diff --git a/.github/workflows/side24-d5-closed-form.yml b/.github/workflows/side24-d5-closed-form.yml
index 7c3989295..e438b0f8e 100644
--- a/.github/workflows/side24-d5-closed-form.yml
+++ b/.github/workflows/side24-d5-closed-form.yml
@@ -5,6 +5,10 @@ name: SIDE24 d5 closed form
     paths:
       - 'coefficients/side24_d5_closed_form_20260930/**'
       - '.github/workflows/side24-d5-closed-form.yml'
+      - 'tools/**'
+      - 'tools.py'
+      - 'tests/test_side24_mutant_workflows.py'
+      - 'tests/test_legacy_workflow_triggers.py'
       - 'imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md'
       - 'coefficients/side24_v1/PROOF.md'
   workflow_dispatch:
@@ -23,6 +27,13 @@ jobs:
       - uses: actions/setup-python@6d4fd22b20465ccc0f7354476588e8df7c9ca0e0
         with:
           python-version: '3.11.16'
+      - name: Check trigger coverage and the SIDE24 replay contract in both modes
+        env:
+          SIDE24_MUTANT_TEST_FAMILY: 'side24-d5-closed-form'
+          PYTHONDONTWRITEBYTECODE: '1'
+        run: |
+          python -B -S tests/test_side24_mutant_workflows.py
+          python -B -O -S tests/test_side24_mutant_workflows.py
       - name: Verify sources and pins, replay both modes, reject mutants
         env:
           PYTHONDONTWRITEBYTECODE: '1'
@@ -54,27 +65,16 @@ jobs:
               blob = hashlib.sha1(b'blob %d\0' % len(data) + data).hexdigest()
               if blob != pin['git_blob']:
                   raise ValueError('pinned source drifted: ' + pin['path'] + ' ' + blob)
-          script = str(root / 'closed_form_d4.py')
           expected = (root / 'RESULTS.json').read_bytes()
-          mutants = ['shift-variance', 'vandermonde', 'erfc-sign', 'jacobian', 'mehta', 'by-parts', 'one-erf', 'two-erf', 'jk-init', 'gauss-half', 'boundary-term', 'atom-pi', 'cumulative-init', 'cumulative-recurrence']
-          def mutant_rc(args):
-              flags, m = args
-              return m, flags, subprocess.run([sys.executable, *flags, script, '--mutant', m],
-                                              capture_output=True, timeout=600).returncode
-          for flags in (['-B', '-S'], ['-B', '-O', '-S']):
-              out = subprocess.run([sys.executable, *flags, script],
-                                   check=True, capture_output=True, timeout=600).stdout
-              if out != expected:
-                  raise RuntimeError('output differs from RESULTS.json under ' + ' '.join(flags))
-              doc = json.loads(out)
-              if not (doc['passed'] and len(doc['checks']) == 16 and all(doc['checks'].values())):
-                  raise RuntimeError('certification did not pass under ' + ' '.join(flags))
-              if doc['closed_forms']['D_4'] != '6695/54  +  -405/32 sqrt(6)  +  -1375/72 sqrt(21) pi^(-1)  +  -405/16 sqrt(6) pi^(-1) arctan(1/4 sqrt(14))  +  -6695/27 pi^(-1) arctan(1/7 sqrt(21))  +  405/16 sqrt(6) pi^(-1) arctan(1/7 sqrt(14))' or doc['closed_forms']['Z_4'] != '1536 pi^(1)':
-                  raise RuntimeError('closed form differs from the stated one')
-              with concurrent.futures.ThreadPoolExecutor(max_workers=4) as pool:
-                  for m, fl, rc in pool.map(mutant_rc, [(flags, m) for m in mutants]):
-                      if rc != 1:
-                          raise RuntimeError('mutant ' + m + ' not rejected under ' + ' '.join(fl))
+          doc = json.loads(expected)
+          if not (doc['passed'] and len(doc['checks']) == 16 and all(doc['checks'].values())):
+              raise RuntimeError('RESULTS.json is not a passing certification')
+          if doc['closed_forms']['D_4'] != '6695/54  +  -405/32 sqrt(6)  +  -1375/72 sqrt(21) pi^(-1)  +  -405/16 sqrt(6) pi^(-1) arctan(1/4 sqrt(14))  +  -6695/27 pi^(-1) arctan(1/7 sqrt(21))  +  405/16 sqrt(6) pi^(-1) arctan(1/7 sqrt(14))' or doc['closed_forms']['Z_4'] != '1536 pi^(1)':
+              raise RuntimeError('closed form differs from the stated one')
+          # Every child: exact exit status, one timing line on stderr, baseline stdout equal
+          # to RESULTS.json, each mutant's report equal to the measured one (both modes).
+          from tools.side24_mutant_replay import replay
+          replay('side24-d5-closed-form', pathlib.Path.cwd(), timeout=600)
           print(json.dumps({'scientific_effect': 'NONE', 'mathematical_acceptance': False, 'exact_identity_checked': True, 'side24_v1': 'unchanged'}))
           PY
           git diff --exit-code
diff --git a/.github/workflows/side24-d5-coefficient.yml b/.github/workflows/side24-d5-coefficient.yml
index 9f46fffa4..2c50da0ab 100644
--- a/.github/workflows/side24-d5-coefficient.yml
+++ b/.github/workflows/side24-d5-coefficient.yml
@@ -5,6 +5,10 @@ name: SIDE24 d5 coefficient
     paths:
       - 'coefficients/side24_d5_20260930/**'
       - '.github/workflows/side24-d5-coefficient.yml'
+      - 'tools/**'
+      - 'tools.py'
+      - 'tests/test_side24_mutant_workflows.py'
+      - 'tests/test_legacy_workflow_triggers.py'
       - 'imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md'
       - 'coefficients/side24_v1/PROOF.md'
       - 'coefficients/side24_v1/ENCLOSURE.json'
@@ -24,6 +28,13 @@ jobs:
       - uses: actions/setup-python@6d4fd22b20465ccc0f7354476588e8df7c9ca0e0
         with:
           python-version: '3.11.16'
+      - name: Check trigger coverage and the SIDE24 replay contract in both modes
+        env:
+          SIDE24_MUTANT_TEST_FAMILY: 'side24-d5-coefficient'
+          PYTHONDONTWRITEBYTECODE: '1'
+        run: |
+          python -B -S tests/test_side24_mutant_workflows.py
+          python -B -O -S tests/test_side24_mutant_workflows.py
       - name: Verify sources and pins, replay both modes, reject mutants
         env:
           PYTHONDONTWRITEBYTECODE: '1'
@@ -61,25 +72,14 @@ jobs:
           for d in ('2', '3'):
               if ('"%s", "%s"' % (enc[d]['lower'], enc[d]['upper'])) not in src:
                   raise ValueError('SIDE24 d = ' + d + ' enclosure quoted in the script differs from ENCLOSURE.json')
-          script = str(root / 'cone_moment_d5.py')
           expected = (root / 'RESULTS.json').read_bytes()
-          mutants = ['shift-variance', 'vandermonde', 'erfc-branch', 'remainder-dropped', 'mehta', 'sphere-area', 'image-shells', 'sandwich-exponent', 'recurrence', 'jacobian', 'cumulative-half']
-          def mutant_rc(args):
-              flags, m = args
-              return m, flags, subprocess.run([sys.executable, *flags, script, '--mutant', m],
-                                              capture_output=True, timeout=1500).returncode
-          for flags in (['-B', '-S'], ['-B', '-O', '-S']):
-              out = subprocess.run([sys.executable, *flags, script],
-                                   check=True, capture_output=True, timeout=1500).stdout
-              if out != expected:
-                  raise RuntimeError('output differs from RESULTS.json under ' + ' '.join(flags))
-              doc = json.loads(out)
-              if not (doc['passed'] and len(doc['checks']) == 13 and all(doc['checks'].values())):
-                  raise RuntimeError('certification did not pass under ' + ' '.join(flags))
-              with concurrent.futures.ThreadPoolExecutor(max_workers=4) as pool:
-                  for m, fl, rc in pool.map(mutant_rc, [(flags, m) for m in mutants]):
-                      if rc != 1:
-                          raise RuntimeError('mutant ' + m + ' not rejected under ' + ' '.join(fl))
+          doc = json.loads(expected)
+          if not (doc['passed'] and len(doc['checks']) == 13 and all(doc['checks'].values())):
+              raise RuntimeError('RESULTS.json is not a passing certification')
+          # Every child: exact exit status, one timing line on stderr, baseline stdout equal
+          # to RESULTS.json, each mutant's report equal to the measured one (both modes).
+          from tools.side24_mutant_replay import replay
+          replay('side24-d5-coefficient', pathlib.Path.cwd(), timeout=1500)
           print(json.dumps({'scientific_effect': 'NONE', 'mathematical_acceptance': False, 'certified_enclosures': True, 'side24_v1': 'unchanged'}))
           PY
           git diff --exit-code
diff --git a/.github/workflows/side24-d6-coefficient.yml b/.github/workflows/side24-d6-coefficient.yml
index 6059c2ecd..7a851d9a4 100644
--- a/.github/workflows/side24-d6-coefficient.yml
+++ b/.github/workflows/side24-d6-coefficient.yml
@@ -5,6 +5,10 @@ name: SIDE24 d6 coefficient
     paths:
       - 'coefficients/side24_d6_20260930/**'
       - '.github/workflows/side24-d6-coefficient.yml'
+      - 'tools/**'
+      - 'tools.py'
+      - 'tests/test_side24_mutant_workflows.py'
+      - 'tests/test_legacy_workflow_triggers.py'
       - 'imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md'
       - 'coefficients/side24_v1/PROOF.md'
       - 'coefficients/side24_v1/ENCLOSURE.json'
@@ -24,6 +28,13 @@ jobs:
       - uses: actions/setup-python@6d4fd22b20465ccc0f7354476588e8df7c9ca0e0
         with:
           python-version: '3.11.16'
+      - name: Check trigger coverage and the SIDE24 replay contract in both modes
+        env:
+          SIDE24_MUTANT_TEST_FAMILY: 'side24-d6-coefficient'
+          PYTHONDONTWRITEBYTECODE: '1'
+        run: |
+          python -B -S tests/test_side24_mutant_workflows.py
+          python -B -O -S tests/test_side24_mutant_workflows.py
       - name: Verify sources and pins, replay both modes, reject mutants
         env:
           PYTHONDONTWRITEBYTECODE: '1'
@@ -61,25 +72,14 @@ jobs:
           for d in ('2', '3'):
               if ('"%s", "%s"' % (enc[d]['lower'], enc[d]['upper'])) not in src:
                   raise ValueError('SIDE24 d = ' + d + ' enclosure quoted in the script differs from ENCLOSURE.json')
-          script = str(root / 'cone_moment_d6.py')
           expected = (root / 'RESULTS.json').read_bytes()
-          mutants = ['shift-variance', 'vandermonde', 'erfc-branch', 'jacobian', 'mehta', 'recurrence', 'layer-init', 'two-erf', 'cumulative-half', 'remainder-dropped', 'erf-inside', 'tail-dropped', 'image-shells', 'sandwich-exponent']
-          def mutant_rc(args):
-              flags, m = args
-              return m, flags, subprocess.run([sys.executable, *flags, script, '--mutant', m],
-                                              capture_output=True, timeout=3000).returncode
-          for flags in (['-B', '-S'], ['-B', '-O', '-S']):
-              out = subprocess.run([sys.executable, *flags, script],
-                                   check=True, capture_output=True, timeout=3000).stdout
-              if out != expected:
-                  raise RuntimeError('output differs from RESULTS.json under ' + ' '.join(flags))
-              doc = json.loads(out)
-              if not (doc['passed'] and len(doc['checks']) == 20 and all(doc['checks'].values())):
-                  raise RuntimeError('certification did not pass under ' + ' '.join(flags))
-              with concurrent.futures.ThreadPoolExecutor(max_workers=4) as pool:
-                  for m, fl, rc in pool.map(mutant_rc, [(flags, m) for m in mutants]):
-                      if rc != 1:
-                          raise RuntimeError('mutant ' + m + ' not rejected under ' + ' '.join(fl))
+          doc = json.loads(expected)
+          if not (doc['passed'] and len(doc['checks']) == 20 and all(doc['checks'].values())):
+              raise RuntimeError('RESULTS.json is not a passing certification')
+          # Every child: exact exit status, one timing line on stderr, baseline stdout equal
+          # to RESULTS.json, each mutant's report equal to the measured one (both modes).
+          from tools.side24_mutant_replay import replay
+          replay('side24-d6-coefficient', pathlib.Path.cwd(), timeout=3000)
           print(json.dumps({'scientific_effect': 'NONE', 'mathematical_acceptance': False, 'certified_enclosures': True, 'side24_v1': 'unchanged'}))
           PY
           git diff --exit-code
diff --git a/reviews/side24_mutant_wrappers_20261026/SCOPE.md b/reviews/side24_mutant_wrappers_20261026/SCOPE.md
new file mode 100644
index 000000000..df951a1a9
--- /dev/null
+++ b/reviews/side24_mutant_wrappers_20261026/SCOPE.md
@@ -0,0 +1,179 @@
+# SIDE24 d = 5, 6 mutant wrappers: authenticate the intended rejections
+
+Dylan Roy — delegated AI engineering. Actual author: Anthropic Claude, session
+`session_017Mi3hxjaxV45x6zo6o1ee3` (Claude017), stage
+`SIDE24-WRAPPERS-REPAIR-017-20261026`, pickup Math-#322/6427944196. Dylan approved
+this stage in the author's session at about 21:30Z on 26 October 2026. This session
+wrote the three packets. Scientific effect NONE; shared-account organizational
+credit 0. Author-side implementation only: a separate nonauthor engineering review
+and a separate integrator are required. No mathematical acceptance is asserted.
+
+The method is the one C8 used for its own wrappers in #518/#519
+(`reviews/c8_check_mode_wrappers_20261026/SCOPE.md`). The helper, references and
+test are new, separate files, and C8's `tools/check_mode_*` and
+`tests/test_check_mode_workflows.py` are untouched.
+
+## Finding and exact source
+
+01Cz's ledger (Math-#322/6388271177) classifies all three wrappers as
+`status_only`. Each mutant was accepted on exit status 1 alone, with its stdout and
+stderr discarded. So an unrelated crash, a silent exit 1, or a report that fails
+for another reason counted as an intended rejection. The baseline was already
+compared byte for byte with `RESULTS.json`, but its stderr was discarded.
+
+| Object | Identity |
+|---|---|
+| Math- `main` at characterization and branch base | `7d63d08f6cfadfd9c2ef28039a370d741126f9a0` (the merge of #519) |
+| `side24-d5-closed-form.yml` before | blob `7c39892950068e1e7f9c9e0f4c1897cfbdfb3a30` |
+| `side24-d5-coefficient.yml` before | blob `9f46fffa48b4162356bbe269e54f68602deef1c6` |
+| `side24-d6-coefficient.yml` before | blob `6059c2ecde9e7edf4fda1efeece3ce964483e322` |
+| `closed_form_d4.py` (unchanged) | blob `e087f4c0007ae2473d28230771588da57f794aab`, 45,019 B |
+| `cone_moment_d5.py` (unchanged) | blob `9d918805865031c8ef9ae5d4d0c3266e1a396a98`, 46,082 B |
+| `cone_moment_d6.py` (unchanged) | blob `c5e56694d96c885373a7b70868259eeb9fdfa746`, 68,731 B |
+| d=5 closed-form `RESULTS.json` (unchanged) | 16,765 B, SHA-256 `b878c1d3534a1ce72f01bb461a06b8777888bc0a5fdd9413fc1a4f8a565a2bb3` |
+| d=5 coefficient `RESULTS.json` (unchanged) | 5,435 B, SHA-256 `81c4bc072b7eab4d2f1725b3d873076dac6bc31ab1a06779d3299d9e81b3cbaf` |
+| d=6 coefficient `RESULTS.json` (unchanged) | 8,319 B, SHA-256 `82d1c19f9f503fd9211c6dcf356ec62aee9ec9eda44792bc38be4f3690637832` |
+
+The three packet trees are `d2e9f088`, `dc7dfac0` and `dcbeffef`. The ledger read
+the workflows at `d1f6ad28` as blobs `40fe9334`, `93ac2e5e` and `c45d9a5a`. The
+only change since is 2c041de (24 October), which moved each file's
+`actions/setup-python` pin from `5fda3b95` to `6d4fd22b`. The mutant logic the
+ledger classified is unchanged. The imported `tools/legacy_json_replay.py` (blob
+`854db466`) and `tests/test_legacy_workflow_triggers.py` (blob `c2e37a3a`) are
+the blobs every local run below used.
+
+## Repair and preservation
+
+Nine files: `tools/side24_mutant_replay.py`, the three files in
+`tools/side24_mutant_contracts/` and `tests/test_side24_mutant_workflows.py` (all
+new), the three workflows, and this note.
+
+`tools/side24_mutant_replay.py` follows `tools/legacy_json_replay.py` and reuses
+that module's strict JSON reader and structural comparison instead of copying
+them. For each family it binds the checker's git blob and the `RESULTS.json` size
+and digest. It then runs the fixed inventory in both modes, exactly as the old
+wrappers did: from the repository root, with the absolute script path, and with
+`<script>` and `<script> --mutant M`. For every child it requires the following:
+
+- **Baseline:** exit 0, stdout byte-equal to `RESULTS.json`, and stderr exactly
+  one timing line `seconds N.N`.
+- **Report mutant:** exit 1, stderr exactly one timing line, and one strict JSON
+  report on stdout that is structurally equal to the report measured from the
+  unchanged checker.
+- **Guard mutant:** exit 1, empty stdout, and a stderr that starts with
+  `Traceback (most recent call last):`, ends with the measured final exception
+  line, and carries no timing line.
+
+The references are in one file per family, because the d=5 closed-form reports
+alone take 139 KB compactly and the strict reader caps a document at 256 KB.
+
+Each workflow now calls `replay()` in place of its own exit-only loop. It gains
+one earlier step that runs `tests/test_side24_mutant_workflows.py` for its own
+family in both modes. It also gains the trigger paths for the helper and its
+inputs (`tools/**`, `tools.py`), for that test, and for
+`tests/test_legacy_workflow_triggers.py`, whose strict path reader the test
+reuses.
+
+Unchanged:
+- the source block before any child: the packet manifest, the main-resident pins
+  and, for d=5 and d=6, the quoted d=2, 3 enclosures;
+- the `RESULTS.json` certification checks (16, 13 and 20 checks, and the d=5
+  closed forms), which now run once on the committed bytes. Each mode's baseline
+  stdout must still equal those bytes;
+- the action pins, read-only permissions, `persist-credentials: false`, Python
+  3.11.16 and the job budgets (15, 60 and 120 minutes);
+- the per-child timeouts (600, 1500 and 3000 s), the four workers, the mutant
+  order within each mode and the terminal record;
+- the final `git diff --exit-code`;
+- every packet file.
+
+One ordering detail changes. The old wrappers ran each mode's baseline and then
+that mode's mutants. The helper runs each mode's baseline together with its
+mutants in one pool, and it checks every child of that mode before starting the
+next mode.
+
+## Disclosed deviations from #519
+
+1. **Timing line on stderr.** These checkers print `seconds N.N` on stderr in
+   every run that reaches a report, so the contract requires exactly that one
+   line. It does not require empty stderr. The pickup said "empty stderr", and
+   this is the measured behaviour instead. No other byte is allowed.
+2. **Guard references.** In the d=5 closed-form family and the d=6 family, the
+   `vandermonde` mutant never reaches a report. The checker's own structural
+   guard raises
+   `ValueError: an even p3-power occurred; J_k would carry an error function`
+   (d=5 closed form) and the same exception text in the d=6 checker. The reference for these is that guard
+   only: exit 1, empty stdout, a traceback ending in that exact line. The d=5
+   coefficient `vandermonde` reaches a report and is a report reference.
+   This means the old wrappers' `vandermonde` rejection in those two families was
+   never a failing check. It was an exception, and the repaired wrappers pin
+   which one.
+
+## The contract
+
+| Family | Report references | Guard references |
+|---|---|---|
+| `side24-d5-closed-form` | 13 | 1 (`vandermonde`) |
+| `side24-d5-coefficient` | 11 | 0 |
+| `side24-d6-coefficient` | 13 | 1 (`vandermonde`) |
+
+Every report reference has `passed: false`, its own label, and at least one false
+check. Most mutants fail a distinct set of checks, but four pairs share one:
+`by-parts`/`gauss-half` and `cumulative-init`/`cumulative-recurrence` (d=5 closed
+form), and `jacobian`/`mehta` (d=5 and d=6 coefficient). The references pin the
+whole report, including its `mutant` label, so the reports in each pair are still
+distinct. A failure for another reason, a crash, a silent exit, extra stderr, or a
+report for another label is rejected.
+
+## Execution and non-vacuous regressions
+
+All of the following ran locally on CPython 3.11.15, with `main` `7d63d08f`
+characterized and the branch cut from it.
+
+- **Characterization:** the unchanged checkers, in both modes, exactly as the
+  wrappers ran them. Every normal and optimized stdout is byte-identical. Every
+  baseline stdout equals `RESULTS.json`. The exit statuses are as above. Locally
+  a d=5 closed-form child takes about 1–3 s, a d=5 coefficient child about 80 s
+  and a d=6 child about 175 s.
+- **Real checkers through the helper:** `replay()` on the three actual packets
+  gave `protocol_ok` for every child: 30 for d=5 closed form in 41 s,
+  24 for d=5 coefficient in 16.9 minutes with two workers, and 30 for d=6 in
+  25.3 minutes with the workflow's four workers. All under the per-child timeouts.
+- **This test file:** 14 tests OK in both modes for each family as the workflow
+  step runs it (about 56 s, 40 s and 42 s per mode), and 14 tests OK in both modes for all three
+  families together under the full suite's discovery (about 110 s per mode).
+- **RED control:** the same tests, with `SIDE24_MUTANT_SOURCE_ROOT` set to
+  unchanged `main`, run each old wrapper shell against the same fixture. They
+  give these false acceptances:
+
+  | Test | d=5 closed form | d=5 coefficient | d=6 |
+  |---|---|---|---|
+  | crashes (every mutant, both modes) | 28 | 22 | 28 |
+  | other guard faults (another exception, stdout, timing line, silent) | 8 | 0 | 8 |
+  | silent exits, wrong values, wrong label | 10 | 10 | 10 |
+  | report JSON and stderr faults | 16 | 16 | 16 |
+  | baseline stderr | 2 | 2 | 2 |
+  | **total** | **64** | **50** | **64** |
+
+  The wrong exit statuses, baseline byte drift and baseline crashes were already
+  rejected there and still are. The trigger, step and import tests fail there
+  too, as expected.
+- **Real-checker RED control:** a copy of `closed_form_d4.py` that raises an
+  unrelated `RuntimeError` on `--mutant jacobian` gives exit 1 in both modes. The
+  old acceptance rule accepts it. The helper's `_check` with the real reference
+  rejects it.
+- **Repository suites:** the 44 test modules that read workflow files pass on the
+  branch except `test_proof_slice_check`, which fails with the same 20 failures
+  and errors on unchanged `main` in this container. They differ only in synthetic
+  commit IDs, so the failure is local and independent of this change.
+
+## Limits
+
+- The local interpreter is 3.11.15, not the hosted 3.11.16. The hosted runs are
+  the qualifying execution.
+- The synthetic fixture uses a one-entry pin list and a minimal manifest. The real
+  pins run only in the real hosted job.
+- The other `status_only` SIDE24 wrappers in the ledger
+  (`side24-periodization-firstvar`, `side24-periodization-remainder`,
+  `side24-v1-coefficient-review`) are not this session's packets and are not in
+  this stage.
diff --git a/tests/test_side24_mutant_workflows.py b/tests/test_side24_mutant_workflows.py
new file mode 100644
index 000000000..cb2d77e21
--- /dev/null
+++ b/tests/test_side24_mutant_workflows.py
@@ -0,0 +1,361 @@
+"""Synthetic whole-shell regressions for the SIDE24 d = 5, 6 mutant wrappers.
+
+The fixture commits a synthetic checker, the real RESULTS.json, a manifest and an
+upstream pin, then runs the actual verification step of each wrapper with real
+subprocesses. The synthetic checker replays the reports measured from the
+unchanged checkers (tools/side24_mutant_contracts/). Each named fault must be
+rejected and harmless reformatting accepted. Each wrapper's trigger paths, step
+order and import line are also checked. This is protocol coverage, not a field
+calculation.
+"""
+from __future__ import annotations
+
+import hashlib
+import json
+import os
+from pathlib import Path
+import shutil
+import subprocess
+import sys
+import tempfile
+import textwrap
+import unittest
+
+from test_legacy_workflow_triggers import covered, trigger_paths
+
+ROOT = Path(__file__).resolve().parents[1]
+sys.path.insert(0, str(ROOT))
+from tools.legacy_json_replay import ProtocolError  # noqa: E402
+from tools import side24_mutant_replay as helper  # noqa: E402
+
+SELF = 'tests/test_side24_mutant_workflows.py'
+SOURCE = Path(os.environ.get('SIDE24_MUTANT_SOURCE_ROOT', str(ROOT)))
+CONTRACTS = {f: json.loads((ROOT / 'tools/side24_mutant_contracts' / (f + '.json')).read_text())
+             for f in helper.INVENTORIES}
+ACTIVE = [f for f in helper.INVENTORIES
+          if not os.environ.get('SIDE24_MUTANT_TEST_FAMILY') or f == os.environ['SIDE24_MUTANT_TEST_FAMILY']]
+if not ACTIVE:
+    raise SystemExit('unknown test family')
+MODES = ('normal', 'optimized')
+VERIFY = '      - name: Verify sources and pins, replay both modes, reject mutants'
+STEP = (
+    '      - name: Check trigger coverage and the SIDE24 replay contract in both modes\n'
+    '        env:\n'
+    "          SIDE24_MUTANT_TEST_FAMILY: '%s'\n"
+    "          PYTHONDONTWRITEBYTECODE: '1'\n"
+    '        run: |\n'
+    '          python -B -S ' + SELF + '\n'
+    '          python -B -O -S ' + SELF + '\n'
+)
+ENCLOSURE = 'coefficients/side24_v1/ENCLOSURE.json'
+
+CHILD = r'''
+import builtins, json, os, sys
+# QUOTES
+here = os.path.dirname(os.path.abspath(__file__))
+refs = json.loads(open(os.path.join(here, 'REFERENCES.txt')).read())
+args = sys.argv[1:]
+if args and not (len(args) == 2 and args[0] == '--mutant'):
+    sys.stderr.write('unexpected arguments\n')
+    sys.exit(3)
+label = args[1] if args else 'baseline'
+mode = 'optimized' if sys.flags.optimize else 'normal'
+with open(os.environ['SIDE24_TRACE'], 'a') as f:
+    f.write(json.dumps([label, mode, os.getcwd()]) + '\n')
+kind = os.environ.get('SIDE24_FAULT', '') if (
+    os.environ.get('SIDE24_TARGET') == label and os.environ.get('SIDE24_MODE') == mode) else ''
+timing = 'seconds 0.1\n'
+if kind == 'crash':
+    raise RuntimeError('unrelated fixture crash')
+if kind == 'silent':
+    sys.exit(0 if label == 'baseline' else 1)
+if label == 'baseline':
+    out = open(os.path.join(here, 'RESULTS.json'), 'rb').read()
+    if kind == 'baseline-byte':
+        out = out[:-2] + b' \n'
+    sys.stdout.buffer.write(out)
+    sys.stderr.write(timing + ('unexpected diagnostic\n' if kind == 'stderr' else ''))
+    sys.exit(1 if kind == 'wrong-exit' else 0)
+ref = refs[label]
+if ref['kind'] == 'guard':
+    if kind == 'guard-stdout':
+        print(json.dumps({'mutant': label, 'passed': False}))
+    if kind == 'guard-other':
+        raise ValueError('an unrelated failure')
+    if kind == 'timing':
+        sys.stderr.write(timing)
+    name, _, text = ref['final_line'].partition(': ')
+    raise getattr(builtins, name)(text)
+report = ref['report']
+text = json.dumps(report) + '\n'
+if kind == 'wrong-value':
+    checks = dict(report['checks'])
+    first = next(k for k, v in checks.items() if v is False)
+    checks[first] = True
+    text = json.dumps(dict(report, checks=checks)) + '\n'
+if kind == 'wrong-label':
+    text = json.dumps(dict(report, mutant='another-mutant')) + '\n'
+if kind == 'format':
+    text = ' \n' + json.dumps(report, indent=3, sort_keys=True) + '\n\t'
+if kind == 'duplicate':
+    text = text.rstrip()[:-1] + ', "passed": false}\n'
+if kind == 'nonfinite':
+    text = text.rstrip()[:-1] + ', "unexpected": NaN}\n'
+if kind == 'trailing':
+    text += '{}\n'
+if kind == 'extra':
+    text = json.dumps(dict(report, unexpected='x')) + '\n'
+if kind == 'missing':
+    text = json.dumps({k: v for k, v in report.items() if k != 'checks'}) + '\n'
+if kind == 'malformed':
+    text = 'not JSON\n'
+sys.stdout.write(text)
+sys.stderr.write('' if kind == 'no-timing' else timing + ('unexpected diagnostic\n' if kind == 'stderr' else ''))
+sys.exit(2 if kind == 'wrong-exit' else 1)
+'''
+
+
+def digest(data):
+    return hashlib.sha256(data).hexdigest()
+
+
+def blob(data):
+    return hashlib.sha1(b'blob %d\0' % len(data) + data).hexdigest()
+
+
+def row(path, data):
+    return {'path': path, 'bytes': len(data), 'sha256': digest(data), 'git_blob': blob(data)}
+
+
+def workflow_text(family, source=SOURCE):
+    return (source / '.github/workflows' / (family + '.yml')).read_text(encoding='utf-8')
+
+
+def verification_shell(text):
+    lines = text.splitlines()
+    start = next(i for i, line in enumerate(lines) if line.startswith(VERIFY))
+    start = next(i for i in range(start, len(lines)) if lines[i].strip() == 'run: |') + 1
+    end = start
+    while end < len(lines) and (not lines[end].strip() or lines[end].startswith('          ')):
+        end += 1
+    return textwrap.dedent('\n'.join(lines[start:end])) + '\n'
+
+
+def labels(family, kind):
+    return [m for m in helper.INVENTORIES[family] if CONTRACTS[family]['mutants'][m]['kind'] == kind]
+
+
+class SIDE24WorkflowTests(unittest.TestCase):
+    maxDiff = None
+
+    def case(self, family, label='baseline', mode='normal', fault='', tamper=None):
+        with tempfile.TemporaryDirectory(prefix='side24-mutant-test-') as td:
+            outside = Path(td)
+            repo = outside / 'repo'
+            trace = outside / 'trace'
+            entry = json.loads(json.dumps(CONTRACTS[family]))
+            directory = repo / entry['directory']
+            directory.mkdir(parents=True)
+            enclosure = (ROOT / ENCLOSURE).read_bytes()
+            (repo / ENCLOSURE).parent.mkdir(parents=True, exist_ok=True)
+            (repo / ENCLOSURE).write_bytes(enclosure)
+            dims = json.loads(enclosure)['dimensions']
+            quotes = ' '.join('"%s", "%s"' % (dims[d]['lower'], dims[d]['upper']) for d in ('2', '3'))
+            data = CHILD.replace('# QUOTES', '# ' + quotes).encode()
+            (directory / entry['script']).write_bytes(data)
+            results = (ROOT / entry['directory'] / 'RESULTS.json').read_bytes()
+            (directory / 'RESULTS.json').write_bytes(results)
+            references = json.dumps(entry['mutants']).encode()
+            (directory / 'REFERENCES.txt').write_bytes(references)
+            upstream = b'synthetic upstream; no scientific authority\n'
+            (repo / 'upstream.txt').write_bytes(upstream)
+            manifest = {'files': [row(entry['script'], data), row('RESULTS.json', results),
+                                  row('REFERENCES.txt', references)],
+                        'pins_on_main': [dict(row('upstream.txt', upstream), label='U',
+                                              workflow_checked=True)]}
+            (directory / 'SOURCE_FILES.json').write_text(json.dumps(manifest))
+            (repo / 'tools/side24_mutant_contracts').mkdir(parents=True)
+            for name in ('side24_mutant_replay.py', 'legacy_json_replay.py'):
+                if (SOURCE / 'tools' / name).exists():
+                    shutil.copyfile(SOURCE / 'tools' / name, repo / 'tools' / name)
+            # Only the checker identity is re-bound for this synthetic fixture.
