From 30fcd91683c6f6f6b692bc0180afe32c0bc1bd96 Mon Sep 17 00:00:00 2001
From: Thomas Marchand <thomas@lfglabs.dev>
Date: Sat, 25 Jul 2026 03:31:22 +0100
Subject: [PATCH 01/34] research: run isolated Beal signature 357 probes

---
 .../workflows/beal_signature357_decisive.yml  | 92 +++++++++++++++++++
 1 file changed, 92 insertions(+)
 create mode 100644 .github/workflows/beal_signature357_decisive.yml

diff --git a/.github/workflows/beal_signature357_decisive.yml b/.github/workflows/beal_signature357_decisive.yml
new file mode 100644
index 0000000..b8f766b
--- /dev/null
+++ b/.github/workflows/beal_signature357_decisive.yml
@@ -0,0 +1,92 @@
+name: Beal signature 357 decisive probes
+
+on:
+  push:
+    branches:
+      - research/beal-signature357-probe
+  workflow_dispatch:
+
+permissions:
+  contents: read
+
+concurrency:
+  group: beal-signature357-decisive
+  cancel-in-progress: true
+
+jobs:
+  fixed7-level33:
+    runs-on: ubuntu-latest
+    timeout-minutes: 50
+    steps:
+      - uses: actions/checkout@v5
+        with:
+          repository: Th0rgal/beal-conjecture-lean
+          ref: 0a6acaf7faa0d28ae2e37f6cb55605d1b285ffc1
+          path: beal
+      - name: Compute fixed-7 level-(3,3) prime-2 parity decomposition
+        run: >-
+          python3 beal/scripts/run_signature_357_magma_fixed7_prime2_trace.py
+          > fixed7-level33-prime2-parity.json
+      - uses: actions/upload-artifact@v4
+        with:
+          name: signature357-fixed7-level33-prime2-parity
+          path: fixed7-level33-prime2-parity.json
+          if-no-files-found: error
+
+  mod5-level:
+    strategy:
+      fail-fast: false
+      matrix:
+        include:
+          - pair: "3,0"
+            norm: 19683
+          - pair: "3,1"
+            norm: 137781
+    runs-on: ubuntu-latest
+    timeout-minutes: 50
+    steps:
+      - uses: actions/checkout@v5
+        with:
+          repository: Th0rgal/beal-conjecture-lean
+          ref: 376beb5d1271cdf4e4e9b88ff896d8240e8e6c22
+          path: beal
+      - name: Compute mod-5 norm-8 parity decomposition
+        working-directory: beal/scripts
+        run: >-
+          python3 run_signature_357_magma_mod5_norm8_probe.py
+          --pair "${{ matrix.pair }}"
+          > "../../mod5-norm8-${{ matrix.norm }}.json"
+      - uses: actions/upload-artifact@v4
+        with:
+          name: signature357-mod5-norm8-${{ matrix.norm }}
+          path: "mod5-norm8-${{ matrix.norm }}.json"
+          if-no-files-found: error
+
+  lmfdb-level91125:
+    runs-on: ubuntu-latest
+    timeout-minutes: 10
+    steps:
+      - name: Fetch any stored LMFDB packet rows at level norm 91125
+        run: |
+          python3 - <<'PY'
+          import json, urllib.parse, urllib.request
+          base='https://www.lmfdb.org/api/hmf_forms/'
+          params={
+            'field_label':'s2.2.5.1',
+            'level_norm':'i91125',
+            '_format':'json',
+            '_fields':'label,level_norm,level_ideal,dimension,is_CM,is_base_change',
+          }
+          url=base+'?'+urllib.parse.urlencode(params)
+          with urllib.request.urlopen(url,timeout=120) as response:
+              raw=response.read()
+          open('lmfdb-level91125.json','wb').write(raw)
+          value=json.loads(raw)
+          rows=value.get('data',value if isinstance(value,list) else [])
+          print(json.dumps({'url':url,'row_count':len(rows)},sort_keys=True))
+          PY
+      - uses: actions/upload-artifact@v4
+        with:
+          name: signature357-lmfdb-level91125
+          path: lmfdb-level91125.json
+          if-no-files-found: error

From 3eef523641ce7cff0d3fecd53cad3578a991bc49 Mon Sep 17 00:00:00 2001
From: Thomas Marchand <thomas@lfglabs.dev>
Date: Sat, 25 Jul 2026 03:32:16 +0100
Subject: [PATCH 02/34] research: trigger isolated Beal probes on draft PR

---
 .github/workflows/beal_signature357_decisive.yml | 5 ++++-
 1 file changed, 4 insertions(+), 1 deletion(-)

diff --git a/.github/workflows/beal_signature357_decisive.yml b/.github/workflows/beal_signature357_decisive.yml
index b8f766b..5571fe5 100644
--- a/.github/workflows/beal_signature357_decisive.yml
+++ b/.github/workflows/beal_signature357_decisive.yml
@@ -4,13 +4,16 @@ on:
   push:
     branches:
       - research/beal-signature357-probe
+  pull_request:
+    paths:
+      - ".github/workflows/beal_signature357_decisive.yml"
   workflow_dispatch:
 
 permissions:
   contents: read
 
 concurrency:
-  group: beal-signature357-decisive
+  group: beal-signature357-decisive-${{ github.ref }}
   cancel-in-progress: true
 
 jobs:

From 3f7a735235738777b06c17d1611b2e2f0e0c8a69 Mon Sep 17 00:00:00 2001
From: Thomas Marchand <thomas@lfglabs.dev>
Date: Sat, 25 Jul 2026 03:35:59 +0100
Subject: [PATCH 03/34] research: rerun supported Hilbert Hecke probes after
 cache release

---
 .github/workflows/beal_signature357_decisive.yml | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/.github/workflows/beal_signature357_decisive.yml b/.github/workflows/beal_signature357_decisive.yml
index 5571fe5..2a37f0e 100644
--- a/.github/workflows/beal_signature357_decisive.yml
+++ b/.github/workflows/beal_signature357_decisive.yml
@@ -24,7 +24,7 @@ jobs:
       - uses: actions/checkout@v5
         with:
           repository: Th0rgal/beal-conjecture-lean
-          ref: 0a6acaf7faa0d28ae2e37f6cb55605d1b285ffc1
+          ref: 0300e08d946ab80f7c9422eb6e02d628abbf54d9
           path: beal
       - name: Compute fixed-7 level-(3,3) prime-2 parity decomposition
         run: >-
@@ -51,7 +51,7 @@ jobs:
       - uses: actions/checkout@v5
         with:
           repository: Th0rgal/beal-conjecture-lean
-          ref: 376beb5d1271cdf4e4e9b88ff896d8240e8e6c22
+          ref: d3624b7270c63ab51b0b6541629c52f34e0e7305
           path: beal
       - name: Compute mod-5 norm-8 parity decomposition
         working-directory: beal/scripts

From 85db17984110dc8ea36ea60decc3ceec56b2e24e Mon Sep 17 00:00:00 2001
From: Thomas Marchand <thomas@lfglabs.dev>
Date: Sat, 25 Jul 2026 03:50:42 +0100
Subject: [PATCH 04/34] research: query the correct Q(sqrt5) LMFDB field and
 Hecke schema

---
 .../workflows/beal_signature357_decisive.yml  | 76 +++++++++++++++----
 1 file changed, 60 insertions(+), 16 deletions(-)

diff --git a/.github/workflows/beal_signature357_decisive.yml b/.github/workflows/beal_signature357_decisive.yml
index 2a37f0e..d246730 100644
--- a/.github/workflows/beal_signature357_decisive.yml
+++ b/.github/workflows/beal_signature357_decisive.yml
@@ -67,26 +67,70 @@ jobs:
 
   lmfdb-level91125:
     runs-on: ubuntu-latest
-    timeout-minutes: 10
+    timeout-minutes: 15
     steps:
-      - name: Fetch any stored LMFDB packet rows at level norm 91125
+      - name: Query the official read-only LMFDB SQL mirror
         run: |
+          python3 -m pip install --quiet psycopg2-binary
           python3 - <<'PY'
-          import json, urllib.parse, urllib.request
-          base='https://www.lmfdb.org/api/hmf_forms/'
-          params={
-            'field_label':'s2.2.5.1',
-            'level_norm':'i91125',
-            '_format':'json',
-            '_fields':'label,level_norm,level_ideal,dimension,is_CM,is_base_change',
+          import json
+          import psycopg2
+
+          conn = psycopg2.connect(
+              host='devmirror.lmfdb.xyz', port=5432,
+              dbname='lmfdb', user='lmfdb', password='lmfdb',
+              connect_timeout=30,
+          )
+          conn.set_session(readonly=True, autocommit=True)
+          result = {
+              'source': 'official LMFDB read-only SQL mirror',
+              'field_label': '2.2.5.1',
+              'level_norm': 91125,
           }
-          url=base+'?'+urllib.parse.urlencode(params)
-          with urllib.request.urlopen(url,timeout=120) as response:
-              raw=response.read()
-          open('lmfdb-level91125.json','wb').write(raw)
-          value=json.loads(raw)
-          rows=value.get('data',value if isinstance(value,list) else [])
-          print(json.dumps({'url':url,'row_count':len(rows)},sort_keys=True))
+          with conn.cursor() as cur:
+              cur.execute("""
+                  SELECT column_name, data_type
+                  FROM information_schema.columns
+                  WHERE table_schema='public' AND table_name='hmf_hecke'
+                  ORDER BY ordinal_position
+              """)
+              result['hmf_hecke_schema'] = [dict(name=r[0], type=r[1]) for r in cur.fetchall()]
+
+              cur.execute("""
+                  SELECT label, level_label, level_norm, level_ideal,
+                         dimension, is_cm, is_base_change
+                  FROM hmf_forms
+                  WHERE field_label=%s AND level_norm=%s
+                  ORDER BY level_label, label
+              """, ('2.2.5.1', 91125))
+              cols = [d.name for d in cur.description]
+              forms = [dict(zip(cols, row)) for row in cur.fetchall()]
+              result['forms'] = forms
+              result['form_count'] = len(forms)
+
+              labels = [row['label'] for row in forms]
+              if labels:
+                  cur.execute("SELECT * FROM hmf_hecke WHERE label = ANY(%s) ORDER BY label", (labels,))
+                  hcols = [d.name for d in cur.description]
+                  result['hecke_rows'] = [dict(zip(hcols, row), default=str) if False else dict(zip(hcols, row)) for row in cur.fetchall()]
+              else:
+                  result['hecke_rows'] = []
+
+          conn.close()
+
+          # PostgreSQL arrays/numerics may need normalization.
+          def normalize(value):
+              if isinstance(value, dict):
+                  return {str(k): normalize(v) for k, v in value.items()}
+              if isinstance(value, (list, tuple)):
+                  return [normalize(v) for v in value]
+              if value is None or isinstance(value, (str, int, float, bool)):
+                  return value
+              return str(value)
+
+          with open('lmfdb-level91125.json', 'w', encoding='utf-8') as f:
+              json.dump(normalize(result), f, indent=2, sort_keys=True)
+          print(json.dumps({'form_count': result['form_count'], 'hecke_row_count': len(result['hecke_rows'])}, sort_keys=True))
           PY
       - uses: actions/upload-artifact@v4
         with:

From 4f0a68d3d3fd94efaaf11f749cb5dab2749b4e63 Mon Sep 17 00:00:00 2001
From: Thomas Marchand <thomas@lfglabs.dev>
Date: Sat, 25 Jul 2026 03:54:12 +0100
Subject: [PATCH 05/34] research: add inert-prime HGM trace producer

