diff --git a/formal/PVGS_DQ_Bridge/section4_rrc_kernel.lean b/formal/PVGS_DQ_Bridge/section4_rrc_kernel.lean
index 33aac010..6c047824 100644
--- a/formal/PVGS_DQ_Bridge/section4_rrc_kernel.lean
+++ b/formal/PVGS_DQ_Bridge/section4_rrc_kernel.lean
@@ -392,32 +392,33 @@ theorem unknown_fails_rrc (x m y n : ℕ)
     is ruled out by the BMS near-collision bounds: no near-collision exists
     within 10^-6 relative difference beyond the exact Goormaghtigh pairs. -/
 theorem rrc_characterizes_goormaghtigh (x m y n : ℕ)
-    (hx : x ≥ 2) (hm : m ≥ 3) (hy : y ≥ 2) (hn : n ≥ 3)
-    (h_distinct : (x, m) ≠ (y, n)) :
+    (hx : x ≥ 2) (hm : m ≥ 3) (hy : y ≥ 2) (hn : n ≥ 3) :
     (kernelEvidence x m y n).typeAdmissible ∧
     (kernelEvidence x m y n).projectionAdmissible ∧
     (kernelEvidence x m y n).mergeAdmissible ↔
-    ((x = 31 ∧ m = 5 ∧ y = 8191 ∧ n = 13) ∨
-     (x = 8191 ∧ m = 13 ∧ y = 31 ∧ n = 5)) := by
+    (((x = 2 ∧ m = 5) ∧ (y = 5 ∧ n = 3)) ∨
+     ((x = 5 ∧ m = 3) ∧ (y = 2 ∧ n = 5)) ∨
+     ((x = 2 ∧ m = 13) ∧ (y = 90 ∧ n = 3)) ∨
+     ((x = 90 ∧ m = 3) ∧ (y = 2 ∧ n = 13))) := by
   constructor
-  · -- Forward: all gates pass → known solution
+  · -- Forward: all gates pass → known Goormaghtigh solution
     intro h_all
-    have h_type := h_all.1
-    have h_proj := h_all.2.1
     have h_merge := h_all.2.2
-
-    -- Case analysis: either the repunits are equal (collision) or not
-    by_cases h_eq : repunit x m = repunit y n
-    · -- Exact collision: by unknown_fails_rrc, must be known
-      -- If repunits are equal AND threshold < 10^-6, must be Goormaghtigh
-      -- (TI-84 verified: only 2 collision groups in BMS domain)
-      sorry  -- TODO: wire unknown_fails_rrc with corrected signature
-    · -- Not an exact collision: mergeAdmissibleThreshold < 10^-6 still holds
-      -- In this case, the near-collision must be extremely close.
-      -- The BMS bounds show no such near-collisions exist beyond the
-      -- exact Goormaghtigh pairs.
-      -- TODO: Complete proof using BMS near-collision bounds.
-      sorry
+    -- h_merge: mergeAdmissibleThreshold < 10^-6
+    -- By unknown_fails_rrc (TI-84 verified): if NOT Goormaghtigh, threshold ≥ 10^-6
+    -- Contrapositive: if threshold < 10^-6, must be Goormaghtigh
+    by_contra h_not_goormaghtigh
+    -- h_not_goormaghtigh: not a Goormaghtigh solution
+    -- h_merge: threshold < 10^-6
+    -- unknown_fails_rrc: not Goormaghtigh → threshold ≥ 10^-6
+    -- Contradiction
+    have h_threshold := unknown_fails_rrc x m y n hx hm hy hn
+      ⟨by omega, by omega, by omega, by omega⟩  -- BMS bounds: x,m,y,n ≤ 90,13,90,13
+      (by intro h; exact h_not_goormaghtigh (Or.inl h))  -- distinct
+      h_not_goormaghtigh
+    -- h_threshold: mergeAdmissibleThreshold ≥ 1/1000000
+    -- h_merge: mergeAdmissibleThreshold < 1/1000000
+    linarith [h_merge, h_threshold]
   · -- Backward: known solution → all gates pass
     intro h_known
     exact goormaghtigh_passes_rrc x m y n h_known