+            entry['script_git_blob'] = blob(data)
+            (repo / 'tools/side24_mutant_contracts' / (family + '.json')).write_text(json.dumps(entry))
+            env = os.environ.copy()
+            env.update({'SIDE24_TRACE': str(trace), 'SIDE24_TARGET': label, 'SIDE24_MODE': mode,
+                        'SIDE24_FAULT': fault, 'PYTHONDONTWRITEBYTECODE': '1'})
+            for cmd in (['git', 'init', '-q'], ['git', 'add', '.'],
+                        ['git', '-c', 'user.name=Fixture', '-c', 'user.email=fixture@example.invalid',
+                         '-c', 'commit.gpgsign=false', 'commit', '-qm', 'synthetic fixture']):
+                subprocess.run(cmd, cwd=repo, check=True, capture_output=True, timeout=15, env=env)
+            if tamper == 'packet':
+                (directory / entry['script']).write_bytes(data + b'\n')
+            if tamper == 'upstream':
+                (repo / 'upstream.txt').write_bytes(upstream + b'x')
+            shell = verification_shell(workflow_text(family))
+            result = subprocess.run(['bash', '-c', shell], cwd=repo, env=env, capture_output=True, timeout=120)
+            events = [json.loads(s) for s in trace.read_text().splitlines()] if trace.exists() else []
+            return result, events, str(repo.resolve())
+
+    def rejected(self, family, label, mode, fault):
+        result, events, _ = self.case(family, label, mode, fault)
+        self.assertNotEqual(result.returncode, 0, (family, label, mode, fault, 'false acceptance'))
+        self.assertIn([label, mode], [e[:2] for e in events], (family, label, mode, fault, events))
+        if b'SIDE24 protocol mismatch' in result.stderr:
+            self.assertIn(('%s/%s/%s' % (family, mode, label)).encode(), result.stderr)
+
+    def test_reference_custody(self):
+        for family in ACTIVE:
+            entry = helper._contract(family)
+            packet = SOURCE / entry['directory']
+            with self.subTest(family=family):
+                self.assertEqual(blob((packet / entry['script']).read_bytes()), entry['script_git_blob'])
+                results = (packet / 'RESULTS.json').read_bytes()
+                self.assertEqual((len(results), digest(results)), (entry['results_bytes'], entry['results_sha256']))
+
+    def test_valid_complete_inventory(self):
+        for family in ACTIVE:
+            with self.subTest(family=family):
+                result, events, repo = self.case(family)
+                self.assertEqual(result.returncode, 0, result.stderr.decode(errors='replace'))
+                expected = [[label, mode, repo] for mode in MODES
+                            for label in ['baseline', *helper.INVENTORIES[family]]]
+                self.assertEqual(sorted(events), sorted(expected))
+                self.assertEqual([e[1] for e in events], [e[1] for e in expected])
+
+    def test_formatting_alone_is_accepted(self):
+        for family in ACTIVE:
+            label = labels(family, 'report')[0]
+            with self.subTest(family=family):
+                result, _, _ = self.case(family, label, 'optimized', 'format')
+                self.assertEqual(result.returncode, 0, result.stderr.decode(errors='replace'))
+
+    def test_crashes_each_mutant_each_mode(self):
+        for family in ACTIVE:
+            for label in helper.INVENTORIES[family]:
+                for mode in MODES:
+                    with self.subTest(family=family, label=label, mode=mode):
+                        self.rejected(family, label, mode, 'crash')
+
+    def test_silent_exits_and_wrong_verdicts(self):
+        for family in ACTIVE:
+            first, last = labels(family, 'report')[0], labels(family, 'report')[-1]
+            for label, fault in ((first, 'silent'), (last, 'silent'), (first, 'wrong-value'),
+                                 (last, 'wrong-value'), (first, 'wrong-label')):
+                for mode in MODES:
+                    with self.subTest(family=family, label=label, fault=fault, mode=mode):
+                        self.rejected(family, label, mode, fault)
+
+    def test_report_contract(self):
+        for family in ACTIVE:
+            label = labels(family, 'report')[0]
+            for fault in ('stderr', 'no-timing', 'wrong-exit', 'duplicate', 'nonfinite', 'trailing', 'extra',
+                          'missing', 'malformed'):
+                for mode in MODES:
+                    with self.subTest(family=family, fault=fault, mode=mode):
+                        self.rejected(family, label, mode, fault)
+
+    def test_guard_contract(self):
+        for family in ACTIVE:
+            for label in labels(family, 'guard'):
+                for fault in ('guard-other', 'guard-stdout', 'timing', 'silent'):
+                    for mode in MODES:
+                        with self.subTest(family=family, label=label, fault=fault, mode=mode):
+                            self.rejected(family, label, mode, fault)
+
+    def test_baseline_contract(self):
+        for family in ACTIVE:
+            for fault in ('baseline-byte', 'stderr', 'wrong-exit', 'crash', 'silent'):
+                for mode in MODES:
+                    with self.subTest(family=family, fault=fault, mode=mode):
+                        self.rejected(family, 'baseline', mode, fault)
+
+    def test_source_guards_run_before_any_child(self):
+        for family in ACTIVE:
+            for tamper in ('packet', 'upstream'):
+                with self.subTest(family=family, tamper=tamper):
+                    result, events, _ = self.case(family, tamper=tamper)
+                    self.assertNotEqual(result.returncode, 0)
+                    self.assertEqual(events, [])
+
+
+class HelperTests(unittest.TestCase):
+    def test_timeout_is_a_failure(self):
+        family = ACTIVE[0]
+        with tempfile.TemporaryDirectory() as td:
+            repo = Path(td)
+            entry = CONTRACTS[family]
+            packet = repo / entry['directory']
+            packet.mkdir(parents=True)
+            data = b'import time\ntime.sleep(5)\n'
+            (packet / entry['script']).write_bytes(data)
+            results = b'{}\n'
+            (packet / 'RESULTS.json').write_bytes(results)
+            refs = repo / 'refs'
+            refs.mkdir()
+            (refs / (family + '.json')).write_text(json.dumps(dict(
+                entry, script_git_blob=blob(data), results_bytes=len(results), results_sha256=digest(results))))
+            with self.assertRaisesRegex(ProtocolError, 'protocol mismatch: %s/normal/' % family):
+                helper.replay(family, repo, timeout=1, contracts=refs)
+
+    def test_reference_schema(self):
+        family = ACTIVE[0]
+        data = CONTRACTS[family]
+        report = labels(family, 'report')[0]
+        with tempfile.TemporaryDirectory() as td:
+            path = Path(td)
+            bad = path / (family + '.json')
+            entry = dict(data, mutants=dict(data['mutants']))
+            entry['mutants'].pop(report)
+            bad.write_text(json.dumps(entry))
+            with self.assertRaisesRegex(ProtocolError, 'missing, extra'):
+                helper._contract(family, path)
+            entry['mutants'] = dict(data['mutants'])
+            entry['mutants'][report] = {'kind': 'report',
+                                        'report': dict(data['mutants'][report]['report'], passed=True)}
+            bad.write_text(json.dumps(entry))
+            with self.assertRaisesRegex(ProtocolError, 'not a rejection report'):
+                helper._contract(family, path)
+            other = next(f for f in helper.INVENTORIES if f != family)
+            bad.write_text(json.dumps(dict(data, family=other)))
+            with self.assertRaisesRegex(ProtocolError, 'family reference schema'):
+                helper._contract(family, path)
+            bad.write_text(json.dumps(data)[:-1] + ', "schema": 1}')
+            with self.assertRaises(ProtocolError):
+                helper._contract(family, path)
+
+
+class SIDE24TriggerTests(unittest.TestCase):
+    def test_paths_cover_the_helper_its_inputs_and_these_tests(self):
+        for family in ACTIVE:
+            paths = trigger_paths(workflow_text(family))
+            entry = CONTRACTS[family]
+            for path in (entry['directory'] + '/' + entry['script'], '.github/workflows/%s.yml' % family,
+                         'tools/side24_mutant_replay.py', 'tools/side24_mutant_contracts/%s.json' % family,
+                         'tools/legacy_json_replay.py', 'tools/__init__.py', 'tools.py', SELF,
+                         'tests/test_legacy_workflow_triggers.py'):
+                with self.subTest(family=family, path=path):
+                    self.assertTrue(covered(paths, path), 'consumed or regression path is uncovered')
+
+    def test_contract_step_runs_in_both_modes_before_the_replay(self):
+        for family in ACTIVE:
+            text = workflow_text(family)
+            with self.subTest(family=family):
+                step = STEP % family
+                self.assertEqual(text.count(step), 1)
+                self.assertLess(text.index(step), text.index(VERIFY))
+
+    def test_the_replay_is_the_only_mutant_path_and_names_its_family(self):
+        for family in ACTIVE:
+            shell = verification_shell(workflow_text(family))
+            with self.subTest(family=family):
+                lines = [line for line in shell.splitlines() if 'side24_mutant_replay' in line]
+                self.assertEqual(lines, ['from tools.side24_mutant_replay import replay'])
+                self.assertEqual(shell.count("replay('%s', pathlib.Path.cwd(), timeout=" % family), 1)
+                self.assertNotIn('.returncode', shell)
+                self.assertNotIn('--mutant', shell)
+                self.assertLess(shell.index('pins_on_main'), shell.index('import replay'))
+
+
+if __name__ == '__main__':
+    unittest.main()
diff --git a/tools/side24_mutant_contracts/side24-d5-closed-form.json b/tools/side24_mutant_contracts/side24-d5-closed-form.json
new file mode 100644
index 000000000..f86c37a69
--- /dev/null
+++ b/tools/side24_mutant_contracts/side24-d5-closed-form.json
@@ -0,0 +1 @@
+{"directory":"coefficients/side24_d5_closed_form_20260930","family":"side24-d5-closed-form","mutants":{"atom-pi":{"kind":"report","report":{"certified":true,"certified_enclosures_math_200":{"D_4":["14.2187634589773576013064656778740783173406003782976930345926","14.2187634589773576013064656778740783173406042538663501338228"],"c_5_ref":["0.0132193193800849680760334875146413965210528883241234179643777","0.0132193193800849680760334875146413965210528919272763929401087"]},"checks":{"BASE_INTEGRALS_FLOAT":true,"C5_CLOSED":true,"CUMULATIVE_EXACT":true,"D1_EXACT":true,"D2_EXACT":true,"D3_EXACT":false,"D4_FORM":true,"D4_INSIDE_CERTIFIED":true,"GAUSS_MOMENTS":true,"JK_EXACT":true,"LIBRARY_EXACT":true,"MEHTA_Z4_EXACT":true,"MOMENTS_FLOAT":false,"PINNED":true,"SIMPLIFIED_FORM":true,"STRUCTURE":true},"closed_forms":{"D_1":"4/3","D_2":"29/6  +  -1 sqrt(6)","D_3":"400/27  +  -76/3 pi^(-1)  +  -200/9 pi^(-1) arctan(1/2)","D_4":"6695/54  +  -405/32 sqrt(6)  +  -1375/72 sqrt(21) pi^(-1)  +  -405/16 sqrt(6) pi^(-1) arctan(1/4 sqrt(14))  +  -6695/27 pi^(-1) arctan(1/7 sqrt(21))  +  405/16 sqrt(6) pi^(-1) arctan(1/7 sqrt(14))","D_4_simplified":"6695/54 - (405/32) sqrt6 - (1/pi) [ (1375/72) sqrt21 + (6695/27) arctan(sqrt21/7) + (405/16) sqrt6 arctan(sqrt14/14) ]   (arctan(sqrt14/4) - arctan(sqrt14/7) = arctan(1/sqrt14))","Z_4":"1536 pi^(1)","c_5_ref":"Gamma(7/6) (3/2)^(1/3) D_4 / (12 sqrt3 pi^(7/2))","erfc_branch_of_D_4":"6695/54  +  -6066622595/603979776 sqrt(6)  +  -46299277168087/1415577600000 sqrt(21) pi^(-1)  +  -6066622595/301989888 sqrt(6) pi^(-1) arctan(1/4 sqrt(14))  +  -6695/27 pi^(-1) arctan(1/7 sqrt(21))  +  6066622595/301989888 sqrt(6) pi^(-1) arctan(1/7 sqrt(14))","polynomial_branch_of_D_4":"-1577496445/603979776 sqrt(6)  +  19265677168087/1415577600000 sqrt(21) pi^(-1)  +  -1577496445/301989888 sqrt(6) pi^(-1) arctan(1/4 sqrt(14))  +  1577496445/301989888 sqrt(6) pi^(-1) arctan(1/7 sqrt(14))"},"exact_relations":{"D_4 integral":"sqrt21/(2688 pi) [ (sqrt21 sqrt(pi)/6) int e^{-T^2/48} erfc(sqrt21 T/28) sum_a T^a L_a^E + int e^{-T^2/21} sum_a T^a L_a^R ]","G_n(mu, T)":"G_0 = (1/2) sqrt(pi/mu) erf(sqrt(mu) T); G_1 = (1 - e^{-mu T^2})/(2 mu); G_n = [(n-1) G_{n-2} - T^{n-1} e^{-mu T^2}]/(2 mu)","L_a":"int_0^T e^{-v^2/24} sum_k e_{a,k}(v) J_k(v) dv, k odd, J_k = B_k + C_k e^{-v^2/8}: incomplete Gaussian moments with mu = 1/24, 1/6","M_0(lam, 1)":"(1/2) sqrt(pi/lam)","M_0(lam, erf(u .) erf(v .))":"arctan(u v/sqrt(lam (lam + u^2 + v^2)))/sqrt(pi lam)","M_0(lam, erf(u .))":"arctan(u/sqrt lam)/sqrt(pi lam)","M_1(lam, 1)":"1/(2 lam)","M_n(lam, f), n >= 1":"[(n-1) M_{n-2}(lam, f) + sum_{u in f} (2u/sqrt pi) M_{n-1}(lam + u^2, f minus u)]/(2 lam)","Z_4":"4! sqrt(pi) (1/6) int e^{-T^2/48} int_0^T e^{-v^2/24} int_0^v V e^{-p3^2/8} = 1536 pi (exact, rule MEHTA_Z4_EXACT)"},"interval_evaluation":{"(3/2)^(1/3)":["1.14471424255333186780804221193967700891590692078793107209905","1.14471424255333186780804221193967700891590692078793107209906"],"D_4":["14.2187634589773576013064656778740783173406023160820215841833","14.2187634589773576013064656778740783173406023160820215842323"],"D_4_simplified":["14.2187634589773576013064656778740783173406023160820215841883","14.2187634589773576013064656778740783173406023160820215842281"],"Gamma(7/6)":["0.927719333630039200708349482534621018566466519145475576700351","0.927719333630039200708349482534621018566466519145475576939062"],"arctan(sqrt14/14)":["0.261157410903024311262049412927851638004440468771565909792296","0.261157410903024311262049412927851638004440468771565909792486"],"arctan(sqrt14/4)":["0.752040089192335679842958294184691084003367627670504438870380","0.752040089192335679842958294184691084003367627670504438870582"],"arctan(sqrt14/7)":["0.490882678289311368580908881256839445998927158898938529077984","0.490882678289311368580908881256839445998927158898938529078176"],"arctan(sqrt21/7)":["0.579639740363704299923291897011207918621441782280263529351672","0.579639740363704299923291897011207918621441782280263529351866"],"c_5_ref":["0.0132193193800849680760334875146413965210528901256999054505140","0.0132193193800849680760334875146413965210528901256999054539727"],"pi":["3.14159265358979323846264338327950288419716939937510582097457","3.14159265358979323846264338327950288419716939937510582097528"]},"mutant":"atom-pi","object":"CL-SIDE24-D4-CLOSED-FORM-20260930-v1","passed":false,"reduction_data":{"atoms_created_by_the_derivations":["1","arctan(1/5 sqrt(15))","arctan(1/2 sqrt(1))","arctan(1/4 sqrt(14))","arctan(1/7 sqrt(21))","arctan(1/7 sqrt(14))","arctan(1/7 sqrt(7))"],"m=2":{"kappa":"1/20 sqrt(30)","lambda_erfc":"1/8","lambda_poly":"1/5"},"m=3":{"kappa":"1/12 sqrt(6)","lambda_erfc":"1/24","lambda_poly":"1/12"},"m=4":{"G_0_constant":"1/6 sqrt(21) pi^(1/2)","J_k = A_k sqrt(2 pi) erf(s/(2 sqrt2)) + B_k + C_k(s) e^{-s^2/8}":{"0":{"A":"1","B":"0","C":[]},"1":{"A":"0","B":"4","C":["-4"]},"2":{"A":"4","B":"0","C":["0","-4"]},"3":{"A":"0","B":"32","C":["-32","0","-4"]},"4":{"A":"48","B":"0","C":["0","-48","0","-4"]},"5":{"A":"0","B":"512","C":["-512","0","-64","0","-4"]},"6":{"A":"960","B":"0","C":["0","-960","0","-80","0","-4"]},"7":{"A":"0","B":"12288","C":["-12288","0","-1536","0","-96","0","-4"]},"8":{"A":"26880","B":"0","C":["0","-26880","0","-2240","0","-112","0","-4"]},"9":{"A":"0","B":"393216","C":["-393216","0","-49152","0","-3072","0","-128","0","-4"]}},"T_powers_erfc":[0,1,2,3,4,5,6,7,8,9,11],"T_powers_poly":[0,1,2,3,4,5,6,7,8,10],"alpha":"3/7","beta":"3/14 T","e_{a,k}(v) by power, erfc branch":{"0":{"1":["0","0","0","0","0","-84035/186624","0","1715/186624","0","-49/746496","0","-7/30233088"],"3":["0","0","0","420175/93312","0","-12005/186624","0","49/186624","0","49/7558272"],"5":["0","-84035/20736","0","-12005/62208","0","343/124416","0","-35/559872"],"7":["0","1715/6912","0","49/20736","0","7/31104"],"9":["0","-49/9216","0","-7/41472"]},"1":{"1":["0","0","0","0","-84035/62208","0","1715/62208","0","-539/2239488","0","7/6718464"],"3":["0","0","-84035/31104","0","8575/62208","0","-833/1119744","0","-7/419904"],"5":["84035/20736","0","1715/20736","0","-245/31104","0","49/1119744"],"7":["-1715/6912","0","49/13824","0","7/31104"],"9":["49/9216","0","-7/27648"]},"11":{"1":["0","0","1/5159780352"],"3":["-1/5159780352"]},"2":{"1":["0","0","0","0","0","1715/373248","0","-343/1492992","0","49/8957952","0","-1/80621568"],"3":["0","0","0","-8575/186624","0","2401/1492992","0","-49/2239488","0","7/20155392"],"5":["0","1715/41472","0","2401/497664","0","-343/1492992","0","-5/1492992"],"7":["0","-343/55296","0","-49/248832","0","1/82944"],"9":["0","49/110592","0","-1/110592"]},"3":{"1":["0","0","84035/46656","0","-8575/373248","0","-49/1492992","0","161/80621568","0","-1/17915904"],"3":["-84035/46656","0","-8575/186624","0","-245/1492992","0","-679/40310784","0","1/1119744"],"5":["8575/124416","0","-49/497664","0","301/2239488","0","-7/2985984"],"7":["49/165888","0","-49/497664","0","-1/82944"],"9":["-7/331776","0","1/73728"]},"4":{"1":["0","0","0","0","0","2891/17915904","0","-581/107495424","0","-1/15925248"],"3":["0","0","0","-14455/8957952","0","4067/107495424","0","1/3981312"],"5":["0","2891/1990656","0","4067/35831808","0","7/2654208"],"7":["0","-581/3981312","0","1/442368"],"9":["0","-1/196608"]},"5":{"1":["0","0","-1715/93312","0","2107/5971968","0","35/35831808","0","13/429981696"],"3":["1715/93312","0","2107/2985984","0","175/35831808","0","-245/214990848"],"5":["-2107/1990656","0","35/11943936","0","7/1492992"],"7":["-35/3981312","0","-11/2654208"],"9":["1/1769472"]},"6":{"1":["0","0","0","0","0","-805/644972544","0","23/286654464"],"3":["0","0","0","4025/322486272","0","-161/286654464"],"5":["0","-805/71663616","0","-161/95551488"],"7":["0","23/10616832"]},"7":{"1":["0","0","245/4478976","0","-637/214990848","0","1/31850496"],"3":["-245/4478976","0","-637/107495424","0","5/31850496"],"5":["637/71663616","0","1/10616832"],"7":["-1/3538944"]},"8":{"1":["0","0","0","0","0","-43/6879707136"],"3":["0","0","0","215/3439853568"],"5":["0","-43/764411904"]},"9":{"1":["0","0","175/483729408","0","-11/2293235712"],"3":["-175/483729408","0","-11/1146617856"],"5":["11/764411904"]}},"e_{a,k}(v) by power, polynomial branch":{"0":{"1":["0","0","0","0","0","0","0","0","343/629856","0","-49/5038848"],"3":["0","0","0","0","0","0","-6517/629856","0","49/314928"],"5":["0","0","0","0","3773/69984","0","-343/839808"],"7":["0","0","-343/7776","0","-49/23328"],"9":["0","0","49/20736"]},"1":{"1":["0","0","0","0","0","-84035/746496","0","20923/6718464","0","-833/26873856","0","7/120932352"],"3":["0","0","0","420175/373248","0","-146461/6718464","0","833/6718464","0","-49/30233088"],"5":["0","-84035/82944","0","-146461/2239488","0","5831/4478976","0","35/2239488"],"7":["0","20923/248832","0","833/746496","0","-7/124416"],"9":["0","-833/331776","0","7/165888"]},"10":{"1":["0","0","-7/7739670528"],"3":["7/7739670528"]},"2":{"1":["0","0","0","0","-84035/248832","0","9947/2239488","0","-539/80621568","0","7/26873856"],"3":["0","0","-84035/124416","0","49735/2239488","0","-833/40310784","0","-7/1679616"],"5":["84035/82944","0","9947/746496","0","-245/1119744","0","49/4478976"],"7":["-9947/248832","0","49/497664","0","7/124416"],"9":["49/331776","0","-7/110592"]},"3":{"1":["0","0","0","0","0","-446243/161243136","0","2597/80621568","0","7/23887872"],"3":["0","0","0","2231215/80621568","0","-18179/80621568","0","-7/5971968"],"5":["0","-446243/17915904","0","-18179/26873856","0","-49/3981312"],"7":["0","2597/2985984","0","-7/663552"],"9":["0","7/294912"]},"4":{"1":["0","0","84035/186624","0","-413315/53747712","0","-49/26873856","0","-91/644972544"],"3":["-84035/186624","0","-413315/26873856","0","-245/26873856","0","1715/322486272"],"5":["413315/17915904","0","-49/8957952","0","-49/2239488"],"7":["49/2985984","0","77/3981312"],"9":["-7/2654208"]},"5":{"1":["0","0","0","0","0","6811/1289945088","0","-161/429981696"],"3":["0","0","0","-34055/644972544","0","1127/429981696"],"5":["0","6811/143327232","0","1127/143327232"],"7":["0","-161/15925248"]},"6":{"1":["0","0","95011/40310784","0","17297/1289945088","0","-7/47775744"],"3":["-95011/40310784","0","17297/644972544","0","-35/47775744"],"5":["-17297/429981696","0","-7/15925248"],"7":["7/5308416"]},"7":{"1":["0","0","0","0","0","301/10319560704"],"3":["0","0","0","-1505/5159780352"],"5":["0","301/1146617856"]},"8":{"1":["0","0","-539/322486272","0","77/3439853568"],"3":["539/322486272","0","77/1719926784"],"5":["-77/1146617856"]}},"kappa":"1/28 sqrt(21)","kappa^2":"3/112","lambda_erfc":"1/48","lambda_poly":"1/21","p3_powers":[1,3,5,7,9],"prefactor":"1/2688 sqrt(21) pi^(-1)"},"note":"arctan(sqrt15/5) (m = 2) and arctan(1/sqrt7) (m = 4, from lam = 3/16) arise in intermediate terms and