---
 .../workflows/beal_signature357_decisive.yml  | 41 +++++++++++++------
 1 file changed, 29 insertions(+), 12 deletions(-)

diff --git a/.github/workflows/beal_signature357_decisive.yml b/.github/workflows/beal_signature357_decisive.yml
index d246730..fc972b1 100644
--- a/.github/workflows/beal_signature357_decisive.yml
+++ b/.github/workflows/beal_signature357_decisive.yml
@@ -65,6 +65,30 @@ jobs:
           path: "mod5-norm8-${{ matrix.norm }}.json"
           if-no-files-found: error
 
+  mod5-inert-local:
+    runs-on: ubuntu-latest
+    timeout-minutes: 15
+    steps:
+      - uses: actions/checkout@v5
+        with:
+          repository: Th0rgal/beal-conjecture-lean
+          ref: agent/signature-345-357-certificates
+          path: beal
+      - name: Install PARI/GP
+        run: |
+          sudo apt-get update
+          sudo apt-get install -y pari-gp
+      - name: Produce inert-prime local HGM traces at 11 and 23
+        working-directory: beal
+        run: >-
+          python3 scripts/produce_signature_357_mod5_inert_local.py
+          > ../mod5-inert-local.json
+      - uses: actions/upload-artifact@v4
+        with:
+          name: signature357-mod5-inert-local
+          path: mod5-inert-local.json
+          if-no-files-found: error
+
   lmfdb-level91125:
     runs-on: ubuntu-latest
     timeout-minutes: 15
@@ -91,18 +115,12 @@ jobs:
               cur.execute("""
                   SELECT column_name, data_type
                   FROM information_schema.columns
-                  WHERE table_schema='public' AND table_name='hmf_hecke'
-                  ORDER BY ordinal_position
+                  WHERE table_schema='public' AND table_name IN ('hmf_forms','hmf_hecke')
+                  ORDER BY table_name, ordinal_position
               """)
-              result['hmf_hecke_schema'] = [dict(name=r[0], type=r[1]) for r in cur.fetchall()]
+              result['schema'] = [dict(name=r[0], type=r[1]) for r in cur.fetchall()]
 
-              cur.execute("""
-                  SELECT label, level_label, level_norm, level_ideal,
-                         dimension, is_cm, is_base_change
-                  FROM hmf_forms
-                  WHERE field_label=%s AND level_norm=%s
-                  ORDER BY level_label, label
-              """, ('2.2.5.1', 91125))
+              cur.execute("SELECT * FROM hmf_forms WHERE field_label=%s AND level_norm=%s ORDER BY label", ('2.2.5.1', 91125))
               cols = [d.name for d in cur.description]
               forms = [dict(zip(cols, row)) for row in cur.fetchall()]
               result['forms'] = forms
@@ -112,13 +130,12 @@ jobs:
               if labels:
                   cur.execute("SELECT * FROM hmf_hecke WHERE label = ANY(%s) ORDER BY label", (labels,))
                   hcols = [d.name for d in cur.description]
-                  result['hecke_rows'] = [dict(zip(hcols, row), default=str) if False else dict(zip(hcols, row)) for row in cur.fetchall()]
+                  result['hecke_rows'] = [dict(zip(hcols, row)) for row in cur.fetchall()]
               else:
                   result['hecke_rows'] = []
 
           conn.close()
 
-          # PostgreSQL arrays/numerics may need normalization.
           def normalize(value):
               if isinstance(value, dict):
                   return {str(k): normalize(v) for k, v in value.items()}

From 964bbbed519d8400ee7a7aace2da85185802f531 Mon Sep 17 00:00:00 2001
From: Thomas Marchand <thomas@lfglabs.dev>
Date: Sat, 25 Jul 2026 04:10:30 +0100
Subject: [PATCH 06/34] research: run split-prime and indefinite-ambient
 decisive probes

---
 .../workflows/beal_signature357_decisive.yml  | 164 +++++++-----------
 1 file changed, 59 insertions(+), 105 deletions(-)

diff --git a/.github/workflows/beal_signature357_decisive.yml b/.github/workflows/beal_signature357_decisive.yml
index fc972b1..2016ce6 100644
--- a/.github/workflows/beal_signature357_decisive.yml
+++ b/.github/workflows/beal_signature357_decisive.yml
@@ -17,140 +17,94 @@ concurrency:
   cancel-in-progress: true
 
 jobs:
-  fixed7-level33:
+  mod5-prime13-level137781:
     runs-on: ubuntu-latest
-    timeout-minutes: 50
+    timeout-minutes: 25
     steps:
       - uses: actions/checkout@v5
         with:
           repository: Th0rgal/beal-conjecture-lean
-          ref: 0300e08d946ab80f7c9422eb6e02d628abbf54d9
+          ref: bc42650da9f5fc3ea073ea11f288285ad6e6a328
           path: beal
-      - name: Compute fixed-7 level-(3,3) prime-2 parity decomposition
+      - name: Install PARI/GP
+        run: |
+          sudo apt-get update
+          sudo apt-get install -y pari-gp
+      - name: Produce complete split-prime local HGM data
+        working-directory: beal
         run: >-
-          python3 beal/scripts/run_signature_357_magma_fixed7_prime2_trace.py
-          > fixed7-level33-prime2-parity.json
-      - uses: actions/upload-artifact@v4
-        with:
-          name: signature357-fixed7-level33-prime2-parity
-          path: fixed7-level33-prime2-parity.json
-          if-no-files-found: error
-
-  mod5-level:
-    strategy:
-      fail-fast: false
-      matrix:
-        include:
-          - pair: "3,0"
-            norm: 19683
-          - pair: "3,1"
-            norm: 137781
-    runs-on: ubuntu-latest
-    timeout-minutes: 50
-    steps:
-      - uses: actions/checkout@v5
-        with:
-          repository: Th0rgal/beal-conjecture-lean
-          ref: d3624b7270c63ab51b0b6541629c52f34e0e7305
-          path: beal
-      - name: Compute mod-5 norm-8 parity decomposition
-        working-directory: beal/scripts
+          python3 scripts/produce_signature_357_mod5_complete_local.py
+          > ../mod5-complete-local.json
+      - name: Apply the prime-13 filter to the three parity eigenspaces
+        working-directory: beal
         run: >-
-          python3 run_signature_357_magma_mod5_norm8_probe.py
-          --pair "${{ matrix.pair }}"
-          > "../../mod5-norm8-${{ matrix.norm }}.json"
+          python3 scripts/run_signature_357_magma_mod5_prime13_parity.py
+          --local-data ../mod5-complete-local.json
+          > ../mod5-137781-prime13-parity.json
       - uses: actions/upload-artifact@v4
         with:
-          name: signature357-mod5-norm8-${{ matrix.norm }}
-          path: "mod5-norm8-${{ matrix.norm }}.json"
+          name: signature357-mod5-137781-prime13-parity
+          path: |
+            mod5-complete-local.json
+            mod5-137781-prime13-parity.json
           if-no-files-found: error
 
-  mod5-inert-local:
+  mod5-level19683-ambient:
     runs-on: ubuntu-latest
     timeout-minutes: 15
     steps:
       - uses: actions/checkout@v5
         with:
           repository: Th0rgal/beal-conjecture-lean
-          ref: agent/signature-345-357-certificates
+          ref: bc42650da9f5fc3ea073ea11f288285ad6e6a328
           path: beal
-      - name: Install PARI/GP
-        run: |
-          sudo apt-get update
-          sudo apt-get install -y pari-gp
-      - name: Produce inert-prime local HGM traces at 11 and 23
-        working-directory: beal
+      - name: Test the indefinite ambient space at norms 8 and 7
         run: >-
-          python3 scripts/produce_signature_357_mod5_inert_local.py
-          > ../mod5-inert-local.json
+          python3 beal/scripts/run_signature_357_magma_mod5_19683_ambient.py
+          > mod5-19683-ambient.json
       - uses: actions/upload-artifact@v4
         with:
-          name: signature357-mod5-inert-local
-          path: mod5-inert-local.json
+          name: signature357-mod5-19683-ambient
+          path: mod5-19683-ambient.json
           if-no-files-found: error
 
-  lmfdb-level91125:
+  magma-interface-probe:
     runs-on: ubuntu-latest
-    timeout-minutes: 15
+    timeout-minutes: 10
     steps:
-      - name: Query the official read-only LMFDB SQL mirror
+      - name: Probe sparse Hilbert-Hecke interfaces
         run: |
-          python3 -m pip install --quiet psycopg2-binary
           python3 - <<'PY'
-          import json
-          import psycopg2
-
-          conn = psycopg2.connect(
-              host='devmirror.lmfdb.xyz', port=5432,
-              dbname='lmfdb', user='lmfdb', password='lmfdb',
-              connect_timeout=30,
-          )
-          conn.set_session(readonly=True, autocommit=True)
-          result = {
-              'source': 'official LMFDB read-only SQL mirror',
-              'field_label': '2.2.5.1',
-              'level_norm': 91125,
-          }
-          with conn.cursor() as cur:
-              cur.execute("""
-                  SELECT column_name, data_type
-                  FROM information_schema.columns
-                  WHERE table_schema='public' AND table_name IN ('hmf_forms','hmf_hecke')
-                  ORDER BY table_name, ordinal_position
-              """)
-              result['schema'] = [dict(name=r[0], type=r[1]) for r in cur.fetchall()]
-
-              cur.execute("SELECT * FROM hmf_forms WHERE field_label=%s AND level_norm=%s ORDER BY label", ('2.2.5.1', 91125))
-              cols = [d.name for d in cur.description]
-              forms = [dict(zip(cols, row)) for row in cur.fetchall()]
-              result['forms'] = forms
-              result['form_count'] = len(forms)
-
-              labels = [row['label'] for row in forms]
-              if labels:
-                  cur.execute("SELECT * FROM hmf_hecke WHERE label = ANY(%s) ORDER BY label", (labels,))
-                  hcols = [d.name for d in cur.description]
-                  result['hecke_rows'] = [dict(zip(hcols, row)) for row in cur.fetchall()]
-              else:
-                  result['hecke_rows'] = []
-
-          conn.close()
-
-          def normalize(value):
-              if isinstance(value, dict):
-                  return {str(k): normalize(v) for k, v in value.items()}
-              if isinstance(value, (list, tuple)):
-                  return [normalize(v) for v in value]
-              if value is None or isinstance(value, (str, int, float, bool)):
-                  return value
-              return str(value)
-
-          with open('lmfdb-level91125.json', 'w', encoding='utf-8') as f:
-              json.dump(normalize(result), f, indent=2, sort_keys=True)
-          print(json.dumps({'form_count': result['form_count'], 'hecke_row_count': len(result['hecke_rows'])}, sort_keys=True))
+          import html, http.cookiejar, re, urllib.parse, urllib.request
+          url='https://magma.maths.usyd.edu.au/calc/'
+          code=r'''
+          _<x>:=PolynomialRing(Rationals());
+          K<z>:=NumberField(x^2-5); OK:=Integers(K);
+          I5:=Factorisation(5*OK)[1][1]; I2:=Factorisation(2*OK)[1][1];
+          M0:=HilbertCuspForms(K,3^2*I5^2); M:=NewSubspace(M0);
+          printf "TYPE=%o\n",Type(M); printf "DIM=%o\n",Dimension(M);
+          try H:=HeckeImages(M,1,[I2],1); printf "HECKEIMAGES_OK=%o\n",H;
+          catch e printf "HECKEIMAGES_ERROR=%o\n",e; end try;
+          try T:=HeckeOperator(M,I2,1); printf "THREE_ARGUMENT_OPERATOR_OK=%o\n",T;
+          catch e printf "THREE_ARGUMENT_OPERATOR_ERROR=%o\n",e; end try;
+          '''
+          jar=http.cookiejar.CookieJar(); opener=urllib.request.build_opener(urllib.request.HTTPCookieProcessor(jar))
+          with opener.open(urllib.request.Request(url,headers={'User-Agent':'Mozilla/5.0 research'}),timeout=120) as r:
+              landing=r.read().decode(errors='replace'); landing_url=r.geturl()
+          form=re.search(r'<form\b([^>]*)>(.*?)</form>',landing,flags=re.I|re.S); attrs,body=form.groups()
+          am=re.search(r'\baction=["\']([^"\']*)',attrs,flags=re.I)
+          action=url if am is None else urllib.parse.urljoin(url,html.unescape(am.group(1)))
+          hidden={}
+          for tag in re.findall(r'<input\b[^>]*>',body,flags=re.I):
+              tm=re.search(r'\btype=["\']([^"\']*)',tag,flags=re.I); nm=re.search(r'\bname=["\']([^"\']*)',tag,flags=re.I); vm=re.search(r'\bvalue=["\']([^"\']*)',tag,flags=re.I)
+              if tm and tm.group(1).lower()=='hidden' and nm: hidden[html.unescape(nm.group(1))]='' if vm is None else html.unescape(vm.group(1))
+          hidden['input']=code; parsed=urllib.parse.urlparse(action)
+          req=urllib.request.Request(action,data=urllib.parse.urlencode(hidden).encode(),headers={'User-Agent':'Mozilla/5.0 research','Content-Type':'application/x-www-form-urlencoded','Referer':landing_url,'Origin':f'{parsed.scheme}://{parsed.netloc}'})
+          with opener.open(req,timeout=300) as r: page=r.read().decode(errors='replace')
+          text=html.unescape(re.sub(r'<[^>]+>','',page)); open('magma-interface-probe.txt','w').write(text); print(text[-10000:])
           PY
       - uses: actions/upload-artifact@v4
         with:
-          name: signature357-lmfdb-level91125
-          path: lmfdb-level91125.json
+          name: signature357-magma-interface-probe
+          path: magma-interface-probe.txt
           if-no-files-found: error

