diff --git a/README.md b/README.md
index b15c5b9..97b0e1f 100644
--- a/README.md
+++ b/README.md
@@ -11,7 +11,7 @@ the same discipline as the four `*-ed25519-verified` subject corpora.
 |---|---|---|
 | L1 | bytes, hleaf/hnode, domain separation (Lemma 1) | **done** (domsep: axiom-free) |
 | L2 | MTH, Root, ConsRec definitions + termination | **done** (cones: propext, LTLAcc.sha256, Quot.sound) |
-| L3 | inclusion completeness (Theorem 1) | pending |
+| L3 | inclusion completeness (Theorem 1) | **done** (incl_complete: propext, Classical.choice, LTLAcc.sha256, Quot.sound) |
 | L4 | frontier hash-fold + root binding (Lemma 2) | pending |
 | L5 | inclusion/consistency soundness as collision extractors (Theorems 2, 3) | pending |
 | L6 | pin-store state machine safety (Proposition 1) | pending |
diff --git a/verification/Proofs/AxiomCheck.lean b/verification/Proofs/AxiomCheck.lean
index 669e2f3..8c0d204 100644
--- a/verification/Proofs/AxiomCheck.lean
+++ b/verification/Proofs/AxiomCheck.lean
@@ -1,5 +1,6 @@
 /- Axiom-cone observation for the audit (Phase 3 of check.sh). -/
 import Proofs.Basic
+import Proofs.Completeness
 #print axioms LTLAcc.domsep
 #print axioms LTLAcc.kbelow_pos
 #print axioms LTLAcc.kbelow_lt
@@ -7,3 +8,5 @@ import Proofs.Basic
 #print axioms LTLAcc.MTH
 #print axioms LTLAcc.Root
 #print axioms LTLAcc.ConsRec
+#print axioms LTLAcc.Path
+#print axioms LTLAcc.incl_complete
diff --git a/verification/Proofs/AxiomCheck.olean b/verification/Proofs/AxiomCheck.olean
index c21ff1c..ad5187d 100644
Binary files a/verification/Proofs/AxiomCheck.olean and b/verification/Proofs/AxiomCheck.olean differ
diff --git a/verification/Proofs/Completeness.lean b/verification/Proofs/Completeness.lean
new file mode 100644
index 0000000..24ae35e
--- /dev/null
+++ b/verification/Proofs/Completeness.lean
@@ -0,0 +1,166 @@
+/- L3 of the accumulator pyramid: the prover-side Path function and
+   **Theorem 1 (inclusion completeness)** — honest receipts always verify:
+
+     Root (hleaf D[m]) m |D| (Path m D) = some (MTH D)
+
+   No hash property is used anywhere (the theorem is about shapes). -/
+import Proofs.Basic
+
+namespace LTLAcc
+
+/-- `Path` (paper §5.3): the operator's inclusion path for index `m`.