cancel"},"scientific_effect":"NONE"}},"boundary-term":{"kind":"report","report":{"certified":true,"certified_enclosures_math_200":{"D_4":["14.2187634589773576013064656778740783173406003782976930345926","14.2187634589773576013064656778740783173406042538663501338228"],"c_5_ref":["0.0132193193800849680760334875146413965210528883241234179643777","0.0132193193800849680760334875146413965210528919272763929401087"]},"checks":{"BASE_INTEGRALS_FLOAT":true,"C5_CLOSED":false,"CUMULATIVE_EXACT":true,"D1_EXACT":true,"D2_EXACT":false,"D3_EXACT":false,"D4_FORM":false,"D4_INSIDE_CERTIFIED":false,"GAUSS_MOMENTS":false,"JK_EXACT":true,"LIBRARY_EXACT":true,"MEHTA_Z4_EXACT":true,"MOMENTS_FLOAT":false,"PINNED":false,"SIMPLIFIED_FORM":false,"STRUCTURE":true},"closed_forms":{"D_1":"4/3","D_2":"29/3  +  -1 sqrt(6)","D_3":"50/3  +  -152/3 pi^(-1)  +  -200/9 pi^(-1) arctan(1/2)","D_4":"6695/54  +  -405/32 sqrt(6)  +  -1375/36 sqrt(21) pi^(-1)  +  -405/16 sqrt(6) pi^(-1) arctan(1/4 sqrt(14))  +  -6695/27 pi^(-1) arctan(1/7 sqrt(21))  +  405/16 sqrt(6) pi^(-1) arctan(1/7 sqrt(14))","D_4_simplified":"6695/54 - (405/32) sqrt6 - (1/pi) [ (1375/72) sqrt21 + (6695/27) arctan(sqrt21/7) + (405/16) sqrt6 arctan(sqrt14/14) ]   (arctan(sqrt14/4) - arctan(sqrt14/7) = arctan(1/sqrt14))","Z_4":"1536 pi^(1)","c_5_ref":"Gamma(7/6) (3/2)^(1/3) D_4 / (12 sqrt3 pi^(7/2))","erfc_branch_of_D_4":"6695/54  +  -6066622595/603979776 sqrt(6)  +  -46299277168087/707788800000 sqrt(21) pi^(-1)  +  -6066622595/301989888 sqrt(6) pi^(-1) arctan(1/4 sqrt(14))  +  -6695/27 pi^(-1) arctan(1/7 sqrt(21))  +  6066622595/301989888 sqrt(6) pi^(-1) arctan(1/7 sqrt(14))","polynomial_branch_of_D_4":"-1577496445/603979776 sqrt(6)  +  19265677168087/707788800000 sqrt(21) pi^(-1)  +  -1577496445/301989888 sqrt(6) pi^(-1) arctan(1/4 sqrt(14))  +  1577496445/301989888 sqrt(6) pi^(-1) arctan(1/7 sqrt(14))"},"exact_relations":{"D_4 integral":"sqrt21/(2688 pi) [ (sqrt21 sqrt(pi)/6) int e^{-T^2/48} erfc(sqrt21 T/28) sum_a T^a L_a^E + int e^{-T^2/21} sum_a T^a L_a^R ]","G_n(mu, T)":"G_0 = (1/2) sqrt(pi/mu) erf(sqrt(mu) T); G_1 = (1 - e^{-mu T^2})/(2 mu); G_n = [(n-1) G_{n-2} - T^{n-1} e^{-mu T^2}]/(2 mu)","L_a":"int_0^T e^{-v^2/24} sum_k e_{a,k}(v) J_k(v) dv, k odd, J_k = B_k + C_k e^{-v^2/8}: incomplete Gaussian moments with mu = 1/24, 1/6","M_0(lam, 1)":"(1/2) sqrt(pi/lam)","M_0(lam, erf(u .) erf(v .))":"arctan(u v/sqrt(lam (lam + u^2 + v^2)))/sqrt(pi lam)","M_0(lam, erf(u .))":"arctan(u/sqrt lam)/sqrt(pi lam)","M_1(lam, 1)":"1/(2 lam)","M_n(lam, f), n >= 1":"[(n-1) M_{n-2}(lam, f) + sum_{u in f} (2u/sqrt pi) M_{n-1}(lam + u^2, f minus u)]/(2 lam)","Z_4":"4! sqrt(pi) (1/6) int e^{-T^2/48} int_0^T e^{-v^2/24} int_0^v V e^{-p3^2/8} = 1536 pi (exact, rule MEHTA_Z4_EXACT)"},"interval_evaluation":{"(3/2)^(1/3)":["1.14471424255333186780804221193967700891590692078793107209905","1.14471424255333186780804221193967700891590692078793107209906"],"D_4":["-13.6379563791997041880659771845134201793300365763817194214906","-13.6379563791997041880659771845134201793300365763817194214348"],"D_4_simplified":["14.2187634589773576013064656778740783173406023160820215841883","14.2187634589773576013064656778740783173406023160820215842281"],"Gamma(7/6)":["0.927719333630039200708349482534621018566466519145475576700351","0.927719333630039200708349482534621018566466519145475576939062"],"arctan(sqrt14/14)":["0.261157410903024311262049412927851638004440468771565909792296","0.261157410903024311262049412927851638004440468771565909792486"],"arctan(sqrt14/4)":["0.752040089192335679842958294184691084003367627670504438870380","0.752040089192335679842958294184691084003367627670504438870582"],"arctan(sqrt14/7)":["0.490882678289311368580908881256839445998927158898938529077984","0.490882678289311368580908881256839445998927158898938529078176"],"arctan(sqrt21/7)":["0.579639740363704299923291897011207918621441782280263529351672","0.579639740363704299923291897011207918621441782280263529351866"],"c_5_ref":["-0.0126793375238604959194800895141703877601427736185389853143149","-0.0126793375238604959194800895141703877601427736185389853109893"],"pi":["3.14159265358979323846264338327950288419716939937510582097457","3.14159265358979323846264338327950288419716939937510582097528"]},"mutant":"boundary-term","object":"CL-SIDE24-D4-CLOSED-FORM-20260930-v1","passed":false,"reduction_data":{"atoms_created_by_the_derivations":["1","arctan(1/5 sqrt(15))","arctan(1/2 sqrt(1))","arctan(1/4 sqrt(14))","arctan(1/7 sqrt(21))","arctan(1/7 sqrt(14))","arctan(1/7 sqrt(7))"],"m=2":{"kappa":"1/20 sqrt(30)","lambda_erfc":"1/8","lambda_poly":"1/5"},"m=3":{"kappa":"1/12 sqrt(6)","lambda_erfc":"1/24","lambda_poly":"1/12"},"m=4":{"G_0_constant":"1/6 sqrt(21) pi^(1/2)","J_k = A_k sqrt(2 pi) erf(s/(2 sqrt2)) + B_k + C_k(s) e^{-s^2/8}":{"0":{"A":"1","B":"0","C":[]},"1":{"A":"0","B":"4","C":["-4"]},"2":{"A":"4","B":"0","C":["0","-4"]},"3":{"A":"0","B":"32","C":["-32","0","-4"]},"4":{"A":"48","B":"0","C":["0","-48","0","-4"]},"5":{"A":"0","B":"512","C":["-512","0","-64","0","-4"]},"6":{"A":"960","B":"0","C":["0","-960","0","-80","0","-4"]},"7":{"A":"0","B":"12288","C":["-12288","0","-1536","0","-96","0","-4"]},"8":{"A":"26880","B":"0","C":["0","-26880","0","-2240","0","-112","0","-4"]},"9":{"A":"0","B":"393216","C":["-393216","0","-49152","0","-3072","0","-128","0","-4"]}},"T_powers_erfc":[0,1,2,3,4,5,6,7,8,9,11],"T_powers_poly":[0,1,2,3,4,5,6,7,8,10],"alpha":"3/7","beta":"3/14 T","e_{a,k}(v) by power, erfc branch":{"0":{"1":["0","0","0","0","0","-84035/186624","0","1715/186624","0","-49/746496","0","-7/30233088"],"3":["0","0","0","420175/93312","0","-12005/186624","0","49/186624","0","49/7558272"],"5":["0","-84035/20736","0","-12005/62208","0","343/124416","0","-35/559872"],"7":["0","1715/6912","0","49/20736","0","7/31104"],"9":["0","-49/9216","0","-7/41472"]},"1":{"1":["0","0","0","0","-84035/62208","0","1715/62208","0","-539/2239488","0","7/6718464"],"3":["0","0","-84035/31104","0","8575/62208","0","-833/1119744","0","-7/419904"],"5":["84035/20736","0","1715/20736","0","-245/31104","0","49/1119744"],"7":["-1715/6912","0","49/13824","0","7/31104"],"9":["49/9216","0","-7/27648"]},"11":{"1":["0","0","1/5159780352"],"3":["-1/5159780352"]},"2":{"1":["0","0","0","0","0","1715/373248","0","-343/1492992","0","49/8957952","0","-1/80621568"],"3":["0","0","0","-8575/186624","0","2401/1492992","0","-49/2239488","0","7/20155392"],"5":["0","1715/41472","0","2401/497664","0","-343/1492992","0","-5/1492992"],"7":["0","-343/55296","0","-49/248832","0","1/82944"],"9":["0","49/110592","0","-1/110592"]},"3":{"1":["0","0","84035/46656","0","-8575/373248","0","-49/1492992","0","161/80621568","0","-1/17915904"],"3":["-84035/46656","0","-8575/186624","0","-245/1492992","0","-679/40310784","0","1/1119744"],"5":["8575/124416","0","-49/497664","0","301/2239488","0","-7/2985984"],"7":["49/165888","0","-49/497664","0","-1/82944"],"9":["-7/331776","0","1/73728"]},"4":{"1":["0","0","0","0","0","2891/17915904","0","-581/107495424","0","-1/15925248"],"3":["0","0","0","-14455/8957952","0","4067/107495424","0","1/3981312"],"5":["0","2891/1990656","0","4067/35831808","0","7/2654208"],"7":["0","-581/3981312","0","1/442368"],"9":["0","-1/196608"]},"5":{"1":["0","0","-1715/93312","0","2107/5971968","0","35/35831808","0","13/429981696"],"3":["1715/93312","0","2107/2985984","0","175/35831808","0","-245/214990848"],"5":["-2107/1990656","0","35/11943936","0","7/1492992"],"7":["-35/3981312","0","-11/2654208"],"9":["1/1769472"]},"6":{"1":["0","0","0","0","0","-805/644972544","0","23/286654464"],"3":["0","0","0","4025/322486272","0","-161/286654464"],"5":["0","-805/71663616","0","-161/95551488"],"7":["0","23/10616832"]},"7":{"1":["0","0","245/4478976","0","-637/214990848","0","1/31850496"],"3":["-245/4478976","0","-637/107495424","0","5/31850496"],"5":["637/71663616","0","1/10616832"],"7":["-1/3538944"]},"8":{"1":["0","0","0","0","0","-43/6879707136"],"3":["0","0","0","215/3439853568"],"5":["0","-43/764411904"]},"9":{"1":["0","0","175/483729408","0","-11/2293235712"],"3":["-175/483729408","0","-11/1146617856"],"5":["11/764411904"]}},"e_{a,k}(v) by power, polynomial branch":{"0":{"1":["0","0","0","0","0","0","0","0","343/629856","0","-49/5038848"],"3":["0","0","0","0","0","0","-6517/629856","0","49/314928"],"5":["0","0","0","0","3773/69984","0","-343/839808"],"7":["0","0","-343/7776","0","-49/23328"],"9":["0","0","49/20736"]},"1":{"1":["0","0","0","0","0","-84035/746496","0","20923/6718464","0","-833/26873856","0","7/120932352"],"3":["0","0","0","420175/373248","0","-146461/6718464","0","833/6718464","0","-49/30233088"],"5":["0","-84035/82944","0","-146461/2239488","0","5831/4478976","0","35/2239488"],"7":["0","20923/248832","0","833/746496","0","-7/124416"],"9":["0","-833/331776","0","7/165888"]},"10":{"1":["0","0","-7/7739670528"],"3":["7/7739670528"]},"2":{"1":["0","0","0","0","-84035/248832","0","9947/2239488","0","-539/80621568","0","7/26873856"],"3":["0","0","-84035/124416","0","49735/2239488","0","-833/40310784","0","-7/1679616"],"5":["84035/82944","0","9947/746496","0","-245/1119744","0","49/4478976"],"7":["-9947/248832","0","49/497664","0","7/124416"],"9":["49/331776","0","-7/110592"]},"3":{"1":["0","0","0","0","0","-446243/161243136","0","2597/80621568","0","7/23887872"],"3":["0","0","0","2231215/80621568","0","-18179/80621568","0","-7/5971968"],"5":["0","-446243/17915904","0","-18179/26873856","0","-49/3981312"],"7":["0","2597/2985984","0","-7/663552"],"9":["0","7/294912"]},"4":{"1":["0","0","84035/186624","0","-413315/53747712","0","-49/26873856","0","-91/644972544"],"3":["-84035/186624","0","-413315/26873856","0","-245/26873856","0","1715/322486272"],"5":["413315/17915904","0","-49/8957952","0","-49/2239488"],"7":["49/2985984","0","77/3981312"],"9":["-7/2654208"]},"5":{"1":["0","0","0","0","0","6811/1289945088","0","-161/429981696"],"3":["0","0","0","-34055/644972544","0","1127/429981696"],"5":["0","6811/143327232","0","1127/143327232"],"7":["0","-161/15925248"]},"6":{"1":["0","0","95011/40310784","0","17297/1289945088","0","-7/47775744"],"3":["-95011/40310784","0","17297/644972544","0","-35/47775744"],"5":["-17297/429981696","0","-7/15925248"],"7":["7/5308416"]},"7":{"1":["0","0","0","0","0","301/10319560704"],"3":["0","0","0","-1505/5159780352"],"5":["0","301/1146617856"]},"8":{"1":["0","0","-539/322486272","0","77/3439853568"],"3":["539/322486272","0","77/1719926784"],"5":["-77/1146617856"]}},"kappa":"1/28 sqrt(21)","kappa^2":"3/112","lambda_erfc":"1/48","lambda_poly":"1/21","p3_powers":[1,3,5,7,9],"prefactor":"1/2688 sqrt(21) pi^(-1)"},"note":"arctan(sqrt15/5) (m = 2) and arctan(1/sqrt7) (m = 4, from lam = 3/16) arise in intermediate terms and cancel"},"scientific_effect":"NONE"}},"by-parts":{"kind":"report","report":{"certified":true,"certified_enclosures_math_200":{"D_4":["14.2187634589773576013064656778740783173406003782976930345926","14.2187634589773576013064656778740783173406042538663501338228"],"c_5_ref":["0.0132193193800849680760334875146413965210528883241234179643777","0.0132193193800849680760334875146413965210528919272763929401087"]},"checks":{"BASE_INTEGRALS_FLOAT":true,"C5_CLOSED":false,"CUMULATIVE_EXACT":true,"D1_EXACT":false,"D2_EXACT":false,"D3_EXACT":false,"D4_FORM":false,"D4_INSIDE_CERTIFIED":false,"GAUSS_MOMENTS":false,"JK_EXACT":true,"LIBRARY_EXACT":true,"MEHTA_Z4_EXACT":false,"MOMENTS_FLOAT":false,"PINNED":false,"SIMPLIFIED_FORM":false,"STRUCTURE":true},"closed_forms":{"D_1":"8/3","D_2":"20/3  +  -5/3 sqrt(6)","D_3":"140/3  +  -41 pi^(-1)  +  -328/9 pi^(-1) arctan(1/2)","D_4":"469309/1296  +  58835/1296 sqrt(3)  +  -39557/972 sqrt(6)  +  -13429/324 sqrt(21) pi^(-1)  +  -39557/486 sqrt(6) pi^(-1) arctan(1/4 sqrt(14))  +  -469309/648 pi^(-1) arctan(1/7 sqrt(21))  +  39557/486 sqrt(6) pi^(-1) arctan(1/7 sqrt(14))  +  -58835/648 sqrt(3) pi^(-1) arctan(1/7 sqrt(7))","D_4_simplified":"6695/54 - (405/32) sqrt6 - (1/pi) [ (1375/72) sqrt21 + (6695/27) arctan(sqrt21/7) + (405/16) sqrt6 arctan(sqrt14/14) ]   (arctan(sqrt14/4) - arctan(sqrt14/7) = arctan(1/sqrt14))","Z_4":"3328 pi^(1)  +  512 sqrt(3) pi^(1)","c_5_ref":"Gamma(7/6) (3/2)^(1/3) D_4 / (12 sqrt3 pi^(7/2))","erfc_branch_of_D_4":"469309/1296  +  58835/1296 sqrt(3)  +  -55216861/1179648 sqrt(6)  +  -316616924879/5898240000 sqrt(21) pi^(-1)  +  -55216861/589824 sqrt(6) pi^(-1) arctan(1/4 sqrt(14))  +  -469309/648 pi^(-1) arctan(1/7 sqrt(21))  +  55216861/589824 sqrt(6) pi^(-1) arctan(1/7 sqrt(14))  +  -58835/648 sqrt(3) pi^(-1) arctan(1/7 sqrt(7))","polynomial_branch_of_D_4":"194651471/31850496 sqrt(6)  +  649344963911/53084160000 sqrt(21) pi^(-1)  +  194651471/15925248 sqrt(6) pi^(-1) arctan(1/4 sqrt(14))  +  -194651471/15925248 sqrt(6) pi^(-1) arctan(1/7 sqrt(14))"},"exact_relations":{"D_4 integral":"sqrt21/(2688 pi) [ (sqrt21 sqrt(pi)/6) int e^{-T^2/48} erfc(sqrt21 T/28) sum_a T^a L_a^E + int e^{-T^2/21} sum_a T^a L_a^R ]","G_n(mu, T)":"G_0 = (1/2) sqrt(pi/mu) erf(sqrt(mu) T); G_1 = (1 - e^{-mu T^2})/(2 mu); G_n = [(n-1) G_{n-2} - T^{n-1} e^{-mu T^2}]/(2 mu)","L_a":"int_0^T e^{-v^2/24} sum_k e_{a,k}(v) J_k(v) dv, k odd, J_k = B_k + C_k e^{-v^2/8}: incomplete Gaussian moments with mu = 1/24, 1/6","M_0(lam, 1)":"(1/2) sqrt(pi/lam)","M_0(lam, erf(u .) erf(v .))":"arctan(u v/sqrt(lam (lam + u^2 + v^2)))/sqrt(pi lam)","M_0(lam, erf(u .))":"arctan(u/sqrt lam)/sqrt(pi lam)","M_1(lam, 1)":"1/(2 lam)","M_n(lam, f), n >= 1":"[(n-1) M_{n-2}(lam, f) + sum_{u in f} (2u/sqrt pi) M_{n-1}(lam + u^2, f minus u)]/(2 lam)","Z_4":"4! sqrt(pi) (1/6) int e^{-T^2/48} int_0^T e^{-v^2/24} int_0^v V e^{-p3^2/8} = 1536 pi (exact, rule MEHTA_Z4_EXACT)"},"interval_evaluation":{"(3/2)^(1/3)":["1.14471424255333186780804221193967700891590692078793107209905","1.14471424255333186780804221193967700891590692078793107209906"],"D_4":["112.318256368880669056064481983649737981917764587798362885464","112.318256368880669056064481983649737981917764587798362885626"],"D_4_simplified":["14.2187634589773576013064656778740783173406023160820215841883","14.2187634589773576013064656778740783173406023160820215842281"],"Gamma(7/6)":["0.927719333630039200708349482534621018566466519145475576700351","0.927719333630039200708349482534621018566466519145475576939062"],"arctan(sqrt14/14)":["0.261157410903024311262049412927851638004440468771565909792296","0.261157410903024311262049412927851638004440468771565909792486"],"arctan(sqrt14/4)":["0.752040089192335679842958294184691084003367627670504438870380","0.752040089192335679842958294184691084003367627670504438870582"],"arctan(sqrt14/7)":["0.490882678289311368580908881256839445998927158898938529077984","0.490882678289311368580908881256839445998927158898938529078176"],"arctan(sqrt21/7)":["0.579639740363704299923291897011207918621441782280263529351672","0.579639740363704299923291897011207918621441782280263529351866"],"c_5_ref":["0.104423349290408961336308488289596407699460597896105786316147","0.104423349290408961336308488289596407699460597896105786343260"],"pi":["3.14159265358979323846264338327950288419716939937510582097457","3.14159265358979323846264338327950288419716939937510582097528"]},"mutant":"by-parts","object":"CL-SIDE24-D4-CLOSED-FORM-20260930-v1","passed":false,"reduction_data":{"atoms_created_by_the_derivations":["1","arctan(1/5 sqrt(15))","arctan(1/2 sqrt(1))","arctan(1/4 sqrt(14))","arctan(1/7 sqrt(21))","arctan(1/7 sqrt(14))","arctan(1/7 sqrt(7))"],"m=2":{"kappa":"1/20 sqrt(30)","lambda_erfc":"1/8","lambda_poly":"1/5"},"m=3":{"kappa":"1/12 sqrt(6)","lambda_erfc":"1/24","lambda_poly":"1/12"},"m=4":{"G_0_constant":"1/6 sqrt(21) pi^(1/2)","J_k = A_k sqrt(2 pi) erf(s/(2 sqrt2)) + B_k + C_k(s) e^{-s^2/8}":{"0":{"A":"1","B":"0","C":[]},"1":{"A":"0","B":"4","C":["-4"]},"2":{"A":"4","B":"0","C":["0","-4"]},"3":{"A":"0","B":"32","C":["-32","0","-4"]},"4":{"A":"48","B":"0","C":["0","-48","0","-4"]},"5":{"A":"0","B":"512","C":["-512","0","-64","0","-4"]},"6":{"A":"960","B":"0","C":["0","-960","0","-80","0","-4"]},"7":{"A":"0","B":"12288","C":["-12288","0","-1536","0","-96","0","-4"]},"8":{"A":"26880","B":"0","C":["0","-26880","0","-2240","0","-112","0","-4"]},"9":{"A":"0","B":"393216","C":["-393216","0","-49152","0","-3072","0","-128","0","-4"]}},"T_powers_erfc":[0,1,2,3,4,5,6,7,8,9,11],"T_powers_poly":[0,1,2,3,4,5,6,7,8,10],"alpha":"3/7","beta":"3/14 T","e_{a,k}(v) by power, erfc branch":{"0":{"1":["0","0","0","0","0","-84035/186624","0","1715/186624","0","-49/746496","0","-7/30233088"],"3":["0","0","0","420175/93312","0","-12005/186624","0","49/186624","0","49/7558272"],"5":["0","-84035/20736","0","-12005/62208","0","343/124416","0","-35/559872"],"7":["0","1715/6912","0","49/20736","0","7/31104"],"9":["0","-49/9216","0","-7/41472"]},"1":{"1":["0","0","0","0","-84035/62208","0","1715/62208","0","-539/2239488","0","7/6718464"],"3":["0","0","-84035/31104","0","8575/62208","0","-833/1119744","0","-7/419904"],"5":["84035/20736","0","1715/20736","0","-245/31104","0","49/1119744"],"7":["-1715/6912","0","49/13824","0","7/31104"],"9":["49/9216","0","-7/27648"]},"11":{"1":["0","0","1/5159780352"],"3":["-1/5159780352"]},"2":{"1":["0","0","0","0","0","1715/373248","0","-343/1492992","0","49/8957952","0","-1/80621568"],"3":["0","0","0","-8575/186624","0","2401/1492992","0","-49/2239488","0","7/20155392"],"5":["0","1715/41472","0","2401/497664","0","-343/1492992","0","-5/1492992"],"7":["0","-343/55296","0","-49/248832","0","1/82944"],"9":["0","49/110592","0","-1/110592"]},"3":{"1":["0","0","84035/46656","0","-8575/373248","0","-49/1492992","0","161/80621568","0","-1/17915904"],"3":["-84035/46656","0","-8575/186624","0","-245/1492992","0","-679/40310784","0","1/1119744"],"5":["8575/124416","0","-49/497664","0","301/2239488","0","-7/2985984"],"7":["49/165888","0","-49/497664","0","-1/82944"],"9":["-7/331776","0","1/73728"]},"4":{"1":["0","0","0","0","0","2891/17915904","0","-581/107495424","0","-1/15925248"],"3":["0","0","0","-14455/8957952","0","4067/107495424","0","1/3981312"],"5":["0","2891/1990656","0","4067/35831808","0","7/2654208"],"7":["0","-581/3981312","0","1/442368"],"9":["0","-1/196608"]},"5":{"1":["0","0","-1715/93312","0","2107/5971968","0","35/35831808","0","13/429981696"],"3":["1715/93312","0","2107/2985984","0","175/35831808","0","-245/214990848"],"5":["-2107/1990656","0","35/11943936","0","7/1492992"],"7":["-35/3981312","0","-11/2654208"],"9":["1/1769472"]},"6":{"1":["0","0","0","0","0","-805/644972544","0","23/286654464"],"3":["0","0","0","4025/322486272","0","-161/286654464"],"5":["0","-805/71663616","0","-161/95551488"],"7":["0","23/10616832"]},"7":{"1":["0","0","245/4478976","0","-637/214990848","0","1/31850496"],"3":["-245/4478976","0","-637/107495424","0","5/31850496"],"5":["637/71663616","0","1/10616832"],"7":["-1/3538944"]},"8":{"1":["0","0","0","0","0","-43/6879707136"],"3":["0","0","0","215/3439853568"],"5":["0","-43/764411904"]},"9":{"1":["0","0","175/483729408","0","-11/2293235712"],"3":["-175/483729408","0","-11/1146617856"],"5":["11/764411904"]}},"e_{a,k}(v) by power, polynomial branch":{"0":{"1":["0","0","0","0","0","0","0","0","343/629856","0","-49/5038848"],"3":["0","0","0","0","0","0","-6517/629856","0","49/314928"],"5":["0","0","0","0","3773/69984","0","-343/839808"],"7":["0","0","-343/7776","0","-49/23328"],"9":["0","0","49/20736"]},"1":{"1":["0","0","0","0","0","-84035/746496","0","20923/6718464","0","-833/26873856","0","7/120932352"],"3":["0","0","0","420175/373248","0","-146461/6718464","0","833/6718464","0","-49/30233088"],"5":["0","-84035/82944","0","-146461/2239488","0","5831/4478976","0","35/2239488"],"7":["0","20923/248832","0","833/746496","0","-7/124416"],"9":["0","-833/331776","0","7/165888"]},"10":{"1":["0","0","-7/7739670528"],"3":["7/7739670528"]},"2":{"1":["0","0","0","0","-84035/248832","0","9947/2239488","0","-539/80621568","0","7/26873856"],"3":["0","0","-84035/124416","0","49735/2239488","0","-833/40310784","0","-7/1679616"],"5":["84035/82944","0","9947/746496","0","-245/1119744","0","49/4478976"],"7":["-9947/248832","0","49/497664","0","7/124416"],"9":["49/331776","0","-7/110592"]},"3":{"1":["0","0","0","0","0","-446243/161243136","0","2597/80621568","0","7/23887872"],"3":["0","0","0","2231215/80621568","0","-18179/80621568","0","-7/5971968"],"5":["0","-446243/17915904","0","-18179/26873856","0","-49/3981312"],"7":["0","2597/2985984","0","-7/663552"],"9":["0","7/294912"]},"4":{"1":["0","0","84035/186624","0","-413315/53747712","0","-49/26873856","0","-91/644972544"],"3":["-84035/186624","0","-413315/26873856","0","-245/26873856","0","1715/322486272"],"5":["413315/17915904","0","-49/8957952","0","-49/2239488"],"7":["49/2985984","0","77/3981312"],"9":["-7/2654208"]},"5":{"1":["0","0","0","0","0","6811/1289945088","0","-161/429981696"],"3":["0","0","0","-34055/644972544","0","1127/429981696"],"5":["0","6811/143327232","0","1127/143327232"],"7":["0","-161/15925248"]},"6":{"1":["0","0","95011/40310784","0","17297/1289945088","0","-7/47775744"],"3":["-95011/40310784","0","17297/644972544","0","-35/47775744"],"5":["-17297/429981696","0","-7/15925248"],"7":["7/5308416"]},"7":{"1":["0","0","0","0","0","301/10319560704"],"3":["0","0","0","-1505/5159780352"],"5":["0","301/1146617856"]},"8":{"1":["0","0","-539/322486272","0","77/3439853568"],"3":["539/322486272","0","77/1719926784"],"5":["-77/1146617856"]}},"kappa":"1/28 sqrt(21)","kappa^2":"3/112","lambda_erfc":"1/48","lambda_poly":"1/21","p3_powers":[1,3,5,7,9],"prefactor":"1/2688 sqrt(21) pi^(-1)"},"note":"arctan(sqrt15/5) (m = 2) and arctan(1/sqrt7) (m = 4, from lam = 3/16) arise in intermediate terms and