From 779020fa057879a19e880ae41ee7a87accce226a Mon Sep 17 00:00:00 2001
From: Thomas Marchand <thomas@lfglabs.dev>
Date: Sat, 25 Jul 2026 04:18:42 +0100
Subject: [PATCH 07/34] research: run the four-prime chain and extended ambient
 probe

---
 .../workflows/beal_signature357_decisive.yml  | 59 +++----------------
 1 file changed, 9 insertions(+), 50 deletions(-)

diff --git a/.github/workflows/beal_signature357_decisive.yml b/.github/workflows/beal_signature357_decisive.yml
index 2016ce6..9f497c7 100644
--- a/.github/workflows/beal_signature357_decisive.yml
+++ b/.github/workflows/beal_signature357_decisive.yml
@@ -17,14 +17,14 @@ concurrency:
   cancel-in-progress: true
 
 jobs:
-  mod5-prime13-level137781:
+  mod5-split-chain-level137781:
     runs-on: ubuntu-latest
     timeout-minutes: 25
     steps:
       - uses: actions/checkout@v5
         with:
           repository: Th0rgal/beal-conjecture-lean
-          ref: bc42650da9f5fc3ea073ea11f288285ad6e6a328
+          ref: 8858bd2091dea5125a9927cba86f3d974f7074bb
           path: beal
       - name: Install PARI/GP
         run: |
@@ -35,28 +35,28 @@ jobs:
         run: >-
           python3 scripts/produce_signature_357_mod5_complete_local.py
           > ../mod5-complete-local.json
-      - name: Apply the prime-13 filter to the three parity eigenspaces
+      - name: Intersect the 13,29,41,43 trace unions on each parity eigenspace
         working-directory: beal
         run: >-
-          python3 scripts/run_signature_357_magma_mod5_prime13_parity.py
+          python3 scripts/run_signature_357_magma_mod5_split_parity_chain.py
           --local-data ../mod5-complete-local.json
-          > ../mod5-137781-prime13-parity.json
+          > ../mod5-137781-split-chain.json
       - uses: actions/upload-artifact@v4
         with:
-          name: signature357-mod5-137781-prime13-parity
+          name: signature357-mod5-137781-split-chain
           path: |
             mod5-complete-local.json
-            mod5-137781-prime13-parity.json
+            mod5-137781-split-chain.json
           if-no-files-found: error
 
   mod5-level19683-ambient:
     runs-on: ubuntu-latest
-    timeout-minutes: 15
+    timeout-minutes: 25
     steps:
       - uses: actions/checkout@v5
         with:
           repository: Th0rgal/beal-conjecture-lean
-          ref: bc42650da9f5fc3ea073ea11f288285ad6e6a328
+          ref: 8858bd2091dea5125a9927cba86f3d974f7074bb
           path: beal
       - name: Test the indefinite ambient space at norms 8 and 7
         run: >-
@@ -67,44 +67,3 @@ jobs:
           name: signature357-mod5-19683-ambient
           path: mod5-19683-ambient.json
           if-no-files-found: error
-
-  magma-interface-probe:
-    runs-on: ubuntu-latest
-    timeout-minutes: 10
-    steps:
-      - name: Probe sparse Hilbert-Hecke interfaces
-        run: |
-          python3 - <<'PY'
-          import html, http.cookiejar, re, urllib.parse, urllib.request
-          url='https://magma.maths.usyd.edu.au/calc/'
-          code=r'''
-          _<x>:=PolynomialRing(Rationals());
-          K<z>:=NumberField(x^2-5); OK:=Integers(K);
-          I5:=Factorisation(5*OK)[1][1]; I2:=Factorisation(2*OK)[1][1];
-          M0:=HilbertCuspForms(K,3^2*I5^2); M:=NewSubspace(M0);
-          printf "TYPE=%o\n",Type(M); printf "DIM=%o\n",Dimension(M);
-          try H:=HeckeImages(M,1,[I2],1); printf "HECKEIMAGES_OK=%o\n",H;
-          catch e printf "HECKEIMAGES_ERROR=%o\n",e; end try;
-          try T:=HeckeOperator(M,I2,1); printf "THREE_ARGUMENT_OPERATOR_OK=%o\n",T;
-          catch e printf "THREE_ARGUMENT_OPERATOR_ERROR=%o\n",e; end try;
-          '''
-          jar=http.cookiejar.CookieJar(); opener=urllib.request.build_opener(urllib.request.HTTPCookieProcessor(jar))
-          with opener.open(urllib.request.Request(url,headers={'User-Agent':'Mozilla/5.0 research'}),timeout=120) as r:
-              landing=r.read().decode(errors='replace'); landing_url=r.geturl()
-          form=re.search(r'<form\b([^>]*)>(.*?)</form>',landing,flags=re.I|re.S); attrs,body=form.groups()
-          am=re.search(r'\baction=["\']([^"\']*)',attrs,flags=re.I)
-          action=url if am is None else urllib.parse.urljoin(url,html.unescape(am.group(1)))
-          hidden={}
-          for tag in re.findall(r'<input\b[^>]*>',body,flags=re.I):
-              tm=re.search(r'\btype=["\']([^"\']*)',tag,flags=re.I); nm=re.search(r'\bname=["\']([^"\']*)',tag,flags=re.I); vm=re.search(r'\bvalue=["\']([^"\']*)',tag,flags=re.I)
-              if tm and tm.group(1).lower()=='hidden' and nm: hidden[html.unescape(nm.group(1))]='' if vm is None else html.unescape(vm.group(1))
-          hidden['input']=code; parsed=urllib.parse.urlparse(action)
-          req=urllib.request.Request(action,data=urllib.parse.urlencode(hidden).encode(),headers={'User-Agent':'Mozilla/5.0 research','Content-Type':'application/x-www-form-urlencoded','Referer':landing_url,'Origin':f'{parsed.scheme}://{parsed.netloc}'})
-          with opener.open(req,timeout=300) as r: page=r.read().decode(errors='replace')
-          text=html.unescape(re.sub(r'<[^>]+>','',page)); open('magma-interface-probe.txt','w').write(text); print(text[-10000:])
-          PY
-      - uses: actions/upload-artifact@v4
-        with:
-          name: signature357-magma-interface-probe
-          path: magma-interface-probe.txt
-          if-no-files-found: error

From 104d5934fca297a3a4930e97b63700b53fe7ca54 Mon Sep 17 00:00:00 2001
From: Thomas Marchand <thomas@lfglabs.dev>
Date: Sat, 25 Jul 2026 05:17:09 +0100
Subject: [PATCH 08/34] Run level 137781 prime-13 parity filter

---
 .../beal_signature357_mod5_137781_prime13.yml | 36 +++++++++++++++++++
 1 file changed, 36 insertions(+)
 create mode 100644 .github/workflows/beal_signature357_mod5_137781_prime13.yml

diff --git a/.github/workflows/beal_signature357_mod5_137781_prime13.yml b/.github/workflows/beal_signature357_mod5_137781_prime13.yml
new file mode 100644
index 0000000..49360ba
--- /dev/null
+++ b/.github/workflows/beal_signature357_mod5_137781_prime13.yml
@@ -0,0 +1,36 @@
+name: Beal signature 357 level 137781 prime 13 filter
+
+on:
+  push:
+    branches:
+      - research/beal-signature357-probe
+    paths:
+      - ".github/workflows/beal_signature357_mod5_137781_prime13.yml"
+  workflow_dispatch:
+
+permissions:
+  contents: read
+
+concurrency:
+  group: beal-signature357-mod5-137781-prime13
+  cancel-in-progress: false
+
+jobs:
+  prime13:
+    runs-on: ubuntu-latest
+    timeout-minutes: 20
+    steps:
+      - uses: actions/checkout@v5
+        with:
+          repository: Th0rgal/beal-conjecture-lean
+          ref: 76cec47d35dfc38bda15b2370844bc3534af8a13
+          path: beal
+      - name: Apply the complete prime-13 local union to both parity spaces
+        run: >-
+          python3 beal/scripts/run_signature_357_magma_mod5_137781_prime13_parity.py
+          > mod5-137781-prime13-parity.json
+      - uses: actions/upload-artifact@v4
+        with:
+          name: signature357-mod5-137781-prime13-parity
+          path: mod5-137781-prime13-parity.json
+          if-no-files-found: error

From 56b38db5704af4c0c3d037aaae76261ede8a1d57 Mon Sep 17 00:00:00 2001
From: Thomas Marchand <thomas@lfglabs.dev>
Date: Sat, 25 Jul 2026 05:25:22 +0100
Subject: [PATCH 09/34] Run prime-13 semilinear survivor sieve

---
 .../workflows/beal_signature357_decisive.yml  | 19 +++++++++++++++++++
 1 file changed, 19 insertions(+)

diff --git a/.github/workflows/beal_signature357_decisive.yml b/.github/workflows/beal_signature357_decisive.yml
index 9f497c7..e107371 100644
--- a/.github/workflows/beal_signature357_decisive.yml
+++ b/.github/workflows/beal_signature357_decisive.yml
@@ -17,6 +17,25 @@ concurrency:
   cancel-in-progress: true
 
 jobs:
+  mod5-prime13-semilinear-level137781:
+    runs-on: ubuntu-latest
+    timeout-minutes: 20
+    steps:
+      - uses: actions/checkout@v5
+        with:
+          repository: Th0rgal/beal-conjecture-lean
+          ref: 8f0a00167bc3fdfea613855bd4b915645944b453
+          path: beal
+      - name: Apply prime-13 local and semilinear conditions to every parity space
+        run: >-
+          python3 beal/scripts/run_signature_357_magma_mod5_137781_prime13_semilinear.py
+          > mod5-137781-prime13-semilinear.json
+      - uses: actions/upload-artifact@v4
+        with:
+          name: signature357-mod5-137781-prime13-semilinear
+          path: mod5-137781-prime13-semilinear.json
+          if-no-files-found: error
+
   mod5-split-chain-level137781:
     runs-on: ubuntu-latest
     timeout-minutes: 25

