diff --git a/verification/Proofs/AxiomCheck.lean b/verification/Proofs/AxiomCheck.lean
index 3d27b90..b587815 100644
--- a/verification/Proofs/AxiomCheck.lean
+++ b/verification/Proofs/AxiomCheck.lean
@@ -6,6 +6,7 @@ import Proofs.Descent
 import Proofs.Consistency
 import Proofs.Binding3
 import Proofs.Refactor
+import Proofs.Theorem3
 #print axioms LTLAcc.domsep
 #print axioms LTLAcc.kbelow_pos
 #print axioms LTLAcc.kbelow_lt
@@ -32,3 +33,6 @@ import Proofs.Refactor
 #print axioms LTLAcc.consRecBinding
 #print axioms LTLAcc.consRec_base_false_eq
 #print axioms LTLAcc.consRec_base_true_eq
+#print axioms LTLAcc.extractCons
+#print axioms LTLAcc.extractCons_correct
+#print axioms LTLAcc.extractCons_nonvacuous
diff --git a/verification/Proofs/AxiomCheck.olean b/verification/Proofs/AxiomCheck.olean
index f2afdfa..08b26f1 100644
Binary files a/verification/Proofs/AxiomCheck.olean and b/verification/Proofs/AxiomCheck.olean differ
diff --git a/verification/Proofs/Theorem3.lean b/verification/Proofs/Theorem3.lean
new file mode 100644
index 0000000..61d78ac
--- /dev/null
+++ b/verification/Proofs/Theorem3.lean
@@ -0,0 +1,67 @@
+/- S5.4 — **Theorem 3 (Consistency soundness)**, assembled: the explicit
+   extractor 𝓔′ for history rewrites. Joins consRecBinding (steps 1-2,
+   S5.3) to extractMTH (step 3, S4).
+
+   Statement design per the corpus discipline: a NAMED function whose
+   correctness is about ITS OUTPUT (pigeonhole/choice cannot discharge
+   it), guarded by a permanent non-vacuity witness. -/
+import Proofs.Binding3
+
+namespace LTLAcc
+
+/-- The consistency extractor 𝓔′ (paper Theorem 3). On a claimed rewrite
+    — an accepted consistency proof `C` between the pinned root of `D₀`
+    and the head of `D₁`, where `D₀` is NOT the real prefix — return the
+    node collision found while walking the fold, or descend into the two
+    same-root prefix trees. -/
+noncomputable def extractCons (n₀ : Nat) (C : List Hash)
+    (D₀ D₁ : List Bytes) : List UInt8 × List UInt8 :=
+  match extractConsNode n₀ D₁.length C true (MTH D₀) D₁ with
+  | some c => c
+  | none => extractMTH D₀ (D₁.take n₀)
+
+/-- **Theorem 3 (Consistency soundness), explicit form**: if the
+    consumer's verifier accepts `C` between the pinned head `MTH D₀`
+    (size `n₀`) and the offered head `MTH D₁`, but `D₀` is not the real
+    prefix of `D₁`, then `extractCons` outputs a genuine SHA-256
+    collision. -/
+theorem extractCons_correct (n₀ : Nat) (C : List Hash) (D₀ D₁ : List Bytes)
+    (hlen0 : D₀.length = n₀) (hn0 : 0 < n₀) (hle : n₀ ≤ D₁.length)
+    (hne : D₀ ≠ D₁.take n₀)
+    (hacc : ConsRec n₀ D₁.length C true (MTH D₀) = some (MTH D₀, MTH D₁)) :
+    IsCollision (extractCons n₀ C D₀ D₁).1 (extractCons n₀ C D₀ D₁).2 := by
+  have hbind := consRecBinding (MTH D₀) n₀ D₁.length C true D₁
+    (MTH D₀) (MTH D₁) rfl hn0 hle hacc rfl
+  rw [extractCons]
+  cases hrec : extractConsNode n₀ D₁.length C true (MTH D₀) D₁ with
+  | some c =>
+      rw [hrec] at hbind
+      exact hbind
+  | none =>
+      rw [hrec] at hbind
+      -- hbind : MTH D₀ = MTH (D₁.take n₀); descend
+      have htklen : (D₁.take n₀).length = n₀ := by
+        rw [List.length_take]; omega
+      exact extractMTH_correct D₀ (D₁.take n₀) (by omega) hne hbind
+
+/-- Permanent non-vacuity witness: on a NON-rewrite input (the pinned
+    list IS the real prefix), the extractor's output is provably NOT a
+    collision — so the correctness conclusion is false for some inputs
+    and cannot be discharged by pigeonhole or choice. Uses the honest
+    n₀ = n base: D₀ = D₁ = [[7]], C = [], where ConsRec accepts and
+    extractConsNode returns none, so extractCons = extractMTH D₀ D₀ =
+    the equal leaf pair. -/
+theorem extractCons_nonvacuous :
+    ¬ IsCollision (extractCons 1 [] [([7] : List UInt8)] [([7] : List UInt8)]).1
+                  (extractCons 1 [] [([7] : List UInt8)] [([7] : List UInt8)]).2 := by
+  rw [extractCons]
+  have hrec : extractConsNode 1 ([([7] : List UInt8)]).length [] true
+      (MTH [([7] : List UInt8)]) [([7] : List UInt8)] = none := by
+    rw [extractConsNode]; simp
+  rw [hrec]
+  rw [extractMTH]
+  simp only [List.length_singleton, if_pos (by omega : (1:Nat) ≤ 1)]
+  intro hcol
+  exact hcol.1 rfl
+
+end LTLAcc
diff --git a/verification/Proofs/Theorem3.olean b/verification/Proofs/Theorem3.olean
new file mode 100644
index 0000000..01527c4
Binary files /dev/null and b/verification/Proofs/Theorem3.olean differ
diff --git a/verification/check.sh b/verification/check.sh
index 6654bd4..06f318e 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 Completeness Extract Descent Consistency Binding3 Refactor )
+PROOFS=( Basic Completeness Extract Descent Consistency Binding3 Refactor Theorem3 )
 
 # Certificates and their exact expected cones (observed at first green
 # compile, 2026-07-10; any drift in EITHER direction is a failure).
@@ -48,6 +48,9 @@ declare -A CONES=(
   [LTLAcc.consRecBinding]="propext, Classical.choice, LTLAcc.sha256, Quot.sound"
   [LTLAcc.consRec_base_false_eq]="propext, Quot.sound"
   [LTLAcc.consRec_base_true_eq]="propext, Quot.sound"
+  [LTLAcc.extractCons]="propext, LTLAcc.sha256, Quot.sound"
+  [LTLAcc.extractCons_correct]="propext, Classical.choice, LTLAcc.sha256, Quot.sound"
+  [LTLAcc.extractCons_nonvacuous]="propext, LTLAcc.sha256, Quot.sound"
 )
 
 free -m | awk '/Mem:/{if($7<2048){print "FATAL: <2GB RAM available — refusing to compile"; exit 1}}'