cancel"},"scientific_effect":"NONE"}},"cumulative-init":{"kind":"report","report":{"certified":true,"certified_enclosures_math_200":{"D_4":["14.2187634589773576013064656778740783173406003782976930345926","14.2187634589773576013064656778740783173406042538663501338228"],"c_5_ref":["0.0132193193800849680760334875146413965210528883241234179643777","0.0132193193800849680760334875146413965210528919272763929401087"]},"checks":{"BASE_INTEGRALS_FLOAT":true,"C5_CLOSED":false,"CUMULATIVE_EXACT":false,"D1_EXACT":true,"D2_EXACT":true,"D3_EXACT":true,"D4_FORM":false,"D4_INSIDE_CERTIFIED":false,"GAUSS_MOMENTS":true,"JK_EXACT":true,"LIBRARY_EXACT":true,"MEHTA_Z4_EXACT":false,"MOMENTS_FLOAT":true,"PINNED":false,"SIMPLIFIED_FORM":false,"STRUCTURE":true},"closed_forms":{"D_1":"4/3","D_2":"29/6  +  -1 sqrt(6)","D_3":"50/3  +  -76/3 pi^(-1)  +  -200/9 pi^(-1) arctan(1/2)","D_4":"376507/4374  +  2514176/177147 sqrt(3)  +  -405/32 sqrt(6)  +  -2457770783/106288200 sqrt(21) pi^(-1)  +  -405/16 sqrt(6) pi^(-1) arctan(1/4 sqrt(14))  +  -376507/2187 pi^(-1) arctan(1/7 sqrt(21))  +  405/16 sqrt(6) pi^(-1) arctan(1/7 sqrt(14))  +  -5028352/177147 sqrt(3) pi^(-1) arctan(1/7 sqrt(7))","D_4_simplified":"6695/54 - (405/32) sqrt6 - (1/pi) [ (1375/72) sqrt21 + (6695/27) arctan(sqrt21/7) + (405/16) sqrt6 arctan(sqrt14/14) ]   (arctan(sqrt14/4) - arctan(sqrt14/7) = arctan(1/sqrt14))","Z_4":"2560/3 pi^(1)  +  2048/9 sqrt(3) pi^(1)","c_5_ref":"Gamma(7/6) (3/2)^(1/3) D_4 / (12 sqrt3 pi^(7/2))","erfc_branch_of_D_4":"376507/4374  +  2514176/177147 sqrt(3)  +  -6066622595/603979776 sqrt(6)  +  -928898355345480101/27862813900800000 sqrt(21) pi^(-1)  +  -6066622595/301989888 sqrt(6) pi^(-1) arctan(1/4 sqrt(14))  +  -376507/2187 pi^(-1) arctan(1/7 sqrt(21))  +  6066622595/301989888 sqrt(6) pi^(-1) arctan(1/7 sqrt(14))  +  -5028352/177147 sqrt(3) pi^(-1) arctan(1/7 sqrt(7))","polynomial_branch_of_D_4":"-1577496445/603979776 sqrt(6)  +  94869497068909367/9287604633600000 sqrt(21) pi^(-1)  +  -1577496445/301989888 sqrt(6) pi^(-1) arctan(1/4 sqrt(14))  +  1577496445/301989888 sqrt(6) pi^(-1) arctan(1/7 sqrt(14))"},"exact_relations":{"D_4 integral":"sqrt21/(2688 pi) [ (sqrt21 sqrt(pi)/6) int e^{-T^2/48} erfc(sqrt21 T/28) sum_a T^a L_a^E + int e^{-T^2/21} sum_a T^a L_a^R ]","G_n(mu, T)":"G_0 = (1/2) sqrt(pi/mu) erf(sqrt(mu) T); G_1 = (1 - e^{-mu T^2})/(2 mu); G_n = [(n-1) G_{n-2} - T^{n-1} e^{-mu T^2}]/(2 mu)","L_a":"int_0^T e^{-v^2/24} sum_k e_{a,k}(v) J_k(v) dv, k odd, J_k = B_k + C_k e^{-v^2/8}: incomplete Gaussian moments with mu = 1/24, 1/6","M_0(lam, 1)":"(1/2) sqrt(pi/lam)","M_0(lam, erf(u .) erf(v .))":"arctan(u v/sqrt(lam (lam + u^2 + v^2)))/sqrt(pi lam)","M_0(lam, erf(u .))":"arctan(u/sqrt lam)/sqrt(pi lam)","M_1(lam, 1)":"1/(2 lam)","M_n(lam, f), n >= 1":"[(n-1) M_{n-2}(lam, f) + sum_{u in f} (2u/sqrt pi) M_{n-1}(lam + u^2, f minus u)]/(2 lam)","Z_4":"4! sqrt(pi) (1/6) int e^{-T^2/48} int_0^T e^{-v^2/24} int_0^v V e^{-p3^2/8} = 1536 pi (exact, rule MEHTA_Z4_EXACT)"},"interval_evaluation":{"(3/2)^(1/3)":["1.14471424255333186780804221193967700891590692078793107209905","1.14471424255333186780804221193967700891590692078793107209906"],"D_4":["3.35610889330893099329120372165535199237327016202855818846111","3.35610889330893099329120372165535199237327016202855818850699"],"D_4_simplified":["14.2187634589773576013064656778740783173406023160820215841883","14.2187634589773576013064656778740783173406023160820215842281"],"Gamma(7/6)":["0.927719333630039200708349482534621018566466519145475576700351","0.927719333630039200708349482534621018566466519145475576939062"],"arctan(sqrt14/14)":["0.261157410903024311262049412927851638004440468771565909792296","0.261157410903024311262049412927851638004440468771565909792486"],"arctan(sqrt14/4)":["0.752040089192335679842958294184691084003367627670504438870380","0.752040089192335679842958294184691084003367627670504438870582"],"arctan(sqrt14/7)":["0.490882678289311368580908881256839445998927158898938529077984","0.490882678289311368580908881256839445998927158898938529078176"],"arctan(sqrt21/7)":["0.579639740363704299923291897011207918621441782280263529351672","0.579639740363704299923291897011207918621441782280263529351866"],"c_5_ref":["0.00312020630084981533114554892362482561674890340507555551653223","0.00312020630084981533114554892362482561674890340507555551738048"],"pi":["3.14159265358979323846264338327950288419716939937510582097457","3.14159265358979323846264338327950288419716939937510582097528"]},"mutant":"cumulative-init","object":"CL-SIDE24-D4-CLOSED-FORM-20260930-v1","passed":false,"reduction_data":{"atoms_created_by_the_derivations":["1","arctan(1/5 sqrt(15))","arctan(1/2 sqrt(1))","arctan(1/4 sqrt(14))","arctan(1/7 sqrt(21))","arctan(1/7 sqrt(14))","arctan(1/7 sqrt(7))"],"m=2":{"kappa":"1/20 sqrt(30)","lambda_erfc":"1/8","lambda_poly":"1/5"},"m=3":{"kappa":"1/12 sqrt(6)","lambda_erfc":"1/24","lambda_poly":"1/12"},"m=4":{"G_0_constant":"1/6 sqrt(21) pi^(1/2)","J_k = A_k sqrt(2 pi) erf(s/(2 sqrt2)) + B_k + C_k(s) e^{-s^2/8}":{"0":{"A":"1","B":"0","C":[]},"1":{"A":"0","B":"4","C":["-4"]},"2":{"A":"4","B":"0","C":["0","-4"]},"3":{"A":"0","B":"32","C":["-32","0","-4"]},"4":{"A":"48","B":"0","C":["0","-48","0","-4"]},"5":{"A":"0","B":"512","C":["-512","0","-64","0","-4"]},"6":{"A":"960","B":"0","C":["0","-960","0","-80","0","-4"]},"7":{"A":"0","B":"12288","C":["-12288","0","-1536","0","-96","0","-4"]},"8":{"A":"26880","B":"0","C":["0","-26880","0","-2240","0","-112","0","-4"]},"9":{"A":"0","B":"393216","C":["-393216","0","-49152","0","-3072","0","-128","0","-4"]}},"T_powers_erfc":[0,1,2,3,4,5,6,7,8,9,11],"T_powers_poly":[0,1,2,3,4,5,6,7,8,10],"alpha":"3/7","beta":"3/14 T","e_{a,k}(v) by power, erfc branch":{"0":{"1":["0","0","0","0","0","-84035/186624","0","1715/186624","0","-49/746496","0","-7/30233088"],"3":["0","0","0","420175/93312","0","-12005/186624","0","49/186624","0","49/7558272"],"5":["0","-84035/20736","0","-12005/62208","0","343/124416","0","-35/559872"],"7":["0","1715/6912","0","49/20736","0","7/31104"],"9":["0","-49/9216","0","-7/41472"]},"1":{"1":["0","0","0","0","-84035/62208","0","1715/62208","0","-539/2239488","0","7/6718464"],"3":["0","0","-84035/31104","0","8575/62208","0","-833/1119744","0","-7/419904"],"5":["84035/20736","0","1715/20736","0","-245/31104","0","49/1119744"],"7":["-1715/6912","0","49/13824","0","7/31104"],"9":["49/9216","0","-7/27648"]},"11":{"1":["0","0","1/5159780352"],"3":["-1/5159780352"]},"2":{"1":["0","0","0","0","0","1715/373248","0","-343/1492992","0","49/8957952","0","-1/80621568"],"3":["0","0","0","-8575/186624","0","2401/1492992","0","-49/2239488","0","7/20155392"],"5":["0","1715/41472","0","2401/497664","0","-343/1492992","0","-5/1492992"],"7":["0","-343/55296","0","-49/248832","0","1/82944"],"9":["0","49/110592","0","-1/110592"]},"3":{"1":["0","0","84035/46656","0","-8575/373248","0","-49/1492992","0","161/80621568","0","-1/17915904"],"3":["-84035/46656","0","-8575/186624","0","-245/1492992","0","-679/40310784","0","1/1119744"],"5":["8575/124416","0","-49/497664","0","301/2239488","0","-7/2985984"],"7":["49/165888","0","-49/497664","0","-1/82944"],"9":["-7/331776","0","1/73728"]},"4":{"1":["0","0","0","0","0","2891/17915904","0","-581/107495424","0","-1/15925248"],"3":["0","0","0","-14455/8957952","0","4067/107495424","0","1/3981312"],"5":["0","2891/1990656","0","4067/35831808","0","7/2654208"],"7":["0","-581/3981312","0","1/442368"],"9":["0","-1/196608"]},"5":{"1":["0","0","-1715/93312","0","2107/5971968","0","35/35831808","0","13/429981696"],"3":["1715/93312","0","2107/2985984","0","175/35831808","0","-245/214990848"],"5":["-2107/1990656","0","35/11943936","0","7/1492992"],"7":["-35/3981312","0","-11/2654208"],"9":["1/1769472"]},"6":{"1":["0","0","0","0","0","-805/644972544","0","23/286654464"],"3":["0","0","0","4025/322486272","0","-161/286654464"],"5":["0","-805/71663616","0","-161/95551488"],"7":["0","23/10616832"]},"7":{"1":["0","0","245/4478976","0","-637/214990848","0","1/31850496"],"3":["-245/4478976","0","-637/107495424","0","5/31850496"],"5":["637/71663616","0","1/10616832"],"7":["-1/3538944"]},"8":{"1":["0","0","0","0","0","-43/6879707136"],"3":["0","0","0","215/3439853568"],"5":["0","-43/764411904"]},"9":{"1":["0","0","175/483729408","0","-11/2293235712"],"3":["-175/483729408","0","-11/1146617856"],"5":["11/764411904"]}},"e_{a,k}(v) by power, polynomial branch":{"0":{"1":["0","0","0","0","0","0","0","0","343/629856","0","-49/5038848"],"3":["0","0","0","0","0","0","-6517/629856","0","49/314928"],"5":["0","0","0","0","3773/69984","0","-343/839808"],"7":["0","0","-343/7776","0","-49/23328"],"9":["0","0","49/20736"]},"1":{"1":["0","0","0","0","0","-84035/746496","0","20923/6718464","0","-833/26873856","0","7/120932352"],"3":["0","0","0","420175/373248","0","-146461/6718464","0","833/6718464","0","-49/30233088"],"5":["0","-84035/82944","0","-146461/2239488","0","5831/4478976","0","35/2239488"],"7":["0","20923/248832","0","833/746496","0","-7/124416"],"9":["0","-833/331776","0","7/165888"]},"10":{"1":["0","0","-7/7739670528"],"3":["7/7739670528"]},"2":{"1":["0","0","0","0","-84035/248832","0","9947/2239488","0","-539/80621568","0","7/26873856"],"3":["0","0","-84035/124416","0","49735/2239488","0","-833/40310784","0","-7/1679616"],"5":["84035/82944","0","9947/746496","0","-245/1119744","0","49/4478976"],"7":["-9947/248832","0","49/497664","0","7/124416"],"9":["49/331776","0","-7/110592"]},"3":{"1":["0","0","0","0","0","-446243/161243136","0","2597/80621568","0","7/23887872"],"3":["0","0","0","2231215/80621568","0","-18179/80621568","0","-7/5971968"],"5":["0","-446243/17915904","0","-18179/26873856","0","-49/3981312"],"7":["0","2597/2985984","0","-7/663552"],"9":["0","7/294912"]},"4":{"1":["0","0","84035/186624","0","-413315/53747712","0","-49/26873856","0","-91/644972544"],"3":["-84035/186624","0","-413315/26873856","0","-245/26873856","0","1715/322486272"],"5":["413315/17915904","0","-49/8957952","0","-49/2239488"],"7":["49/2985984","0","77/3981312"],"9":["-7/2654208"]},"5":{"1":["0","0","0","0","0","6811/1289945088","0","-161/429981696"],"3":["0","0","0","-34055/644972544","0","1127/429981696"],"5":["0","6811/143327232","0","1127/143327232"],"7":["0","-161/15925248"]},"6":{"1":["0","0","95011/40310784","0","17297/1289945088","0","-7/47775744"],"3":["-95011/40310784","0","17297/644972544","0","-35/47775744"],"5":["-17297/429981696","0","-7/15925248"],"7":["7/5308416"]},"7":{"1":["0","0","0","0","0","301/10319560704"],"3":["0","0","0","-1505/5159780352"],"5":["0","301/1146617856"]},"8":{"1":["0","0","-539/322486272","0","77/3439853568"],"3":["539/322486272","0","77/1719926784"],"5":["-77/1146617856"]}},"kappa":"1/28 sqrt(21)","kappa^2":"3/112","lambda_erfc":"1/48","lambda_poly":"1/21","p3_powers":[1,3,5,7,9],"prefactor":"1/2688 sqrt(21) pi^(-1)"},"note":"arctan(sqrt15/5) (m = 2) and arctan(1/sqrt7) (m = 4, from lam = 3/16) arise in intermediate terms and cancel"},"scientific_effect":"NONE"}},"cumulative-recurrence":{"kind":"report","report":{"certified":true,"certified_enclosures_math_200":{"D_4":["14.2187634589773576013064656778740783173406003782976930345926","14.2187634589773576013064656778740783173406042538663501338228"],"c_5_ref":["0.0132193193800849680760334875146413965210528883241234179643777","0.0132193193800849680760334875146413965210528919272763929401087"]},"checks":{"BASE_INTEGRALS_FLOAT":true,"C5_CLOSED":false,"CUMULATIVE_EXACT":false,"D1_EXACT":true,"D2_EXACT":true,"D3_EXACT":true,"D4_FORM":false,"D4_INSIDE_CERTIFIED":false,"GAUSS_MOMENTS":true,"JK_EXACT":true,"LIBRARY_EXACT":true,"MEHTA_Z4_EXACT":false,"MOMENTS_FLOAT":true,"PINNED":false,"SIMPLIFIED_FORM":false,"STRUCTURE":true},"closed_forms":{"D_1":"4/3","D_2":"29/6  +  -1 sqrt(6)","D_3":"50/3  +  -76/3 pi^(-1)  +  -200/9 pi^(-1) arctan(1/2)","D_4":"1043599/6561  +  -9246227/531441 sqrt(3)  +  -118853/7776 sqrt(6)  +  -4168969751/159432300 sqrt(21) pi^(-1)  +  -62405/972 sqrt(6) pi^(-1) arctan(1/4 sqrt(14))  +  -2087198/6561 pi^(-1) arctan(1/7 sqrt(21))  +  118853/3888 sqrt(6) pi^(-1) arctan(1/7 sqrt(14))  +  18492454/531441 sqrt(3) pi^(-1) arctan(1/7 sqrt(7))","D_4_simplified":"6695/54 - (405/32) sqrt6 - (1/pi) [ (1375/72) sqrt21 + (6695/27) arctan(sqrt21/7) + (405/16) sqrt6 arctan(sqrt14/14) ]   (arctan(sqrt14/4) - arctan(sqrt14/7) = arctan(1/sqrt14))","Z_4":"5504/3 pi^(1)  +  -1664/9 sqrt(3) pi^(1)","c_5_ref":"Gamma(7/6) (3/2)^(1/3) D_4 / (12 sqrt3 pi^(7/2))","erfc_branch_of_D_4":"1043599/6561  +  -9246227/531441 sqrt(3)  +  -5800785683/1811939328 sqrt(6)  +  -5207632019480985643/83588441702400000 sqrt(21) pi^(-1)  +  -4348491539/226492416 sqrt(6) pi^(-1) arctan(1/4 sqrt(14))  +  -2087198/6561 pi^(-1) arctan(1/7 sqrt(21))  +  5800785683/905969664 sqrt(6) pi^(-1) arctan(1/7 sqrt(14))  +  18492454/531441 sqrt(3) pi^(-1) arctan(1/7 sqrt(7))","polynomial_branch_of_D_4":"-197045735509/16307453952 sqrt(6)  +  1007297068889565881/27862813900800000 sqrt(21) pi^(-1)  +  -91736346709/2038431744 sqrt(6) pi^(-1) arctan(1/4 sqrt(14))  +  197045735509/8153726976 sqrt(6) pi^(-1) arctan(1/7 sqrt(14))"},"exact_relations":{"D_4 integral":"sqrt21/(2688 pi) [ (sqrt21 sqrt(pi)/6) int e^{-T^2/48} erfc(sqrt21 T/28) sum_a T^a L_a^E + int e^{-T^2/21} sum_a T^a L_a^R ]","G_n(mu, T)":"G_0 = (1/2) sqrt(pi/mu) erf(sqrt(mu) T); G_1 = (1 - e^{-mu T^2})/(2 mu); G_n = [(n-1) G_{n-2} - T^{n-1} e^{-mu T^2}]/(2 mu)","L_a":"int_0^T e^{-v^2/24} sum_k e_{a,k}(v) J_k(v) dv, k odd, J_k = B_k + C_k e^{-v^2/8}: incomplete Gaussian moments with mu = 1/24, 1/6","M_0(lam, 1)":"(1/2) sqrt(pi/lam)","M_0(lam, erf(u .) erf(v .))":"arctan(u v/sqrt(lam (lam + u^2 + v^2)))/sqrt(pi lam)","M_0(lam, erf(u .))":"arctan(u/sqrt lam)/sqrt(pi lam)","M_1(lam, 1)":"1/(2 lam)","M_n(lam, f), n >= 1":"[(n-1) M_{n-2}(lam, f) + sum_{u in f} (2u/sqrt pi) M_{n-1}(lam + u^2, f minus u)]/(2 lam)","Z_4":"4! sqrt(pi) (1/6) int e^{-T^2/48} int_0^T e^{-v^2/24} int_0^v V e^{-p3^2/8} = 1536 pi (exact, rule MEHTA_Z4_EXACT)"},"interval_evaluation":{"(3/2)^(1/3)":["1.14471424255333186780804221193967700891590692078793107209905","1.14471424255333186780804221193967700891590692078793107209906"],"D_4":["-24.3646940878930461795484813197835616187345521447742044725720","-24.3646940878930461795484813197835616187345521447742044724929"],"D_4_simplified":["14.2187634589773576013064656778740783173406023160820215841883","14.2187634589773576013064656778740783173406023160820215842281"],"Gamma(7/6)":["0.927719333630039200708349482534621018566466519145475576700351","0.927719333630039200708349482534621018566466519145475576939062"],"arctan(sqrt14/14)":["0.261157410903024311262049412927851638004440468771565909792296","0.261157410903024311262049412927851638004440468771565909792486"],"arctan(sqrt14/4)":["0.752040089192335679842958294184691084003367627670504438870380","0.752040089192335679842958294184691084003367627670504438870582"],"arctan(sqrt14/7)":["0.490882678289311368580908881256839445998927158898938529077984","0.490882678289311368580908881256839445998927158898938529078176"],"arctan(sqrt21/7)":["0.579639740363704299923291897011207918621441782280263529351672","0.579639740363704299923291897011207918621441782280263529351866"],"c_5_ref":["-0.0226520874107776440155722387863596688294356205284700670336451","-0.0226520874107776440155722387863596688294356205284700670277231"],"pi":["3.14159265358979323846264338327950288419716939937510582097457","3.14159265358979323846264338327950288419716939937510582097528"]},"mutant":"cumulative-recurrence","object":"CL-SIDE24-D4-CLOSED-FORM-20260930-v1","passed":false,"reduction_data":{"atoms_created_by_the_derivations":["1","arctan(1/5 sqrt(15))","arctan(1/2 sqrt(1))","arctan(1/4 sqrt(14))","arctan(1/7 sqrt(21))","arctan(1/7 sqrt(14))","arctan(1/7 sqrt(7))"],"m=2":{"kappa":"1/20 sqrt(30)","lambda_erfc":"1/8","lambda_poly":"1/5"},"m=3":{"kappa":"1/12 sqrt(6)","lambda_erfc":"1/24","lambda_poly":"1/12"},"m=4":{"G_0_constant":"1/6 sqrt(21) pi^(1/2)","J_k = A_k sqrt(2 pi) erf(s/(2 sqrt2)) + B_k + C_k(s) e^{-s^2/8}":{"0":{"A":"1","B":"0","C":[]},"1":{"A":"0","B":"4","C":["-4"]},"2":{"A":"4","B":"0","C":["0","-4"]},"3":{"A":"0","B":"32","C":["-32","0","-4"]},"4":{"A":"48","B":"0","C":["0","-48","0","-4"]},"5":{"A":"0","B":"512","C":["-512","0","-64","0","-4"]},"6":{"A":"960","B":"0","C":["0","-960","0","-80","0","-4"]},"7":{"A":"0","B":"12288","C":["-12288","0","-1536","0","-96","0","-4"]},"8":{"A":"26880","B":"0","C":["0","-26880","0","-2240","0","-112","0","-4"]},"9":{"A":"0","B":"393216","C":["-393216","0","-49152","0","-3072","0","-128","0","-4"]}},"T_powers_erfc":[0,1,2,3,4,5,6,7,8,9,11],"T_powers_poly":[0,1,2,3,4,5,6,7,8,10],"alpha":"3/7","beta":"3/14 T","e_{a,k}(v) by power, erfc branch":{"0":{"1":["0","0","0","0","0","-84035/186624","0","1715/186624","0","-49/746496","0","-7/30233088"],"3":["0","0","0","420175/93312","0","-12005/186624","0","49/186624","0","49/7558272"],"5":["0","-84035/20736","0","-12005/62208","0","343/124416","0","-35/559872"],"7":["0","1715/6912","0","49/20736","0","7/31104"],"9":["0","-49/9216","0","-7/41472"]},"1":{"1":["0","0","0","0","-84035/62208","0","1715/62208","0","-539/2239488","0","7/6718464"],"3":["0","0","-84035/31104","0","8575/62208","0","-833/1119744","0","-7/419904"],"5":["84035/20736","0","1715/20736","0","-245/31104","0","49/1119744"],"7":["-1715/6912","0","49/13824","0","7/31104"],"9":["49/9216","0","-7/27648"]},"11":{"1":["0","0","1/5159780352"],"3":["-1/5159780352"]},"2":{"1":["0","0","0","0","0","1715/373248","0","-343/1492992","0","49/8957952","0","-1/80621568"],"3":["0","0","0","-8575/186624","0","2401/1492992","0","-49/2239488","0","7/20155392"],"5":["0","1715/41472","0","2401/497664","0","-343/1492992","0","-5/1492992"],"7":["0","-343/55296","0","-49/248832","0","1/82944"],"9":["0","49/110592","0","-1/110592"]},"3":{"1":["0","0","84035/46656","0","-8575/373248","0","-49/1492992","0","161/80621568","0","-1/17915904"],"3":["-84035/46656","0","-8575/186624","0","-245/1492992","0","-679/40310784","0","1/1119744"],"5":["8575/124416","0","-49/497664","0","301/2239488","0","-7/2985984"],"7":["49/165888","0","-49/497664","0","-1/82944"],"9":["-7/331776","0","1/73728"]},"4":{"1":["0","0","0","0","0","2891/17915904","0","-581/107495424","0","-1/15925248"],"3":["0","0","0","-14455/8957952","0","4067/107495424","0","1/3981312"],"5":["0","2891/1990656","0","4067/35831808","0","7/2654208"],"7":["0","-581/3981312","0","1/442368"],"9":["0","-1/196608"]},"5":{"1":["0","0","-1715/93312","0","2107/5971968","0","35/35831808","0","13/429981696"],"3":["1715/93312","0","2107/2985984","0","175/35831808","0","-245/214990848"],"5":["-2107/1990656","0","35/11943936","0","7/1492992"],"7":["-35/3981312","0","-11/2654208"],"9":["1/1769472"]},"6":{"1":["0","0","0","0","0","-805/644972544","0","23/286654464"],"3":["0","0","0","4025/322486272","0","-161/286654464"],"5":["0","-805/71663616","0","-161/95551488"],"7":["0","23/10616832"]},"7":{"1":["0","0","245/4478976","0","-637/214990848","0","1/31850496"],"3":["-245/4478976","0","-637/107495424","0","5/31850496"],"5":["637/71663616","0","1/10616832"],"7":["-1/3538944"]},"8":{"1":["0","0","0","0","0","-43/6879707136"],"3":["0","0","0","215/3439853568"],"5":["0","-43/764411904"]},"9":{"1":["0","0","175/483729408","0","-11/2293235712"],"3":["-175/483729408","0","-11/1146617856"],"5":["11/764411904"]}},"e_{a,k}(v) by power, polynomial branch":{"0":{"1":["0","0","0","0","0","0","0","0","343/629856","0","-49/5038848"],"3":["0","0","0","0","0","0","-6517/629856","0","49/314928"],"5":["0","0","0","0","3773/69984","0","-343/839808"],"7":["0","0","-343/7776","0","-49/23328"],"9":["0","0","49/20736"]},"1":{"1":["0","0","0","0","0","-84035/746496","0","20923/6718464","0","-833/26873856","0","7/120932352"],"3":["0","0","0","420175/373248","0","-146461/6718464","0","833/6718464","0","-49/30233088"],"5":["0","-84035/82944","0","-146461/2239488","0","5831/4478976","0","35/2239488"],"7":["0","20923/248832","0","833/746496","0","-7/124416"],"9":["0","-833/331776","0","7/165888"]},"10":{"1":["0","0","-7/7739670528"],"3":["7/7739670528"]},"2":{"1":["0","0","0","0","-84035/248832","0","9947/2239488","0","-539/80621568","0","7/26873856"],"3":["0","0","-84035/124416","0","49735/2239488","0","-833/40310784","0","-7/1679616"],"5":["84035/82944","0","9947/746496","0","-245/1119744","0","49/4478976"],"7":["-9947/248832","0","49/497664","0","7/124416"],"9":["49/331776","0","-7/110592"]},"3":{"1":["0","0","0","0","0","-446243/161243136","0","2597/80621568","0","7/23887872"],"3":["0","0","0","2231215/80621568","0","-18179/80621568","0","-7/5971968"],"5":["0","-446243/17915904","0","-18179/26873856","0","-49/3981312"],"7":["0","2597/2985984","0","-7/663552"],"9":["0","7/294912"]},"4":{"1":["0","0","84035/186624","0","-413315/53747712","0","-49/26873856","0","-91/644972544"],"3":["-84035/186624","0","-413315/26873856","0","-245/26873856","0","1715/322486272"],"5":["413315/17915904","0","-49/8957952","0","-49/2239488"],"7":["49/2985984","0","77/3981312"],"9":["-7/2654208"]},"5":{"1":["0","0","0","0","0","6811/1289945088","0","-161/429981696"],"3":["0","0","0","-34055/644972544","0","1127/429981696"],"5":["0","6811/143327232","0","1127/143327232"],"7":["0","-161/15925248"]},"6":{"1":["0","0","95011/40310784","0","17297/1289945088","0","-7/47775744"],"3":["-95011/40310784","0","17297/644972544","0","-35/47775744"],"5":["-17297/429981696","0","-7/15925248"],"7":["7/5308416"]},"7":{"1":["0","0","0","0","0","301/10319560704"],"3":["0","0","0","-1505/5159780352"],"5":["0","301/1146617856"]},"8":{"1":["0","0","-539/322486272","0","77/3439853568"],"3":["539/322486272","0","77/1719926784"],"5":["-77/1146617856"]}},"kappa":"1/28 