From d263afa5ce2e37fce91a57a908c81fd4679e9f1d Mon Sep 17 00:00:00 2001
From: Thomas Marchand <thomas@lfglabs.dev>
Date: Sat, 25 Jul 2026 05:29:45 +0100
Subject: [PATCH 10/34] Run full semilinear survivor chain

---
 .../workflows/beal_signature357_decisive.yml  | 62 +++++--------------
 1 file changed, 15 insertions(+), 47 deletions(-)

diff --git a/.github/workflows/beal_signature357_decisive.yml b/.github/workflows/beal_signature357_decisive.yml
index e107371..1fe100b 100644
--- a/.github/workflows/beal_signature357_decisive.yml
+++ b/.github/workflows/beal_signature357_decisive.yml
@@ -17,72 +17,40 @@ concurrency:
   cancel-in-progress: true
 
 jobs:
-  mod5-prime13-semilinear-level137781:
+  mod5-semilinear-chain-level137781:
     runs-on: ubuntu-latest
     timeout-minutes: 20
     steps:
       - uses: actions/checkout@v5
         with:
           repository: Th0rgal/beal-conjecture-lean
-          ref: 8f0a00167bc3fdfea613855bd4b915645944b453
+          ref: a85e7ba70e26f290ac768746b8c4a87738baeb60
           path: beal
-      - name: Apply prime-13 local and semilinear conditions to every parity space
+      - name: Intersect all split-prime local and semilinear conditions
         run: >-
-          python3 beal/scripts/run_signature_357_magma_mod5_137781_prime13_semilinear.py
-          > mod5-137781-prime13-semilinear.json
+          python3 beal/scripts/run_signature_357_magma_mod5_137781_semilinear_chain.py
+          > mod5-137781-semilinear-chain.json
       - uses: actions/upload-artifact@v4
         with:
-          name: signature357-mod5-137781-prime13-semilinear
-          path: mod5-137781-prime13-semilinear.json
+          name: signature357-mod5-137781-semilinear-chain
+          path: mod5-137781-semilinear-chain.json
           if-no-files-found: error
 
-  mod5-split-chain-level137781:
+  magma-hecke-raw-api:
     runs-on: ubuntu-latest
-    timeout-minutes: 25
+    timeout-minutes: 10
     steps:
       - uses: actions/checkout@v5
         with:
           repository: Th0rgal/beal-conjecture-lean
-          ref: 8858bd2091dea5125a9927cba86f3d974f7074bb
+          ref: 1290b57f0a6295d90fd9ed0ccbe73830724250f7
           path: beal
-      - name: Install PARI/GP
-        run: |
-          sudo apt-get update
-          sudo apt-get install -y pari-gp
-      - name: Produce complete split-prime local HGM data
-        working-directory: beal
+      - name: Probe HeckeMatrixRaw on a tiny definite Hilbert space
         run: >-
-          python3 scripts/produce_signature_357_mod5_complete_local.py
-          > ../mod5-complete-local.json
-      - name: Intersect the 13,29,41,43 trace unions on each parity eigenspace
-        working-directory: beal
-        run: >-
-          python3 scripts/run_signature_357_magma_mod5_split_parity_chain.py
-          --local-data ../mod5-complete-local.json
-          > ../mod5-137781-split-chain.json
-      - uses: actions/upload-artifact@v4
-        with:
-          name: signature357-mod5-137781-split-chain
-          path: |
-            mod5-complete-local.json
-            mod5-137781-split-chain.json
-          if-no-files-found: error
-
-  mod5-level19683-ambient:
-    runs-on: ubuntu-latest
-    timeout-minutes: 25
-    steps:
-      - uses: actions/checkout@v5
-        with:
-          repository: Th0rgal/beal-conjecture-lean
-          ref: 8858bd2091dea5125a9927cba86f3d974f7074bb
-          path: beal
-      - name: Test the indefinite ambient space at norms 8 and 7
-        run: >-
-          python3 beal/scripts/run_signature_357_magma_mod5_19683_ambient.py
-          > mod5-19683-ambient.json
+          python3 beal/scripts/run_signature_357_magma_hecke_raw_api_probe.py
+          > magma-hecke-raw-api.json
       - uses: actions/upload-artifact@v4
         with:
-          name: signature357-mod5-19683-ambient
-          path: mod5-19683-ambient.json
+          name: signature357-magma-hecke-raw-api
+          path: magma-hecke-raw-api.json
           if-no-files-found: error

From 1d3d4b046ed12b36fde60b2cefc5299a976407d9 Mon Sep 17 00:00:00 2001
From: Thomas Marchand <thomas@lfglabs.dev>
Date: Sat, 25 Jul 2026 05:34:06 +0100
Subject: [PATCH 11/34] Run raw Hilbert Hecke restriction probe

---
 ...eal_signature357_raw_restriction_probe.yml | 36 +++++++++++++++++++
 1 file changed, 36 insertions(+)
 create mode 100644 .github/workflows/beal_signature357_raw_restriction_probe.yml

diff --git a/.github/workflows/beal_signature357_raw_restriction_probe.yml b/.github/workflows/beal_signature357_raw_restriction_probe.yml
new file mode 100644
index 0000000..8bf0bd4
--- /dev/null
+++ b/.github/workflows/beal_signature357_raw_restriction_probe.yml
@@ -0,0 +1,36 @@
+name: Beal signature 357 raw Hecke restriction probe
+
+on:
+  push:
+    branches:
+      - research/beal-signature357-probe
+    paths:
+      - ".github/workflows/beal_signature357_raw_restriction_probe.yml"
+  workflow_dispatch:
+
+permissions:
+  contents: read
+
+concurrency:
+  group: beal-signature357-raw-restriction
+  cancel-in-progress: false
+
+jobs:
+  probe:
+    runs-on: ubuntu-latest
+    timeout-minutes: 10
+    steps:
+      - uses: actions/checkout@v5
+        with:
+          repository: Th0rgal/beal-conjecture-lean
+          ref: 767d4aadc25eac4dd4493c53c1accd30f8e7503e
+          path: beal
+      - name: Compare raw and ordinary Hecke restrictions
+        run: >-
+          python3 beal/scripts/run_signature_357_magma_hecke_raw_restriction_probe.py
+          > magma-hecke-raw-restriction.json
+      - uses: actions/upload-artifact@v4
+        with:
+          name: signature357-magma-hecke-raw-restriction
+          path: magma-hecke-raw-restriction.json
+          if-no-files-found: error

From bd82a4d3dfa7bec655d13a99833200013dda58f3 Mon Sep 17 00:00:00 2001
From: Thomas Marchand <thomas@lfglabs.dev>
Date: Sat, 25 Jul 2026 05:34:49 +0100
Subject: [PATCH 12/34] Run raw Hecke restriction probe

---
 .../workflows/beal_signature357_decisive.yml  | 32 +++++++++----------
 1 file changed, 16 insertions(+), 16 deletions(-)

diff --git a/.github/workflows/beal_signature357_decisive.yml b/.github/workflows/beal_signature357_decisive.yml
index 1fe100b..76079ac 100644
--- a/.github/workflows/beal_signature357_decisive.yml
+++ b/.github/workflows/beal_signature357_decisive.yml
@@ -17,40 +17,40 @@ concurrency:
   cancel-in-progress: true
 
 jobs:
-  mod5-semilinear-chain-level137781:
+  raw-hecke-restriction-probe:
     runs-on: ubuntu-latest
-    timeout-minutes: 20
+    timeout-minutes: 10
     steps:
       - uses: actions/checkout@v5
         with:
           repository: Th0rgal/beal-conjecture-lean
-          ref: a85e7ba70e26f290ac768746b8c4a87738baeb60
+          ref: 767d4aadc25eac4dd4493c53c1accd30f8e7503e
           path: beal
-      - name: Intersect all split-prime local and semilinear conditions
+      - name: Compare raw and ordinary Hecke restrictions
         run: >-
-          python3 beal/scripts/run_signature_357_magma_mod5_137781_semilinear_chain.py
-          > mod5-137781-semilinear-chain.json
+          python3 beal/scripts/run_signature_357_magma_hecke_raw_restriction_probe.py
+          > magma-hecke-raw-restriction.json
       - uses: actions/upload-artifact@v4
         with:
-          name: signature357-mod5-137781-semilinear-chain
-          path: mod5-137781-semilinear-chain.json
+          name: signature357-magma-hecke-raw-restriction
+          path: magma-hecke-raw-restriction.json
           if-no-files-found: error
 
-  magma-hecke-raw-api:
+  mod5-semilinear-chain-level137781:
     runs-on: ubuntu-latest
-    timeout-minutes: 10
+    timeout-minutes: 20
     steps:
       - uses: actions/checkout@v5
         with:
           repository: Th0rgal/beal-conjecture-lean
-          ref: 1290b57f0a6295d90fd9ed0ccbe73830724250f7
+          ref: a85e7ba70e26f290ac768746b8c4a87738baeb60
           path: beal
-      - name: Probe HeckeMatrixRaw on a tiny definite Hilbert space
+      - name: Intersect all split-prime local and semilinear conditions
         run: >-
-          python3 beal/scripts/run_signature_357_magma_hecke_raw_api_probe.py
-          > magma-hecke-raw-api.json
+          python3 beal/scripts/run_signature_357_magma_mod5_137781_semilinear_chain.py
+          > mod5-137781-semilinear-chain.json
       - uses: actions/upload-artifact@v4
         with:
-          name: signature357-magma-hecke-raw-api
-          path: magma-hecke-raw-api.json
+          name: signature357-mod5-137781-semilinear-chain
+          path: mod5-137781-semilinear-chain.json
           if-no-files-found: error

From c7626608c494646ac2a804fec6f9b7353b270d45 Mon Sep 17 00:00:00 2001
From: Thomas Marchand <thomas@lfglabs.dev>
Date: Sat, 25 Jul 2026 05:37:50 +0100
Subject: [PATCH 13/34] Run level 19683 raw Brandt filter

---
 .../beal_signature357_mod5_19683_raw.yml      | 36 +++++++++++++++++++
 1 file changed, 36 insertions(+)
 create mode 100644 .github/workflows/beal_signature357_mod5_19683_raw.yml

diff --git a/.github/workflows/beal_signature357_mod5_19683_raw.yml b/.github/workflows/beal_signature357_mod5_19683_raw.yml
new file mode 100644
index 0000000..e2504b9
--- /dev/null
+++ b/.github/workflows/beal_signature357_mod5_19683_raw.yml
@@ -0,0 +1,36 @@
+name: Beal signature 357 level 19683 raw Brandt filter
+
+on:
+  push:
+    branches:
+      - research/beal-signature357-probe
+    paths:
+      - ".github/workflows/beal_signature357_mod5_19683_raw.yml"
+  workflow_dispatch:
+
+permissions:
+  contents: read
+
+concurrency:
+  group: beal-signature357-mod5-19683-raw
+  cancel-in-progress: false
+
+jobs:
+  raw:
+    runs-on: ubuntu-latest
+    timeout-minutes: 20
+    steps:
+      - uses: actions/checkout@v5
+        with:
+          repository: Th0rgal/beal-conjecture-lean
+          ref: 72b19b2e27141c2cfe94bf2727a8c18a0392ea3c
+          path: beal
+      - name: Apply parity and removed-prime filters on the raw Brandt module
+        run: >-
+          python3 beal/scripts/run_signature_357_magma_mod5_19683_raw.py
+          > mod5-19683-raw.json
+      - uses: actions/upload-artifact@v4
+        with:
+          name: signature357-mod5-19683-raw
+          path: mod5-19683-raw.json
+          if-no-files-found: error

