diff --git a/CompPoly/Data/Polynomial/RabinCertificate.lean b/CompPoly/Data/Polynomial/RabinCertificate.lean
index cc060bc5..f5df9fdd 100644
--- a/CompPoly/Data/Polynomial/RabinCertificate.lean
+++ b/CompPoly/Data/Polynomial/RabinCertificate.lean
@@ -389,4 +389,37 @@ theorem irreducible_of_rabin_degree_six {F : Type*} [Field F] [Fintype F] {f : F
   irreducible_of_rabin_two_prime_factors h_deg (by norm_num) primeFactors_six h_trace
     (by simpa using h_cop₃) (by simpa using h_cop₂)
 
+/-- Explicit-cardinality form of the prime-degree Rabin test.
+
+Identical content to `irreducible_of_rabin_prime_degree`, but the field size is
+a caller-supplied numeral `q` (with `Fintype.card F = q`) rather than
+`Fintype.card F` itself. This lets a caller state the trace and coprimality
+certificates with `q` a concrete literal. That matters for **re-checking**: a
+kernel replay of the resulting proof from an empty environment then never has to
+reduce `Fintype.card F` (for a `ZMod p` field, an enumeration of ~`p` elements)
+to reconcile it with the literal through a rewrite cast. `q` stays universally
+quantified here, so this declaration itself carries no concrete large numeral,
+and the two forms are definitionally equal, so nothing is weakened. -/
+theorem irreducible_of_rabin_prime_degree_of_card {F : Type*} [Field F] [Fintype F]
+    {f : F[X]} {d : ℕ} (q : ℕ) (hq : Fintype.card F = q)
+    (hd : d.Prime) (h_deg : f.natDegree = d)
+    (h_trace : f ∣ X ^ (q ^ d) - X)
+    (h_cop : IsCoprime f (X ^ q - X)) :
+    Irreducible f := by
+  subst hq
+  exact irreducible_of_rabin_prime_degree hd h_deg h_trace h_cop
+
+/-- Explicit-cardinality form of the degree-6 Rabin test. See
+`irreducible_of_rabin_prime_degree_of_card` for why the numeral form matters for
+re-checking; the same reasoning applies here at composite degree 6. -/
+theorem irreducible_of_rabin_degree_six_of_card {F : Type*} [Field F] [Fintype F]
+    {f : F[X]} (q : ℕ) (hq : Fintype.card F = q)
+    (h_deg : f.natDegree = 6)
+    (h_trace : f ∣ X ^ (q ^ 6) - X)
+    (h_cop₃ : IsCoprime f (X ^ (q ^ 3) - X))
+    (h_cop₂ : IsCoprime f (X ^ (q ^ 2) - X)) :
+    Irreducible f := by
+  subst hq
+  exact irreducible_of_rabin_degree_six h_deg h_trace h_cop₃ h_cop₂
+
 end CompPoly.RabinCert
diff --git a/CompPoly/Fields/KoalaBear/Ext5/QuinticIrreducible.lean b/CompPoly/Fields/KoalaBear/Ext5/QuinticIrreducible.lean
index b93f5029..cf5ef709 100644
--- a/CompPoly/Fields/KoalaBear/Ext5/QuinticIrreducible.lean
+++ b/CompPoly/Fields/KoalaBear/Ext5/QuinticIrreducible.lean
@@ -110,12 +110,11 @@ theorem quintic_bezout_check :
 kernel-checked certificates for both conditions. -/
 theorem quinticPoly_irreducible : Irreducible quinticPoly := by
   have hcard : Fintype.card Field = fieldSize := ZMod.card _
-  refine irreducible_of_rabin_prime_degree (by norm_num) quinticPoly_natDegree ?_ ?_
-  · rw [hcard]
-    exact dvd_X_pow_sub_X_of_runChain toPoly_quinticL quinticPoly_ne_zero
+  refine irreducible_of_rabin_prime_degree_of_card fieldSize hcard (by norm_num)
+    quinticPoly_natDegree ?_ ?_
+  · exact dvd_X_pow_sub_X_of_runChain toPoly_quinticL quinticPoly_ne_zero
       quintic_trace_chain quintic_trace_exp
-  · rw [hcard]
-    exact isCoprime_X_pow_sub_X_of_runChain toPoly_quinticL quinticPoly_ne_zero
+  · exact isCoprime_X_pow_sub_X_of_runChain toPoly_quinticL quinticPoly_ne_zero
       quintic_frob_chain quintic_frob_exp quintic_w_check quintic_bezout_check
 
 instance : Fact (Irreducible quinticPoly) := ⟨quinticPoly_irreducible⟩
diff --git a/CompPoly/Fields/KoalaBear/Ext6/SexticIrreducible.lean b/CompPoly/Fields/KoalaBear/Ext6/SexticIrreducible.lean
index e566afd9..bd5a293f 100644
--- a/CompPoly/Fields/KoalaBear/Ext6/SexticIrreducible.lean
+++ b/CompPoly/Fields/KoalaBear/Ext6/SexticIrreducible.lean
@@ -138,15 +138,12 @@ theorem sextic_cop2_bezout_check :
 factors, with kernel-checked certificates for all three conditions. -/
 theorem sexticPoly_irreducible : Irreducible sexticPoly := by
   have hcard : Fintype.card Field = fieldSize := ZMod.card _
-  refine irreducible_of_rabin_degree_six sexticPoly_natDegree ?_ ?_ ?_
-  · rw [hcard]
-    exact dvd_X_pow_sub_X_of_runChain toPoly_sexticL sexticPoly_ne_zero
+  refine irreducible_of_rabin_degree_six_of_card fieldSize hcard sexticPoly_natDegree ?_ ?_ ?_
+  · exact dvd_X_pow_sub_X_of_runChain toPoly_sexticL sexticPoly_ne_zero
       sextic_trace_chain sextic_trace_exp
-  · rw [hcard]
-    exact isCoprime_X_pow_sub_X_of_runChain toPoly_sexticL sexticPoly_ne_zero
+  · exact isCoprime_X_pow_sub_X_of_runChain toPoly_sexticL sexticPoly_ne_zero
       sextic_cop3_chain sextic_cop3_exp sextic_cop3_w_check sextic_cop3_bezout_check
-  · rw [hcard]
-    exact isCoprime_X_pow_sub_X_of_runChain toPoly_sexticL sexticPoly_ne_zero
+  · exact isCoprime_X_pow_sub_X_of_runChain toPoly_sexticL sexticPoly_ne_zero
       sextic_cop2_chain sextic_cop2_exp sextic_cop2_w_check sextic_cop2_bezout_check
 
 instance : Fact (Irreducible sexticPoly) := ⟨sexticPoly_irreducible⟩