sqrt(21)","kappa^2":"3/112","lambda_erfc":"1/48","lambda_poly":"1/21","p3_powers":[1,3,5,7,9],"prefactor":"1/2688 sqrt(21) pi^(-1)"},"note":"arctan(sqrt15/5) (m = 2) and arctan(1/sqrt7) (m = 4, from lam = 3/16) arise in intermediate terms and cancel"},"scientific_effect":"NONE"}},"erfc-sign":{"kind":"report","report":{"certified":true,"certified_enclosures_math_200":{"D_4":["14.2187634589773576013064656778740783173406003782976930345926","14.2187634589773576013064656778740783173406042538663501338228"],"c_5_ref":["0.0132193193800849680760334875146413965210528883241234179643777","0.0132193193800849680760334875146413965210528919272763929401087"]},"checks":{"BASE_INTEGRALS_FLOAT":true,"C5_CLOSED":false,"CUMULATIVE_EXACT":true,"D1_EXACT":true,"D2_EXACT":false,"D3_EXACT":false,"D4_FORM":false,"D4_INSIDE_CERTIFIED":false,"GAUSS_MOMENTS":true,"JK_EXACT":true,"LIBRARY_EXACT":true,"MEHTA_Z4_EXACT":true,"MOMENTS_FLOAT":true,"PINNED":false,"SIMPLIFIED_FORM":false,"STRUCTURE":true},"closed_forms":{"D_1":"4/3","D_2":"29/6  +  1393/768 sqrt(6)","D_3":"250/9  +  18416/375 pi^(-1)  +  200/9 pi^(-1) arctan(1/2)","D_4":"6695/54  +  2244563075/301989888 sqrt(6)  +  32782477168087/707788800000 sqrt(21) pi^(-1)  +  2244563075/150994944 sqrt(6) pi^(-1) arctan(1/4 sqrt(14))  +  6695/27 pi^(-1) arctan(1/7 sqrt(21))  +  -2244563075/150994944 sqrt(6) pi^(-1) arctan(1/7 sqrt(14))","D_4_simplified":"6695/54 - (405/32) sqrt6 - (1/pi) [ (1375/72) sqrt21 + (6695/27) arctan(sqrt21/7) + (405/16) sqrt6 arctan(sqrt14/14) ]   (arctan(sqrt14/4) - arctan(sqrt14/7) = arctan(1/sqrt14))","Z_4":"1536 pi^(1)","c_5_ref":"Gamma(7/6) (3/2)^(1/3) D_4 / (12 sqrt3 pi^(7/2))","erfc_branch_of_D_4":"6695/54  +  6066622595/603979776 sqrt(6)  +  46299277168087/1415577600000 sqrt(21) pi^(-1)  +  6066622595/301989888 sqrt(6) pi^(-1) arctan(1/4 sqrt(14))  +  6695/27 pi^(-1) arctan(1/7 sqrt(21))  +  -6066622595/301989888 sqrt(6) pi^(-1) arctan(1/7 sqrt(14))","polynomial_branch_of_D_4":"-1577496445/603979776 sqrt(6)  +  19265677168087/1415577600000 sqrt(21) pi^(-1)  +  -1577496445/301989888 sqrt(6) pi^(-1) arctan(1/4 sqrt(14))  +  1577496445/301989888 sqrt(6) pi^(-1) arctan(1/7 sqrt(14))"},"exact_relations":{"D_4 integral":"sqrt21/(2688 pi) [ (sqrt21 sqrt(pi)/6) int e^{-T^2/48} erfc(sqrt21 T/28) sum_a T^a L_a^E + int e^{-T^2/21} sum_a T^a L_a^R ]","G_n(mu, T)":"G_0 = (1/2) sqrt(pi/mu) erf(sqrt(mu) T); G_1 = (1 - e^{-mu T^2})/(2 mu); G_n = [(n-1) G_{n-2} - T^{n-1} e^{-mu T^2}]/(2 mu)","L_a":"int_0^T e^{-v^2/24} sum_k e_{a,k}(v) J_k(v) dv, k odd, J_k = B_k + C_k e^{-v^2/8}: incomplete Gaussian moments with mu = 1/24, 1/6","M_0(lam, 1)":"(1/2) sqrt(pi/lam)","M_0(lam, erf(u .) erf(v .))":"arctan(u v/sqrt(lam (lam + u^2 + v^2)))/sqrt(pi lam)","M_0(lam, erf(u .))":"arctan(u/sqrt lam)/sqrt(pi lam)","M_1(lam, 1)":"1/(2 lam)","M_n(lam, f), n >= 1":"[(n-1) M_{n-2}(lam, f) + sum_{u in f} (2u/sqrt pi) M_{n-1}(lam + u^2, f minus u)]/(2 lam)","Z_4":"4! sqrt(pi) (1/6) int e^{-T^2/48} int_0^T e^{-v^2/24} int_0^v V e^{-p3^2/8} = 1536 pi (exact, rule MEHTA_Z4_EXACT)"},"interval_evaluation":{"(3/2)^(1/3)":["1.14471424255333186780804221193967700891590692078793107209905","1.14471424255333186780804221193967700891590692078793107209906"],"D_4":["258.526097825635161722382810352942415767344410393558422632713","258.526097825635161722382810352942415767344410393558422632770"],"D_4_simplified":["14.2187634589773576013064656778740783173406023160820215841883","14.2187634589773576013064656778740783173406023160820215842281"],"Gamma(7/6)":["0.927719333630039200708349482534621018566466519145475576700351","0.927719333630039200708349482534621018566466519145475576939062"],"arctan(sqrt14/14)":["0.261157410903024311262049412927851638004440468771565909792296","0.261157410903024311262049412927851638004440468771565909792486"],"arctan(sqrt14/4)":["0.752040089192335679842958294184691084003367627670504438870380","0.752040089192335679842958294184691084003367627670504438870582"],"arctan(sqrt14/7)":["0.490882678289311368580908881256839445998927158898938529077984","0.490882678289311368580908881256839445998927158898938529078176"],"arctan(sqrt21/7)":["0.579639740363704299923291897011207918621441782280263529351672","0.579639740363704299923291897011207918621441782280263529351866"],"c_5_ref":["0.240354167583146332316582131163970225741131865630949641134803","0.240354167583146332316582131163970225741131865630949641196913"],"pi":["3.14159265358979323846264338327950288419716939937510582097457","3.14159265358979323846264338327950288419716939937510582097528"]},"mutant":"erfc-sign","object":"CL-SIDE24-D4-CLOSED-FORM-20260930-v1","passed":false,"reduction_data":{"atoms_created_by_the_derivations":["1","arctan(1/5 sqrt(15))","arctan(1/2 sqrt(1))","arctan(1/4 sqrt(14))","arctan(1/7 sqrt(21))","arctan(1/7 sqrt(14))","arctan(1/7 sqrt(7))"],"m=2":{"kappa":"1/20 sqrt(30)","lambda_erfc":"1/8","lambda_poly":"1/5"},"m=3":{"kappa":"1/12 sqrt(6)","lambda_erfc":"1/24","lambda_poly":"1/12"},"m=4":{"G_0_constant":"1/6 sqrt(21) pi^(1/2)","J_k = A_k sqrt(2 pi) erf(s/(2 sqrt2)) + B_k + C_k(s) e^{-s^2/8}":{"0":{"A":"1","B":"0","C":[]},"1":{"A":"0","B":"4","C":["-4"]},"2":{"A":"4","B":"0","C":["0","-4"]},"3":{"A":"0","B":"32","C":["-32","0","-4"]},"4":{"A":"48","B":"0","C":["0","-48","0","-4"]},"5":{"A":"0","B":"512","C":["-512","0","-64","0","-4"]},"6":{"A":"960","B":"0","C":["0","-960","0","-80","0","-4"]},"7":{"A":"0","B":"12288","C":["-12288","0","-1536","0","-96","0","-4"]},"8":{"A":"26880","B":"0","C":["0","-26880","0","-2240","0","-112","0","-4"]},"9":{"A":"0","B":"393216","C":["-393216","0","-49152","0","-3072","0","-128","0","-4"]}},"T_powers_erfc":[0,1,2,3,4,5,6,7,8,9,11],"T_powers_poly":[0,1,2,3,4,5,6,7,8,10],"alpha":"3/7","beta":"3/14 T","e_{a,k}(v) by power, erfc branch":{"0":{"1":["0","0","0","0","0","-84035/186624","0","1715/186624","0","-49/746496","0","-7/30233088"],"3":["0","0","0","420175/93312","0","-12005/186624","0","49/186624","0","49/7558272"],"5":["0","-84035/20736","0","-12005/62208","0","343/124416","0","-35/559872"],"7":["0","1715/6912","0","49/20736","0","7/31104"],"9":["0","-49/9216","0","-7/41472"]},"1":{"1":["0","0","0","0","-84035/62208","0","1715/62208","0","-539/2239488","0","7/6718464"],"3":["0","0","-84035/31104","0","8575/62208","0","-833/1119744","0","-7/419904"],"5":["84035/20736","0","1715/20736","0","-245/31104","0","49/1119744"],"7":["-1715/6912","0","49/13824","0","7/31104"],"9":["49/9216","0","-7/27648"]},"11":{"1":["0","0","1/5159780352"],"3":["-1/5159780352"]},"2":{"1":["0","0","0","0","0","1715/373248","0","-343/1492992","0","49/8957952","0","-1/80621568"],"3":["0","0","0","-8575/186624","0","2401/1492992","0","-49/2239488","0","7/20155392"],"5":["0","1715/41472","0","2401/497664","0","-343/1492992","0","-5/1492992"],"7":["0","-343/55296","0","-49/248832","0","1/82944"],"9":["0","49/110592","0","-1/110592"]},"3":{"1":["0","0","84035/46656","0","-8575/373248","0","-49/1492992","0","161/80621568","0","-1/17915904"],"3":["-84035/46656","0","-8575/186624","0","-245/1492992","0","-679/40310784","0","1/1119744"],"5":["8575/124416","0","-49/497664","0","301/2239488","0","-7/2985984"],"7":["49/165888","0","-49/497664","0","-1/82944"],"9":["-7/331776","0","1/73728"]},"4":{"1":["0","0","0","0","0","2891/17915904","0","-581/107495424","0","-1/15925248"],"3":["0","0","0","-14455/8957952","0","4067/107495424","0","1/3981312"],"5":["0","2891/1990656","0","4067/35831808","0","7/2654208"],"7":["0","-581/3981312","0","1/442368"],"9":["0","-1/196608"]},"5":{"1":["0","0","-1715/93312","0","2107/5971968","0","35/35831808","0","13/429981696"],"3":["1715/93312","0","2107/2985984","0","175/35831808","0","-245/214990848"],"5":["-2107/1990656","0","35/11943936","0","7/1492992"],"7":["-35/3981312","0","-11/2654208"],"9":["1/1769472"]},"6":{"1":["0","0","0","0","0","-805/644972544","0","23/286654464"],"3":["0","0","0","4025/322486272","0","-161/286654464"],"5":["0","-805/71663616","0","-161/95551488"],"7":["0","23/10616832"]},"7":{"1":["0","0","245/4478976","0","-637/214990848","0","1/31850496"],"3":["-245/4478976","0","-637/107495424","0","5/31850496"],"5":["637/71663616","0","1/10616832"],"7":["-1/3538944"]},"8":{"1":["0","0","0","0","0","-43/6879707136"],"3":["0","0","0","215/3439853568"],"5":["0","-43/764411904"]},"9":{"1":["0","0","175/483729408","0","-11/2293235712"],"3":["-175/483729408","0","-11/1146617856"],"5":["11/764411904"]}},"e_{a,k}(v) by power, polynomial branch":{"0":{"1":["0","0","0","0","0","0","0","0","343/629856","0","-49/5038848"],"3":["0","0","0","0","0","0","-6517/629856","0","49/314928"],"5":["0","0","0","0","3773/69984","0","-343/839808"],"7":["0","0","-343/7776","0","-49/23328"],"9":["0","0","49/20736"]},"1":{"1":["0","0","0","0","0","-84035/746496","0","20923/6718464","0","-833/26873856","0","7/120932352"],"3":["0","0","0","420175/373248","0","-146461/6718464","0","833/6718464","0","-49/30233088"],"5":["0","-84035/82944","0","-146461/2239488","0","5831/4478976","0","35/2239488"],"7":["0","20923/248832","0","833/746496","0","-7/124416"],"9":["0","-833/331776","0","7/165888"]},"10":{"1":["0","0","-7/7739670528"],"3":["7/7739670528"]},"2":{"1":["0","0","0","0","-84035/248832","0","9947/2239488","0","-539/80621568","0","7/26873856"],"3":["0","0","-84035/124416","0","49735/2239488","0","-833/40310784","0","-7/1679616"],"5":["84035/82944","0","9947/746496","0","-245/1119744","0","49/4478976"],"7":["-9947/248832","0","49/497664","0","7/124416"],"9":["49/331776","0","-7/110592"]},"3":{"1":["0","0","0","0","0","-446243/161243136","0","2597/80621568","0","7/23887872"],"3":["0","0","0","2231215/80621568","0","-18179/80621568","0","-7/5971968"],"5":["0","-446243/17915904","0","-18179/26873856","0","-49/3981312"],"7":["0","2597/2985984","0","-7/663552"],"9":["0","7/294912"]},"4":{"1":["0","0","84035/186624","0","-413315/53747712","0","-49/26873856","0","-91/644972544"],"3":["-84035/186624","0","-413315/26873856","0","-245/26873856","0","1715/322486272"],"5":["413315/17915904","0","-49/8957952","0","-49/2239488"],"7":["49/2985984","0","77/3981312"],"9":["-7/2654208"]},"5":{"1":["0","0","0","0","0","6811/1289945088","0","-161/429981696"],"3":["0","0","0","-34055/644972544","0","1127/429981696"],"5":["0","6811/143327232","0","1127/143327232"],"7":["0","-161/15925248"]},"6":{"1":["0","0","95011/40310784","0","17297/1289945088","0","-7/47775744"],"3":["-95011/40310784","0","17297/644972544","0","-35/47775744"],"5":["-17297/429981696","0","-7/15925248"],"7":["7/5308416"]},"7":{"1":["0","0","0","0","0","301/10319560704"],"3":["0","0","0","-1505/5159780352"],"5":["0","301/1146617856"]},"8":{"1":["0","0","-539/322486272","0","77/3439853568"],"3":["539/322486272","0","77/1719926784"],"5":["-77/1146617856"]}},"kappa":"1/28 sqrt(21)","kappa^2":"3/112","lambda_erfc":"1/48","lambda_poly":"1/21","p3_powers":[1,3,5,7,9],"prefactor":"1/2688 sqrt(21) pi^(-1)"},"note":"arctan(sqrt15/5) (m = 2) and arctan(1/sqrt7) (m = 4, from lam = 3/16) arise in intermediate terms and cancel"},"scientific_effect":"NONE"}},"gauss-half":{"kind":"report","report":{"certified":true,"certified_enclosures_math_200":{"D_4":["14.2187634589773576013064656778740783173406003782976930345926","14.2187634589773576013064656778740783173406042538663501338228"],"c_5_ref":["0.0132193193800849680760334875146413965210528883241234179643777","0.0132193193800849680760334875146413965210528919272763929401087"]},"checks":{"BASE_INTEGRALS_FLOAT":true,"C5_CLOSED":false,"CUMULATIVE_EXACT":true,"D1_EXACT":false,"D2_EXACT":false,"D3_EXACT":false,"D4_FORM":false,"D4_INSIDE_CERTIFIED":false,"GAUSS_MOMENTS":false,"JK_EXACT":true,"LIBRARY_EXACT":true,"MEHTA_Z4_EXACT":false,"MOMENTS_FLOAT":false,"PINNED":false,"SIMPLIFIED_FORM":false,"STRUCTURE":true},"closed_forms":{"D_1":"8/3","D_2":"29/6  +  -2 sqrt(6)","D_3":"350/9  +  -76/3 pi^(-1)  +  -200/9 pi^(-1) arctan(1/2)","D_4":"6695/27  +  -405/32 sqrt(6)  +  -1375/72 sqrt(21) pi^(-1)  +  -405/16 sqrt(6) pi^(-1) arctan(1/4 sqrt(14))  +  -6695/27 pi^(-1) arctan(1/7 sqrt(21))  +  405/16 sqrt(6) pi^(-1) arctan(1/7 sqrt(14))","D_4_simplified":"6695/54 - (405/32) sqrt6 - (1/pi) [ (1375/72) sqrt21 + (6695/27) arctan(sqrt21/7) + (405/16) sqrt6 arctan(sqrt14/14) ]   (arctan(sqrt14/4) - arctan(sqrt14/7) = arctan(1/sqrt14))","Z_4":"3072 pi^(1)","c_5_ref":"Gamma(7/6) (3/2)^(1/3) D_4 / (12 sqrt3 pi^(7/2))","erfc_branch_of_D_4":"6695/27  +  -6066622595/603979776 sqrt(6)  +  -46299277168087/1415577600000 sqrt(21) pi^(-1)  +  -6066622595/301989888 sqrt(6) pi^(-1) arctan(1/4 sqrt(14))  +  -6695/27 pi^(-1) arctan(1/7 sqrt(21))  +  6066622595/301989888 sqrt(6) pi^(-1) arctan(1/7 sqrt(14))","polynomial_branch_of_D_4":"-1577496445/603979776 sqrt(6)  +  19265677168087/1415577600000 sqrt(21) pi^(-1)  +  -1577496445/301989888 sqrt(6) pi^(-1) arctan(1/4 sqrt(14))  +  1577496445/301989888 sqrt(6) pi^(-1) arctan(1/7 sqrt(14))"},"exact_relations":{"D_4 integral":"sqrt21/(2688 pi) [ (sqrt21 sqrt(pi)/6) int e^{-T^2/48} erfc(sqrt21 T/28) sum_a T^a L_a^E + int e^{-T^2/21} sum_a T^a L_a^R ]","G_n(mu, T)":"G_0 = (1/2) sqrt(pi/mu) erf(sqrt(mu) T); G_1 = (1 - e^{-mu T^2})/(2 mu); G_n = [(n-1) G_{n-2} - T^{n-1} e^{-mu T^2}]/(2 mu)","L_a":"int_0^T e^{-v^2/24} sum_k e_{a,k}(v) J_k(v) dv, k odd, J_k = B_k + C_k e^{-v^2/8}: incomplete Gaussian moments with mu = 1/24, 1/6","M_0(lam, 1)":"(1/2) sqrt(pi/lam)","M_0(lam, erf(u .) erf(v .))":"arctan(u v/sqrt(lam (lam + u^2 + v^2)))/sqrt(pi lam)","M_0(lam, erf(u .))":"arctan(u/sqrt lam)/sqrt(pi lam)","M_1(lam, 1)":"1/(2 lam)","M_n(lam, f), n >= 1":"[(n-1) M_{n-2}(lam, f) + sum_{u in f} (2u/sqrt pi) M_{n-1}(lam + u^2, f minus u)]/(2 lam)","Z_4":"4! sqrt(pi) (1/6) int e^{-T^2/48} int_0^T e^{-v^2/24} int_0^v V e^{-p3^2/8} = 1536 pi (exact, rule MEHTA_Z4_EXACT)"},"interval_evaluation":{"(3/2)^(1/3)":["1.14471424255333186780804221193967700891590692078793107209905","1.14471424255333186780804221193967700891590692078793107209906"],"D_4":["138.200244940458839082787947159355559798822083797563503065662","138.200244940458839082787947159355559798822083797563503065714"],"D_4_simplified":["14.2187634589773576013064656778740783173406023160820215841883","14.2187634589773576013064656778740783173406023160820215842281"],"Gamma(7/6)":["0.927719333630039200708349482534621018566466519145475576700351","0.927719333630039200708349482534621018566466519145475576939062"],"arctan(sqrt14/14)":["0.261157410903024311262049412927851638004440468771565909792296","0.261157410903024311262049412927851638004440468771565909792486"],"arctan(sqrt14/4)":["0.752040089192335679842958294184691084003367627670504438870380","0.752040089192335679842958294184691084003367627670504438870582"],"arctan(sqrt14/7)":["0.490882678289311368580908881256839445998927158898938529077984","0.490882678289311368580908881256839445998927158898938529078176"],"arctan(sqrt21/7)":["0.579639740363704299923291897011207918621441782280263529351672","0.579639740363704299923291897011207918621441782280263529351866"],"c_5_ref":["0.128486079787791363914584977343703277037270555443291973968648","0.128486079787791363914584977343703277037270555443291974001870"],"pi":["3.14159265358979323846264338327950288419716939937510582097457","3.14159265358979323846264338327950288419716939937510582097528"]},"mutant":"gauss-half","object":"CL-SIDE24-D4-CLOSED-FORM-20260930-v1","passed":false,"reduction_data":{"atoms_created_by_the_derivations":["1","arctan(1/5 sqrt(15))","arctan(1/2 sqrt(1))","arctan(1/4 sqrt(14))","arctan(1/7 sqrt(21))","arctan(1/7 sqrt(14))","arctan(1/7 sqrt(7))"],"m=2":{"kappa":"1/20 sqrt(30)","lambda_erfc":"1/8","lambda_poly":"1/5"},"m=3":{"kappa":"1/12 sqrt(6)","lambda_erfc":"1/24","lambda_poly":"1/12"},"m=4":{"G_0_constant":"1/6 sqrt(21) pi^(1/2)","J_k = A_k sqrt(2 pi) erf(s/(2 sqrt2)) + B_k + C_k(s) e^{-s^2/8}":{"0":{"A":"1","B":"0","C":[]},"1":{"A":"0","B":"4","C":["-4"]},"2":{"A":"4","B":"0","C":["0","-4"]},"3":{"A":"0","B":"32","C":["-32","0","-4"]},"4":{"A":"48","B":"0","C":["0","-48","0","-4"]},"5":{"A":"0","B":"512","C":["-512","0","-64","0","-4"]},"6":{"A":"960","B":"0","C":["0","-960","0","-80","0","-4"]},"7":{"A":"0","B":"12288","C":["-12288","0","-1536","0","-96","0","-4"]},"8":{"A":"26880","B":"0","C":["0","-26880","0","-2240","0","-112","0","-4"]},"9":{"A":"0","B":"393216","C":["-393216","0","-49152","0","-3072","0","-128","0","-4"]}},"T_powers_erfc":[0,1,2,3,4,5,6,7,8,9,11],"T_powers_poly":[0,1,2,3,4,5,6,7,8,10],"alpha":"3/7","beta":"3/14 T","e_{a,k}(v) by power, erfc branch":{"0":{"1":["0","0","0","0","0","-84035/186624","0","1715/186624","0","-49/746496","0","-7/30233088"],"3":["0","0","0","420175/93312","0","-12005/186624","0","49/186624","0","49/7558272"],"5":["0","-84035/20736","0","-12005/62208","0","343/124416","0","-35/559872"],"7":["0","1715/6912","0","49/20736","0","7/31104"],"9":["0","-49/9216","0","-7/41472"]},"1":{"1":["0","0","0","0","-84035/62208","0","1715/62208","0","-539/2239488","0","7/6718464"],"3":["0","0","-84035/31104","0","8575/62208","0","-833/1119744","0","-7/419904"],"5":["84035/20736","0","1715/20736","0","-245/31104","0","49/1119744"],"7":["-1715/6912","0","49/13824","0","7/31104"],"9":["49/9216","0","-7/27648"]},"11":{"1":["0","0","1/5159780352"],"3":["-1/5159780352"]},"2":{"1":["0","0","0","0","0","1715/373248","0","-343/1492992","0","49/8957952","0","-1/80621568"],"3":["0","0","0","-8575/186624","0","2401/1492992","0","-49/2239488","0","7/20155392"],"5":["0","1715/41472","0","2401/497664","0","-343/1492992","0","-5/1492992"],"7":["0","-343/55296","0","-49/248832","0","1/82944"],"9":["0","49/110592","0","-1/110592"]},"3":{"1":["0","0","84035/46656","0","-8575/373248","0","-49/1492992","0","161/80621568","0","-1/17915904"],"3":["-84035/46656","0","-8575/186624","0","-245/1492992","0","-679/40310784","0","1/1119744"],"5":["8575/124416","0","-49/497664","0","301/2239488","0","-7/2985984"],"7":["49/165888","0","-49/497664","0","-1/82944"],"9":["-7/331776","0","1/73728"]},"4":{"1":["0","0","0","0","0","2891/17915904","0","-581/107495424","0","-1/15925248"],"3":["0","0","0","-14455/8957952","0","4067/107495424","0","1/3981312"],"5":["0","2891/1990656","0","4067/35831808","0","7/2654208"],"7":["0","-581/3981312","0","1/442368"],"9":["0","-1/196608"]},"5":{"1":["0","0","-1715/93312","0","2107/5971968","0","35/35831808","0","13/429981696"],"3":["1715/93312","0","2107/2985984","0","175/35831808","0","-245/214990848"],"5":["-2107/1990656","0","35/11943936","0","7/1492992"],"7":["-35/3981312","0","-11/2654208"],"9":["1/1769472"]},"6":{"1":["0","0","0","0","0","-805/644972544","0","23/286654464"],"3":["0","0","0","4025/322486272","0","-161/286654464"],"5":["0","-805/71663616","0","-161/95551488"],"7":["0","23/10616832"]},"7":{"1":["0","0","245/4478976","0","-637/214990848","0","1/31850496"],"3":["-245/4478976","0","-637/107495424","0","5/31850496"],"5":["637/71663616","0","1/10616832"],"7":["-1/3538944"]},"8":{"1":["0","0","0","0","0","-43/6879707136"],"3":["0","0","0","215/3439853568"],"5":["0","-43/764411904"]},"9":{"1":["0","0","175/483729408","0","-11/2293235712"],"3":["-175/483729408","0","-11/1146617856"],"5":["11/764411904"]}},"e_{a,k}(v) by power, polynomial branch":{"0":{"1":["0","0","0","0","0","0","0","0","343/629856","0","-49/5038848"],"3":["0","0","0","0","0","0","-6517/629856","0","49/314928"],"5":["0","0","0","0","3773/69984","0","-343/839808"],"7":["0","0","-343/7776","0","-49/23328"],"9":["0","0","49/20736"]},"1":{"1":["0","0","0","0","0","-84035/746496","0","20923/6718464","0","-833/26873856","0","7/120932352"],"3":["0","0","0","420175/373248","0","-146461/6718464","0","833/6718464","0","-49/30233088"],"5":["0","-84035/82944","0","-146461/2239488","0","5831/4478976","0","35/2239488"],"7":["0","20923/248832","0","833/746496","0","-7/124416"],"9":["0","-833/331776","0","7/165888"]},"10":{"1":["0","0","-7/7739670528"],"3":["7/7739670528"]},"2":{"1":["0","0","0","0","-84035/248832","0","9947/2239488","0","-539/80621568","0","7/26873856"],"3":["0","0","-84035/124416","0","49735/2239488","0","-833/40310784","0","-7/1679616"],"5":["84035/82944","0","9947/746496","0","-245/1119744","0","49/4478976"],"7":["-9947/248832","0","49/497664","0","7/124416"],"9":["49/331776","0","-7/110592"]},"3":{"1":["0","0","0","0","0","-446243/161243136","0","2597/80621568","0","7/23887872"],"3":["0","0","0","2231215/80621568","0","-18179/80621568","0","-7/5971968"],"5":["0","-446243/17915904","0","-18179/26873856","0","-49/3981312"],"7":["0","2597/2985984","0","-7/663552"],"9":["0","7/294912"]},"4":{"1":["0","0","84035/186624","0","-413315/53747712","0","-49/26873856","0","-91/644972544"],"3":["-84035/186624","0","-413315/26873856","0","-245/26873856","0","1715/322486272"],"5":["413315/17915904","0","-49/8957952","0","-49/2239488"],"7":["49/2985984","0","77/3981312"],"9":["-7/2654208"]},"5":{"1":["0","0","0","0","0","6811/1289945088","0","-161/429981696"],"3":["0","0","0","-34055/644972544","0","1127/429981696"],"5":["0","6811/143327232","0","1127/143327232"],"7":["0","-161/15925248"]},"6":{"1":["0","0","95011/40310784","0","17297/1289945088","0","-7/47775744"],"3":["-95011/40310784","0","17297/644972544","0","-35/47775744"],"5":["-17297/429981696","0","-7/15925248"],"7":["7/5308416"]},"7":{"1":["0","0","0","0","0","301/10319560704"],"3":["0","0","0","-1505/5159780352"],"5":["0","301/1146617856"]},"8":{"1":["0","0","-539/322486272","0","77/3439853568"],"3":["539/322486272","0","77/1719926784"],"5":["-77/1146617856"]}},"kappa":"1/28 