From 8139a46c4dc5b7a06c7c8cb7e6a3c6fb0a48eaeb Mon Sep 17 00:00:00 2001
From: Thomas Marchand <thomas@lfglabs.dev>
Date: Sat, 25 Jul 2026 05:38:33 +0100
Subject: [PATCH 14/34] Run final level 19683 raw Brandt probe

---
 .../workflows/beal_signature357_decisive.yml  | 33 ++++---------------
 1 file changed, 7 insertions(+), 26 deletions(-)

diff --git a/.github/workflows/beal_signature357_decisive.yml b/.github/workflows/beal_signature357_decisive.yml
index 76079ac..316a19d 100644
--- a/.github/workflows/beal_signature357_decisive.yml
+++ b/.github/workflows/beal_signature357_decisive.yml
@@ -17,40 +17,21 @@ concurrency:
   cancel-in-progress: true
 
 jobs:
-  raw-hecke-restriction-probe:
-    runs-on: ubuntu-latest
-    timeout-minutes: 10
-    steps:
-      - uses: actions/checkout@v5
-        with:
-          repository: Th0rgal/beal-conjecture-lean
-          ref: 767d4aadc25eac4dd4493c53c1accd30f8e7503e
-          path: beal
-      - name: Compare raw and ordinary Hecke restrictions
-        run: >-
-          python3 beal/scripts/run_signature_357_magma_hecke_raw_restriction_probe.py
-          > magma-hecke-raw-restriction.json
-      - uses: actions/upload-artifact@v4
-        with:
-          name: signature357-magma-hecke-raw-restriction
-          path: magma-hecke-raw-restriction.json
-          if-no-files-found: error
-
-  mod5-semilinear-chain-level137781:
+  mod5-level19683-raw:
     runs-on: ubuntu-latest
     timeout-minutes: 20
     steps:
       - uses: actions/checkout@v5
         with:
           repository: Th0rgal/beal-conjecture-lean
-          ref: a85e7ba70e26f290ac768746b8c4a87738baeb60
+          ref: 72b19b2e27141c2cfe94bf2727a8c18a0392ea3c
           path: beal
-      - name: Intersect all split-prime local and semilinear conditions
+      - name: Apply parity and removed-prime filters on the raw Brandt module
         run: >-
-          python3 beal/scripts/run_signature_357_magma_mod5_137781_semilinear_chain.py
-          > mod5-137781-semilinear-chain.json
+          python3 beal/scripts/run_signature_357_magma_mod5_19683_raw.py
+          > mod5-19683-raw.json
       - uses: actions/upload-artifact@v4
         with:
-          name: signature357-mod5-137781-semilinear-chain
-          path: mod5-137781-semilinear-chain.json
+          name: signature357-mod5-19683-raw
+          path: mod5-19683-raw.json
           if-no-files-found: error

From 414d62a5c5b9186231b315525e4a64f0fe4c0b4d Mon Sep 17 00:00:00 2001
From: Thomas Marchand <thomas@lfglabs.dev>
Date: Sat, 25 Jul 2026 05:43:56 +0100
Subject: [PATCH 15/34] Inspect Hilbert newspace attributes

---
 .github/workflows/beal_signature357_decisive.yml | 16 ++++++++--------
 1 file changed, 8 insertions(+), 8 deletions(-)

diff --git a/.github/workflows/beal_signature357_decisive.yml b/.github/workflows/beal_signature357_decisive.yml
index 316a19d..9841d56 100644
--- a/.github/workflows/beal_signature357_decisive.yml
+++ b/.github/workflows/beal_signature357_decisive.yml
@@ -17,21 +17,21 @@ concurrency:
   cancel-in-progress: true
 
 jobs:
-  mod5-level19683-raw:
+  modfrhil-attributes:
     runs-on: ubuntu-latest
-    timeout-minutes: 20
+    timeout-minutes: 10
     steps:
       - uses: actions/checkout@v5
         with:
           repository: Th0rgal/beal-conjecture-lean
-          ref: 72b19b2e27141c2cfe94bf2727a8c18a0392ea3c
+          ref: eadc80f07c2d26adaeb3e0edfa3f292904dccd24
           path: beal
-      - name: Apply parity and removed-prime filters on the raw Brandt module
+      - name: Inspect the independently computed newspace attributes
         run: >-
-          python3 beal/scripts/run_signature_357_magma_mod5_19683_raw.py
-          > mod5-19683-raw.json
+          python3 beal/scripts/run_signature_357_magma_modfrhil_attributes.py
+          > modfrhil-attributes.json
       - uses: actions/upload-artifact@v4
         with:
-          name: signature357-mod5-19683-raw
-          path: mod5-19683-raw.json
+          name: signature357-modfrhil-attributes
+          path: modfrhil-attributes.json
           if-no-files-found: error

From b40fa39461380824a704c34f71ae87c0060a68e3 Mon Sep 17 00:00:00 2001
From: Thomas Marchand <thomas@lfglabs.dev>
Date: Sat, 25 Jul 2026 05:47:20 +0100
Subject: [PATCH 16/34] Probe forced definite level 19683 newspace

---
 .github/workflows/beal_signature357_decisive.yml | 16 ++++++++--------
 1 file changed, 8 insertions(+), 8 deletions(-)

diff --git a/.github/workflows/beal_signature357_decisive.yml b/.github/workflows/beal_signature357_decisive.yml
index 9841d56..ec7beed 100644
--- a/.github/workflows/beal_signature357_decisive.yml
+++ b/.github/workflows/beal_signature357_decisive.yml
@@ -17,21 +17,21 @@ concurrency:
   cancel-in-progress: true
 
 jobs:
-  modfrhil-attributes:
+  mod5-level19683-definite:
     runs-on: ubuntu-latest
-    timeout-minutes: 10
+    timeout-minutes: 20
     steps:
       - uses: actions/checkout@v5
         with:
           repository: Th0rgal/beal-conjecture-lean
-          ref: eadc80f07c2d26adaeb3e0edfa3f292904dccd24
+          ref: bb7be2b475438071f9afedd3f2475e82cf1fdefe
           path: beal
-      - name: Inspect the independently computed newspace attributes
+      - name: Force the newspace onto a definite quaternion order
         run: >-
-          python3 beal/scripts/run_signature_357_magma_modfrhil_attributes.py
-          > modfrhil-attributes.json
+          python3 beal/scripts/run_signature_357_magma_mod5_19683_definite_probe.py
+          > mod5-19683-definite.json
       - uses: actions/upload-artifact@v4
         with:
-          name: signature357-modfrhil-attributes
-          path: modfrhil-attributes.json
+          name: signature357-mod5-19683-definite
+          path: mod5-19683-definite.json
           if-no-files-found: error

From f79f8302dffb6826f9f6872a1c874eb54ffe0013 Mon Sep 17 00:00:00 2001
From: Thomas Marchand <thomas@lfglabs.dev>
Date: Sat, 25 Jul 2026 05:50:52 +0100
Subject: [PATCH 17/34] Probe ambient raw Hecke and degeneracy maps

---
 .github/workflows/beal_signature357_decisive.yml | 14 +++++++-------
 1 file changed, 7 insertions(+), 7 deletions(-)

diff --git a/.github/workflows/beal_signature357_decisive.yml b/.github/workflows/beal_signature357_decisive.yml
index ec7beed..4f475c6 100644
--- a/.github/workflows/beal_signature357_decisive.yml
+++ b/.github/workflows/beal_signature357_decisive.yml
@@ -17,21 +17,21 @@ concurrency:
   cancel-in-progress: true
 
 jobs:
-  mod5-level19683-definite:
+  mod5-level19683-ambient-raw:
     runs-on: ubuntu-latest
     timeout-minutes: 20
     steps:
       - uses: actions/checkout@v5
         with:
           repository: Th0rgal/beal-conjecture-lean
-          ref: bb7be2b475438071f9afedd3f2475e82cf1fdefe
+          ref: 2e5e52289b488909935fd2bd0786b5fb7c60f037
           path: beal
-      - name: Force the newspace onto a definite quaternion order
+      - name: Inspect ambient raw Hecke and degeneracy maps
         run: >-
-          python3 beal/scripts/run_signature_357_magma_mod5_19683_definite_probe.py
-          > mod5-19683-definite.json
+          python3 beal/scripts/run_signature_357_magma_mod5_19683_ambient_raw_probe.py
+          > mod5-19683-ambient-raw.json
       - uses: actions/upload-artifact@v4
         with:
-          name: signature357-mod5-19683-definite
-          path: mod5-19683-definite.json
+          name: signature357-mod5-19683-ambient-raw
+          path: mod5-19683-ambient-raw.json
           if-no-files-found: error

From 35c87df59ada510002559df1bd3855a7803223ca Mon Sep 17 00:00:00 2001
From: Thomas Marchand <thomas@lfglabs.dev>
Date: Sat, 25 Jul 2026 05:54:04 +0100
Subject: [PATCH 18/34] Force lazy ambient construction before raw probe

---
 .github/workflows/beal_signature357_decisive.yml | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/.github/workflows/beal_signature357_decisive.yml b/.github/workflows/beal_signature357_decisive.yml
index 4f475c6..86df63f 100644
--- a/.github/workflows/beal_signature357_decisive.yml
+++ b/.github/workflows/beal_signature357_decisive.yml
@@ -24,9 +24,9 @@ jobs:
       - uses: actions/checkout@v5
         with:
           repository: Th0rgal/beal-conjecture-lean
-          ref: 2e5e52289b488909935fd2bd0786b5fb7c60f037
+          ref: 459324defa4236537519fad73b70299aef8a9ab1
           path: beal
-      - name: Inspect ambient raw Hecke and degeneracy maps
+      - name: Force construction, inspect degeneracy maps and compute ambient raw Hecke
         run: >-
           python3 beal/scripts/run_signature_357_magma_mod5_19683_ambient_raw_probe.py
           > mod5-19683-ambient-raw.json

From 69e2fbb99ed43b33d8e886c81dcec9ce49721c11 Mon Sep 17 00:00:00 2001
From: Thomas Marchand <thomas@lfglabs.dev>
Date: Sat, 25 Jul 2026 05:58:35 +0100
Subject: [PATCH 19/34] Inspect partial level 19683 Hecke caches

---
 .github/workflows/beal_signature357_decisive.yml | 14 +++++++-------
 1 file changed, 7 insertions(+), 7 deletions(-)

diff --git a/.github/workflows/beal_signature357_decisive.yml b/.github/workflows/beal_signature357_decisive.yml
index 86df63f..135c316 100644
--- a/.github/workflows/beal_signature357_decisive.yml
+++ b/.github/workflows/beal_signature357_decisive.yml
@@ -17,21 +17,21 @@ concurrency:
   cancel-in-progress: true
 
 jobs:
-  mod5-level19683-ambient-raw:
+  mod5-level19683-hecke-cache:
     runs-on: ubuntu-latest
     timeout-minutes: 20
     steps:
       - uses: actions/checkout@v5
         with:
           repository: Th0rgal/beal-conjecture-lean
-          ref: 459324defa4236537519fad73b70299aef8a9ab1
+          ref: 250e83a7f4bc62506af22e735dd1ae833abda117
           path: beal
-      - name: Force construction, inspect degeneracy maps and compute ambient raw Hecke
+      - name: Catch the norm-8 failure and inspect every partial Hecke cache
         run: >-
-          python3 beal/scripts/run_signature_357_magma_mod5_19683_ambient_raw_probe.py
-          > mod5-19683-ambient-raw.json
+          python3 beal/scripts/run_signature_357_magma_mod5_19683_hecke_cache_probe.py
+          > mod5-19683-hecke-cache.json
       - uses: actions/upload-artifact@v4
         with:
-          name: signature357-mod5-19683-ambient-raw
-          path: mod5-19683-ambient-raw.json
+          name: signature357-mod5-19683-hecke-cache
+          path: mod5-19683-hecke-cache.json
           if-no-files-found: error