+    Sibling subtree heads, leaf-to-root order (the paper's `‖ [s]`). -/
+noncomputable def Path (m : Nat) (D : List Bytes) : List Bytes :=
+  if D.length ≤ 1 then []
+  else
+    let k := kbelow D.length
+    if m < k then Path m (D.take k) ++ [MTH (D.drop k)]
+    else Path (m - k) (D.drop k) ++ [MTH (D.take k)]
+termination_by D.length
+decreasing_by
+  · simp only [List.length_take]
+    have h2 : 2 ≤ D.length := by omega
+    have hk := kbelow_lt D.length h2
+    omega
+  · simp only [List.length_drop]
+    have hp := kbelow_pos D.length
+    omega
+
+/-! ### List helper lemmas (self-contained; no stdlib-name dependence) -/
+
+theorem getD_take (l : List Bytes) (k m : Nat) (h : m < k) :
+    (l.take k).getD m [] = l.getD m [] := by
+  induction l generalizing k m with
+  | nil => simp
+  | cons a t ih =>
+      cases k with
+      | zero => omega
+      | succ k' =>
+          cases m with
+          | zero => simp
+          | succ m' =>
+              simp only [List.take_succ_cons, List.getD_cons_succ]
+              exact ih k' m' (by omega)
+
+theorem getD_drop (l : List Bytes) (k i : Nat) :
+    (l.drop k).getD i [] = l.getD (k + i) [] := by
+  induction l generalizing k with
+  | nil => simp
+  | cons a t ih =>
+      cases k with
+      | zero => simp
+      | succ k' =>
+          have hidx : k' + 1 + i = (k' + i) + 1 := by omega
+          simp only [List.drop_succ_cons, hidx, List.getD_cons_succ]
+          exact ih k'
+
+theorem exists_singleton_of_length_one (l : List Bytes)
+    (h : l.length = 1) : ∃ d, l = [d] := by
+  cases l with
+  | nil => simp at h
+  | cons a t =>
+      cases t with
+      | nil => exact ⟨a, rfl⟩
+      | cons b u => simp at h
+
+/-! ### Equation lemmas -/
+
+theorem MTH_single (d : Bytes) : MTH [d] = hleaf d := by
+  rw [MTH]; rfl
+
+theorem MTH_split (D : List Bytes) (h : 2 ≤ D.length) :
+    MTH D = hnode (MTH (D.take (kbelow D.length)))
+                  (MTH (D.drop (kbelow D.length))) := by
+  rw [MTH]
+  have h0 : ¬ D.length = 0 := by omega
+  have h1 : ¬ D.length = 1 := by omega
+  simp only [h0, h1, dite_false]
+
+theorem Root_one (v : Bytes) (m : Nat) : Root v m 1 [] = some v := by
+  rw [Root]; rfl
+
+/-- `Root` at a composite size, left branch (`m < k`). -/
+theorem Root_left (v : Bytes) (m n : Nat) (P : List Bytes) (s : Bytes)
+    (hn : 2 ≤ n) (hm : m < kbelow n) :
+    Root v m n (P ++ [s]) = (Root v m (kbelow n) P).map (hnode · s) := by
+  have h1 : ¬ n = 1 := by omega
+  have h0 : ¬ n = 0 := by omega
+  have hne : ¬ (P ++ [s] = []) := by simp
+  cases hR : Root v m (kbelow n) P with
+  | none =>
+      rw [Root]; simp [h0, h1, hm, hR]
+      exact fun hh => absurd hh hne
+  | some x =>
+      rw [Root]; simp [h0, h1, hm, hR]
+      exact fun hh => absurd hh hne
+
+/-- `Root` at a composite size, right branch (`m ≥ k`). -/
+theorem Root_right (v : Bytes) (m n : Nat) (P : List Bytes) (s : Bytes)
+    (hn : 2 ≤ n) (hm : ¬ m < kbelow n) :
+    Root v m n (P ++ [s]) =
+      (Root v (m - kbelow n) (n - kbelow n) P).map (hnode s ·) := by
+  have h1 : ¬ n = 1 := by omega
+  have h0 : ¬ n = 0 := by omega
+  have hne : ¬ (P ++ [s] = []) := by simp
+  cases hR : Root v (m - kbelow n) (n - kbelow n) P with
+  | none =>
+      rw [Root]; simp [h0, h1, hm, hR]
+      exact fun hh => absurd hh hne
+  | some x =>
+      rw [Root]; simp [h0, h1, hm, hR]
+      exact fun hh => absurd hh hne
+
