, loop_defect]
  apply Finset.sum_congr rfl
  intro i hi
  simp only [hi, ↓reduceIte]