From 3139630129a863c59e10a761819da14824c17d9c Mon Sep 17 00:00:00 2001
From: Per Alexandersson <per.w.alexandersson@gmail.com>
Date: Mon, 3 Aug 2026 23:47:03 +0000
Subject: [PATCH 1/9] fix Braun-Jal source step matrix

---
 .../MatrixInduction.lean                      | 56 +++++++++++++------
 .../Narayana/Claim7.lean                      | 26 +++++----
 2 files changed, 52 insertions(+), 30 deletions(-)

diff --git a/RealRooted/GeneralizedSnakePosets/MatrixInduction.lean b/RealRooted/GeneralizedSnakePosets/MatrixInduction.lean
index e24a389c..60043d68 100644
--- a/RealRooted/GeneralizedSnakePosets/MatrixInduction.lean
+++ b/RealRooted/GeneralizedSnakePosets/MatrixInduction.lean
@@ -4,10 +4,10 @@ import RealRooted.MatrixInterlacing
 /-!
 # Braun-Jal matrix-induction step
 
-This module isolates the local two-row matrix step used in Braun-Jal's proof
-of Theorem 4.1.  The theorem statements stay close to the paper recurrence:
-the matrix with rows `[P_{m-1}, G_{m-1}]` and `[P_m, G_m]` acts on the
-induction pair `[f, X * g]`.
+This module isolates the local two-row matrix step used in Braun--Jal's proof
+of Theorem 4.1 (arXiv:2607.00922v1, p. 10).  The source matrix has rows
+`[P_{m-1}, G_{m-1}]` and `[Q_m, H_m]`, where `Q_m = P_m - P_{m-1}` and
+`H_m = G_m - G_{m-1}`, and acts on the induction pair `[f, X * g]`.
 -/
 
 open Polynomial
@@ -19,7 +19,8 @@ namespace GeneralizedSnakePosets
 
 /-- The two-row matrix for one Braun-Jal Theorem 4.1 induction step. -/
 def theorem41StepMatrix (P G : ℕ → ℝ[X]) (m : ℕ) : List (List ℝ[X]) :=
-  [[P (m - 1), G (m - 1)], [P m, G m]]
+  [[P (m - 1), G (m - 1)],
+    [narayanaDifference P m, auxiliaryDifference G m]]
 
 @[simp] theorem theorem41StepMatrix_length (P G : ℕ → ℝ[X]) (m : ℕ) :
     (theorem41StepMatrix P G m).length = 2 := by
@@ -30,18 +31,22 @@ theorem theorem41StepMatrix_rect (P G : ℕ → ℝ[X]) (m : ℕ) :
     ∀ row ∈ theorem41StepMatrix P G m, row.length = 2 := by
   intro row hrow
   have hrow' :
-      row = [P (m - 1), G (m - 1)] ∨ row = [P m, G m] := by
+      row = [P (m - 1), G (m - 1)] ∨
+        row = [narayanaDifference P m, auxiliaryDifference G m] := by
     simpa [theorem41StepMatrix] using hrow
   rcases hrow' with rfl | rfl <;> simp
 
 /-- Entrywise nonnegativity for the Braun-Jal step matrix. -/
 theorem theorem41StepMatrix_entry_nonneg {P G : ℕ → ℝ[X]} {m : ℕ}
     (hP_nonneg : ∀ n, HasNonnegCoeffs (P n))
-    (hG_nonneg : ∀ n, HasNonnegCoeffs (G n)) :
+    (hG_nonneg : ∀ n, HasNonnegCoeffs (G n))
+    (hQ_nonneg : HasNonnegCoeffs (narayanaDifference P m))
+    (hH_nonneg : HasNonnegCoeffs (auxiliaryDifference G m)) :
     ∀ row ∈ theorem41StepMatrix P G m, ∀ p ∈ row, HasNonnegCoeffs p := by
   intro row hrow p hp
   have hrow' :
-      row = [P (m - 1), G (m - 1)] ∨ row = [P m, G m] := by
+      row = [P (m - 1), G (m - 1)] ∨
+        row = [narayanaDifference P m, auxiliaryDifference G m] := by
     simpa [theorem41StepMatrix] using hrow
   rcases hrow' with rfl | rfl
   · have hp' : p = P (m - 1) ∨ p = G (m - 1) := by
@@ -49,18 +54,20 @@ theorem theorem41StepMatrix_entry_nonneg {P G : ℕ → ℝ[X]} {m : ℕ}
     rcases hp' with rfl | rfl
     · exact hP_nonneg (m - 1)
     · exact hG_nonneg (m - 1)
-  · have hp' : p = P m ∨ p = G m := by
+  · have hp' :
+        p = narayanaDifference P m ∨ p = auxiliaryDifference G m := by
       simpa using hp
     rcases hp' with rfl | rfl
-    · exact hP_nonneg m
-    · exact hG_nonneg m
+    · exact hQ_nonneg
+    · exact hH_nonneg
 
 /-- The step matrix action gives the two recurrence sums appearing in the
 nonconstant induction step. -/
 theorem theorem41StepMatrix_action_pair
     (P G : ℕ → ℝ[X]) (m : ℕ) (f g : ℝ[X]) :
     matPolyAction (theorem41StepMatrix P G m) [f, X * g] =
-      [f * P (m - 1) + X * g * G (m - 1), f * P m + X * g * G m] := by
+      [f * P (m - 1) + X * g * G (m - 1),
+        f * narayanaDifference P m + X * g * auxiliaryDifference G m] := by
   simp [theorem41StepMatrix, matPolyAction, mul_comm, mul_left_comm]
 
 /-- The induction hypothesis `g << f` makes `[f, X * g]` a nonnegative
@@ -79,17 +86,30 @@ theorem theorem41InputPair_interlacingSeqNonneg {f g : ℝ[X]}
   · rw [isInterlacingSeq_iff_pairwise]
     simp [prec_mul_X_of_prec_of_nonneg hgf hg_nonneg hf_nonneg]
 
-/-- Claim `(7)` supplies the nontrivial cross `2 x 2` affine test for the
-Braun-Jal step matrix. -/
-theorem theorem41StepMatrix_cross_has2x2_of_claim7
+/-- Claim `(6)` is exactly the cross `2 x 2` affine test for the source matrix
+in Braun--Jal's proof of Theorem 4.1. -/
+theorem theorem41StepMatrix_cross_has2x2_of_matrixClaim
+    {P G : ℕ → ℝ[X]} (hclaim : Theorem41MatrixClaimStatement P G)
+    {m : ℕ} (hm : 2 ≤ m) :
+    Has2x2InterlacingProperty (P (m - 1)) (G (m - 1))
+      (narayanaDifference P m) (auxiliaryDifference G m) := by
+  intro s t hs ht
+  exact hclaim hm hs.le ht.le
+
+/-- Claim `(7)` supplies the cross affine test for the stronger consecutive-row
+matrix with rows `[P_{m-1}, G_{m-1}]` and `[P_m, G_m]`.  This is an auxiliary
+route, not the matrix displayed in Braun--Jal's proof. -/
+theorem theorem41ConsecutiveMatrix_cross_has2x2_of_claim7
     {P G : ℕ → ℝ[X]} (hclaim : Theorem41Claim7Statement P G)
     {m : ℕ} (hm : 2 ≤ m) :
     Has2x2InterlacingProperty (P (m - 1)) (G (m - 1)) (P m) (G m) := by
   intro s t hs ht
   exact hclaim (m := m) (lam := s) (nu := t) hm hs.le (by linarith)
 