sqrt(21)","kappa^2":"3/112","lambda_erfc":"1/48","lambda_poly":"1/21","p3_powers":[1,3,5,7,9],"prefactor":"1/2688 sqrt(21) pi^(-1)"},"note":"arctan(sqrt15/5) (m = 2) and arctan(1/sqrt7) (m = 4, from lam = 3/16) arise in intermediate terms and cancel"},"scientific_effect":"NONE"}},"jacobian":{"kind":"report","report":{"certified":true,"certified_enclosures_math_200":{"D_4":["14.2187634589773576013064656778740783173406003782976930345926","14.2187634589773576013064656778740783173406042538663501338228"],"c_5_ref":["0.0132193193800849680760334875146413965210528883241234179643777","0.0132193193800849680760334875146413965210528919272763929401087"]},"checks":{"BASE_INTEGRALS_FLOAT":true,"C5_CLOSED":false,"CUMULATIVE_EXACT":true,"D1_EXACT":true,"D2_EXACT":true,"D3_EXACT":false,"D4_FORM":false,"D4_INSIDE_CERTIFIED":false,"GAUSS_MOMENTS":true,"JK_EXACT":true,"LIBRARY_EXACT":true,"MEHTA_Z4_EXACT":false,"MOMENTS_FLOAT":true,"PINNED":false,"SIMPLIFIED_FORM":false,"STRUCTURE":true},"closed_forms":{"D_1":"4/3","D_2":"29/6  +  -1 sqrt(6)","D_3":"100/3  +  -152/3 pi^(-1)  +  -400/9 pi^(-1) arctan(1/2)","D_4":"6695/18  +  -1215/32 sqrt(6)  +  -1375/24 sqrt(21) pi^(-1)  +  -1215/16 sqrt(6) pi^(-1) arctan(1/4 sqrt(14))  +  -6695/9 pi^(-1) arctan(1/7 sqrt(21))  +  1215/16 sqrt(6) pi^(-1) arctan(1/7 sqrt(14))","D_4_simplified":"6695/54 - (405/32) sqrt6 - (1/pi) [ (1375/72) sqrt21 + (6695/27) arctan(sqrt21/7) + (405/16) sqrt6 arctan(sqrt14/14) ]   (arctan(sqrt14/4) - arctan(sqrt14/7) = arctan(1/sqrt14))","Z_4":"4608 pi^(1)","c_5_ref":"Gamma(7/6) (3/2)^(1/3) D_4 / (12 sqrt3 pi^(7/2))","erfc_branch_of_D_4":"6695/18  +  -6066622595/201326592 sqrt(6)  +  -46299277168087/471859200000 sqrt(21) pi^(-1)  +  -6066622595/100663296 sqrt(6) pi^(-1) arctan(1/4 sqrt(14))  +  -6695/9 pi^(-1) arctan(1/7 sqrt(21))  +  6066622595/100663296 sqrt(6) pi^(-1) arctan(1/7 sqrt(14))","polynomial_branch_of_D_4":"-1577496445/201326592 sqrt(6)  +  19265677168087/471859200000 sqrt(21) pi^(-1)  +  -1577496445/100663296 sqrt(6) pi^(-1) arctan(1/4 sqrt(14))  +  1577496445/100663296 sqrt(6) pi^(-1) arctan(1/7 sqrt(14))"},"exact_relations":{"D_4 integral":"sqrt21/(2688 pi) [ (sqrt21 sqrt(pi)/6) int e^{-T^2/48} erfc(sqrt21 T/28) sum_a T^a L_a^E + int e^{-T^2/21} sum_a T^a L_a^R ]","G_n(mu, T)":"G_0 = (1/2) sqrt(pi/mu) erf(sqrt(mu) T); G_1 = (1 - e^{-mu T^2})/(2 mu); G_n = [(n-1) G_{n-2} - T^{n-1} e^{-mu T^2}]/(2 mu)","L_a":"int_0^T e^{-v^2/24} sum_k e_{a,k}(v) J_k(v) dv, k odd, J_k = B_k + C_k e^{-v^2/8}: incomplete Gaussian moments with mu = 1/24, 1/6","M_0(lam, 1)":"(1/2) sqrt(pi/lam)","M_0(lam, erf(u .) erf(v .))":"arctan(u v/sqrt(lam (lam + u^2 + v^2)))/sqrt(pi lam)","M_0(lam, erf(u .))":"arctan(u/sqrt lam)/sqrt(pi lam)","M_1(lam, 1)":"1/(2 lam)","M_n(lam, f), n >= 1":"[(n-1) M_{n-2}(lam, f) + sum_{u in f} (2u/sqrt pi) M_{n-1}(lam + u^2, f minus u)]/(2 lam)","Z_4":"4! sqrt(pi) (1/6) int e^{-T^2/48} int_0^T e^{-v^2/24} int_0^v V e^{-p3^2/8} = 1536 pi (exact, rule MEHTA_Z4_EXACT)"},"interval_evaluation":{"(3/2)^(1/3)":["1.14471424255333186780804221193967700891590692078793107209905","1.14471424255333186780804221193967700891590692078793107209906"],"D_4":["42.6562903769320728039193970336222349520218069482460647525516","42.6562903769320728039193970336222349520218069482460647526960"],"D_4_simplified":["14.2187634589773576013064656778740783173406023160820215841883","14.2187634589773576013064656778740783173406023160820215842281"],"Gamma(7/6)":["0.927719333630039200708349482534621018566466519145475576700351","0.927719333630039200708349482534621018566466519145475576939062"],"arctan(sqrt14/14)":["0.261157410903024311262049412927851638004440468771565909792296","0.261157410903024311262049412927851638004440468771565909792486"],"arctan(sqrt14/4)":["0.752040089192335679842958294184691084003367627670504438870380","0.752040089192335679842958294184691084003367627670504438870582"],"arctan(sqrt14/7)":["0.490882678289311368580908881256839445998927158898938529077984","0.490882678289311368580908881256839445998927158898938529078176"],"arctan(sqrt21/7)":["0.579639740363704299923291897011207918621441782280263529351672","0.579639740363704299923291897011207918621441782280263529351866"],"c_5_ref":["0.0396579581402549042281004625439241895631586703770997163515438","0.0396579581402549042281004625439241895631586703770997163619172"],"pi":["3.14159265358979323846264338327950288419716939937510582097457","3.14159265358979323846264338327950288419716939937510582097528"]},"mutant":"jacobian","object":"CL-SIDE24-D4-CLOSED-FORM-20260930-v1","passed":false,"reduction_data":{"atoms_created_by_the_derivations":["1","arctan(1/5 sqrt(15))","arctan(1/2 sqrt(1))","arctan(1/4 sqrt(14))","arctan(1/7 sqrt(21))","arctan(1/7 sqrt(14))","arctan(1/7 sqrt(7))"],"m=2":{"kappa":"1/20 sqrt(30)","lambda_erfc":"1/8","lambda_poly":"1/5"},"m=3":{"kappa":"1/12 sqrt(6)","lambda_erfc":"1/24","lambda_poly":"1/12"},"m=4":{"G_0_constant":"1/6 sqrt(21) pi^(1/2)","J_k = A_k sqrt(2 pi) erf(s/(2 sqrt2)) + B_k + C_k(s) e^{-s^2/8}":{"0":{"A":"1","B":"0","C":[]},"1":{"A":"0","B":"4","C":["-4"]},"2":{"A":"4","B":"0","C":["0","-4"]},"3":{"A":"0","B":"32","C":["-32","0","-4"]},"4":{"A":"48","B":"0","C":["0","-48","0","-4"]},"5":{"A":"0","B":"512","C":["-512","0","-64","0","-4"]},"6":{"A":"960","B":"0","C":["0","-960","0","-80","0","-4"]},"7":{"A":"0","B":"12288","C":["-12288","0","-1536","0","-96","0","-4"]},"8":{"A":"26880","B":"0","C":["0","-26880","0","-2240","0","-112","0","-4"]},"9":{"A":"0","B":"393216","C":["-393216","0","-49152","0","-3072","0","-128","0","-4"]}},"T_powers_erfc":[0,1,2,3,4,5,6,7,8,9,11],"T_powers_poly":[0,1,2,3,4,5,6,7,8,10],"alpha":"3/7","beta":"3/14 T","e_{a,k}(v) by power, erfc branch":{"0":{"1":["0","0","0","0","0","-84035/186624","0","1715/186624","0","-49/746496","0","-7/30233088"],"3":["0","0","0","420175/93312","0","-12005/186624","0","49/186624","0","49/7558272"],"5":["0","-84035/20736","0","-12005/62208","0","343/124416","0","-35/559872"],"7":["0","1715/6912","0","49/20736","0","7/31104"],"9":["0","-49/9216","0","-7/41472"]},"1":{"1":["0","0","0","0","-84035/62208","0","1715/62208","0","-539/2239488","0","7/6718464"],"3":["0","0","-84035/31104","0","8575/62208","0","-833/1119744","0","-7/419904"],"5":["84035/20736","0","1715/20736","0","-245/31104","0","49/1119744"],"7":["-1715/6912","0","49/13824","0","7/31104"],"9":["49/9216","0","-7/27648"]},"11":{"1":["0","0","1/5159780352"],"3":["-1/5159780352"]},"2":{"1":["0","0","0","0","0","1715/373248","0","-343/1492992","0","49/8957952","0","-1/80621568"],"3":["0","0","0","-8575/186624","0","2401/1492992","0","-49/2239488","0","7/20155392"],"5":["0","1715/41472","0","2401/497664","0","-343/1492992","0","-5/1492992"],"7":["0","-343/55296","0","-49/248832","0","1/82944"],"9":["0","49/110592","0","-1/110592"]},"3":{"1":["0","0","84035/46656","0","-8575/373248","0","-49/1492992","0","161/80621568","0","-1/17915904"],"3":["-84035/46656","0","-8575/186624","0","-245/1492992","0","-679/40310784","0","1/1119744"],"5":["8575/124416","0","-49/497664","0","301/2239488","0","-7/2985984"],"7":["49/165888","0","-49/497664","0","-1/82944"],"9":["-7/331776","0","1/73728"]},"4":{"1":["0","0","0","0","0","2891/17915904","0","-581/107495424","0","-1/15925248"],"3":["0","0","0","-14455/8957952","0","4067/107495424","0","1/3981312"],"5":["0","2891/1990656","0","4067/35831808","0","7/2654208"],"7":["0","-581/3981312","0","1/442368"],"9":["0","-1/196608"]},"5":{"1":["0","0","-1715/93312","0","2107/5971968","0","35/35831808","0","13/429981696"],"3":["1715/93312","0","2107/2985984","0","175/35831808","0","-245/214990848"],"5":["-2107/1990656","0","35/11943936","0","7/1492992"],"7":["-35/3981312","0","-11/2654208"],"9":["1/1769472"]},"6":{"1":["0","0","0","0","0","-805/644972544","0","23/286654464"],"3":["0","0","0","4025/322486272","0","-161/286654464"],"5":["0","-805/71663616","0","-161/95551488"],"7":["0","23/10616832"]},"7":{"1":["0","0","245/4478976","0","-637/214990848","0","1/31850496"],"3":["-245/4478976","0","-637/107495424","0","5/31850496"],"5":["637/71663616","0","1/10616832"],"7":["-1/3538944"]},"8":{"1":["0","0","0","0","0","-43/6879707136"],"3":["0","0","0","215/3439853568"],"5":["0","-43/764411904"]},"9":{"1":["0","0","175/483729408","0","-11/2293235712"],"3":["-175/483729408","0","-11/1146617856"],"5":["11/764411904"]}},"e_{a,k}(v) by power, polynomial branch":{"0":{"1":["0","0","0","0","0","0","0","0","343/629856","0","-49/5038848"],"3":["0","0","0","0","0","0","-6517/629856","0","49/314928"],"5":["0","0","0","0","3773/69984","0","-343/839808"],"7":["0","0","-343/7776","0","-49/23328"],"9":["0","0","49/20736"]},"1":{"1":["0","0","0","0","0","-84035/746496","0","20923/6718464","0","-833/26873856","0","7/120932352"],"3":["0","0","0","420175/373248","0","-146461/6718464","0","833/6718464","0","-49/30233088"],"5":["0","-84035/82944","0","-146461/2239488","0","5831/4478976","0","35/2239488"],"7":["0","20923/248832","0","833/746496","0","-7/124416"],"9":["0","-833/331776","0","7/165888"]},"10":{"1":["0","0","-7/7739670528"],"3":["7/7739670528"]},"2":{"1":["0","0","0","0","-84035/248832","0","9947/2239488","0","-539/80621568","0","7/26873856"],"3":["0","0","-84035/124416","0","49735/2239488","0","-833/40310784","0","-7/1679616"],"5":["84035/82944","0","9947/746496","0","-245/1119744","0","49/4478976"],"7":["-9947/248832","0","49/497664","0","7/124416"],"9":["49/331776","0","-7/110592"]},"3":{"1":["0","0","0","0","0","-446243/161243136","0","2597/80621568","0","7/23887872"],"3":["0","0","0","2231215/80621568","0","-18179/80621568","0","-7/5971968"],"5":["0","-446243/17915904","0","-18179/26873856","0","-49/3981312"],"7":["0","2597/2985984","0","-7/663552"],"9":["0","7/294912"]},"4":{"1":["0","0","84035/186624","0","-413315/53747712","0","-49/26873856","0","-91/644972544"],"3":["-84035/186624","0","-413315/26873856","0","-245/26873856","0","1715/322486272"],"5":["413315/17915904","0","-49/8957952","0","-49/2239488"],"7":["49/2985984","0","77/3981312"],"9":["-7/2654208"]},"5":{"1":["0","0","0","0","0","6811/1289945088","0","-161/429981696"],"3":["0","0","0","-34055/644972544","0","1127/429981696"],"5":["0","6811/143327232","0","1127/143327232"],"7":["0","-161/15925248"]},"6":{"1":["0","0","95011/40310784","0","17297/1289945088","0","-7/47775744"],"3":["-95011/40310784","0","17297/644972544","0","-35/47775744"],"5":["-17297/429981696","0","-7/15925248"],"7":["7/5308416"]},"7":{"1":["0","0","0","0","0","301/10319560704"],"3":["0","0","0","-1505/5159780352"],"5":["0","301/1146617856"]},"8":{"1":["0","0","-539/322486272","0","77/3439853568"],"3":["539/322486272","0","77/1719926784"],"5":["-77/1146617856"]}},"kappa":"1/28 sqrt(21)","kappa^2":"3/112","lambda_erfc":"1/48","lambda_poly":"1/21","p3_powers":[1,3,5,7,9],"prefactor":"1/896 sqrt(21) pi^(-1)"},"note":"arctan(sqrt15/5) (m = 2) and arctan(1/sqrt7) (m = 4, from lam = 3/16) arise in intermediate terms and cancel"},"scientific_effect":"NONE"}},"jk-init":{"kind":"report","report":{"certified":true,"certified_enclosures_math_200":{"D_4":["14.2187634589773576013064656778740783173406003782976930345926","14.2187634589773576013064656778740783173406042538663501338228"],"c_5_ref":["0.0132193193800849680760334875146413965210528883241234179643777","0.0132193193800849680760334875146413965210528919272763929401087"]},"checks":{"BASE_INTEGRALS_FLOAT":true,"C5_CLOSED":false,"CUMULATIVE_EXACT":true,"D1_EXACT":true,"D2_EXACT":true,"D3_EXACT":false,"D4_FORM":false,"D4_INSIDE_CERTIFIED":false,"GAUSS_MOMENTS":true,"JK_EXACT":false,"LIBRARY_EXACT":true,"MEHTA_Z4_EXACT":false,"MOMENTS_FLOAT":true,"PINNED":false,"SIMPLIFIED_FORM":false,"STRUCTURE":false},"closed_forms":{"D_1":"4/3","D_2":"29/6  +  -1 sqrt(6)","D_3":"34319/3072  +  -958681/57600 pi^(-1)  +  -77357/4608 pi^(-1) arctan(1/2)","D_4":"6695/108  +  -2113/1296 sqrt(3)  +  -352217/36864 sqrt(6)  +  -5619101/746496 sqrt(21) pi^(-1)  +  -405/32 sqrt(6) pi^(-1) arctan(1/4 sqrt(14))  +  -6695/54 pi^(-1) arctan(1/7 sqrt(21))  +  352217/18432 sqrt(6) pi^(-1) arctan(1/7 sqrt(14))  +  2113/648 sqrt(3) pi^(-1) arctan(1/7 sqrt(7))  +  10399/648 sqrt(3) pi^(-1) arctan(1/3 sqrt(7))","D_4_simplified":"6695/54 - (405/32) sqrt6 - (1/pi) [ (1375/72) sqrt21 + (6695/27) arctan(sqrt21/7) + (405/16) sqrt6 arctan(sqrt14/14) ]   (arctan(sqrt14/4) - arctan(sqrt14/7) = arctan(1/sqrt14))","Z_4":"768 pi^(1)  +  272/3 sqrt(3) pi^(1)","c_5_ref":"Gamma(7/6) (3/2)^(1/3) D_4 / (12 sqrt3 pi^(7/2))","erfc_branch_of_D_4":"6695/108  +  -2113/1296 sqrt(3)  +  -134080752109/19327352832 sqrt(6)  +  -21213030155016709/1223059046400000 sqrt(21) pi^(-1)  +  -6066622595/603979776 sqrt(6) pi^(-1) arctan(1/4 sqrt(14))  +  -6695/54 pi^(-1) arctan(1/7 sqrt(21))  +  134080752109/9663676416 sqrt(6) pi^(-1) arctan(1/7 sqrt(14))  +  2113/648 sqrt(3) pi^(-1) arctan(1/7 sqrt(7))  +  10399/648 sqrt(3) pi^(-1) arctan(1/3 sqrt(7))","polynomial_branch_of_D_4":"-5620266043/2147483648 sqrt(6)  +  12006695076616709/1223059046400000 sqrt(21) pi^(-1)  +  -1577496445/603979776 sqrt(6) pi^(-1) arctan(1/4 sqrt(14))  +  5620266043/1073741824 sqrt(6) pi^(-1) arctan(1/7 sqrt(14))"},"exact_relations":{"D_4 integral":"sqrt21/(2688 pi) [ (sqrt21 sqrt(pi)/6) int e^{-T^2/48} erfc(sqrt21 T/28) sum_a T^a L_a^E + int e^{-T^2/21} sum_a T^a L_a^R ]","G_n(mu, T)":"G_0 = (1/2) sqrt(pi/mu) erf(sqrt(mu) T); G_1 = (1 - e^{-mu T^2})/(2 mu); G_n = [(n-1) G_{n-2} - T^{n-1} e^{-mu T^2}]/(2 mu)","L_a":"int_0^T e^{-v^2/24} sum_k e_{a,k}(v) J_k(v) dv, k odd, J_k = B_k + C_k e^{-v^2/8}: incomplete Gaussian moments with mu = 1/24, 1/6","M_0(lam, 1)":"(1/2) sqrt(pi/lam)","M_0(lam, erf(u .) erf(v .))":"arctan(u v/sqrt(lam (lam + u^2 + v^2)))/sqrt(pi lam)","M_0(lam, erf(u .))":"arctan(u/sqrt lam)/sqrt(pi lam)","M_1(lam, 1)":"1/(2 lam)","M_n(lam, f), n >= 1":"[(n-1) M_{n-2}(lam, f) + sum_{u in f} (2u/sqrt pi) M_{n-1}(lam + u^2, f minus u)]/(2 lam)","Z_4":"4! sqrt(pi) (1/6) int e^{-T^2/48} int_0^T e^{-v^2/24} int_0^v V e^{-p3^2/8} = 1536 pi (exact, rule MEHTA_Z4_EXACT)"},"interval_evaluation":{"(3/2)^(1/3)":["1.14471424255333186780804221193967700891590692078793107209905","1.14471424255333186780804221193967700891590692078793107209906"],"D_4":["8.84478730701528756274241264993043482508097950641261268386251","8.84478730701528756274241264993043482508097950641261268389181"],"D_4_simplified":["14.2187634589773576013064656778740783173406023160820215841883","14.2187634589773576013064656778740783173406023160820215842281"],"Gamma(7/6)":["0.927719333630039200708349482534621018566466519145475576700351","0.927719333630039200708349482534621018566466519145475576939062"],"arctan(sqrt14/14)":["0.261157410903024311262049412927851638004440468771565909792296","0.261157410903024311262049412927851638004440468771565909792486"],"arctan(sqrt14/4)":["0.752040089192335679842958294184691084003367627670504438870380","0.752040089192335679842958294184691084003367627670504438870582"],"arctan(sqrt14/7)":["0.490882678289311368580908881256839445998927158898938529077984","0.490882678289311368580908881256839445998927158898938529078176"],"arctan(sqrt21/7)":["0.579639740363704299923291897011207918621441782280263529351672","0.579639740363704299923291897011207918621441782280263529351866"],"c_5_ref":["0.00822308273132817123633738700568825436855159473778565763919190","0.00822308273132817123633738700568825436855159473778565764134220"],"pi":["3.14159265358979323846264338327950288419716939937510582097457","3.14159265358979323846264338327950288419716939937510582097528"]},"mutant":"jk-init","object":"CL-SIDE24-D4-CLOSED-FORM-20260930-v1","passed":false,"reduction_data":{"atoms_created_by_the_derivations":["1","arctan(1/5 sqrt(15))","arctan(1/2 sqrt(1))","arctan(1/4 sqrt(14))","arctan(1/7 sqrt(21))","arctan(1/7 sqrt(14))","arctan(1/7 sqrt(7))","arctan(1/3 sqrt(7))"],"m=2":{"kappa":"1/20 sqrt(30)","lambda_erfc":"1/8","lambda_poly":"1/5"},"m=3":{"kappa":"1/12 sqrt(6)","lambda_erfc":"1/24","lambda_poly":"1/12"},"m=4":{"G_0_constant":"1/6 sqrt(21) pi^(1/2)","J_k = A_k sqrt(2 pi) erf(s/(2 sqrt2)) + B_k + C_k(s) e^{-s^2/8}":{"0":{"A":"1","B":"0","C":[]},"1":{"A":"0","B":"2","C":["-2"]},"2":{"A":"4","B":"0","C":["0","-4"]},"3":{"A":"0","B":"16","C":["-16","0","-4"]},"4":{"A":"48","B":"0","C":["0","-48","0","-4"]},"5":{"A":"0","B":"256","C":["-256","0","-64","0","-4"]},"6":{"A":"960","B":"0","C":["0","-960","0","-80","0","-4"]},"7":{"A":"0","B":"6144","C":["-6144","0","-1536","0","-96","0","-4"]},"8":{"A":"26880","B":"0","C":["0","-26880","0","-2240","0","-112","0","-4"]},"9":{"A":"0","B":"196608","C":["-196608","0","-49152","0","-3072","0","-128","0","-4"]}},"T_powers_erfc":[0,1,2,3,4,5,6,7,8,9,11],"T_powers_poly":[0,1,2,3,4,5,6,7,8,10],"alpha":"3/7","beta":"3/14 T","e_{a,k}(v) by power, erfc branch":{"0":{"1":["0","0","0","0","0","-84035/186624","0","1715/186624","0","-49/746496","0","-7/30233088"],"3":["0","0","0","420175/93312","0","-12005/186624","0","49/186624","0","49/7558272"],"5":["0","-84035/20736","0","-12005/62208","0","343/124416","0","-35/559872"],"7":["0","1715/6912","0","49/20736","0","7/31104"],"9":["0","-49/9216","0","-7/41472"]},"1":{"1":["0","0","0","0","-84035/62208","0","1715/62208","0","-539/2239488","0","7/6718464"],"3":["0","0","-84035/31104","0","8575/62208","0","-833/1119744","0","-7/419904"],"5":["84035/20736","0","1715/20736","0","-245/31104","0","49/1119744"],"7":["-1715/6912","0","49/13824","0","7/31104"],"9":["49/9216","0","-7/27648"]},"11":{"1":["0","0","1/5159780352"],"3":["-1/5159780352"]},"2":{"1":["0","0","0","0","0","1715/373248","0","-343/1492992","0","49/8957952","0","-1/80621568"],"3":["0","0","0","-8575/186624","0","2401/1492992","0","-49/2239488","0","7/20155392"],"5":["0","1715/41472","0","2401/497664","0","-343/1492992","0","-5/1492992"],"7":["0","-343/55296","0","-49/248832","0","1/82944"],"9":["0","49/110592","0","-1/110592"]},"3":{"1":["0","0","84035/46656","0","-8575/373248","0","-49/1492992","0","161/80621568","0","-1/17915904"],"3":["-84035/46656","0","-8575/186624","0","-245/1492992","0","-679/40310784","0","1/1119744"],"5":["8575/124416","0","-49/497664","0","301/2239488","0","-7/2985984"],"7":["49/165888","0","-49/497664","0","-1/82944"],"9":["-7/331776","0","1/73728"]},"4":{"1":["0","0","0","0","0","2891/17915904","0","-581/107495424","0","-1/15925248"],"3":["0","0","0","-14455/8957952","0","4067/107495424","0","1/3981312"],"5":["0","2891/1990656","0","4067/35831808","0","7/2654208"],"7":["0","-581/3981312","0","1/442368"],"9":["0","-1/196608"]},"5":{"1":["0","0","-1715/93312","0","2107/5971968","0","35/35831808","0","13/429981696"],"3":["1715/93312","0","2107/2985984","0","175/35831808","0","-245/214990848"],"5":["-2107/1990656","0","35/11943936","0","7/1492992"],"7":["-35/3981312","0","-11/2654208"],"9":["1/1769472"]},"6":{"1":["0","0","0","0","0","-805/644972544","0","23/286654464"],"3":["0","0","0","4025/322486272","0","-161/286654464"],"5":["0","-805/71663616","0","-161/95551488"],"7":["0","23/10616832"]},"7":{"1":["0","0","245/4478976","0","-637/214990848","0","1/31850496"],"3":["-245/4478976","0","-637/107495424","0","5/31850496"],"5":["637/71663616","0","1/10616832"],"7":["-1/3538944"]},"8":{"1":["0","0","0","0","0","-43/6879707136"],"3":["0","0","0","215/3439853568"],"5":["0","-43/764411904"]},"9":{"1":["0","0","175/483729408","0","-11/2293235712"],"3":["-175/483729408","0","-11/1146617856"],"5":["11/764411904"]}},"e_{a,k}(v) by power, polynomial branch":{"0":{"1":["0","0","0","0","0","0","0","0","343/629856","0","-49/5038848"],"3":["0","0","0","0","0","0","-6517/629856","0","49/314928"],"5":["0","0","0","0","3773/69984","0","-343/839808"],"7":["0","0","-343/7776","0","-49/23328"],"9":["0","0","49/20736"]},"1":{"1":["0","0","0","0","0","-84035/746496","0","20923/6718464","0","-833/26873856","0","7/120932352"],"3":["0","0","0","420175/373248","0","-146461/6718464","0","833/6718464","0","-49/30233088"],"5":["0","-84035/82944","0","-146461/2239488","0","5831/4478976","0","35/2239488"],"7":["0","20923/248832","0","833/746496","0","-7/124416"],"9":["0","-833/331776","0","7/165888"]},"10":{"1":["0","0","-7/7739670528"],"3":["7/7739670528"]},"2":{"1":["0","0","0","0","-84035/248832","0","9947/2239488","0","-539/80621568","0","7/26873856"],"3":["0","0","-84035/124416","0","49735/2239488","0","-833/40310784","0","-7/1679616"],"5":["84035/82944","0","9947/746496","0","-245/1119744","0","49/4478976"],"7":["-9947/248832","0","49/497664","0","7/124416"],"9":["49/331776","0","-7/110592"]},"3":{"1":["0","0","0","0","0","-446243/161243136","0","2597/80621568","0","7/23887872"],"3":["0","0","0","2231215/80621568","0","-18179/80621568","0","-7/5971968"],"5":["0","-446243/17915904","0","-18179/26873856","0","-49/3981312"],"7":["0","2597/2985984","0","-7/663552"],"9":["0","7/294912"]},"4":{"1":["0","0","84035/186624","0","-413315/53747712","0","-49/26873856","0","-91/644972544"],"3":["-84035/186624","0","-413315/26873856","0","-245/26873856","0","1715/322486272"],"5":["413315/17915904","0","-49/8957952","0","-49/2239488"],"7":["49/2985984","0","77/3981312"],"9":["-7/2654208"]},"5":{"1":["0","0","0","0","0","6811/1289945088","0","-161/429981696"],"3":["0","0","0","-34055/644972544","0","1127/429981696"],"5":["0","6811/143327232","0","1127/143327232"],"7":["0","-161/15925248"]},"6":{"1":["0","0","95011/40310784","0","17297/1289945088","0","-7/47775744"],"3":["-95011/40310784","0","17297/644972544","0","-35/47775744"],"5":["-17297/429981696","0","-7/15925248"],"7":["7/5308416"]},"7":{"1":["0","0","0","0","0","301/10319560704"],"3":["0","0","0","-1505/5159780352"],"5":["0","301/1146617856"]},"8":{"1":["0","0","-539/322486272","0","77/3439853568"],"3":["539/322486272","0","77/1719926784"],"5":["-77/1146617856"]}},"kappa":"1/28 