+/-! ### Theorem 1 -/
+
+/-- **Theorem 1 (Inclusion completeness)**, paper §6: for every non-empty
+    leaf list `D` and every `m < |D|`, the honestly produced receipt
+    verifies to the honest root. -/
+theorem incl_complete (m : Nat) (D : List Bytes) (hm : m < D.length) :
+    Root (hleaf (D.getD m [])) m D.length (Path m D) = some (MTH D) := by
+  induction m, D using Path.induct with
+  | case1 m D hle =>
+      -- |D| ≤ 1 and m < |D| force D = [d], m = 0
+      have h1 : D.length = 1 := by omega
+      obtain ⟨d, rfl⟩ := exists_singleton_of_length_one D h1
+      have hm0 : m = 0 := by simpa using hm
+      subst hm0
+      rw [Path]
+      simp only [List.length_singleton, if_pos (by omega : (1:Nat) ≤ 1)]
+      rw [MTH_single]
+      simpa using Root_one (hleaf d) 0
+  | case2 m D hgt k hmk ih =>
+      have h2 : 2 ≤ D.length := by omega
+      have hkeq : k = kbelow D.length := rfl
+      have hkl : k < D.length := by rw [hkeq]; exact kbelow_lt D.length h2
+      have hmk' : m < kbelow D.length := by rw [← hkeq]; exact hmk
+      rw [Path]
+      simp only [if_neg hgt, ← hkeq, if_pos hmk]
+      rw [Root_left _ _ _ _ _ h2 hmk', ← hkeq]
+      have htklen : (D.take k).length = k := by
+        simp [List.length_take]; omega
+      have ihm : m < (D.take k).length := by omega
+      have hrec := ih ihm
+      rw [getD_take D k m hmk, htklen] at hrec
+      rw [hrec, MTH_split D h2, ← hkeq]
+      rfl
+  | case3 m D hgt k hmk ih =>
+      have h2 : 2 ≤ D.length := by omega
+      have hkeq : k = kbelow D.length := rfl
+      have hkl : k < D.length := by rw [hkeq]; exact kbelow_lt D.length h2
+      have hkp : 0 < k := by rw [hkeq]; exact kbelow_pos D.length
+      have hmk' : ¬ m < kbelow D.length := by rw [← hkeq]; exact hmk
+      rw [Path]
+      simp only [if_neg hgt, ← hkeq, if_neg hmk]
+      rw [Root_right _ _ _ _ _ h2 hmk', ← hkeq]
+      have hidx : k + (m - k) = m := by omega
+      have hget : (D.drop k).getD (m - k) [] = D.getD m [] := by
+        rw [getD_drop, hidx]
+      have hdplen : (D.drop k).length = D.length - k := by
+        simp [List.length_drop]
+      have ihm : m - k < (D.drop k).length := by omega
+      have hrec := ih ihm
+      rw [hget, hdplen] at hrec
+      rw [hrec, MTH_split D h2, ← hkeq]
+      rfl
+
+end LTLAcc
diff --git a/verification/Proofs/Completeness.olean b/verification/Proofs/Completeness.olean
new file mode 100644
index 0000000..95ee8a9
Binary files /dev/null and b/verification/Proofs/Completeness.olean differ
diff --git a/verification/check.sh b/verification/check.sh
index 8e86693..637b463 100755
--- a/verification/check.sh
+++ b/verification/check.sh
@@ -17,7 +17,7 @@ export LEAN_MEM_MB="${LEAN_MEM_MB:-4096}"
 CORES="${LEAN_MAX_CORES:-0-3}"
 
 GEN_MODULES=( LTLAcc/HashExternal )
-PROOFS=( Basic )
+PROOFS=( Basic Completeness )
 
 # Certificates and their exact expected cones (observed at first green
 # compile, 2026-07-10; any drift in EITHER direction is a failure).
@@ -29,6 +29,8 @@ declare -A CONES=(
   [LTLAcc.MTH]="propext, LTLAcc.sha256, Quot.sound"
   [LTLAcc.Root]="propext, LTLAcc.sha256, Quot.sound"
   [LTLAcc.ConsRec]="propext, LTLAcc.sha256, Quot.sound"
+  [LTLAcc.Path]="propext, LTLAcc.sha256, Quot.sound"
+  [LTLAcc.incl_complete]="propext, Classical.choice, LTLAcc.sha256, Quot.sound"
 )
 
 free -m | awk '/Mem:/{if($7<2048){print "FATAL: <2GB RAM available — refusing to compile"; exit 1}}'