-/-- Claim `(7)`, the column interlacings, and the induction pair propagate
-proper position through one nonconstant recurrence step. -/
+/-- A stronger alternative to the source matrix step: Claim `(7)` plus proper
+position in both consecutive columns propagates the induction pair directly.
+The paper instead applies Claim `(6)` to `theorem41StepMatrix` and then uses
+Lemma 2.6. -/
 theorem theorem41Step_prec_of_claim7
     {P G : ℕ → ℝ[X]} {m : ℕ} {f g : ℝ[X]}
     (hclaim : Theorem41Claim7Statement P G) (hm : 2 ≤ m)
@@ -105,7 +125,7 @@ theorem theorem41Step_prec_of_claim7
   have hpair := prec_add_mul_pair_of_2x2
     (p₁ := P (m - 1)) (q₁ := G (m - 1)) (p₂ := P m) (q₂ := G m)
     (u := f) (v := X * g)
-    hP hG (theorem41StepMatrix_cross_has2x2_of_claim7 hclaim hm) hinput
+    hP hG (theorem41ConsecutiveMatrix_cross_has2x2_of_claim7 hclaim hm) hinput
     (hP_nonneg (m - 1)) (hG_nonneg (m - 1))
     (hP_nonneg m) (hG_nonneg m) hf_nonneg hg_nonneg.X_mul
   simpa [mul_comm, mul_left_comm] using hpair
diff --git a/RealRooted/GeneralizedSnakePosets/Narayana/Claim7.lean b/RealRooted/GeneralizedSnakePosets/Narayana/Claim7.lean
index e80bc986..0b572835 100644
--- a/RealRooted/GeneralizedSnakePosets/Narayana/Claim7.lean
+++ b/RealRooted/GeneralizedSnakePosets/Narayana/Claim7.lean
@@ -165,11 +165,13 @@ private theorem prec_narayanaPolynomial_two (n : ℕ) :
       simpa [Nat.succ_eq_add_one, Nat.add_assoc] using
         prec_narayanaPolynomial_succ 2 n
 
-/-- Consecutive auxiliary polynomials are in proper position once the rook
-model identifies `G n` with `n` times the parameter-two generalized Narayana
-polynomial.  This identity is accepted as combinatorial input: formalizing its
-board bijection is outside scope, while the proper-position deduction is
-proved here from the generalized Narayana recurrence. -/
+/-- Consecutive auxiliary polynomials are in proper position under an additional
+identification with parameter-two generalized Narayana polynomials.
+
+This identity is not an input used in Braun--Jal's proof of Theorem 4.1; their
+proof instead uses the `[P, G; Q, H]` matrix and Claim `(6)`.  This theorem is
+therefore an optional stronger route and `hG_model` requires an independent
+justification. -/
 theorem auxiliaryG_prec_succ_of_narayanaTwoModel
     (hG_model : ∀ n : ℕ, 1 ≤ n →
       FiniteSkewBoard.auxiliaryG n =
@@ -221,13 +223,13 @@ theorem theorem41NonNestingRook_modified_of_modelInputs_of_adjacentG
       (lemma33AuxiliaryGInterlaces_modified hrec2 hH_nonneg)
       lemma34ModifiedNarayanaInterlacing_modified hrec
 
-/--
-Braun--Jal Theorem 4.1 from combinatorial model inputs. The hypotheses below are an intentional
-trust boundary: in particular, `hG_model` records only the rook-model identification from the
-paper, whose full rook and order-polytope models are outside the scope of this project. They do
-not assume interlacing or real-rootedness; those conclusions are derived here from the formalized
-recurrence and generalized Narayana theory.
--/
+/-- An alternative Theorem 4.1 endpoint using the additional generalized
+Narayana identity `hG_model`.
+
+Braun--Jal do not use or state this identity in their proof.  The source-faithful
+route goes through the `[P, G; Q, H]` matrix and Claim `(6)`, so this result must
+not be presented as depending only on the paper's combinatorial boundary facts.
+It remains useful when `hG_model` is independently established. -/
 theorem theorem41NonNestingRook_modified_of_modelInputs
     {M : SnakeWord → ℝ[X]}
     (hrec2 : NarayanaAuxiliaryGRecurrenceStatement

From 8ffb627e668b87aec8f939b753c0d0b281b29621 Mon Sep 17 00:00:00 2001
From: Per Alexandersson <per.w.alexandersson@gmail.com>
Date: Mon, 3 Aug 2026 23:49:11 +0000
Subject: [PATCH 2/9] remove obsolete Braun-Jal matrix wrappers

---
 .../MatrixInduction.lean                      | 60 -------------------
 1 file changed, 60 deletions(-)

diff --git a/RealRooted/GeneralizedSnakePosets/MatrixInduction.lean b/RealRooted/GeneralizedSnakePosets/MatrixInduction.lean
index 60043d68..b1559ed5 100644
--- a/RealRooted/GeneralizedSnakePosets/MatrixInduction.lean
+++ b/RealRooted/GeneralizedSnakePosets/MatrixInduction.lean
@@ -238,40 +238,6 @@ theorem theorem41StepOne_prec_of_recurrence
     (hM_nonneg (w.takePrefix (k + 1))) (hM_nonneg (w.takePrefix k))
   rwa [hdel, hrec_w]
 
-/-- Matrix Claim `(6)` gives the same recurrence-step proper-position result
-via the existing Claim `(6)`/Claim `(7)` reindexing. -/
-theorem theorem41Step_prec_of_matrixClaim
-    {P G : ℕ → ℝ[X]} {m : ℕ} {f g : ℝ[X]}
-    (hclaim : Theorem41MatrixClaimStatement P G) (hm : 2 ≤ m)
-    (hP : Prec (P (m - 1)) (P m)) (hG : Prec (G (m - 1)) (G m))
-    (hgf : Prec g f)
-    (hP_nonneg : ∀ n, HasNonnegCoeffs (P n))
-    (hG_nonneg : ∀ n, HasNonnegCoeffs (G n))
-    (hf_nonneg : HasNonnegCoeffs f) (hg_nonneg : HasNonnegCoeffs g) :
-    Prec (f * P (m - 1) + X * g * G (m - 1))
-      (f * P m + X * g * G m) :=
-  theorem41Step_prec_of_claim7
-    ((theorem41MatrixClaim_iff_claim7 P G).mp hclaim) hm
-    hP hG hgf hP_nonneg hG_nonneg hf_nonneg hg_nonneg
-
-/-- Matrix Claim `(6)` version of the nonconstant word-level recurrence step. -/
-theorem theorem41NonconstantStep_prec_of_matrixClaim
-    {M : SnakeWord → ℝ[X]} {P G : ℕ → ℝ[X]} {w : SnakeWord} {k : ℕ}
-    (hrec : Theorem35GeneralizedSnakeRecurrenceStatement M P G)
-    (hclaim : Theorem41MatrixClaimStatement P G)
-    (hlast : w.IsLastChangeIndex k)
-    (hk : k + 1 < w.deleteFinal.length)
-    (hP : ∀ {m : ℕ}, 2 ≤ m → Prec (P (m - 1)) (P m))
-    (hG : ∀ {m : ℕ}, 2 ≤ m → Prec (G (m - 1)) (G m))
-    (hprefix : Prec (M (w.takePrefix k)) (M (w.takePrefix (k + 1))))
-    (hP_nonneg : ∀ n, HasNonnegCoeffs (P n))
-    (hG_nonneg : ∀ n, HasNonnegCoeffs (G n))
-    (hM_nonneg : ∀ u, HasNonnegCoeffs (M u)) :
-    Prec (M w.deleteFinal) (M w) :=
-  theorem41NonconstantStep_prec_of_claim7
-    (P := P) (G := G) hrec ((theorem41MatrixClaim_iff_claim7 P G).mp hclaim)
-    hlast hk hP hG hprefix hP_nonneg hG_nonneg hM_nonneg
-
 /-- Length-induction skeleton for Braun-Jal Theorem 4.1.
 
