diff --git a/backlog/ggeom-pred-pow-nine-sub-one.md b/backlog/ggeom-pred-pow-nine-sub-one.md
new file mode 100644
index 000000000..6d20f72e7
--- /dev/null
+++ b/backlog/ggeom-pred-pow-nine-sub-one.md
@@ -0,0 +1,7 @@
+# ggeom-pred-pow-nine-sub-one
+
+n minus one divides n to the 9 minus one.
+
+- **Source:** self-seeded polynomial-divisibility identity family.
+- **Reference:** n minus one divides n to the 9 minus one. Provable by exhibiting the cofactor and `ring`.
+- **Difficulty:** 1
diff --git a/goals/ggeom-pred-pow-nine-sub-one.aisp b/goals/ggeom-pred-pow-nine-sub-one.aisp
new file mode 100644
index 000000000..e15f6ed11
--- /dev/null
+++ b/goals/ggeom-pred-pow-nine-sub-one.aisp
@@ -0,0 +1,20 @@
+𝔸5.1.goal.ggeom-pred-pow-nine-sub-one@2026-06-17
+γ≔unsorry.goal
+⟦Ω:Goal⟧{
+  id≜ggeom-pred-pow-nine-sub-one
+  phase≜prove
+  status≜proved
+  difficulty≜1
+}
+⟦Σ:Source⟧{
+  src≜backlog/ggeom-pred-pow-nine-sub-one.md
+}
+⟦Γ:Deps⟧{
+  deps≜⟨⟩
+}
+⟦Λ:Artifact⟧{
+  lean≜goals/ggeom-pred-pow-nine-sub-one.lean
+  sha≜798b57ee4a27dff905570d051bd1bf6f4fec08db0bbcca61def335ee33b6b85c
+  aff≜0
+}
+⟦Ε⟧⟨δ≜0.60;τ≜◊⁺⟩
diff --git a/goals/ggeom-pred-pow-nine-sub-one.lean b/goals/ggeom-pred-pow-nine-sub-one.lean
new file mode 100644
index 000000000..1d9aae498
--- /dev/null
+++ b/goals/ggeom-pred-pow-nine-sub-one.lean
@@ -0,0 +1,4 @@
+import Mathlib
+
+theorem ggeom_pred_pow_nine_sub_one (n : ℤ) : (n - 1) ∣ (n^9 - 1) := by
+  sorry
diff --git a/library/Unsorry/GgeomPredPowNineSubOne.lean b/library/Unsorry/GgeomPredPowNineSubOne.lean
new file mode 100644
index 000000000..8c2c1e169
--- /dev/null
+++ b/library/Unsorry/GgeomPredPowNineSubOne.lean
@@ -0,0 +1,4 @@
+import Mathlib
+
+theorem ggeom_pred_pow_nine_sub_one (n : ℤ) : (n - 1) ∣ (n^9 - 1) := by
+  exact ⟨n^8 + n^7 + n^6 + n^5 + n^4 + n^3 + n^2 + n + 1, by ring⟩
diff --git a/library/index/798b57ee4a27dff905570d051bd1bf6f4fec08db0bbcca61def335ee33b6b85c.aisp b/library/index/798b57ee4a27dff905570d051bd1bf6f4fec08db0bbcca61def335ee33b6b85c.aisp
new file mode 100644
index 000000000..fd77fbf22
--- /dev/null
+++ b/library/index/798b57ee4a27dff905570d051bd1bf6f4fec08db0bbcca61def335ee33b6b85c.aisp
@@ -0,0 +1,8 @@
+𝔸5.1.lemma.798b57ee4a27@2026-06-17
+γ≔unsorry.lemma.index
+⟦Ω:Lemma⟧{sha≜798b57ee4a27dff905570d051bd1bf6f4fec08db0bbcca61def335ee33b6b85c; goal≜ggeom-pred-pow-nine-sub-one; name≜ggeom_pred_pow_nine_sub_one}
+⟦Σ:Source⟧{src≜goals/ggeom-pred-pow-nine-sub-one.lean}
+⟦Γ:Tags⟧{tags≜⟨⟩}
+⟦Λ:Meta⟧{use≜0; aff≜0}
+⟦Π:Provenance⟧{solver≜ohdearquant; agent≜mac-158f; provider≜claude; model≜template-ring-cofactor; attempts≜1}
+⟦Ε⟧⟨δ≜0.60;τ≜◊⁺⟩
diff --git a/proof-runs/ggeom-pred-pow-nine-sub-one.mac-158f.20260617t181945112606z-7bfb09b2.aisp b/proof-runs/ggeom-pred-pow-nine-sub-one.mac-158f.20260617t181945112606z-7bfb09b2.aisp
new file mode 100644
index 000000000..3f953875e
--- /dev/null
+++ b/proof-runs/ggeom-pred-pow-nine-sub-one.mac-158f.20260617t181945112606z-7bfb09b2.aisp
@@ -0,0 +1,8 @@
+𝔸5.1.run.ggeom-pred-pow-nine-sub-one.mac-158f.20260617t181945112606z-7bfb09b2@2026-06-17
+γ≔unsorry.proof.run
+⟦Ω:Run⟧{id≜20260617t181945112606z-7bfb09b2; goal≜ggeom-pred-pow-nine-sub-one; agent≜mac-158f; outcome≜proved}
+⟦Π:Provenance⟧{solver≜ohdearquant; provider≜claude; model≜template-ring-cofactor}
+⟦Γ:Goal⟧{goal≜ggeom-pred-pow-nine-sub-one}
+⟦Λ:Metrics⟧{attempts≜1; solve_s≜0; ended≜2026-06-17T18:19:45Z}
+⟦Σ:Artifact⟧{sha≜798b57ee4a27dff905570d051bd1bf6f4fec08db0bbcca61def335ee33b6b85c}
+⟦Ε⟧⟨δ≜0.60;τ≜◊⁺⟩