sqrt(21)","kappa^2":"3/112","lambda_erfc":"1/48","lambda_poly":"1/21","p3_powers":[1,3,5,7,9],"prefactor":"1/2688 sqrt(21) pi^(-1)"},"note":"arctan(sqrt15/5) (m = 2) and arctan(1/sqrt7) (m = 4, from lam = 3/16) arise in intermediate terms and cancel"},"scientific_effect":"NONE"}},"mehta":{"kind":"report","report":{"certified":true,"certified_enclosures_math_200":{"D_4":["14.2187634589773576013064656778740783173406003782976930345926","14.2187634589773576013064656778740783173406042538663501338228"],"c_5_ref":["0.0132193193800849680760334875146413965210528883241234179643777","0.0132193193800849680760334875146413965210528919272763929401087"]},"checks":{"BASE_INTEGRALS_FLOAT":true,"C5_CLOSED":false,"CUMULATIVE_EXACT":true,"D1_EXACT":true,"D2_EXACT":true,"D3_EXACT":true,"D4_FORM":false,"D4_INSIDE_CERTIFIED":false,"GAUSS_MOMENTS":true,"JK_EXACT":true,"LIBRARY_EXACT":true,"MEHTA_Z4_EXACT":false,"MOMENTS_FLOAT":true,"PINNED":false,"SIMPLIFIED_FORM":false,"STRUCTURE":true},"closed_forms":{"D_1":"4/3","D_2":"29/6  +  -1 sqrt(6)","D_3":"50/3  +  -76/3 pi^(-1)  +  -200/9 pi^(-1) arctan(1/2)","D_4":"1339/324 pi^(-1/2)  +  -27/64 sqrt(6) pi^(-1/2)  +  -275/432 sqrt(21) pi^(-3/2)  +  -27/32 sqrt(6) pi^(-3/2) arctan(1/4 sqrt(14))  +  -1339/162 pi^(-3/2) arctan(1/7 sqrt(21))  +  27/32 sqrt(6) pi^(-3/2) arctan(1/7 sqrt(14))","D_4_simplified":"6695/54 - (405/32) sqrt6 - (1/pi) [ (1375/72) sqrt21 + (6695/27) arctan(sqrt21/7) + (405/16) sqrt6 arctan(sqrt14/14) ]   (arctan(sqrt14/4) - arctan(sqrt14/7) = arctan(1/sqrt14))","Z_4":"1536 pi^(1)","c_5_ref":"Gamma(7/6) (3/2)^(1/3) D_4 / (12 sqrt3 pi^(7/2))","erfc_branch_of_D_4":"1339/324 pi^(-1/2)  +  -1213324519/3623878656 sqrt(6) pi^(-1/2)  +  -46299277168087/42467328000000 sqrt(21) pi^(-3/2)  +  -1213324519/1811939328 sqrt(6) pi^(-3/2) arctan(1/4 sqrt(14))  +  -1339/162 pi^(-3/2) arctan(1/7 sqrt(21))  +  1213324519/1811939328 sqrt(6) pi^(-3/2) arctan(1/7 sqrt(14))","polynomial_branch_of_D_4":"-315499289/3623878656 sqrt(6) pi^(-1/2)  +  19265677168087/42467328000000 sqrt(21) pi^(-3/2)  +  -315499289/1811939328 sqrt(6) pi^(-3/2) arctan(1/4 sqrt(14))  +  315499289/1811939328 sqrt(6) pi^(-3/2) arctan(1/7 sqrt(14))"},"exact_relations":{"D_4 integral":"sqrt21/(2688 pi) [ (sqrt21 sqrt(pi)/6) int e^{-T^2/48} erfc(sqrt21 T/28) sum_a T^a L_a^E + int e^{-T^2/21} sum_a T^a L_a^R ]","G_n(mu, T)":"G_0 = (1/2) sqrt(pi/mu) erf(sqrt(mu) T); G_1 = (1 - e^{-mu T^2})/(2 mu); G_n = [(n-1) G_{n-2} - T^{n-1} e^{-mu T^2}]/(2 mu)","L_a":"int_0^T e^{-v^2/24} sum_k e_{a,k}(v) J_k(v) dv, k odd, J_k = B_k + C_k e^{-v^2/8}: incomplete Gaussian moments with mu = 1/24, 1/6","M_0(lam, 1)":"(1/2) sqrt(pi/lam)","M_0(lam, erf(u .) erf(v .))":"arctan(u v/sqrt(lam (lam + u^2 + v^2)))/sqrt(pi lam)","M_0(lam, erf(u .))":"arctan(u/sqrt lam)/sqrt(pi lam)","M_1(lam, 1)":"1/(2 lam)","M_n(lam, f), n >= 1":"[(n-1) M_{n-2}(lam, f) + sum_{u in f} (2u/sqrt pi) M_{n-1}(lam + u^2, f minus u)]/(2 lam)","Z_4":"4! sqrt(pi) (1/6) int e^{-T^2/48} int_0^T e^{-v^2/24} int_0^v V e^{-p3^2/8} = 1536 pi (exact, rule MEHTA_Z4_EXACT)"},"interval_evaluation":{"(3/2)^(1/3)":["1.14471424255333186780804221193967700891590692078793107209905","1.14471424255333186780804221193967700891590692078793107209906"],"D_4":["0.267402607816149668960181444861623130944715166382400137684633","0.267402607816149668960181444861623130944715166382400137686207"],"D_4_simplified":["14.2187634589773576013064656778740783173406023160820215841883","14.2187634589773576013064656778740783173406023160820215842281"],"Gamma(7/6)":["0.927719333630039200708349482534621018566466519145475576700351","0.927719333630039200708349482534621018566466519145475576939062"],"arctan(sqrt14/14)":["0.261157410903024311262049412927851638004440468771565909792296","0.261157410903024311262049412927851638004440468771565909792486"],"arctan(sqrt14/4)":["0.752040089192335679842958294184691084003367627670504438870380","0.752040089192335679842958294184691084003367627670504438870582"],"arctan(sqrt14/7)":["0.490882678289311368580908881256839445998927158898938529077984","0.490882678289311368580908881256839445998927158898938529078176"],"arctan(sqrt21/7)":["0.579639740363704299923291897011207918621441782280263529351672","0.579639740363704299923291897011207918621441782280263529351866"],"c_5_ref":["0.000248606743194497398100059172018518446971493259712893083403502","0.000248606743194497398100059172018518446971493259712893083469152"],"pi":["3.14159265358979323846264338327950288419716939937510582097457","3.14159265358979323846264338327950288419716939937510582097528"]},"mutant":"mehta","object":"CL-SIDE24-D4-CLOSED-FORM-20260930-v1","passed":false,"reduction_data":{"atoms_created_by_the_derivations":["1","arctan(1/5 sqrt(15))","arctan(1/2 sqrt(1))","arctan(1/4 sqrt(14))","arctan(1/7 sqrt(21))","arctan(1/7 sqrt(14))","arctan(1/7 sqrt(7))"],"m=2":{"kappa":"1/20 sqrt(30)","lambda_erfc":"1/8","lambda_poly":"1/5"},"m=3":{"kappa":"1/12 sqrt(6)","lambda_erfc":"1/24","lambda_poly":"1/12"},"m=4":{"G_0_constant":"1/6 sqrt(21) pi^(1/2)","J_k = A_k sqrt(2 pi) erf(s/(2 sqrt2)) + B_k + C_k(s) e^{-s^2/8}":{"0":{"A":"1","B":"0","C":[]},"1":{"A":"0","B":"4","C":["-4"]},"2":{"A":"4","B":"0","C":["0","-4"]},"3":{"A":"0","B":"32","C":["-32","0","-4"]},"4":{"A":"48","B":"0","C":["0","-48","0","-4"]},"5":{"A":"0","B":"512","C":["-512","0","-64","0","-4"]},"6":{"A":"960","B":"0","C":["0","-960","0","-80","0","-4"]},"7":{"A":"0","B":"12288","C":["-12288","0","-1536","0","-96","0","-4"]},"8":{"A":"26880","B":"0","C":["0","-26880","0","-2240","0","-112","0","-4"]},"9":{"A":"0","B":"393216","C":["-393216","0","-49152","0","-3072","0","-128","0","-4"]}},"T_powers_erfc":[0,1,2,3,4,5,6,7,8,9,11],"T_powers_poly":[0,1,2,3,4,5,6,7,8,10],"alpha":"3/7","beta":"3/14 T","e_{a,k}(v) by power, erfc branch":{"0":{"1":["0","0","0","0","0","-84035/186624","0","1715/186624","0","-49/746496","0","-7/30233088"],"3":["0","0","0","420175/93312","0","-12005/186624","0","49/186624","0","49/7558272"],"5":["0","-84035/20736","0","-12005/62208","0","343/124416","0","-35/559872"],"7":["0","1715/6912","0","49/20736","0","7/31104"],"9":["0","-49/9216","0","-7/41472"]},"1":{"1":["0","0","0","0","-84035/62208","0","1715/62208","0","-539/2239488","0","7/6718464"],"3":["0","0","-84035/31104","0","8575/62208","0","-833/1119744","0","-7/419904"],"5":["84035/20736","0","1715/20736","0","-245/31104","0","49/1119744"],"7":["-1715/6912","0","49/13824","0","7/31104"],"9":["49/9216","0","-7/27648"]},"11":{"1":["0","0","1/5159780352"],"3":["-1/5159780352"]},"2":{"1":["0","0","0","0","0","1715/373248","0","-343/1492992","0","49/8957952","0","-1/80621568"],"3":["0","0","0","-8575/186624","0","2401/1492992","0","-49/2239488","0","7/20155392"],"5":["0","1715/41472","0","2401/497664","0","-343/1492992","0","-5/1492992"],"7":["0","-343/55296","0","-49/248832","0","1/82944"],"9":["0","49/110592","0","-1/110592"]},"3":{"1":["0","0","84035/46656","0","-8575/373248","0","-49/1492992","0","161/80621568","0","-1/17915904"],"3":["-84035/46656","0","-8575/186624","0","-245/1492992","0","-679/40310784","0","1/1119744"],"5":["8575/124416","0","-49/497664","0","301/2239488","0","-7/2985984"],"7":["49/165888","0","-49/497664","0","-1/82944"],"9":["-7/331776","0","1/73728"]},"4":{"1":["0","0","0","0","0","2891/17915904","0","-581/107495424","0","-1/15925248"],"3":["0","0","0","-14455/8957952","0","4067/107495424","0","1/3981312"],"5":["0","2891/1990656","0","4067/35831808","0","7/2654208"],"7":["0","-581/3981312","0","1/442368"],"9":["0","-1/196608"]},"5":{"1":["0","0","-1715/93312","0","2107/5971968","0","35/35831808","0","13/429981696"],"3":["1715/93312","0","2107/2985984","0","175/35831808","0","-245/214990848"],"5":["-2107/1990656","0","35/11943936","0","7/1492992"],"7":["-35/3981312","0","-11/2654208"],"9":["1/1769472"]},"6":{"1":["0","0","0","0","0","-805/644972544","0","23/286654464"],"3":["0","0","0","4025/322486272","0","-161/286654464"],"5":["0","-805/71663616","0","-161/95551488"],"7":["0","23/10616832"]},"7":{"1":["0","0","245/4478976","0","-637/214990848","0","1/31850496"],"3":["-245/4478976","0","-637/107495424","0","5/31850496"],"5":["637/71663616","0","1/10616832"],"7":["-1/3538944"]},"8":{"1":["0","0","0","0","0","-43/6879707136"],"3":["0","0","0","215/3439853568"],"5":["0","-43/764411904"]},"9":{"1":["0","0","175/483729408","0","-11/2293235712"],"3":["-175/483729408","0","-11/1146617856"],"5":["11/764411904"]}},"e_{a,k}(v) by power, polynomial branch":{"0":{"1":["0","0","0","0","0","0","0","0","343/629856","0","-49/5038848"],"3":["0","0","0","0","0","0","-6517/629856","0","49/314928"],"5":["0","0","0","0","3773/69984","0","-343/839808"],"7":["0","0","-343/7776","0","-49/23328"],"9":["0","0","49/20736"]},"1":{"1":["0","0","0","0","0","-84035/746496","0","20923/6718464","0","-833/26873856","0","7/120932352"],"3":["0","0","0","420175/373248","0","-146461/6718464","0","833/6718464","0","-49/30233088"],"5":["0","-84035/82944","0","-146461/2239488","0","5831/4478976","0","35/2239488"],"7":["0","20923/248832","0","833/746496","0","-7/124416"],"9":["0","-833/331776","0","7/165888"]},"10":{"1":["0","0","-7/7739670528"],"3":["7/7739670528"]},"2":{"1":["0","0","0","0","-84035/248832","0","9947/2239488","0","-539/80621568","0","7/26873856"],"3":["0","0","-84035/124416","0","49735/2239488","0","-833/40310784","0","-7/1679616"],"5":["84035/82944","0","9947/746496","0","-245/1119744","0","49/4478976"],"7":["-9947/248832","0","49/497664","0","7/124416"],"9":["49/331776","0","-7/110592"]},"3":{"1":["0","0","0","0","0","-446243/161243136","0","2597/80621568","0","7/23887872"],"3":["0","0","0","2231215/80621568","0","-18179/80621568","0","-7/5971968"],"5":["0","-446243/17915904","0","-18179/26873856","0","-49/3981312"],"7":["0","2597/2985984","0","-7/663552"],"9":["0","7/294912"]},"4":{"1":["0","0","84035/186624","0","-413315/53747712","0","-49/26873856","0","-91/644972544"],"3":["-84035/186624","0","-413315/26873856","0","-245/26873856","0","1715/322486272"],"5":["413315/17915904","0","-49/8957952","0","-49/2239488"],"7":["49/2985984","0","77/3981312"],"9":["-7/2654208"]},"5":{"1":["0","0","0","0","0","6811/1289945088","0","-161/429981696"],"3":["0","0","0","-34055/644972544","0","1127/429981696"],"5":["0","6811/143327232","0","1127/143327232"],"7":["0","-161/15925248"]},"6":{"1":["0","0","95011/40310784","0","17297/1289945088","0","-7/47775744"],"3":["-95011/40310784","0","17297/644972544","0","-35/47775744"],"5":["-17297/429981696","0","-7/15925248"],"7":["7/5308416"]},"7":{"1":["0","0","0","0","0","301/10319560704"],"3":["0","0","0","-1505/5159780352"],"5":["0","301/1146617856"]},"8":{"1":["0","0","-539/322486272","0","77/3439853568"],"3":["539/322486272","0","77/1719926784"],"5":["-77/1146617856"]}},"kappa":"1/28 sqrt(21)","kappa^2":"3/112","lambda_erfc":"1/48","lambda_poly":"1/21","p3_powers":[1,3,5,7,9],"prefactor":"1/80640 sqrt(21) pi^(-3/2)"},"note":"arctan(sqrt15/5) (m = 2) and arctan(1/sqrt7) (m = 4, from lam = 3/16) arise in intermediate terms and cancel"},"scientific_effect":"NONE"}},"one-erf":{"kind":"report","report":{"certified":true,"certified_enclosures_math_200":{"D_4":["14.2187634589773576013064656778740783173406003782976930345926","14.2187634589773576013064656778740783173406042538663501338228"],"c_5_ref":["0.0132193193800849680760334875146413965210528883241234179643777","0.0132193193800849680760334875146413965210528919272763929401087"]},"checks":{"BASE_INTEGRALS_FLOAT":true,"C5_CLOSED":false,"CUMULATIVE_EXACT":true,"D1_EXACT":true,"D2_EXACT":true,"D3_EXACT":false,"D4_FORM":false,"D4_INSIDE_CERTIFIED":false,"GAUSS_MOMENTS":true,"JK_EXACT":true,"LIBRARY_EXACT":true,"MEHTA_Z4_EXACT":true,"MOMENTS_FLOAT":false,"PINNED":false,"SIMPLIFIED_FORM":false,"STRUCTURE":false},"closed_forms":{"D_1":"4/3","D_2":"29/6  +  -1 sqrt(6)","D_3":"-76/3 pi^(-1)  +  200/9 pi^(-1) arctan(1/12 sqrt(6))  +  200/9 pi^(-1) arctan(1/3 sqrt(6))","D_4":"-405/16 sqrt(6)  +  -1375/72 sqrt(21) pi^(-1)  +  405/16 sqrt(6) pi^(-1) arctan(2/21 sqrt(6))  +  6695/27 pi^(-1) arctan(1/12 sqrt(21))  +  405/16 sqrt(6) pi^(-1) arctan(1/21 sqrt(6))","D_4_simplified":"6695/54 - (405/32) sqrt6 - (1/pi) [ (1375/72) sqrt21 + (6695/27) arctan(sqrt21/7) + (405/16) sqrt6 arctan(sqrt14/14) ]   (arctan(sqrt14/4) - arctan(sqrt14/7) = arctan(1/sqrt14))","Z_4":"1536 pi^(1)","c_5_ref":"Gamma(7/6) (3/2)^(1/3) D_4 / (12 sqrt3 pi^(7/2))","erfc_branch_of_D_4":"-6066622595/301989888 sqrt(6)  +  -46299277168087/1415577600000 sqrt(21) pi^(-1)  +  6066622595/301989888 sqrt(6) pi^(-1) arctan(2/21 sqrt(6))  +  6695/27 pi^(-1) arctan(1/12 sqrt(21))  +  6066622595/301989888 sqrt(6) pi^(-1) arctan(1/21 sqrt(6))","polynomial_branch_of_D_4":"-1577496445/301989888 sqrt(6)  +  19265677168087/1415577600000 sqrt(21) pi^(-1)  +  1577496445/301989888 sqrt(6) pi^(-1) arctan(2/21 sqrt(6))  +  1577496445/301989888 sqrt(6) pi^(-1) arctan(1/21 sqrt(6))"},"exact_relations":{"D_4 integral":"sqrt21/(2688 pi) [ (sqrt21 sqrt(pi)/6) int e^{-T^2/48} erfc(sqrt21 T/28) sum_a T^a L_a^E + int e^{-T^2/21} sum_a T^a L_a^R ]","G_n(mu, T)":"G_0 = (1/2) sqrt(pi/mu) erf(sqrt(mu) T); G_1 = (1 - e^{-mu T^2})/(2 mu); G_n = [(n-1) G_{n-2} - T^{n-1} e^{-mu T^2}]/(2 mu)","L_a":"int_0^T e^{-v^2/24} sum_k e_{a,k}(v) J_k(v) dv, k odd, J_k = B_k + C_k e^{-v^2/8}: incomplete Gaussian moments with mu = 1/24, 1/6","M_0(lam, 1)":"(1/2) sqrt(pi/lam)","M_0(lam, erf(u .) erf(v .))":"arctan(u v/sqrt(lam (lam + u^2 + v^2)))/sqrt(pi lam)","M_0(lam, erf(u .))":"arctan(u/sqrt lam)/sqrt(pi lam)","M_1(lam, 1)":"1/(2 lam)","M_n(lam, f), n >= 1":"[(n-1) M_{n-2}(lam, f) + sum_{u in f} (2u/sqrt pi) M_{n-1}(lam + u^2, f minus u)]/(2 lam)","Z_4":"4! sqrt(pi) (1/6) int e^{-T^2/48} int_0^T e^{-v^2/24} int_0^v V e^{-p3^2/8} = 1536 pi (exact, rule MEHTA_Z4_EXACT)"},"interval_evaluation":{"(3/2)^(1/3)":["1.14471424255333186780804221193967700891590692078793107209905","1.14471424255333186780804221193967700891590692078793107209906"],"D_4":["-54.2519676602371235765747951797627856893696837993340662541381","-54.2519676602371235765747951797627856893696837993340662541032"],"D_4_simplified":["14.2187634589773576013064656778740783173406023160820215841883","14.2187634589773576013064656778740783173406023160820215842281"],"Gamma(7/6)":["0.927719333630039200708349482534621018566466519145475576700351","0.927719333630039200708349482534621018566466519145475576939062"],"arctan(sqrt14/14)":["0.261157410903024311262049412927851638004440468771565909792296","0.261157410903024311262049412927851638004440468771565909792486"],"arctan(sqrt14/4)":["0.752040089192335679842958294184691084003367627670504438870380","0.752040089192335679842958294184691084003367627670504438870582"],"arctan(sqrt14/7)":["0.490882678289311368580908881256839445998927158898938529077984","0.490882678289311368580908881256839445998927158898938529078176"],"arctan(sqrt21/7)":["0.579639740363704299923291897011207918621441782280263529351672","0.579639740363704299923291897011207918621441782280263529351866"],"c_5_ref":["-0.0504385693993601443596715824630349784990297699826173858756195","-0.0504385693993601443596715824630349784990297699826173858625645"],"pi":["3.14159265358979323846264338327950288419716939937510582097457","3.14159265358979323846264338327950288419716939937510582097528"]},"mutant":"one-erf","object":"CL-SIDE24-D4-CLOSED-FORM-20260930-v1","passed":false,"reduction_data":{"atoms_created_by_the_derivations":["1","arctan(1/12 sqrt(30))","arctan(1/12 sqrt(6))","arctan(1/3 sqrt(6))","arctan(2/21 sqrt(6))","arctan(1/12 sqrt(21))","arctan(1/21 sqrt(6))","arctan(4/21 sqrt(21))"],"m=2":{"kappa":"1/20 sqrt(30)","lambda_erfc":"1/8","lambda_poly":"1/5"},"m=3":{"kappa":"1/12 sqrt(6)","lambda_erfc":"1/24","lambda_poly":"1/12"},"m=4":{"G_0_constant":"1/6 sqrt(21) pi^(1/2)","J_k = A_k sqrt(2 pi) erf(s/(2 sqrt2)) + B_k + C_k(s) e^{-s^2/8}":{"0":{"A":"1","B":"0","C":[]},"1":{"A":"0","B":"4","C":["-4"]},"2":{"A":"4","B":"0","C":["0","-4"]},"3":{"A":"0","B":"32","C":["-32","0","-4"]},"4":{"A":"48","B":"0","C":["0","-48","0","-4"]},"5":{"A":"0","B":"512","C":["-512","0","-64","0","-4"]},"6":{"A":"960","B":"0","C":["0","-960","0","-80","0","-4"]},"7":{"A":"0","B":"12288","C":["-12288","0","-1536","0","-96","0","-4"]},"8":{"A":"26880","B":"0","C":["0","-26880","0","-2240","0","-112","0","-4"]},"9":{"A":"0","B":"393216","C":["-393216","0","-49152","0","-3072","0","-128","0","-4"]}},"T_powers_erfc":[0,1,2,3,4,5,6,7,8,9,11],"T_powers_poly":[0,1,2,3,4,5,6,7,8,10],"alpha":"3/7","beta":"3/14 T","e_{a,k}(v) by power, erfc branch":{"0":{"1":["0","0","0","0","0","-84035/186624","0","1715/186624","0","-49/746496","0","-7/30233088"],"3":["0","0","0","420175/93312","0","-12005/186624","0","49/186624","0","49/7558272"],"5":["0","-84035/20736","0","-12005/62208","0","343/124416","0","-35/559872"],"7":["0","1715/6912","0","49/20736","0","7/31104"],"9":["0","-49/9216","0","-7/41472"]},"1":{"1":["0","0","0","0","-84035/62208","0","1715/62208","0","-539/2239488","0","7/6718464"],"3":["0","0","-84035/31104","0","8575/62208","0","-833/1119744","0","-7/419904"],"5":["84035/20736","0","1715/20736","0","-245/31104","0","49/1119744"],"7":["-1715/6912","0","49/13824","0","7/31104"],"9":["49/9216","0","-7/27648"]},"11":{"1":["0","0","1/5159780352"],"3":["-1/5159780352"]},"2":{"1":["0","0","0","0","0","1715/373248","0","-343/1492992","0","49/8957952","0","-1/80621568"],"3":["0","0","0","-8575/186624","0","2401/1492992","0","-49/2239488","0","7/20155392"],"5":["0","1715/41472","0","2401/497664","0","-343/1492992","0","-5/1492992"],"7":["0","-343/55296","0","-49/248832","0","1/82944"],"9":["0","49/110592","0","-1/110592"]},"3":{"1":["0","0","84035/46656","0","-8575/373248","0","-49/1492992","0","161/80621568","0","-1/17915904"],"3":["-84035/46656","0","-8575/186624","0","-245/1492992","0","-679/40310784","0","1/1119744"],"5":["8575/124416","0","-49/497664","0","301/2239488","0","-7/2985984"],"7":["49/165888","0","-49/497664","0","-1/82944"],"9":["-7/331776","0","1/73728"]},"4":{"1":["0","0","0","0","0","2891/17915904","0","-581/107495424","0","-1/15925248"],"3":["0","0","0","-14455/8957952","0","4067/107495424","0","1/3981312"],"5":["0","2891/1990656","0","4067/35831808","0","7/2654208"],"7":["0","-581/3981312","0","1/442368"],"9":["0","-1/196608"]},"5":{"1":["0","0","-1715/93312","0","2107/5971968","0","35/35831808","0","13/429981696"],"3":["1715/93312","0","2107/2985984","0","175/35831808","0","-245/214990848"],"5":["-2107/1990656","0","35/11943936","0","7/1492992"],"7":["-35/3981312","0","-11/2654208"],"9":["1/1769472"]},"6":{"1":["0","0","0","0","0","-805/644972544","0","23/286654464"],"3":["0","0","0","4025/322486272","0","-161/286654464"],"5":["0","-805/71663616","0","-161/95551488"],"7":["0","23/10616832"]},"7":{"1":["0","0","245/4478976","0","-637/214990848","0","1/31850496"],"3":["-245/4478976","0","-637/107495424","0","5/31850496"],"5":["637/71663616","0","1/10616832"],"7":["-1/3538944"]},"8":{"1":["0","0","0","0","0","-43/6879707136"],"3":["0","0","0","215/3439853568"],"5":["0","-43/764411904"]},"9":{"1":["0","0","175/483729408","0","-11/2293235712"],"3":["-175/483729408","0","-11/1146617856"],"5":["11/764411904"]}},"e_{a,k}(v) by power, polynomial branch":{"0":{"1":["0","0","0","0","0","0","0","0","343/629856","0","-49/5038848"],"3":["0","0","0","0","0","0","-6517/629856","0","49/314928"],"5":["0","0","0","0","3773/69984","0","-343/839808"],"7":["0","0","-343/7776","0","-49/23328"],"9":["0","0","49/20736"]},"1":{"1":["0","0","0","0","0","-84035/746496","0","20923/6718464","0","-833/26873856","0","7/120932352"],"3":["0","0","0","420175/373248","0","-146461/6718464","0","833/6718464","0","-49/30233088"],"5":["0","-84035/82944","0","-146461/2239488","0","5831/4478976","0","35/2239488"],"7":["0","20923/248832","0","833/746496","0","-7/124416"],"9":["0","-833/331776","0","7/165888"]},"10":{"1":["0","0","-7/7739670528"],"3":["7/7739670528"]},"2":{"1":["0","0","0","0","-84035/248832","0","9947/2239488","0","-539/80621568","0","7/26873856"],"3":["0","0","-84035/124416","0","49735/2239488","0","-833/40310784","0","-7/1679616"],"5":["84035/82944","0","9947/746496","0","-245/1119744","0","49/4478976"],"7":["-9947/248832","0","49/497664","0","7/124416"],"9":["49/331776","0","-7/110592"]},"3":{"1":["0","0","0","0","0","-446243/161243136","0","2597/80621568","0","7/23887872"],"3":["0","0","0","2231215/80621568","0","-18179/80621568","0","-7/5971968"],"5":["0","-446243/17915904","0","-18179/26873856","0","-49/3981312"],"7":["0","2597/2985984","0","-7/663552"],"9":["0","7/294912"]},"4":{"1":["0","0","84035/186624","0","-413315/53747712","0","-49/26873856","0","-91/644972544"],"3":["-84035/186624","0","-413315/26873856","0","-245/26873856","0","1715/322486272"],"5":["413315/17915904","0","-49/8957952","0","-49/2239488"],"7":["49/2985984","0","77/3981312"],"9":["-7/2654208"]},"5":{"1":["0","0","0","0","0","6811/1289945088","0","-161/429981696"],"3":["0","0","0","-34055/644972544","0","1127/429981696"],"5":["0","6811/143327232","0","1127/143327232"],"7":["0","-161/15925248"]},"6":{"1":["0","0","95011/40310784","0","17297/1289945088","0","-7/47775744"],"3":["-95011/40310784","0","17297/644972544","0","-35/47775744"],"5":["-17297/429981696","0","-7/15925248"],"7":["7/5308416"]},"7":{"1":["0","0","0","0","0","301/10319560704"],"3":["0","0","0","-1505/5159780352"],"5":["0","301/1146617856"]},"8":{"1":["0","0","-539/322486272","0","77/3439853568"],"3":["539/322486272","0","77/1719926784"],"5":["-77/1146617856"]}},"kappa":"1/28 