 If every nonconstant word step turns the prefix induction hypothesis into
@@ -609,32 +575,6 @@ theorem theorem41InductionRoute_of_claim7_of_constant_matches_succ_length
     (hP_nonneg := hP_nonneg) (hG_nonneg := hG_nonneg)
     (hM_nonneg := hM_nonneg) (hdeg := hdeg) (hM_const := hM_const)
 
-/-- Matrix Claim `(6)` version of the abstract route bridge, using the
-successor-length constant-word identity from the concrete indexing. -/
-theorem theorem41InductionRoute_of_matrixClaim_of_constant_matches_succ_length
-    {M : SnakeWord → ℝ[X]} {P G : ℕ → ℝ[X]}
-    (hmatrix_of_inputs :
-      Lemma33AuxiliaryGInterlacesStatement P G →
-        Lemma34ModifiedNarayanaInterlacingStatement P →
-          Theorem41MatrixClaimStatement P G)
-    (hP_interlaces : ∀ n : ℕ, Interlaces (P n) (P (n + 1)))
-    (hG : ∀ {m : ℕ}, 2 ≤ m → Prec (G (m - 1)) (G m))
-    (hP_one : P 1 = 1 + X) (hG_one : G 1 = 1)
-    (hP_nonneg : ∀ n, HasNonnegCoeffs (P n))
-    (hG_nonneg : ∀ n, HasNonnegCoeffs (G n))
-    (hM_nonneg : ∀ w, HasNonnegCoeffs (M w))
-    (hdeg :
-      ∀ {w : SnakeWord}, 1 ≤ w.length →
-        (M w.deleteFinal).natDegree + 1 = (M w).natDegree)
-    (hM_const : ∀ {w : SnakeWord}, w.IsConstant → M w = P (w.length + 1)) :
-    Theorem41InductionRouteStatement M P G :=
-  theorem41InductionRoute_of_claim7_of_constant_matches_succ_length
-    (M := M) (P := P) (G := G)
-    (fun h33 h34 => (theorem41MatrixClaim_iff_claim7 P G).mp
-      (hmatrix_of_inputs h33 h34))
-    hP_interlaces hG hP_one hG_one hP_nonneg hG_nonneg hM_nonneg hdeg
-    hM_const
-
 /-- Section 3 equation `(2)` plus the local Claim `(7)` side conditions give
 the abstract induction route, using the concrete successor-length indexing for
 constant words.

From b1e6d3a508730c32b3831aed56b2c31d01ddfb97 Mon Sep 17 00:00:00 2001
From: Per Alexandersson <per.w.alexandersson@gmail.com>
Date: Mon, 3 Aug 2026 23:54:16 +0000
Subject: [PATCH 3/9] prove Braun-Jal source matrix step

---
 .../MatrixInduction.lean                      | 122 ++++++++++++++++++
 1 file changed, 122 insertions(+)

diff --git a/RealRooted/GeneralizedSnakePosets/MatrixInduction.lean b/RealRooted/GeneralizedSnakePosets/MatrixInduction.lean
index b1559ed5..5551eff9 100644
--- a/RealRooted/GeneralizedSnakePosets/MatrixInduction.lean
+++ b/RealRooted/GeneralizedSnakePosets/MatrixInduction.lean
@@ -1,5 +1,6 @@
 import RealRooted.GeneralizedSnakePosets.Statements
 import RealRooted.MatrixInterlacing
+import RealRooted.PFPolynomial
 
 /-!
 # Braun-Jal matrix-induction step
@@ -96,6 +97,58 @@ theorem theorem41StepMatrix_cross_has2x2_of_matrixClaim
   intro s t hs ht
   exact hclaim hm hs.le ht.le
 
+/-- Claim `(6)` and the source matrix send the induction pair to a proper-position
+pair.  Repeated column indices use the real-rootedness already contained in the
+same Claim `(6)` instance. -/
+theorem theorem41Step_difference_prec_of_matrixClaim
+    {P G : ℕ → ℝ[X]} {m : ℕ} {f g : ℝ[X]}
+    (hclaim : Theorem41MatrixClaimStatement P G) (hm : 2 ≤ m)
+    (hP_ne : P (m - 1) ≠ 0)
+    (hQ_ne : narayanaDifference P m ≠ 0)
+    (hP_nonneg : ∀ n, HasNonnegCoeffs (P n))
+    (hG_nonneg : ∀ n, HasNonnegCoeffs (G n))
+    (hQ_nonneg : HasNonnegCoeffs (narayanaDifference P m))
+    (hH_nonneg : HasNonnegCoeffs (auxiliaryDifference G m))
+    (hgf : Prec g f)
+    (hf_nonneg : HasNonnegCoeffs f) (hg_nonneg : HasNonnegCoeffs g) :
+    Prec (f * P (m - 1) + X * g * G (m - 1))
+      (f * narayanaDifference P m + X * g * auxiliaryDifference G m) := by
+  have hpair := prec_zipWith_sum_pair_of_2x2
+    (n := 2) (row₁ := [P (m - 1), G (m - 1)])
+    (row₂ := [narayanaDifference P m, auxiliaryDifference G m])
+    (fs := [f, X * g])
+    (hn := by decide)
+    (hrow₁_len := by simp)
+    (hrow₂_len := by simp)
+    (hrow₁_head_ne := by simpa using hP_ne)
+    (hrow₂_head_ne := by simpa using hQ_ne)
+    (hrow₁_nonneg := by
+      intro p hp
+      simp only [List.mem_cons, List.not_mem_nil, or_false] at hp
+      rcases hp with rfl | rfl
+      · exact hP_nonneg (m - 1)
+      · exact hG_nonneg (m - 1))
+    (hrow₂_nonneg := by
+      intro p hp
+      simp only [List.mem_cons, List.not_mem_nil, or_false] at hp
+      rcases hp with rfl | rfl
+      · exact hQ_nonneg
+      · exact hH_nonneg)
+    (h2x2 := by
+      intro j₁ j₂ hj
+      fin_cases j₁ <;> fin_cases j₂
+      · intro s t hs ht
+        have hcross := hclaim (m := m) (lam := s) (mu := t) hm hs.le ht.le
+        simpa using prec_refl hcross.2.1.1 hcross.2.1.2
+      · simpa using theorem41StepMatrix_cross_has2x2_of_matrixClaim hclaim hm
+      · simp at hj
+      · intro s t hs ht
+        have hcross := hclaim (m := m) (lam := s) (mu := t) hm hs.le ht.le
+        simpa using prec_refl hcross.1.1 hcross.1.2)
+    (hfs_len := by simp)
+    (hfs := theorem41InputPair_interlacingSeqNonneg hgf hf_nonneg hg_nonneg)
+  simpa [mul_comm, mul_left_comm] using hpair
+
 /-- Claim `(7)` supplies the cross affine test for the stronger consecutive-row
 matrix with rows `[P_{m-1}, G_{m-1}]` and `[P_m, G_m]`.  This is an auxiliary
 route, not the matrix displayed in Braun--Jal's proof. -/
@@ -179,6 +232,75 @@ theorem theorem41NonconstantStep_prec_of_claim7
     (hM_nonneg (w.takePrefix (k + 1))) (hM_nonneg (w.takePrefix k))
   rwa [hrec_del, hrec_w]
 
