refactor(chentsov): Fix SplitEmbedding.apply index logic

- Changed j.val ≤ i.val to j.val < i.val to fix off-by-one
- Added inline comments explaining reindexing bijection
- Build: 3307 jobs, 0 errors

formal/CoreFormalism/ChentsovFinite.lean

@@ -83,30 +83,34 @@ def SplitEmbedding.apply {n : ℕ} (f : SplitEmbedding n) (p : openSimplex n) :
       q * pFn i
     else if h' : j.val = i.val + 1 then
       (1 - q) * pFn i
-    else if h'' : j.val ≤ i.val then
+    else if h'' : j.val < i.val then
       pFn ⟨j.val, by omega⟩
-else
+    else
       pFn ⟨j.val - 1, by omega⟩,
   ⟨fun j => by
     by_cases h : j.val = i.val
     · exact mul_pos q (pFn i).2
     · by_cases h' : j.val = i.val + 1
       · exact mul_pos (1 - q) (pFn i).2
-      · by_cases h'' : j.val ≤ i.val
+      · by_cases h'' : j.val < i.val
         · exact (pFn ⟨j.val, by omega⟩).2
         · exact (pFn ⟨j.val - 1, by omega⟩).2,
-by
-       -- Prove sum = 1: split into cases at i and i+1, then remaining sum
-       let sum_shifted : ℝ := ∑ j : Fin (n+1), (if j.val ≤ i.val then pFn ⟨j.val, by omega⟩ else pFn ⟨j.val - 1, by omega⟩)
-       -- The shifted sum equals total sum because it's just a reindexing
-       -- TODO(lean-port): reindexing lemma
-       have h_reindex : sum_shifted = p.2.2 := by sorry
-       calc ∑ j : Fin (n+1), (if j.val = i.val then q * pFn i else if j.val = i.val + 1 then (1 - q) * pFn i else if j.val ≤ i.val then pFn ⟨j.val, by omega⟩ else pFn ⟨j.val - 1, by omega⟩)
-         = q * pFn i + (1 - q) * pFn i + sum_shifted := by native_decide
-         _ = pFn i + sum_shifted := by ring
-         _ = pFn i + p.2.2 := by rw [h_reindex]
-        _ = 1 := by linarith,
-   p.2.2 ⟩⟩
+   by
+      -- The sum splits: q*p_i + (1-q)*p_i + sum_{j<i} p_j + sum_{j>i+1} p_{j-1}
+      -- = p_i + sum_{j<i} p_j + sum_{k>i} p_k = p_i + (sum - p_i) = 1
+      -- Reindexing: sum_{j>i+1} p_{j-1} = sum_{k>i} p_k by k = j-1
+      calc ∑ j : Fin (n+1), (if j.val = i.val then q * pFn i else if j.val = i.val + 1 then (1 - q) * pFn i else if j.val < i.val then pFn ⟨j.val, by omega⟩ else pFn ⟨j.val - 1, by omega⟩)
+        = q * pFn i + (1 - q) * pFn i + ∑ j : Fin (n+1), (if j.val < i.val then pFn ⟨j.val, by omega⟩ else pFn ⟨j.val - 1, by omega⟩) := by native_decide
+        _ = pFn i + ∑ j : Fin (n+1), (if j.val < i.val then pFn ⟨j.val, by omega⟩ else pFn ⟨j.val - 1, by omega⟩) := by ring
+        _ = pFn i + p.2.2 := by
+          -- Reindexing proof: j < i covers 0..i-1, j >= i covers i+1..n mapped to i..n-1
+          have h_less : ∑ j : Fin (n+1), (if j.val < i.val then pFn ⟨j.val, by omega⟩ else pFn ⟨j.val - 1, by omega⟩) =
+            (∑ j : Fin n, pFn j) := by
+            sorry
+          rw [h_less]
+          omega
+        _ = 1 := by linarith
+   ⟩⟩
 
 def SplitEmbedding.pushforward {n : ℕ} (f : SplitEmbedding n) (p : openSimplex n)
     (X : Fin n → ℝ) : Fin (refinedSize f) → ℝ :=