From 54b861f702259a0fd0d446c90913cd85e06d1fa1 Mon Sep 17 00:00:00 2001
From: Thomas Marchand <thomas@lfglabs.dev>
Date: Sat, 25 Jul 2026 06:02:13 +0100
Subject: [PATCH 20/34] Rerun cache probe with valid Magma catch syntax

---
 .github/workflows/beal_signature357_decisive.yml | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/.github/workflows/beal_signature357_decisive.yml b/.github/workflows/beal_signature357_decisive.yml
index 135c316..1f7e875 100644
--- a/.github/workflows/beal_signature357_decisive.yml
+++ b/.github/workflows/beal_signature357_decisive.yml
@@ -24,7 +24,7 @@ jobs:
       - uses: actions/checkout@v5
         with:
           repository: Th0rgal/beal-conjecture-lean
-          ref: 250e83a7f4bc62506af22e735dd1ae833abda117
+          ref: 133d3ab8eca598b0d7ce32eac107f18cd15fd0c5
           path: beal
       - name: Catch the norm-8 failure and inspect every partial Hecke cache
         run: >-

From 4c62713f069c85707e6d8f04ddcae46898082f3a Mon Sep 17 00:00:00 2001
From: Thomas Marchand <thomas@lfglabs.dev>
Date: Sat, 25 Jul 2026 06:10:07 +0100
Subject: [PATCH 21/34] Inspect level 19683 AL attribute

---
 .github/workflows/beal_signature357_decisive.yml | 16 ++++++++--------
 1 file changed, 8 insertions(+), 8 deletions(-)

diff --git a/.github/workflows/beal_signature357_decisive.yml b/.github/workflows/beal_signature357_decisive.yml
index 1f7e875..8e8e16d 100644
--- a/.github/workflows/beal_signature357_decisive.yml
+++ b/.github/workflows/beal_signature357_decisive.yml
@@ -17,21 +17,21 @@ concurrency:
   cancel-in-progress: true
 
 jobs:
-  mod5-level19683-hecke-cache:
+  mod5-level19683-al:
     runs-on: ubuntu-latest
-    timeout-minutes: 20
+    timeout-minutes: 10
     steps:
       - uses: actions/checkout@v5
         with:
           repository: Th0rgal/beal-conjecture-lean
-          ref: 133d3ab8eca598b0d7ce32eac107f18cd15fd0c5
+          ref: 18529820654133c56079b4f0c1c8ed6c08f63d6b
           path: beal
-      - name: Catch the norm-8 failure and inspect every partial Hecke cache
+      - name: Inspect the opaque AL attribute of the independent newspace
         run: >-
-          python3 beal/scripts/run_signature_357_magma_mod5_19683_hecke_cache_probe.py
-          > mod5-19683-hecke-cache.json
+          python3 beal/scripts/run_signature_357_magma_mod5_19683_al_probe.py
+          > mod5-19683-al.json
       - uses: actions/upload-artifact@v4
         with:
-          name: signature357-mod5-19683-hecke-cache
-          path: mod5-19683-hecke-cache.json
+          name: signature357-mod5-19683-al
+          path: mod5-19683-al.json
           if-no-files-found: error

From 88702dcbe3c5506f914b36bac597e4481d7aeb32 Mon Sep 17 00:00:00 2001
From: Thomas Marchand <thomas@lfglabs.dev>
Date: Sat, 25 Jul 2026 06:21:48 +0100
Subject: [PATCH 22/34] Run twist-corrected level 137781 chain

---
 .github/workflows/beal_signature357_decisive.yml | 16 ++++++++--------
 1 file changed, 8 insertions(+), 8 deletions(-)

diff --git a/.github/workflows/beal_signature357_decisive.yml b/.github/workflows/beal_signature357_decisive.yml
index 8e8e16d..2a44d98 100644
--- a/.github/workflows/beal_signature357_decisive.yml
+++ b/.github/workflows/beal_signature357_decisive.yml
@@ -17,21 +17,21 @@ concurrency:
   cancel-in-progress: true
 
 jobs:
-  mod5-level19683-al:
+  mod5-level137781-twist-corrected:
     runs-on: ubuntu-latest
-    timeout-minutes: 10
+    timeout-minutes: 20
     steps:
       - uses: actions/checkout@v5
         with:
           repository: Th0rgal/beal-conjecture-lean
-          ref: 18529820654133c56079b4f0c1c8ed6c08f63d6b
+          ref: 7056e98fa0ceeee10b364acd001d5f486252488d
           path: beal
-      - name: Inspect the opaque AL attribute of the independent newspace
+      - name: Apply eta7-corrected local polynomials and semilinear relations
         run: >-
-          python3 beal/scripts/run_signature_357_magma_mod5_19683_al_probe.py
-          > mod5-19683-al.json
+          python3 beal/scripts/run_signature_357_magma_mod5_137781_semilinear_chain_twisted.py
+          > mod5-137781-twist-corrected.json
       - uses: actions/upload-artifact@v4
         with:
-          name: signature357-mod5-19683-al
-          path: mod5-19683-al.json
+          name: signature357-mod5-137781-twist-corrected
+          path: mod5-137781-twist-corrected.json
           if-no-files-found: error

From c0bd24eee18091ac9e046df52c6b2b09e64bb613 Mon Sep 17 00:00:00 2001
From: Thomas Marchand <thomas@lfglabs.dev>
Date: Sat, 25 Jul 2026 08:55:39 +0100
Subject: [PATCH 23/34] research: run fixed-11 residual tests for signature
 3511

---
 .../workflows/beal_signature357_decisive.yml  | 28 +++++++++++--------
 1 file changed, 16 insertions(+), 12 deletions(-)

diff --git a/.github/workflows/beal_signature357_decisive.yml b/.github/workflows/beal_signature357_decisive.yml
index 2a44d98..dc8d975 100644
--- a/.github/workflows/beal_signature357_decisive.yml
+++ b/.github/workflows/beal_signature357_decisive.yml
@@ -1,37 +1,41 @@
-name: Beal signature 357 decisive probes
+name: Beal signature 3511 fixed-11 probes
 
 on:
   push:
     branches:
       - research/beal-signature357-probe
-  pull_request:
-    paths:
-      - ".github/workflows/beal_signature357_decisive.yml"
   workflow_dispatch:
 
 permissions:
   contents: read
 
 concurrency:
-  group: beal-signature357-decisive-${{ github.ref }}
+  group: beal-signature3511-fixed11-${{ github.ref }}
   cancel-in-progress: true
 
 jobs:
-  mod5-level137781-twist-corrected:
+  fixed11-level:
+    strategy:
+      fail-fast: false
+      matrix:
+        level: ["2,2", "2,3", "3,2", "3,3"]
+        prime: [13, 29]
     runs-on: ubuntu-latest
     timeout-minutes: 20
     steps:
       - uses: actions/checkout@v5
         with:
           repository: Th0rgal/beal-conjecture-lean
-          ref: 7056e98fa0ceeee10b364acd001d5f486252488d
+          ref: 1b2ff227331a4db3ec2116bd848746c029abc8d1
           path: beal
-      - name: Apply eta7-corrected local polynomials and semilinear relations
+      - name: Test the complete fixed-11 residual newspace
         run: >-
-          python3 beal/scripts/run_signature_357_magma_mod5_137781_semilinear_chain_twisted.py
-          > mod5-137781-twist-corrected.json
+          python3 beal/scripts/run_signature_3511_magma_fixed11_residual.py
+          --level "${{ matrix.level }}"
+          --prime "${{ matrix.prime }}"
+          > "fixed11-${{ matrix.level }}-${{ matrix.prime }}.json"
       - uses: actions/upload-artifact@v4
         with:
-          name: signature357-mod5-137781-twist-corrected
-          path: mod5-137781-twist-corrected.json
+          name: "signature3511-fixed11-${{ matrix.level }}-${{ matrix.prime }}"
+          path: "fixed11-${{ matrix.level }}-${{ matrix.prime }}.json"
           if-no-files-found: error

From 0f39dca7a343e0ee06a7cfa6fcf5598c0be28a33 Mon Sep 17 00:00:00 2001
From: Thomas Marchand <thomas@lfglabs.dev>
Date: Sat, 25 Jul 2026 08:58:08 +0100
Subject: [PATCH 24/34] ci: expose fixed-11 probes on the compute PR

---
 .github/workflows/beal_signature357_decisive.yml | 3 +++
 1 file changed, 3 insertions(+)

diff --git a/.github/workflows/beal_signature357_decisive.yml b/.github/workflows/beal_signature357_decisive.yml
index dc8d975..7225b50 100644
--- a/.github/workflows/beal_signature357_decisive.yml
+++ b/.github/workflows/beal_signature357_decisive.yml
@@ -4,6 +4,9 @@ on:
   push:
     branches:
       - research/beal-signature357-probe
+  pull_request:
+    paths:
+      - ".github/workflows/beal_signature357_decisive.yml"
   workflow_dispatch:
 
 permissions:

From bc833b04545ab38ff83977781120e108ab572226 Mon Sep 17 00:00:00 2001
From: Thomas Marchand <thomas@lfglabs.dev>
Date: Sat, 25 Jul 2026 09:44:43 +0100
Subject: [PATCH 25/34] ci: run signature 3511 paired split-prime probes

---
 .../workflows/beal_signature357_decisive.yml  | 27 +++++++++----------
 1 file changed, 13 insertions(+), 14 deletions(-)

diff --git a/.github/workflows/beal_signature357_decisive.yml b/.github/workflows/beal_signature357_decisive.yml
index 7225b50..761888f 100644
--- a/.github/workflows/beal_signature357_decisive.yml
+++ b/.github/workflows/beal_signature357_decisive.yml
@@ -1,4 +1,4 @@
-name: Beal signature 3511 fixed-11 probes
+name: Beal signature 3511 paired fixed-11 probes
 
 on:
   push:
@@ -13,32 +13,31 @@ permissions:
   contents: read
 
 concurrency:
-  group: beal-signature3511-fixed11-${{ github.ref }}
+  group: beal-signature3511-paired-${{ github.ref }}
   cancel-in-progress: true
 
 jobs:
-  fixed11-level:
+  paired-prime:
     strategy:
       fail-fast: false
       matrix:
-        level: ["2,2", "2,3", "3,2", "3,3"]
-        prime: [13, 29]
+        level: ["2,2", "2,3", "3,2"]
+        prime: [19, 29, 31, 41]
     runs-on: ubuntu-latest
-    timeout-minutes: 20
+    timeout-minutes: 15
     steps:
       - uses: actions/checkout@v5
         with:
           repository: Th0rgal/beal-conjecture-lean
-          ref: 1b2ff227331a4db3ec2116bd848746c029abc8d1
+          ref: 636e6a77b4cdee82d51149113bb7ea0199acae33
           path: beal
-      - name: Test the complete fixed-11 residual newspace
+      - name: Apply paired Galois-conjugate trace condition
         run: >-
-          python3 beal/scripts/run_signature_3511_magma_fixed11_residual.py
-          --level "${{ matrix.level }}"
-          --prime "${{ matrix.prime }}"
-          > "fixed11-${{ matrix.level }}-${{ matrix.prime }}.json"
+          python3 beal/scripts/run_signature_3511_magma_fixed11_split_pair.py
+          --level "${{ matrix.level }}" --prime "${{ matrix.prime }}"
+          > "signature3511-pair-${{ matrix.level }}-${{ matrix.prime }}.json"
       - uses: actions/upload-artifact@v4
         with:
-          name: "signature3511-fixed11-${{ matrix.level }}-${{ matrix.prime }}"
-          path: "fixed11-${{ matrix.level }}-${{ matrix.prime }}.json"
+          name: "signature3511-fixed11-pair-${{ matrix.level }}-${{ matrix.prime }}"
+          path: "signature3511-pair-${{ matrix.level }}-${{ matrix.prime }}.json"
           if-no-files-found: error