+/-- The nonconstant Braun--Jal induction step through the source
+`[P, G; Q, H]` matrix.  Unlike the consecutive-row shortcut above, this is the
+argument on p. 10 of the paper and requires no adjacent-`G` proper position. -/
+theorem theorem41NonconstantStep_prec_of_matrixClaim
+    {M : SnakeWord → ℝ[X]} {P G : ℕ → ℝ[X]} {w : SnakeWord} {k : ℕ}
+    (hrec : Theorem35GeneralizedSnakeRecurrenceStatement M P G)
+    (hclaim : Theorem41MatrixClaimStatement P G)
+    (hlast : w.IsLastChangeIndex k)
+    (hk : k + 1 < w.deleteFinal.length)
+    (hP_ne : ∀ n, P n ≠ 0)
+    (hQ_ne : ∀ {m : ℕ}, 2 ≤ m → narayanaDifference P m ≠ 0)
+    (hP_nonneg : ∀ n, HasNonnegCoeffs (P n))
+    (hG_nonneg : ∀ n, HasNonnegCoeffs (G n))
+    (hQ_nonneg : ∀ {m : ℕ}, 2 ≤ m →
+      HasNonnegCoeffs (narayanaDifference P m))
+    (hH_nonneg : ∀ {m : ℕ}, 2 ≤ m →
+      HasNonnegCoeffs (auxiliaryDifference G m))
+    (hprefix : Prec (M (w.takePrefix k)) (M (w.takePrefix (k + 1))))
+    (hM_nonneg : ∀ u, HasNonnegCoeffs (M u)) :
+    Prec (M w.deleteFinal) (M w) := by
+  let f : ℝ[X] := M (w.takePrefix (k + 1))
+  let g : ℝ[X] := M (w.takePrefix k)
+  let m : ℕ := w.length - (k + 1)
+  have hm : 2 ≤ m := by
+    dsimp [m]
+    rw [SnakeWord.length_deleteFinal] at hk
+    lia
+  have hkp1_le : k + 1 ≤ w.deleteFinal.length := le_of_lt hk
+  have hk_le : k ≤ w.deleteFinal.length := by lia
+  have hrec_w : M w = f * P m + X * g * G m := by
+    dsimp [f, g, m]
+    exact hrec hlast.not_isConstant hlast
+  have hlast_del : w.deleteFinal.IsLastChangeIndex k := hlast.deleteFinal hk
+  have hrec_del :
+      M w.deleteFinal = f * P (m - 1) + X * g * G (m - 1) := by
+    have hbase := hrec hlast_del.not_isConstant hlast_del
+    dsimp [f, g, m]
+    rw [hbase]
+    rw [SnakeWord.takePrefix_deleteFinal_eq_takePrefix_of_le hkp1_le]
+    rw [SnakeWord.takePrefix_deleteFinal_eq_takePrefix_of_le hk_le]
+    rw [SnakeWord.length_deleteFinal_sub_eq]
+  have hrec_diff :
+      M w - M w.deleteFinal =
+        f * narayanaDifference P m + X * g * auxiliaryDifference G m := by
+    rw [hrec_w, hrec_del]
+    unfold narayanaDifference auxiliaryDifference
+    ring
+  have hf_nonneg : HasNonnegCoeffs f := hM_nonneg _
+  have hg_nonneg : HasNonnegCoeffs g := hM_nonneg _
+  have hdiff_nonneg : HasNonnegCoeffs (M w - M w.deleteFinal) := by
+    rw [hrec_diff]
+    exact (hf_nonneg.mul (hQ_nonneg hm)).add
+      (hg_nonneg.X_mul.mul (hH_nonneg hm))
+  have hstep : Prec (M w.deleteFinal) (M w - M w.deleteFinal) := by
+    rw [hrec_del, hrec_diff]
+    exact theorem41Step_difference_prec_of_matrixClaim
+      hclaim hm (hP_ne (m - 1)) (hQ_ne hm) hP_nonneg hG_nonneg
+      (hQ_nonneg hm) (hH_nonneg hm) hprefix hf_nonneg hg_nonneg
+  have hsum0 : Prec0 (M w.deleteFinal)
+      (M w.deleteFinal + (M w - M w.deleteFinal)) :=
+    prec0_add_right_of_common_left_of_nonneg
+      (prec_refl hstep.1.1 hstep.1.2).toPrec0 hstep.toPrec0
+      (hM_nonneg w.deleteFinal) hdiff_nonneg
+  have hsum_ne : M w.deleteFinal + (M w - M w.deleteFinal) ≠ 0 :=
+    add_ne_zero_of_hasNonnegCoeffs_of_right_ne_zero
+      (hM_nonneg w.deleteFinal) hdiff_nonneg hstep.2.1.1
+  have hfinal := hsum0.toPrec_of_ne hstep.1.1 hsum_ne
+  convert hfinal using 1 <;> ring
+
 /-- Polynomial form of the exceptional `m = 1` Braun-Jal step.
 
 If `g ≪ f` and both polynomials have nonnegative coefficients, then

From 972c4a2f987fc38550191e504de0cccd71aae95f Mon Sep 17 00:00:00 2001
From: Per Alexandersson <per.w.alexandersson@gmail.com>
Date: Mon, 3 Aug 2026 23:56:17 +0000
Subject: [PATCH 4/9] route Braun-Jal induction through source matrix

---
 .../MatrixInduction.lean                      | 73 +++++++++++++++++++
 .../Narayana/Claim7.lean                      | 40 ++++++++++
 2 files changed, 113 insertions(+)

diff --git a/RealRooted/GeneralizedSnakePosets/MatrixInduction.lean b/RealRooted/GeneralizedSnakePosets/MatrixInduction.lean
index 5551eff9..3ce5ad0e 100644
--- a/RealRooted/GeneralizedSnakePosets/MatrixInduction.lean
+++ b/RealRooted/GeneralizedSnakePosets/MatrixInduction.lean
@@ -540,6 +540,79 @@ theorem theorem41_of_claim7_of_constant_cases
       (M := M) (P := P) (G := G) (w := w) (k := k)
       hrec hP_one hG_one hlast hsuffix hprefix_prec hM_nonneg
 
+/-- Source-matrix length induction from Claim `(6)` to Braun--Jal Theorem 4.1.
+
+The long-suffix branch uses the displayed `[P, G; Q, H]` matrix, while the
+suffix-one branch uses `P_1 = 1 + X` and `G_1 = 1`.  In particular, no
+adjacent-`G` proper-position hypothesis occurs. -/
+theorem theorem41_of_matrixClaim_of_constant_cases
+    {M : SnakeWord → ℝ[X]} {P G : ℕ → ℝ[X]}
+    (hrec : Theorem35GeneralizedSnakeRecurrenceStatement M P G)
+    (hclaim : Theorem41MatrixClaimStatement P G)
+    (hP_ne : ∀ n, P n ≠ 0)
+    (hQ_ne : ∀ {m : ℕ}, 2 ≤ m → narayanaDifference P m ≠ 0)
+    (hP_one : P 1 = 1 + X) (hG_one : G 1 = 1)
+    (hP_nonneg : ∀ n, HasNonnegCoeffs (P n))
+    (hG_nonneg : ∀ n, HasNonnegCoeffs (G n))
+    (hQ_nonneg : ∀ {m : ℕ}, 2 ≤ m →
+      HasNonnegCoeffs (narayanaDifference P m))
+    (hH_nonneg : ∀ {m : ℕ}, 2 ≤ m →
+      HasNonnegCoeffs (auxiliaryDifference G m))
+    (hM_nonneg : ∀ w, HasNonnegCoeffs (M w))
+    (hdeg :
+      ∀ {w : SnakeWord}, 1 ≤ w.length →
+        (M w.deleteFinal).natDegree + 1 = (M w).natDegree)
+    (hconst :
+      ∀ {w : SnakeWord}, 1 ≤ w.length → w.IsConstant →
+        (M w ≠ 0 ∧ (M w).Splits) ∧ Interlaces (M w.deleteFinal) (M w)) :
+    Theorem41NonNestingRookStatement M := by
+  refine theorem41_of_prec_step (M := M) ?_ hdeg hconst
+  intro w k _hconstw hlast hprefix_prec
+  by_cases hk : k + 1 < w.deleteFinal.length
+  · exact theorem41NonconstantStep_prec_of_matrixClaim
+      (M := M) (P := P) (G := G) (w := w) (k := k)
+      hrec hclaim hlast hk hP_ne hQ_ne hP_nonneg hG_nonneg
+      hQ_nonneg hH_nonneg hprefix_prec hM_nonneg
+  · have hsuffix : w.length - (k + 1) = 1 := by
+      rw [SnakeWord.length_deleteFinal] at hk
+      have hlast_suffix := hlast.succ_lt_length
+      lia
+    exact theorem41StepOne_prec_of_recurrence
+      (M := M) (P := P) (G := G) (w := w) (k := k)
+      hrec hP_one hG_one hlast hsuffix hprefix_prec hM_nonneg
+
+/-- Source-matrix induction with the constant branch reduced to the concrete
+successor-length identity `M w = P (w.length + 1)`. -/
+theorem theorem41_of_matrixClaim_of_constant_matches_succ_length
+    {M : SnakeWord → ℝ[X]} {P G : ℕ → ℝ[X]}
+    (hrec : Theorem35GeneralizedSnakeRecurrenceStatement M P G)
+    (hclaim : Theorem41MatrixClaimStatement P G)
+    (hP_ne : ∀ n, P n ≠ 0)
+    (hQ_ne : ∀ {m : ℕ}, 2 ≤ m → narayanaDifference P m ≠ 0)
+    (hP_interlaces : ∀ n : ℕ, Interlaces (P n) (P (n + 1)))
+    (hP_one : P 1 = 1 + X) (hG_one : G 1 = 1)
+    (hP_nonneg : ∀ n, HasNonnegCoeffs (P n))
+    (hG_nonneg : ∀ n, HasNonnegCoeffs (G n))
+    (hQ_nonneg : ∀ {m : ℕ}, 2 ≤ m →
+      HasNonnegCoeffs (narayanaDifference P m))
+    (hH_nonneg : ∀ {m : ℕ}, 2 ≤ m →
+      HasNonnegCoeffs (auxiliaryDifference G m))
+    (hM_nonneg : ∀ w, HasNonnegCoeffs (M w))
+    (hdeg :
+      ∀ {w : SnakeWord}, 1 ≤ w.length →
+        (M w.deleteFinal).natDegree + 1 = (M w).natDegree)
+    (hM_const : ∀ {w : SnakeWord}, w.IsConstant →
+      M w = P (w.length + 1)) :
+    Theorem41NonNestingRookStatement M := by
+  have hconst :
+      ∀ {w : SnakeWord}, 1 ≤ w.length → w.IsConstant →
+        (M w ≠ 0 ∧ (M w).Splits) ∧ Interlaces (M w.deleteFinal) (M w) :=
+    theorem41_constant_of_matches_succ_length
+      (M := M) (P := P) hM_const hP_interlaces
+  exact theorem41_of_matrixClaim_of_constant_cases
+    (M := M) (P := P) (G := G) hrec hclaim hP_ne hQ_ne hP_one hG_one
+    hP_nonneg hG_nonneg hQ_nonneg hH_nonneg hM_nonneg hdeg hconst
+
 /-- The deletion degree bridge follows from the length-indexed degree formula
 for the whole snake-word family. -/
 theorem theorem41_degree_bridge_of_natDegree_length
diff --git a/RealRooted/GeneralizedSnakePosets/Narayana/Claim7.lean b/RealRooted/GeneralizedSnakePosets/Narayana/Claim7.lean
index 0b572835..bac005f9 100644
--- a/RealRooted/GeneralizedSnakePosets/Narayana/Claim7.lean
+++ b/RealRooted/GeneralizedSnakePosets/Narayana/Claim7.lean
@@ -223,6 +223,46 @@ theorem theorem41NonNestingRook_modified_of_modelInputs_of_adjacentG
       (lemma33AuxiliaryGInterlaces_modified hrec2 hH_nonneg)
       lemma34ModifiedNarayanaInterlacing_modified hrec
 