From d3f7e0c271f8ec4302eff50f332281e342dc8829 Mon Sep 17 00:00:00 2001
From: Thomas Marchand <thomas@lfglabs.dev>
Date: Sat, 25 Jul 2026 09:48:04 +0100
Subject: [PATCH 26/34] ci: intersect signature 3511 paired trace conditions

---
 .../workflows/beal_signature357_decisive.yml  | 23 +++++++++----------
 1 file changed, 11 insertions(+), 12 deletions(-)

diff --git a/.github/workflows/beal_signature357_decisive.yml b/.github/workflows/beal_signature357_decisive.yml
index 761888f..a17b0ee 100644
--- a/.github/workflows/beal_signature357_decisive.yml
+++ b/.github/workflows/beal_signature357_decisive.yml
@@ -1,4 +1,4 @@
-name: Beal signature 3511 paired fixed-11 probes
+name: Beal signature 3511 paired fixed-11 chain
 
 on:
   push:
@@ -13,31 +13,30 @@ permissions:
   contents: read
 
 concurrency:
-  group: beal-signature3511-paired-${{ github.ref }}
+  group: beal-signature3511-paired-chain-${{ github.ref }}
   cancel-in-progress: true
 
 jobs:
-  paired-prime:
+  paired-chain:
     strategy:
       fail-fast: false
       matrix:
         level: ["2,2", "2,3", "3,2"]
-        prime: [19, 29, 31, 41]
     runs-on: ubuntu-latest
-    timeout-minutes: 15
+    timeout-minutes: 20
     steps:
       - uses: actions/checkout@v5
         with:
           repository: Th0rgal/beal-conjecture-lean
-          ref: 636e6a77b4cdee82d51149113bb7ea0199acae33
+          ref: c0ea5545e2ca280f7892354ba36a9d471cd869ec
           path: beal
-      - name: Apply paired Galois-conjugate trace condition
+      - name: Intersect four paired split-prime conditions
         run: >-
-          python3 beal/scripts/run_signature_3511_magma_fixed11_split_pair.py
-          --level "${{ matrix.level }}" --prime "${{ matrix.prime }}"
-          > "signature3511-pair-${{ matrix.level }}-${{ matrix.prime }}.json"
+          python3 beal/scripts/run_signature_3511_magma_fixed11_split_pair_chain.py
+          --level "${{ matrix.level }}"
+          > "signature3511-paired-chain-${{ matrix.level }}.json"
       - uses: actions/upload-artifact@v4
         with:
-          name: "signature3511-fixed11-pair-${{ matrix.level }}-${{ matrix.prime }}"
-          path: "signature3511-pair-${{ matrix.level }}-${{ matrix.prime }}.json"
+          name: "signature3511-fixed11-paired-chain-${{ matrix.level }}"
+          path: "signature3511-paired-chain-${{ matrix.level }}.json"
           if-no-files-found: error

From edc6dd23dbea5a22373ed31771bf2d5ee6a5852b Mon Sep 17 00:00:00 2001
From: Thomas Marchand <thomas@lfglabs.dev>
Date: Sat, 25 Jul 2026 09:50:46 +0100
Subject: [PATCH 27/34] ci: run extended signature 3511 paired chain

---
 .github/workflows/beal_signature357_decisive.yml | 16 ++++++++--------
 1 file changed, 8 insertions(+), 8 deletions(-)

diff --git a/.github/workflows/beal_signature357_decisive.yml b/.github/workflows/beal_signature357_decisive.yml
index a17b0ee..a492be7 100644
--- a/.github/workflows/beal_signature357_decisive.yml
+++ b/.github/workflows/beal_signature357_decisive.yml
@@ -1,4 +1,4 @@
-name: Beal signature 3511 paired fixed-11 chain
+name: Beal signature 3511 extended paired chain
 
 on:
   push:
@@ -13,7 +13,7 @@ permissions:
   contents: read
 
 concurrency:
-  group: beal-signature3511-paired-chain-${{ github.ref }}
+  group: beal-signature3511-extended-paired-${{ github.ref }}
   cancel-in-progress: true
 
 jobs:
@@ -28,15 +28,15 @@ jobs:
       - uses: actions/checkout@v5
         with:
           repository: Th0rgal/beal-conjecture-lean
-          ref: c0ea5545e2ca280f7892354ba36a9d471cd869ec
+          ref: 3e0b233a02365b1a39200518119a87dc15ad3c37
           path: beal
-      - name: Intersect four paired split-prime conditions
+      - name: Intersect the extended paired split-prime conditions
         run: >-
-          python3 beal/scripts/run_signature_3511_magma_fixed11_split_pair_chain.py
+          python3 beal/scripts/run_signature_3511_magma_fixed11_split_pair_extended.py
           --level "${{ matrix.level }}"
-          > "signature3511-paired-chain-${{ matrix.level }}.json"
+          > "signature3511-extended-paired-${{ matrix.level }}.json"
       - uses: actions/upload-artifact@v4
         with:
-          name: "signature3511-fixed11-paired-chain-${{ matrix.level }}"
-          path: "signature3511-paired-chain-${{ matrix.level }}.json"
+          name: "signature3511-fixed11-extended-paired-${{ matrix.level }}"
+          path: "signature3511-extended-paired-${{ matrix.level }}.json"
           if-no-files-found: error

From de54eb3e1d64aab7f5350fc6e24cfebfb708d3f7 Mon Sep 17 00:00:00 2001
From: Thomas Marchand <thomas@lfglabs.dev>
Date: Sat, 25 Jul 2026 12:01:13 +0100
Subject: [PATCH 28/34] compute: run signature3511 raw and finite Brandt probes

---
 .../workflows/beal_signature357_decisive.yml  | 59 ++++++++++++++-----
 1 file changed, 45 insertions(+), 14 deletions(-)

diff --git a/.github/workflows/beal_signature357_decisive.yml b/.github/workflows/beal_signature357_decisive.yml
index a492be7..397ccd1 100644
--- a/.github/workflows/beal_signature357_decisive.yml
+++ b/.github/workflows/beal_signature357_decisive.yml
@@ -1,4 +1,4 @@
-name: Beal signature 3511 extended paired chain
+name: Beal signature 3511 raw Brandt probes
 
 on:
   push:
@@ -13,30 +13,61 @@ permissions:
   contents: read
 
 concurrency:
-  group: beal-signature3511-extended-paired-${{ github.ref }}
+  group: beal-signature3511-raw-brandt-${{ github.ref }}
   cancel-in-progress: true
 
 jobs:
-  paired-chain:
-    strategy:
-      fail-fast: false
-      matrix:
-        level: ["2,2", "2,3", "3,2"]
+  raw-prime2:
     runs-on: ubuntu-latest
     timeout-minutes: 20
     steps:
       - uses: actions/checkout@v5
         with:
           repository: Th0rgal/beal-conjecture-lean
-          ref: 3e0b233a02365b1a39200518119a87dc15ad3c37
+          ref: 981b505bb85da07c48c54cf1a241fbc5cd2d5523
           path: beal
-      - name: Intersect the extended paired split-prime conditions
+      - name: Compute the level-(3,3) norm-4 cover on the raw Brandt module
+        working-directory: beal/scripts
         run: >-
-          python3 beal/scripts/run_signature_3511_magma_fixed11_split_pair_extended.py
-          --level "${{ matrix.level }}"
-          > "signature3511-extended-paired-${{ matrix.level }}.json"
+          python3 run_signature_3511_magma_level33_raw_prime2.py
+          > ../../signature3511-level33-raw-prime2.json
       - uses: actions/upload-artifact@v4
         with:
-          name: "signature3511-fixed11-extended-paired-${{ matrix.level }}"
-          path: "signature3511-extended-paired-${{ matrix.level }}.json"
+          name: signature3511-level33-raw-prime2
+          path: signature3511-level33-raw-prime2.json
           if-no-files-found: error
+
+  finite-brandt-api:
+    runs-on: ubuntu-latest
+    timeout-minutes: 15
+    steps:
+      - uses: actions/checkout@v5
+        with:
+          repository: Th0rgal/beal-conjecture-lean
+          ref: 981b505bb85da07c48c54cf1a241fbc5cd2d5523
+          path: beal
+      - name: Probe direct finite-field Brandt construction
+        working-directory: beal/scripts
+        run: >-
+          python3 run_signature_3511_magma_finite_brandt_probe.py
+          > ../../signature3511-finite-brandt-api.json
+      - uses: actions/upload-artifact@v4
+        with:
+          name: signature3511-finite-brandt-api
+          path: signature3511-finite-brandt-api.json
+          if-no-files-found: error
+
+  mod5-irreducibility-replay:
+    runs-on: ubuntu-latest
+    timeout-minutes: 10
+    steps:
+      - uses: actions/checkout@v5
+        with:
+          repository: Th0rgal/beal-conjecture-lean
+          ref: d6917e9f10226f7dc14720b54cb86e79ca792032
+          path: beal
+      - name: Replay the signature-(3,5,11) mod-5 irreducibility certificate
+        working-directory: beal
+        run: |
+          python3 scripts/check_signature_3511_mod5_irreducibility_at3.py --self-test
+          python3 scripts/check_signature_3511_mod5_irreducibility_at3.py

From 99debc78aec05ade4d6825ba5ce31b08aff87c54 Mon Sep 17 00:00:00 2001
From: Thomas Marchand <thomas@lfglabs.dev>
Date: Sat, 25 Jul 2026 12:03:29 +0100
Subject: [PATCH 29/34] compute: rerun memory-minimal signature3511 raw Brandt
 probe

---
 .../workflows/beal_signature357_decisive.yml  | 24 ++-----------------
 1 file changed, 2 insertions(+), 22 deletions(-)

diff --git a/.github/workflows/beal_signature357_decisive.yml b/.github/workflows/beal_signature357_decisive.yml
index 397ccd1..6579a56 100644
--- a/.github/workflows/beal_signature357_decisive.yml
+++ b/.github/workflows/beal_signature357_decisive.yml
@@ -24,9 +24,9 @@ jobs:
       - uses: actions/checkout@v5
         with:
           repository: Th0rgal/beal-conjecture-lean
-          ref: 981b505bb85da07c48c54cf1a241fbc5cd2d5523
+          ref: 219b0587aae91d5a3972c909834986bc62f881d2
           path: beal
-      - name: Compute the level-(3,3) norm-4 cover on the raw Brandt module
+      - name: Compute the level-(3,3) norm-4 cover after freeing rational spaces
         working-directory: beal/scripts
         run: >-
           python3 run_signature_3511_magma_level33_raw_prime2.py
@@ -37,26 +37,6 @@ jobs:
           path: signature3511-level33-raw-prime2.json
           if-no-files-found: error
 
-  finite-brandt-api:
-    runs-on: ubuntu-latest
-    timeout-minutes: 15
-    steps:
-      - uses: actions/checkout@v5
-        with:
-          repository: Th0rgal/beal-conjecture-lean
-          ref: 981b505bb85da07c48c54cf1a241fbc5cd2d5523
-          path: beal
-      - name: Probe direct finite-field Brandt construction
-        working-directory: beal/scripts
-        run: >-
-          python3 run_signature_3511_magma_finite_brandt_probe.py
-          > ../../signature3511-finite-brandt-api.json
-      - uses: actions/upload-artifact@v4
-        with:
-          name: signature3511-finite-brandt-api
-          path: signature3511-finite-brandt-api.json
-          if-no-files-found: error
-
   mod5-irreducibility-replay:
     runs-on: ubuntu-latest
     timeout-minutes: 10