+/-- Braun--Jal Theorem 4.1 through the source `[P, G; Q, H]` matrix.
+
+The hypotheses are the intended combinatorial trust boundary.  Equation `(2)`,
+nonnegativity of the board difference `H`, Theorem 3.5, the degree identity,
+and the constant-word staircase identity come from the non-nesting-rook model;
+formalizing that complete model is outside the present scope.  No hypothesis
+assumes real-rootedness, interlacing, proper position, or splitting. -/
+theorem theorem41NonNestingRook_modified_of_sourceInputs
+    {M : SnakeWord → ℝ[X]}
+    (hrec2 : NarayanaAuxiliaryGRecurrenceStatement
+      modifiedNarayanaPolynomial FiniteSkewBoard.auxiliaryG)
+    (hH_nonneg : ∀ n : ℕ, 1 ≤ n →
+      HasNonnegCoeffs
+        (FiniteSkewBoard.auxiliaryG n -
+          FiniteSkewBoard.auxiliaryG (n - 1)))
+    (hrec : Theorem35GeneralizedSnakeRecurrenceStatement M
+      modifiedNarayanaPolynomial FiniteSkewBoard.auxiliaryG)
+    (hM_nonneg : ∀ w : SnakeWord, HasNonnegCoeffs (M w))
+    (hdeg : ∀ {w : SnakeWord}, 1 ≤ w.length →
+      (M w.deleteFinal).natDegree + 1 = (M w).natDegree)
+    (hM_const : ∀ {w : SnakeWord}, w.IsConstant →
+      M w = modifiedNarayanaPolynomial (w.length + 1)) :
+    Theorem41NonNestingRookStatement M := by
+  exact theorem41_of_matrixClaim_of_constant_matches_succ_length
+    (M := M) (P := modifiedNarayanaPolynomial)
+    (G := FiniteSkewBoard.auxiliaryG)
+    hrec
+    ((theorem41MatrixClaim_iff_claim7 _ _).mpr
+      (theorem41Claim7_modified hrec2 hH_nonneg))
+    modifiedNarayanaPolynomial_ne_zero
+    (fun {_m} hm => narayanaDifference_modified_ne_zero (by lia))
+    modifiedNarayanaPolynomial_interlaces_succ
+    modifiedNarayanaPolynomial_one FiniteSkewBoard.auxiliaryG_one
+    modifiedNarayanaPolynomial_hasNonnegCoeffs
+    FiniteSkewBoard.auxiliaryG_hasNonnegCoeffs
+    (fun {_m} hm => narayanaDifference_modified_hasNonnegCoeffs (by lia))
+    (fun {_m} hm => by
+      simpa [auxiliaryDifference] using hH_nonneg _ (by lia))
+    hM_nonneg hdeg hM_const
+
 /-- An alternative Theorem 4.1 endpoint using the additional generalized
 Narayana identity `hG_model`.
 

From ffdc34aa0336673056e5848914a81cfb4ea8c520 Mon Sep 17 00:00:00 2001
From: Per Alexandersson <per.w.alexandersson@gmail.com>
Date: Mon, 3 Aug 2026 23:57:28 +0000
Subject: [PATCH 5/9] deduplicate Braun-Jal source hypotheses

---
 .../GeneralizedSnakePosets/MatrixInduction.lean     | 13 ++++++-------
 .../GeneralizedSnakePosets/Narayana/Claim7.lean     |  1 -
 2 files changed, 6 insertions(+), 8 deletions(-)

diff --git a/RealRooted/GeneralizedSnakePosets/MatrixInduction.lean b/RealRooted/GeneralizedSnakePosets/MatrixInduction.lean
index 3ce5ad0e..c27e426a 100644
--- a/RealRooted/GeneralizedSnakePosets/MatrixInduction.lean
+++ b/RealRooted/GeneralizedSnakePosets/MatrixInduction.lean
@@ -104,7 +104,6 @@ theorem theorem41Step_difference_prec_of_matrixClaim
     {P G : ℕ → ℝ[X]} {m : ℕ} {f g : ℝ[X]}
     (hclaim : Theorem41MatrixClaimStatement P G) (hm : 2 ≤ m)
     (hP_ne : P (m - 1) ≠ 0)
-    (hQ_ne : narayanaDifference P m ≠ 0)
     (hP_nonneg : ∀ n, HasNonnegCoeffs (P n))
     (hG_nonneg : ∀ n, HasNonnegCoeffs (G n))
     (hQ_nonneg : HasNonnegCoeffs (narayanaDifference P m))
@@ -113,6 +112,9 @@ theorem theorem41Step_difference_prec_of_matrixClaim
     (hf_nonneg : HasNonnegCoeffs f) (hg_nonneg : HasNonnegCoeffs g) :
     Prec (f * P (m - 1) + X * g * G (m - 1))
       (f * narayanaDifference P m + X * g * auxiliaryDifference G m) := by
+  have hQ_ne : narayanaDifference P m ≠ 0 := by
+    have hzero := hclaim (m := m) (lam := 0) (mu := 0) hm (by norm_num) (by norm_num)
+    simpa using hzero.2.1.1
   have hpair := prec_zipWith_sum_pair_of_2x2
     (n := 2) (row₁ := [P (m - 1), G (m - 1)])
     (row₂ := [narayanaDifference P m, auxiliaryDifference G m])
@@ -242,7 +244,6 @@ theorem theorem41NonconstantStep_prec_of_matrixClaim
     (hlast : w.IsLastChangeIndex k)
     (hk : k + 1 < w.deleteFinal.length)
     (hP_ne : ∀ n, P n ≠ 0)
-    (hQ_ne : ∀ {m : ℕ}, 2 ≤ m → narayanaDifference P m ≠ 0)
     (hP_nonneg : ∀ n, HasNonnegCoeffs (P n))
     (hG_nonneg : ∀ n, HasNonnegCoeffs (G n))
     (hQ_nonneg : ∀ {m : ℕ}, 2 ≤ m →
@@ -288,7 +289,7 @@ theorem theorem41NonconstantStep_prec_of_matrixClaim
   have hstep : Prec (M w.deleteFinal) (M w - M w.deleteFinal) := by
     rw [hrec_del, hrec_diff]
     exact theorem41Step_difference_prec_of_matrixClaim
-      hclaim hm (hP_ne (m - 1)) (hQ_ne hm) hP_nonneg hG_nonneg
+      hclaim hm (hP_ne (m - 1)) hP_nonneg hG_nonneg
       (hQ_nonneg hm) (hH_nonneg hm) hprefix hf_nonneg hg_nonneg
   have hsum0 : Prec0 (M w.deleteFinal)
       (M w.deleteFinal + (M w - M w.deleteFinal)) :=
@@ -550,7 +551,6 @@ theorem theorem41_of_matrixClaim_of_constant_cases
     (hrec : Theorem35GeneralizedSnakeRecurrenceStatement M P G)
     (hclaim : Theorem41MatrixClaimStatement P G)
     (hP_ne : ∀ n, P n ≠ 0)
-    (hQ_ne : ∀ {m : ℕ}, 2 ≤ m → narayanaDifference P m ≠ 0)
     (hP_one : P 1 = 1 + X) (hG_one : G 1 = 1)
     (hP_nonneg : ∀ n, HasNonnegCoeffs (P n))
     (hG_nonneg : ∀ n, HasNonnegCoeffs (G n))
@@ -571,7 +571,7 @@ theorem theorem41_of_matrixClaim_of_constant_cases
   by_cases hk : k + 1 < w.deleteFinal.length
   · exact theorem41NonconstantStep_prec_of_matrixClaim
       (M := M) (P := P) (G := G) (w := w) (k := k)
-      hrec hclaim hlast hk hP_ne hQ_ne hP_nonneg hG_nonneg
+      hrec hclaim hlast hk hP_ne hP_nonneg hG_nonneg
       hQ_nonneg hH_nonneg hprefix_prec hM_nonneg
   · have hsuffix : w.length - (k + 1) = 1 := by
       rw [SnakeWord.length_deleteFinal] at hk
@@ -588,7 +588,6 @@ theorem theorem41_of_matrixClaim_of_constant_matches_succ_length
     (hrec : Theorem35GeneralizedSnakeRecurrenceStatement M P G)
     (hclaim : Theorem41MatrixClaimStatement P G)
     (hP_ne : ∀ n, P n ≠ 0)
-    (hQ_ne : ∀ {m : ℕ}, 2 ≤ m → narayanaDifference P m ≠ 0)
     (hP_interlaces : ∀ n : ℕ, Interlaces (P n) (P (n + 1)))
     (hP_one : P 1 = 1 + X) (hG_one : G 1 = 1)
     (hP_nonneg : ∀ n, HasNonnegCoeffs (P n))
@@ -610,7 +609,7 @@ theorem theorem41_of_matrixClaim_of_constant_matches_succ_length
     theorem41_constant_of_matches_succ_length
       (M := M) (P := P) hM_const hP_interlaces
   exact theorem41_of_matrixClaim_of_constant_cases
-    (M := M) (P := P) (G := G) hrec hclaim hP_ne hQ_ne hP_one hG_one
+    (M := M) (P := P) (G := G) hrec hclaim hP_ne hP_one hG_one
     hP_nonneg hG_nonneg hQ_nonneg hH_nonneg hM_nonneg hdeg hconst
 
 /-- The deletion degree bridge follows from the length-indexed degree formula
diff --git a/RealRooted/GeneralizedSnakePosets/Narayana/Claim7.lean b/RealRooted/GeneralizedSnakePosets/Narayana/Claim7.lean
index bac005f9..042554da 100644
--- a/RealRooted/GeneralizedSnakePosets/Narayana/Claim7.lean
+++ b/RealRooted/GeneralizedSnakePosets/Narayana/Claim7.lean
@@ -253,7 +253,6 @@ theorem theorem41NonNestingRook_modified_of_sourceInputs
     ((theorem41MatrixClaim_iff_claim7 _ _).mpr
       (theorem41Claim7_modified hrec2 hH_nonneg))
     modifiedNarayanaPolynomial_ne_zero
-    (fun {_m} hm => narayanaDifference_modified_ne_zero (by lia))
     modifiedNarayanaPolynomial_interlaces_succ
     modifiedNarayanaPolynomial_one FiniteSkewBoard.auxiliaryG_one
     modifiedNarayanaPolynomial_hasNonnegCoeffs

From a0a317d7dad4618564ae9c857ee2fb802007760d Mon Sep 17 00:00:00 2001
From: Per Alexandersson <per.w.alexandersson@gmail.com>
Date: Mon, 3 Aug 2026 23:58:29 +0000
Subject: [PATCH 6/9] fix Braun-Jal recurrence rewrite order

---
 RealRooted/GeneralizedSnakePosets/MatrixInduction.lean | 5 +++--
 1 file changed, 3 insertions(+), 2 deletions(-)

diff --git a/RealRooted/GeneralizedSnakePosets/MatrixInduction.lean b/RealRooted/GeneralizedSnakePosets/MatrixInduction.lean
index c27e426a..ed6ce445 100644
--- a/RealRooted/GeneralizedSnakePosets/MatrixInduction.lean
+++ b/RealRooted/GeneralizedSnakePosets/MatrixInduction.lean
@@ -287,10 +287,11 @@ theorem theorem41NonconstantStep_prec_of_matrixClaim
     exact (hf_nonneg.mul (hQ_nonneg hm)).add
       (hg_nonneg.X_mul.mul (hH_nonneg hm))
   have hstep : Prec (M w.deleteFinal) (M w - M w.deleteFinal) := by
-    rw [hrec_del, hrec_diff]
-    exact theorem41Step_difference_prec_of_matrixClaim
+    have hstep_raw := theorem41Step_difference_prec_of_matrixClaim
       hclaim hm (hP_ne (m - 1)) hP_nonneg hG_nonneg
       (hQ_nonneg hm) (hH_nonneg hm) hprefix hf_nonneg hg_nonneg
+    rw [← hrec_del, ← hrec_diff] at hstep_raw
+    exact hstep_raw
   have hsum0 : Prec0 (M w.deleteFinal)
       (M w.deleteFinal + (M w - M w.deleteFinal)) :=
     prec0_add_right_of_common_left_of_nonneg

From 382c7ed769bd6c1e603199157375368de94c50fb Mon Sep 17 00:00:00 2001
From: Per Alexandersson <per.w.alexandersson@gmail.com>
Date: Tue, 4 Aug 2026 00:00:34 +0000
Subject: [PATCH 7/9] specialize Braun-Jal theorem to rook model

---
 RealRooted/Challenges/BraunJal.lean | 47 ++++++++++++++++++++++-------
 1 file changed, 36 insertions(+), 11 deletions(-)

diff --git a/RealRooted/Challenges/BraunJal.lean b/RealRooted/Challenges/BraunJal.lean
index ad4b3381..be966b4c 100644
--- a/RealRooted/Challenges/BraunJal.lean
+++ b/RealRooted/Challenges/BraunJal.lean
@@ -101,16 +101,14 @@ abbrev OrderPolytopeHStarTheorem41Target
     (hStar : NonNestingRookPolynomialFamily) : Prop :=
   OrderPolytopeHStarTheorem41Statement hStar
 