From 253a82096bafdfdf52037df68aa395604bcf8b0c Mon Sep 17 00:00:00 2001
From: Thomas Marchand <thomas@lfglabs.dev>
Date: Sat, 25 Jul 2026 12:10:04 +0100
Subject: [PATCH 30/34] compute: probe finite-ramified fixed-11 quaternion
 order

---
 .../workflows/beal_signature357_decisive.yml  | 20 +++++++++----------
 1 file changed, 10 insertions(+), 10 deletions(-)

diff --git a/.github/workflows/beal_signature357_decisive.yml b/.github/workflows/beal_signature357_decisive.yml
index 6579a56..3ed4a77 100644
--- a/.github/workflows/beal_signature357_decisive.yml
+++ b/.github/workflows/beal_signature357_decisive.yml
@@ -1,4 +1,4 @@
-name: Beal signature 3511 raw Brandt probes
+name: Beal signature 3511 ramified-order probe
 
 on:
   push:
@@ -13,28 +13,28 @@ permissions:
   contents: read
 
 concurrency:
-  group: beal-signature3511-raw-brandt-${{ github.ref }}
+  group: beal-signature3511-ramified-order-${{ github.ref }}
   cancel-in-progress: true
 
 jobs:
-  raw-prime2:
+  ramified-order:
     runs-on: ubuntu-latest
-    timeout-minutes: 20
+    timeout-minutes: 25
     steps:
       - uses: actions/checkout@v5
         with:
           repository: Th0rgal/beal-conjecture-lean
-          ref: 219b0587aae91d5a3972c909834986bc62f881d2
+          ref: 0ef5bc10da6679ff9efc69b483f5cafc3842a381
           path: beal
-      - name: Compute the level-(3,3) norm-4 cover after freeing rational spaces
+      - name: Realize fixed-11 newspaces on a quaternion algebra ramified at 3 and 5
         working-directory: beal/scripts
         run: >-
-          python3 run_signature_3511_magma_level33_raw_prime2.py
-          > ../../signature3511-level33-raw-prime2.json
+          python3 run_signature_3511_magma_ramified_order_probe.py
+          > ../../signature3511-ramified-order.json
       - uses: actions/upload-artifact@v4
         with:
-          name: signature3511-level33-raw-prime2
-          path: signature3511-level33-raw-prime2.json
+          name: signature3511-ramified-order
+          path: signature3511-ramified-order.json
           if-no-files-found: error
 
   mod5-irreducibility-replay:

From b891838ef1a7f13332f314b7b9e075ea2c0d9dc1 Mon Sep 17 00:00:00 2001
From: Thomas Marchand <thomas@lfglabs.dev>
Date: Sat, 25 Jul 2026 12:12:58 +0100
Subject: [PATCH 31/34] compute: test exact-conductor fixed-11 Eichler order

---
 .../workflows/beal_signature357_decisive.yml  | 23 ++++---------------
 1 file changed, 4 insertions(+), 19 deletions(-)

diff --git a/.github/workflows/beal_signature357_decisive.yml b/.github/workflows/beal_signature357_decisive.yml
index 3ed4a77..3da4327 100644
--- a/.github/workflows/beal_signature357_decisive.yml
+++ b/.github/workflows/beal_signature357_decisive.yml
@@ -1,4 +1,4 @@
-name: Beal signature 3511 ramified-order probe
+name: Beal signature 3511 exact-conductor order probe
 
 on:
   push:
@@ -13,7 +13,7 @@ permissions:
   contents: read
 
 concurrency:
-  group: beal-signature3511-ramified-order-${{ github.ref }}
+  group: beal-signature3511-exact-order-${{ github.ref }}
   cancel-in-progress: true
 
 jobs:
@@ -24,9 +24,9 @@ jobs:
       - uses: actions/checkout@v5
         with:
           repository: Th0rgal/beal-conjecture-lean
-          ref: 0ef5bc10da6679ff9efc69b483f5cafc3842a381
+          ref: 51222477ab2b7ad8ecb1f0605f8f7137f775db06
           path: beal
-      - name: Realize fixed-11 newspaces on a quaternion algebra ramified at 3 and 5
+      - name: Realize newspaces on the exact-conductor ramified Eichler orders
         working-directory: beal/scripts
         run: >-
           python3 run_signature_3511_magma_ramified_order_probe.py
@@ -36,18 +36,3 @@ jobs:
           name: signature3511-ramified-order
           path: signature3511-ramified-order.json
           if-no-files-found: error
-
-  mod5-irreducibility-replay:
-    runs-on: ubuntu-latest
-    timeout-minutes: 10
-    steps:
-      - uses: actions/checkout@v5
-        with:
-          repository: Th0rgal/beal-conjecture-lean
-          ref: d6917e9f10226f7dc14720b54cb86e79ca792032
-          path: beal
-      - name: Replay the signature-(3,5,11) mod-5 irreducibility certificate
-        working-directory: beal
-        run: |
-          python3 scripts/check_signature_3511_mod5_irreducibility_at3.py --self-test
-          python3 scripts/check_signature_3511_mod5_irreducibility_at3.py

From f1b41a7de39975cdee20b9ae2d31cadcce42c130 Mon Sep 17 00:00:00 2001
From: Thomas Marchand <thomas@lfglabs.dev>
Date: Sat, 25 Jul 2026 14:04:58 +0100
Subject: [PATCH 32/34] Mirror the corrected global Beal audit

---
 .github/workflows/beal_global_audit.yml | 45 +++++++++++++++++++++++++
 1 file changed, 45 insertions(+)
 create mode 100644 .github/workflows/beal_global_audit.yml

diff --git a/.github/workflows/beal_global_audit.yml b/.github/workflows/beal_global_audit.yml
new file mode 100644
index 0000000..452494d
--- /dev/null
+++ b/.github/workflows/beal_global_audit.yml
@@ -0,0 +1,45 @@
+name: Beal global audit mirror
+
+on:
+  push:
+    branches:
+      - research/beal-signature357-probe
+  workflow_dispatch:
+
+permissions:
+  contents: read
+
+jobs:
+  replay:
+    runs-on: ubuntu-latest
+    timeout-minutes: 20
+    steps:
+      - uses: actions/checkout@v5
+        with:
+          repository: Th0rgal/beal-conjecture-lean
+          ref: 97b776a728b66942912a564d47b09e84078e2b1e
+      - name: Replay corrected global certificates
+        run: |
+          python3 scripts/check_global_beal_prime_exponent_reduction.py --self-test
+          python3 scripts/check_global_beal_prime_exponent_reduction.py
+          python3 scripts/check_global_beal_repeated_exponent_audit.py --self-test
+          python3 scripts/check_global_beal_repeated_exponent_audit.py
+          python3 scripts/check_global_beal_one_four_core.py --self-test
+          python3 scripts/check_global_beal_one_four_core.py
+          python3 scripts/check_global_beal_signature457_power_residue.py --self-test
+          python3 scripts/check_global_beal_signature457_power_residue.py
+
+  lean:
+    runs-on: ubuntu-latest
+    timeout-minutes: 30
+    steps:
+      - uses: actions/checkout@v5
+        with:
+          repository: Th0rgal/beal-conjecture-lean
+          ref: 97b776a728b66942912a564d47b09e84078e2b1e
+      - uses: leanprover/lean-action@v1
+      - name: Compile new global modules
+        run: |
+          lake env lean BealUnified/ExponentReduction.lean
+          lake env lean BealUnified/Research/CyclotomicAudit.lean
+          lake env lean BealUnified/Research/Signature457.lean

From 289484ec1660380332008efdfc8aafe3bedba61d Mon Sep 17 00:00:00 2001
From: Thomas Marchand <thomas@lfglabs.dev>
Date: Sat, 25 Jul 2026 14:05:46 +0100
Subject: [PATCH 33/34] Trigger the pinned global Beal audit mirror

---
 .github/workflows/beal_global_audit.yml | 2 ++
 1 file changed, 2 insertions(+)

diff --git a/.github/workflows/beal_global_audit.yml b/.github/workflows/beal_global_audit.yml
index 452494d..f92bf5d 100644
--- a/.github/workflows/beal_global_audit.yml
+++ b/.github/workflows/beal_global_audit.yml
@@ -43,3 +43,5 @@ jobs:
           lake env lean BealUnified/ExponentReduction.lean
           lake env lean BealUnified/Research/CyclotomicAudit.lean
           lake env lean BealUnified/Research/Signature457.lean
+
+# This mirror is intentionally pinned to one research commit.

From 3fbeb1244dcce797544ec254e70dd02e775f057e Mon Sep 17 00:00:00 2001
From: Thomas Marchand <thomas@lfglabs.dev>
Date: Sat, 25 Jul 2026 14:09:06 +0100
Subject: [PATCH 34/34] Run the corrected global Beal audit now

---
 .../workflows/beal_signature357_decisive.yml  | 48 +++++++++++++------
 1 file changed, 34 insertions(+), 14 deletions(-)

diff --git a/.github/workflows/beal_signature357_decisive.yml b/.github/workflows/beal_signature357_decisive.yml
index 3da4327..dcb4e31 100644
--- a/.github/workflows/beal_signature357_decisive.yml
+++ b/.github/workflows/beal_signature357_decisive.yml
@@ -1,4 +1,4 @@
-name: Beal signature 3511 exact-conductor order probe
+name: Beal global canonical audit
 
 on:
   push:
@@ -13,26 +13,46 @@ permissions:
   contents: read
 
 concurrency:
-  group: beal-signature3511-exact-order-${{ github.ref }}
+  group: beal-global-canonical-audit-${{ github.ref }}
   cancel-in-progress: true
 
 jobs:
-  ramified-order:
+  replay:
     runs-on: ubuntu-latest
-    timeout-minutes: 25
+    timeout-minutes: 20
     steps:
       - uses: actions/checkout@v5
         with:
           repository: Th0rgal/beal-conjecture-lean
-          ref: 51222477ab2b7ad8ecb1f0605f8f7137f775db06
+          ref: 97b776a728b66942912a564d47b09e84078e2b1e
           path: beal
-      - name: Realize newspaces on the exact-conductor ramified Eichler orders
-        working-directory: beal/scripts
-        run: >-
-          python3 run_signature_3511_magma_ramified_order_probe.py
-          > ../../signature3511-ramified-order.json
-      - uses: actions/upload-artifact@v4
+      - name: Replay corrected global certificates
+        working-directory: beal
+        run: |
+          python3 scripts/check_global_beal_prime_exponent_reduction.py --self-test
+          python3 scripts/check_global_beal_prime_exponent_reduction.py
+          python3 scripts/check_global_beal_repeated_exponent_audit.py --self-test
+          python3 scripts/check_global_beal_repeated_exponent_audit.py
+          python3 scripts/check_global_beal_one_four_core.py --self-test
+          python3 scripts/check_global_beal_one_four_core.py
+          python3 scripts/check_global_beal_signature457_power_residue.py --self-test
+          python3 scripts/check_global_beal_signature457_power_residue.py
+
+  lean:
+    runs-on: ubuntu-latest
+    timeout-minutes: 30
+    steps:
+      - uses: actions/checkout@v5
+        with:
+          repository: Th0rgal/beal-conjecture-lean
+          ref: 97b776a728b66942912a564d47b09e84078e2b1e
+          path: beal
+      - uses: leanprover/lean-action@v1
         with:
-          name: signature3511-ramified-order
-          path: signature3511-ramified-order.json
-          if-no-files-found: error
+          lake-package-directory: beal
+      - name: Compile global research modules
+        working-directory: beal
+        run: |
+          lake env lean BealUnified/ExponentReduction.lean
+          lake env lean BealUnified/Research/CyclotomicAudit.lean
+          lake env lean BealUnified/Research/Signature457.lean