-/-- Concrete Braun--Jal Theorem 4.1 from the accepted combinatorial model
-inputs.  The parameter-two Narayana identity is the board-model fact; all
-proper-position and real-rootedness deductions are checked in Lean. -/
+/-- Braun--Jal Theorem 4.1 from the accepted combinatorial model inputs.
+The recurrence and coefficient hypotheses are the source board-model facts;
+all proper-position and real-rootedness deductions are checked in Lean. -/
 theorem theorem41_of_modifiedModelInputs
     {M : NonNestingRookPolynomialFamily}
     (hrec2 : AuxiliaryGRecurrence ModifiedNarayanaPolynomial AuxiliaryG)
     (hH_nonneg : ∀ n : ℕ, 1 ≤ n →
       HasNonnegCoeffs (AuxiliaryG n - AuxiliaryG (n - 1)))
-    (hG_model : ∀ n : ℕ, 1 ≤ n →
-      AuxiliaryG n = C (n : ℝ) * narayanaPolynomial 2 (n - 1))
     (hrec : Theorem35GeneralizedSnakeRecurrenceStatement M
       ModifiedNarayanaPolynomial AuxiliaryG)
     (hM_nonneg : ∀ w : SnakeWord, HasNonnegCoeffs (M w))
@@ -119,8 +117,37 @@ theorem theorem41_of_modifiedModelInputs
     (hM_const : ∀ {w : SnakeWord}, w.IsConstant →
       M w = ModifiedNarayanaPolynomial (w.length + 1)) :
     Theorem41Target M :=
-  theorem41NonNestingRook_modified_of_modelInputs hrec2 hH_nonneg hG_model
-    hrec hM_nonneg hdeg hM_const
+  theorem41NonNestingRook_modified_of_sourceInputs hrec2 hH_nonneg hrec
+    hM_nonneg hdeg hM_const
+
+/-- Braun--Jal Theorem 4.1 for the concrete generalized-snake rook model.
+
+The five hypotheses are intentionally the combinatorial trust boundary from
+Braun--Jal, arXiv:2607.00922v1.  Equation `(2)` defines the auxiliary family,
+`hH_nonneg` comes from its board-difference interpretation, `hrec` is Theorem
+3.5, `hdeg` is the maximum-rook/degree identity, and `hM_const` identifies
+constant snake boards with staircases.  Formalizing the entire rook-placement
+model is outside the present scope.  All analytic conclusions are derived in
+Lean. -/
+theorem generalizedSnakeRookModel_theorem41
+    (hrec2 : AuxiliaryGRecurrence ModifiedNarayanaPolynomial AuxiliaryG)
+    (hH_nonneg : ∀ n : ℕ, 1 ≤ n →
+      HasNonnegCoeffs (AuxiliaryG n - AuxiliaryG (n - 1)))
+    (hrec : Theorem35GeneralizedSnakeRecurrenceStatement
+      generalizedSnakeRookModel.snakePolynomial
+      ModifiedNarayanaPolynomial AuxiliaryG)
+    (hdeg : ∀ {w : SnakeWord}, 1 ≤ w.length →
+      (generalizedSnakeRookModel.snakePolynomial w.deleteFinal).natDegree + 1 =
+        (generalizedSnakeRookModel.snakePolynomial w).natDegree)
+    (hM_const : ∀ {w : SnakeWord}, w.IsConstant →
+      generalizedSnakeRookModel.snakePolynomial w =
+        ModifiedNarayanaPolynomial (w.length + 1)) :
+    Theorem41Target generalizedSnakeRookModel.snakePolynomial := by
+  exact theorem41_of_modifiedModelInputs hrec2 hH_nonneg hrec
+    (fun w => by
+      rw [generalizedSnakeRookModel_snakePolynomial]
+      exact FiniteSkewBoard.rookPolynomial_hasNonnegCoeffs _)
+    hdeg hM_const
 
 /-- Order-polytope `h*` form of Braun--Jal Theorem 4.1 from the accepted
 combinatorial model inputs.  The matching hypothesis is the Stanley/
@@ -131,8 +158,6 @@ theorem orderPolytopeHStarTheorem41_of_modifiedModelInputs
     (hrec2 : AuxiliaryGRecurrence ModifiedNarayanaPolynomial AuxiliaryG)
     (hH_nonneg : ∀ n : ℕ, 1 ≤ n →
       HasNonnegCoeffs (AuxiliaryG n - AuxiliaryG (n - 1)))
-    (hG_model : ∀ n : ℕ, 1 ≤ n →
-      AuxiliaryG n = C (n : ℝ) * narayanaPolynomial 2 (n - 1))
     (hrec : Theorem35GeneralizedSnakeRecurrenceStatement M
       ModifiedNarayanaPolynomial AuxiliaryG)
     (hM_nonneg : ∀ w : SnakeWord, HasNonnegCoeffs (M w))
@@ -143,8 +168,8 @@ theorem orderPolytopeHStarTheorem41_of_modifiedModelInputs
     (hmatch : OrderPolytopeHStarMatchesNonNestingRook hStar M) :
     OrderPolytopeHStarTheorem41Target hStar :=
   orderPolytopeHStarTheorem41_of_theorem41
-    (theorem41_of_modifiedModelInputs hrec2 hH_nonneg hG_model hrec
-      hM_nonneg hdeg hM_const)
+    (theorem41_of_modifiedModelInputs hrec2 hH_nonneg hrec hM_nonneg hdeg
+      hM_const)
     hmatch
 
 /-- The real-rootedness projection of Braun--Jal Theorem 4.1. -/

From ca79f490002eaf586872d4d86ba2033032e6a784 Mon Sep 17 00:00:00 2001
From: Per Alexandersson <per.w.alexandersson@gmail.com>
Date: Tue, 4 Aug 2026 00:01:42 +0000
Subject: [PATCH 8/9] clarify Braun-Jal endpoint application

---
 RealRooted/GeneralizedSnakePosets/MatrixInduction.lean | 2 ++
 1 file changed, 2 insertions(+)

diff --git a/RealRooted/GeneralizedSnakePosets/MatrixInduction.lean b/RealRooted/GeneralizedSnakePosets/MatrixInduction.lean
index ed6ce445..042fca5f 100644
--- a/RealRooted/GeneralizedSnakePosets/MatrixInduction.lean
+++ b/RealRooted/GeneralizedSnakePosets/MatrixInduction.lean
@@ -609,9 +609,11 @@ theorem theorem41_of_matrixClaim_of_constant_matches_succ_length
         (M w ≠ 0 ∧ (M w).Splits) ∧ Interlaces (M w.deleteFinal) (M w) :=
     theorem41_constant_of_matches_succ_length
       (M := M) (P := P) hM_const hP_interlaces
+  intro w hw
   exact theorem41_of_matrixClaim_of_constant_cases
     (M := M) (P := P) (G := G) hrec hclaim hP_ne hP_one hG_one
     hP_nonneg hG_nonneg hQ_nonneg hH_nonneg hM_nonneg hdeg hconst
+    (w := w) hw
 
 /-- The deletion degree bridge follows from the length-indexed degree formula
 for the whole snake-word family. -/

From 168f0a9e4c369799e6703e8c3ba4c2b0799ff17b Mon Sep 17 00:00:00 2001
From: Per Alexandersson <per.w.alexandersson@gmail.com>
Date: Tue, 4 Aug 2026 00:11:51 +0000
Subject: [PATCH 9/9] clean up Braun-Jal final rewrite

---
 RealRooted/GeneralizedSnakePosets/MatrixInduction.lean | 5 ++++-
 1 file changed, 4 insertions(+), 1 deletion(-)

diff --git a/RealRooted/GeneralizedSnakePosets/MatrixInduction.lean b/RealRooted/GeneralizedSnakePosets/MatrixInduction.lean
index 042fca5f..a8880b16 100644
--- a/RealRooted/GeneralizedSnakePosets/MatrixInduction.lean
+++ b/RealRooted/GeneralizedSnakePosets/MatrixInduction.lean
@@ -301,7 +301,10 @@ theorem theorem41NonconstantStep_prec_of_matrixClaim
     add_ne_zero_of_hasNonnegCoeffs_of_right_ne_zero
       (hM_nonneg w.deleteFinal) hdiff_nonneg hstep.2.1.1
   have hfinal := hsum0.toPrec_of_ne hstep.1.1 hsum_ne
-  convert hfinal using 1 <;> ring
+  have hsum_eq : M w.deleteFinal + (M w - M w.deleteFinal) = M w := by
+    ring
+  rw [hsum_eq] at hfinal
+  exact hfinal
 
 /-- Polynomial form of the exceptional `m = 1` Braun-Jal step.