sqrt(21)","kappa^2":"3/112","lambda_erfc":"1/48","lambda_poly":"1/21","p3_powers":[1,3,5,7,9],"prefactor":"1/2688 sqrt(21) pi^(-1)"},"note":"arctan(sqrt15/5) (m = 2) and arctan(1/sqrt7) (m = 4, from lam = 3/16) arise in intermediate terms and cancel"},"scientific_effect":"NONE"}},"shift-variance":{"kind":"report","report":{"certified":true,"certified_enclosures_math_200":{"D_4":["14.2187634589773576013064656778740783173406003782976930345926","14.2187634589773576013064656778740783173406042538663501338228"],"c_5_ref":["0.0132193193800849680760334875146413965210528883241234179643777","0.0132193193800849680760334875146413965210528919272763929401087"]},"checks":{"BASE_INTEGRALS_FLOAT":true,"C5_CLOSED":false,"CUMULATIVE_EXACT":true,"D1_EXACT":true,"D2_EXACT":true,"D3_EXACT":true,"D4_FORM":false,"D4_INSIDE_CERTIFIED":false,"GAUSS_MOMENTS":true,"JK_EXACT":true,"LIBRARY_EXACT":true,"MEHTA_Z4_EXACT":true,"MOMENTS_FLOAT":true,"PINNED":false,"SIMPLIFIED_FORM":false,"STRUCTURE":false},"closed_forms":{"D_1":"4/3","D_2":"29/6  +  -1 sqrt(6)","D_3":"50/3  +  -76/3 pi^(-1)  +  -200/9 pi^(-1) arctan(1/2)","D_4":"360/7 sqrt(7)  +  -625/42 sqrt(21)  +  -254/21 sqrt(21) pi^(-1)  +  1250/63 sqrt(21) pi^(-1) arctan(1/2)","D_4_simplified":"6695/54 - (405/32) sqrt6 - (1/pi) [ (1375/72) sqrt21 + (6695/27) arctan(sqrt21/7) + (405/16) sqrt6 arctan(sqrt14/14) ]   (arctan(sqrt14/4) - arctan(sqrt14/7) = arctan(1/sqrt14))","Z_4":"1536 pi^(1)","c_5_ref":"Gamma(7/6) (3/2)^(1/3) D_4 / (12 sqrt3 pi^(7/2))","erfc_branch_of_D_4":"360/7 sqrt(7)  +  -1318295/86016 sqrt(21)  +  -65979371/1344000 sqrt(21) pi^(-1)  +  1318295/64512 sqrt(21) pi^(-1) arctan(1/2)","polynomial_branch_of_D_4":"12765/28672 sqrt(21)  +  16574457/448000 sqrt(21) pi^(-1)  +  -4255/7168 sqrt(21) pi^(-1) arctan(1/2)"},"exact_relations":{"D_4 integral":"sqrt21/(2688 pi) [ (sqrt21 sqrt(pi)/6) int e^{-T^2/48} erfc(sqrt21 T/28) sum_a T^a L_a^E + int e^{-T^2/21} sum_a T^a L_a^R ]","G_n(mu, T)":"G_0 = (1/2) sqrt(pi/mu) erf(sqrt(mu) T); G_1 = (1 - e^{-mu T^2})/(2 mu); G_n = [(n-1) G_{n-2} - T^{n-1} e^{-mu T^2}]/(2 mu)","L_a":"int_0^T e^{-v^2/24} sum_k e_{a,k}(v) J_k(v) dv, k odd, J_k = B_k + C_k e^{-v^2/8}: incomplete Gaussian moments with mu = 1/24, 1/6","M_0(lam, 1)":"(1/2) sqrt(pi/lam)","M_0(lam, erf(u .) erf(v .))":"arctan(u v/sqrt(lam (lam + u^2 + v^2)))/sqrt(pi lam)","M_0(lam, erf(u .))":"arctan(u/sqrt lam)/sqrt(pi lam)","M_1(lam, 1)":"1/(2 lam)","M_n(lam, f), n >= 1":"[(n-1) M_{n-2}(lam, f) + sum_{u in f} (2u/sqrt pi) M_{n-1}(lam + u^2, f minus u)]/(2 lam)","Z_4":"4! sqrt(pi) (1/6) int e^{-T^2/48} int_0^T e^{-v^2/24} int_0^v V e^{-p3^2/8} = 1536 pi (exact, rule MEHTA_Z4_EXACT)"},"interval_evaluation":{"(3/2)^(1/3)":["1.14471424255333186780804221193967700891590692078793107209905","1.14471424255333186780804221193967700891590692078793107209906"],"D_4":["63.6499586850660065518550616607172288346109634344411227864797","63.6499586850660065518550616607172288346109634344411227864956"],"D_4_simplified":["14.2187634589773576013064656778740783173406023160820215841883","14.2187634589773576013064656778740783173406023160820215842281"],"Gamma(7/6)":["0.927719333630039200708349482534621018566466519145475576700351","0.927719333630039200708349482534621018566466519145475576939062"],"arctan(sqrt14/14)":["0.261157410903024311262049412927851638004440468771565909792296","0.261157410903024311262049412927851638004440468771565909792486"],"arctan(sqrt14/4)":["0.752040089192335679842958294184691084003367627670504438870380","0.752040089192335679842958294184691084003367627670504438870582"],"arctan(sqrt14/7)":["0.490882678289311368580908881256839445998927158898938529077984","0.490882678289311368580908881256839445998927158898938529078176"],"arctan(sqrt21/7)":["0.579639740363704299923291897011207918621441782280263529351672","0.579639740363704299923291897011207918621441782280263529351866"],"c_5_ref":["0.0591759708792297785593923857978417427883205945032350706076337","0.0591759708792297785593923857978417427883205945032350706229268"],"pi":["3.14159265358979323846264338327950288419716939937510582097457","3.14159265358979323846264338327950288419716939937510582097528"]},"mutant":"shift-variance","object":"CL-SIDE24-D4-CLOSED-FORM-20260930-v1","passed":false,"reduction_data":{"atoms_created_by_the_derivations":["1","arctan(1/5 sqrt(15))","arctan(1/2 sqrt(1))","arctan(1/3 sqrt(1))"],"m=2":{"kappa":"1/20 sqrt(30)","lambda_erfc":"1/8","lambda_poly":"1/5"},"m=3":{"kappa":"1/12 sqrt(6)","lambda_erfc":"1/24","lambda_poly":"1/12"},"m=4":{"G_0_constant":"1/2 sqrt(3) pi^(1/2)","J_k = A_k sqrt(2 pi) erf(s/(2 sqrt2)) + B_k + C_k(s) e^{-s^2/8}":{"0":{"A":"1","B":"0","C":[]},"1":{"A":"0","B":"4","C":["-4"]},"2":{"A":"4","B":"0","C":["0","-4"]},"3":{"A":"0","B":"32","C":["-32","0","-4"]},"4":{"A":"48","B":"0","C":["0","-48","0","-4"]},"5":{"A":"0","B":"512","C":["-512","0","-64","0","-4"]},"6":{"A":"960","B":"0","C":["0","-960","0","-80","0","-4"]},"7":{"A":"0","B":"12288","C":["-12288","0","-1536","0","-96","0","-4"]},"8":{"A":"26880","B":"0","C":["0","-26880","0","-2240","0","-112","0","-4"]},"9":{"A":"0","B":"393216","C":["-393216","0","-49152","0","-3072","0","-128","0","-4"]}},"T_powers_erfc":[0,1,2,3,4,5,6,7,8,9,11],"T_powers_poly":[0,1,2,3,4,5,6,7,8,10],"alpha":"1/3","beta":"1/6 T","e_{a,k}(v) by power, erfc branch":{"0":{"1":["0","0","0","0","0","-315/256","0","5/256","0","-1/9216","0","-1/3359232"],"3":["0","0","0","1575/128","0","-35/256","0","1/2304","0","7/839808"],"5":["0","-2835/256","0","-105/256","0","7/1536","0","-5/62208"],"7":["0","135/256","0","1/256","0","1/3456"],"9":["0","-9/1024","0","-1/4608"]},"1":{"1":["0","0","0","0","-945/256","0","15/256","0","-11/27648","0","1/746496"],"3":["0","0","-945/128","0","75/256","0","-17/13824","0","-1/46656"],"5":["2835/256","0","45/256","0","-5/384","0","7/124416"],"7":["-135/256","0","3/512","0","1/3456"],"9":["9/1024","0","-1/3072"]},"11":{"1":["0","0","1/5159780352"],"3":["-1/5159780352"]},"2":{"1":["0","0","0","0","0","5/512","0","-7/18432","0","7/995328","0","-1/80621568"],"3":["0","0","0","-25/256","0","49/18432","0","-7/248832","0","7/20155392"],"5":["0","45/512","0","49/6144","0","-49/165888","0","-5/1492992"],"7":["0","-21/2048","0","-7/27648","0","1/82944"],"9":["0","7/12288","0","-1/110592"]},"3":{"1":["0","0","315/64","0","-25/512","0","-1/18432","0","23/8957952","0","-1/17915904"],"3":["-315/64","0","-25/256","0","-5/18432","0","-97/4478976","0","1/1119744"],"5":["75/512","0","-1/6144","0","43/248832","0","-7/2985984"],"7":["1/2048","0","-7/55296","0","-1/82944"],"9":["-1/36864","0","1/73728"]},"4":{"1":["0","0","0","0","0","59/221184","0","-83/11943936","0","-1/15925248"],"3":["0","0","0","-295/110592","0","581/11943936","0","1/3981312"],"5":["0","59/24576","0","581/3981312","0","7/2654208"],"7":["0","-83/442368","0","1/442368"],"9":["0","-1/196608"]},"5":{"1":["0","0","-5/128","0","43/73728","0","5/3981312","0","13/429981696"],"3":["5/128","0","43/36864","0","25/3981312","0","-245/214990848"],"5":["-43/24576","0","5/1327104","0","7/1492992"],"7":["-5/442368","0","-11/2654208"],"9":["1/1769472"]},"6":{"1":["0","0","0","0","0","-115/71663616","0","23/286654464"],"3":["0","0","0","575/35831808","0","-161/286654464"],"5":["0","-115/7962624","0","-161/95551488"],"7":["0","23/10616832"]},"7":{"1":["0","0","5/55296","0","-91/23887872","0","1/31850496"],"3":["-5/55296","0","-91/11943936","0","5/31850496"],"5":["91/7962624","0","1/10616832"],"7":["-1/3538944"]},"8":{"1":["0","0","0","0","0","-43/6879707136"],"3":["0","0","0","215/3439853568"],"5":["0","-43/764411904"]},"9":{"1":["0","0","25/53747712","0","-11/2293235712"],"3":["-25/53747712","0","-11/1146617856"],"5":["11/764411904"]}},"e_{a,k}(v) by power, polynomial branch":{"0":{"1":["0","0","0","0","0","0","0","0","1/864","0","-1/62208"],"3":["0","0","0","0","0","0","-19/864","0","1/3888"],"5":["0","0","0","0","11/96","0","-7/10368"],"7":["0","0","-3/32","0","-1/288"],"9":["0","0","1/256"]},"1":{"1":["0","0","0","0","0","-315/1024","0","61/9216","0","-17/331776","0","1/13436928"],"3":["0","0","0","1575/512","0","-427/9216","0","17/82944","0","-7/3359232"],"5":["0","-2835/1024","0","-427/3072","0","119/55296","0","5/248832"],"7":["0","183/1024","0","17/9216","0","-1/13824"],"9":["0","-17/4096","0","1/18432"]},"10":{"1":["0","0","-1/859963392"],"3":["1/859963392"]},"2":{"1":["0","0","0","0","-945/1024","0","29/3072","0","-11/995328","0","1/2985984"],"3":["0","0","-945/512","0","145/3072","0","-17/497664","0","-1/186624"],"5":["2835/1024","0","29/1024","0","-5/13824","0","7/497664"],"7":["-87/1024","0","1/6144","0","1/13824"],"9":["1/4096","0","-1/12288"]},"3":{"1":["0","0","0","0","0","-1301/221184","0","53/995328","0","1/2654208"],"3":["0","0","0","6505/110592","0","-371/995328","0","-1/663552"],"5":["0","-1301/24576","0","-371/331776","0","-7/442368"],"7":["0","53/36864","0","-1/73728"],"9":["0","1/32768"]},"4":{"1":["0","0","315/256","0","-1205/73728","0","-1/331776","0","-13/71663616"],"3":["-315/256","0","-1205/36864","0","-5/331776","0","245/35831808"],"5":["1205/24576","0","-1/110592","0","-7/248832"],"7":["1/36864","0","11/442368"],"9":["-1/294912"]},"5":{"1":["0","0","0","0","0","139/15925248","0","-23/47775744"],"3":["0","0","0","-695/7962624","0","161/47775744"],"5":["0","139/1769472","0","161/15925248"],"7":["0","-23/1769472"]},"6":{"1":["0","0","277/55296","0","353/15925248","0","-1/5308416"],"3":["-277/55296","0","353/7962624","0","-5/5308416"],"5":["-353/5308416","0","-1/1769472"],"7":["1/589824"]},"7":{"1":["0","0","0","0","0","43/1146617856"],"3":["0","0","0","-215/573308928"],"5":["0","43/127401984"]},"8":{"1":["0","0","-11/3981312","0","11/382205952"],"3":["11/3981312","0","11/191102976"],"5":["-11/127401984"]}},"kappa":"1/12 sqrt(3)","kappa^2":"1/48","lambda_erfc":"1/48","lambda_poly":"1/24","p3_powers":[1,3,5,7,9],"prefactor":"1/2688 sqrt(21) pi^(-1)"},"note":"arctan(sqrt15/5) (m = 2) and arctan(1/sqrt7) (m = 4, from lam = 3/16) arise in intermediate terms and cancel"},"scientific_effect":"NONE"}},"two-erf":{"kind":"report","report":{"certified":true,"certified_enclosures_math_200":{"D_4":["14.2187634589773576013064656778740783173406003782976930345926","14.2187634589773576013064656778740783173406042538663501338228"],"c_5_ref":["0.0132193193800849680760334875146413965210528883241234179643777","0.0132193193800849680760334875146413965210528919272763929401087"]},"checks":{"BASE_INTEGRALS_FLOAT":true,"C5_CLOSED":true,"CUMULATIVE_EXACT":true,"D1_EXACT":true,"D2_EXACT":true,"D3_EXACT":true,"D4_FORM":true,"D4_INSIDE_CERTIFIED":true,"GAUSS_MOMENTS":true,"JK_EXACT":true,"LIBRARY_EXACT":true,"MEHTA_Z4_EXACT":true,"MOMENTS_FLOAT":false,"PINNED":true,"SIMPLIFIED_FORM":true,"STRUCTURE":true},"closed_forms":{"D_1":"4/3","D_2":"29/6  +  -1 sqrt(6)","D_3":"50/3  +  -76/3 pi^(-1)  +  -200/9 pi^(-1) arctan(1/2)","D_4":"6695/54  +  -405/32 sqrt(6)  +  -1375/72 sqrt(21) pi^(-1)  +  -405/16 sqrt(6) pi^(-1) arctan(1/4 sqrt(14))  +  -6695/27 pi^(-1) arctan(1/7 sqrt(21))  +  405/16 sqrt(6) pi^(-1) arctan(1/7 sqrt(14))","D_4_simplified":"6695/54 - (405/32) sqrt6 - (1/pi) [ (1375/72) sqrt21 + (6695/27) arctan(sqrt21/7) + (405/16) sqrt6 arctan(sqrt14/14) ]   (arctan(sqrt14/4) - arctan(sqrt14/7) = arctan(1/sqrt14))","Z_4":"1536 pi^(1)","c_5_ref":"Gamma(7/6) (3/2)^(1/3) D_4 / (12 sqrt3 pi^(7/2))","erfc_branch_of_D_4":"6695/54  +  -6066622595/603979776 sqrt(6)  +  -46299277168087/1415577600000 sqrt(21) pi^(-1)  +  -6066622595/301989888 sqrt(6) pi^(-1) arctan(1/4 sqrt(14))  +  -6695/27 pi^(-1) arctan(1/7 sqrt(21))  +  6066622595/301989888 sqrt(6) pi^(-1) arctan(1/7 sqrt(14))","polynomial_branch_of_D_4":"-1577496445/603979776 sqrt(6)  +  19265677168087/1415577600000 sqrt(21) pi^(-1)  +  -1577496445/301989888 sqrt(6) pi^(-1) arctan(1/4 sqrt(14))  +  1577496445/301989888 sqrt(6) pi^(-1) arctan(1/7 sqrt(14))"},"exact_relations":{"D_4 integral":"sqrt21/(2688 pi) [ (sqrt21 sqrt(pi)/6) int e^{-T^2/48} erfc(sqrt21 T/28) sum_a T^a L_a^E + int e^{-T^2/21} sum_a T^a L_a^R ]","G_n(mu, T)":"G_0 = (1/2) sqrt(pi/mu) erf(sqrt(mu) T); G_1 = (1 - e^{-mu T^2})/(2 mu); G_n = [(n-1) G_{n-2} - T^{n-1} e^{-mu T^2}]/(2 mu)","L_a":"int_0^T e^{-v^2/24} sum_k e_{a,k}(v) J_k(v) dv, k odd, J_k = B_k + C_k e^{-v^2/8}: incomplete Gaussian moments with mu = 1/24, 1/6","M_0(lam, 1)":"(1/2) sqrt(pi/lam)","M_0(lam, erf(u .) erf(v .))":"arctan(u v/sqrt(lam (lam + u^2 + v^2)))/sqrt(pi lam)","M_0(lam, erf(u .))":"arctan(u/sqrt lam)/sqrt(pi lam)","M_1(lam, 1)":"1/(2 lam)","M_n(lam, f), n >= 1":"[(n-1) M_{n-2}(lam, f) + sum_{u in f} (2u/sqrt pi) M_{n-1}(lam + u^2, f minus u)]/(2 lam)","Z_4":"4! sqrt(pi) (1/6) int e^{-T^2/48} int_0^T e^{-v^2/24} int_0^v V e^{-p3^2/8} = 1536 pi (exact, rule MEHTA_Z4_EXACT)"},"interval_evaluation":{"(3/2)^(1/3)":["1.14471424255333186780804221193967700891590692078793107209905","1.14471424255333186780804221193967700891590692078793107209906"],"D_4":["14.2187634589773576013064656778740783173406023160820215841833","14.2187634589773576013064656778740783173406023160820215842323"],"D_4_simplified":["14.2187634589773576013064656778740783173406023160820215841883","14.2187634589773576013064656778740783173406023160820215842281"],"Gamma(7/6)":["0.927719333630039200708349482534621018566466519145475576700351","0.927719333630039200708349482534621018566466519145475576939062"],"arctan(sqrt14/14)":["0.261157410903024311262049412927851638004440468771565909792296","0.261157410903024311262049412927851638004440468771565909792486"],"arctan(sqrt14/4)":["0.752040089192335679842958294184691084003367627670504438870380","0.752040089192335679842958294184691084003367627670504438870582"],"arctan(sqrt14/7)":["0.490882678289311368580908881256839445998927158898938529077984","0.490882678289311368580908881256839445998927158898938529078176"],"arctan(sqrt21/7)":["0.579639740363704299923291897011207918621441782280263529351672","0.579639740363704299923291897011207918621441782280263529351866"],"c_5_ref":["0.0132193193800849680760334875146413965210528901256999054505140","0.0132193193800849680760334875146413965210528901256999054539727"],"pi":["3.14159265358979323846264338327950288419716939937510582097457","3.14159265358979323846264338327950288419716939937510582097528"]},"mutant":"two-erf","object":"CL-SIDE24-D4-CLOSED-FORM-20260930-v1","passed":false,"reduction_data":{"atoms_created_by_the_derivations":["1","arctan(1/5 sqrt(15))","arctan(1/2 sqrt(1))","arctan(1/4 sqrt(14))","arctan(1/7 sqrt(21))","arctan(1/7 sqrt(14))","arctan(1/7 sqrt(7))"],"m=2":{"kappa":"1/20 sqrt(30)","lambda_erfc":"1/8","lambda_poly":"1/5"},"m=3":{"kappa":"1/12 sqrt(6)","lambda_erfc":"1/24","lambda_poly":"1/12"},"m=4":{"G_0_constant":"1/6 sqrt(21) pi^(1/2)","J_k = A_k sqrt(2 pi) erf(s/(2 sqrt2)) + B_k + C_k(s) e^{-s^2/8}":{"0":{"A":"1","B":"0","C":[]},"1":{"A":"0","B":"4","C":["-4"]},"2":{"A":"4","B":"0","C":["0","-4"]},"3":{"A":"0","B":"32","C":["-32","0","-4"]},"4":{"A":"48","B":"0","C":["0","-48","0","-4"]},"5":{"A":"0","B":"512","C":["-512","0","-64","0","-4"]},"6":{"A":"960","B":"0","C":["0","-960","0","-80","0","-4"]},"7":{"A":"0","B":"12288","C":["-12288","0","-1536","0","-96","0","-4"]},"8":{"A":"26880","B":"0","C":["0","-26880","0","-2240","0","-112","0","-4"]},"9":{"A":"0","B":"393216","C":["-393216","0","-49152","0","-3072","0","-128","0","-4"]}},"T_powers_erfc":[0,1,2,3,4,5,6,7,8,9,11],"T_powers_poly":[0,1,2,3,4,5,6,7,8,10],"alpha":"3/7","beta":"3/14 T","e_{a,k}(v) by power, erfc branch":{"0":{"1":["0","0","0","0","0","-84035/186624","0","1715/186624","0","-49/746496","0","-7/30233088"],"3":["0","0","0","420175/93312","0","-12005/186624","0","49/186624","0","49/7558272"],"5":["0","-84035/20736","0","-12005/62208","0","343/124416","0","-35/559872"],"7":["0","1715/6912","0","49/20736","0","7/31104"],"9":["0","-49/9216","0","-7/41472"]},"1":{"1":["0","0","0","0","-84035/62208","0","1715/62208","0","-539/2239488","0","7/6718464"],"3":["0","0","-84035/31104","0","8575/62208","0","-833/1119744","0","-7/419904"],"5":["84035/20736","0","1715/20736","0","-245/31104","0","49/1119744"],"7":["-1715/6912","0","49/13824","0","7/31104"],"9":["49/9216","0","-7/27648"]},"11":{"1":["0","0","1/5159780352"],"3":["-1/5159780352"]},"2":{"1":["0","0","0","0","0","1715/373248","0","-343/1492992","0","49/8957952","0","-1/80621568"],"3":["0","0","0","-8575/186624","0","2401/1492992","0","-49/2239488","0","7/20155392"],"5":["0","1715/41472","0","2401/497664","0","-343/1492992","0","-5/1492992"],"7":["0","-343/55296","0","-49/248832","0","1/82944"],"9":["0","49/110592","0","-1/110592"]},"3":{"1":["0","0","84035/46656","0","-8575/373248","0","-49/1492992","0","161/80621568","0","-1/17915904"],"3":["-84035/46656","0","-8575/186624","0","-245/1492992","0","-679/40310784","0","1/1119744"],"5":["8575/124416","0","-49/497664","0","301/2239488","0","-7/2985984"],"7":["49/165888","0","-49/497664","0","-1/82944"],"9":["-7/331776","0","1/73728"]},"4":{"1":["0","0","0","0","0","2891/17915904","0","-581/107495424","0","-1/15925248"],"3":["0","0","0","-14455/8957952","0","4067/107495424","0","1/3981312"],"5":["0","2891/1990656","0","4067/35831808","0","7/2654208"],"7":["0","-581/3981312","0","1/442368"],"9":["0","-1/196608"]},"5":{"1":["0","0","-1715/93312","0","2107/5971968","0","35/35831808","0","13/429981696"],"3":["1715/93312","0","2107/2985984","0","175/35831808","0","-245/214990848"],"5":["-2107/1990656","0","35/11943936","0","7/1492992"],"7":["-35/3981312","0","-11/2654208"],"9":["1/1769472"]},"6":{"1":["0","0","0","0","0","-805/644972544","0","23/286654464"],"3":["0","0","0","4025/322486272","0","-161/286654464"],"5":["0","-805/71663616","0","-161/95551488"],"7":["0","23/10616832"]},"7":{"1":["0","0","245/4478976","0","-637/214990848","0","1/31850496"],"3":["-245/4478976","0","-637/107495424","0","5/31850496"],"5":["637/71663616","0","1/10616832"],"7":["-1/3538944"]},"8":{"1":["0","0","0","0","0","-43/6879707136"],"3":["0","0","0","215/3439853568"],"5":["0","-43/764411904"]},"9":{"1":["0","0","175/483729408","0","-11/2293235712"],"3":["-175/483729408","0","-11/1146617856"],"5":["11/764411904"]}},"e_{a,k}(v) by power, polynomial branch":{"0":{"1":["0","0","0","0","0","0","0","0","343/629856","0","-49/5038848"],"3":["0","0","0","0","0","0","-6517/629856","0","49/314928"],"5":["0","0","0","0","3773/69984","0","-343/839808"],"7":["0","0","-343/7776","0","-49/23328"],"9":["0","0","49/20736"]},"1":{"1":["0","0","0","0","0","-84035/746496","0","20923/6718464","0","-833/26873856","0","7/120932352"],"3":["0","0","0","420175/373248","0","-146461/6718464","0","833/6718464","0","-49/30233088"],"5":["0","-84035/82944","0","-146461/2239488","0","5831/4478976","0","35/2239488"],"7":["0","20923/248832","0","833/746496","0","-7/124416"],"9":["0","-833/331776","0","7/165888"]},"10":{"1":["0","0","-7/7739670528"],"3":["7/7739670528"]},"2":{"1":["0","0","0","0","-84035/248832","0","9947/2239488","0","-539/80621568","0","7/26873856"],"3":["0","0","-84035/124416","0","49735/2239488","0","-833/40310784","0","-7/1679616"],"5":["84035/82944","0","9947/746496","0","-245/1119744","0","49/4478976"],"7":["-9947/248832","0","49/497664","0","7/124416"],"9":["49/331776","0","-7/110592"]},"3":{"1":["0","0","0","0","0","-446243/161243136","0","2597/80621568","0","7/23887872"],"3":["0","0","0","2231215/80621568","0","-18179/80621568","0","-7/5971968"],"5":["0","-446243/17915904","0","-18179/26873856","0","-49/3981312"],"7":["0","2597/2985984","0","-7/663552"],"9":["0","7/294912"]},"4":{"1":["0","0","84035/186624","0","-413315/53747712","0","-49/26873856","0","-91/644972544"],"3":["-84035/186624","0","-413315/26873856","0","-245/26873856","0","1715/322486272"],"5":["413315/17915904","0","-49/8957952","0","-49/2239488"],"7":["49/2985984","0","77/3981312"],"9":["-7/2654208"]},"5":{"1":["0","0","0","0","0","6811/1289945088","0","-161/429981696"],"3":["0","0","0","-34055/644972544","0","1127/429981696"],"5":["0","6811/143327232","0","1127/143327232"],"7":["0","-161/15925248"]},"6":{"1":["0","0","95011/40310784","0","17297/1289945088","0","-7/47775744"],"3":["-95011/40310784","0","17297/644972544","0","-35/47775744"],"5":["-17297/429981696","0","-7/15925248"],"7":["7/5308416"]},"7":{"1":["0","0","0","0","0","301/10319560704"],"3":["0","0","0","-1505/5159780352"],"5":["0","301/1146617856"]},"8":{"1":["0","0","-539/322486272","0","77/3439853568"],"3":["539/322486272","0","77/1719926784"],"5":["-77/1146617856"]}},"kappa":"1/28 sqrt(21)","kappa^2":"3/112","lambda_erfc":"1/48","lambda_poly":"1/21","p3_powers":[1,3,5,7,9],"prefactor":"1/2688 sqrt(21) pi^(-1)"},"note":"arctan(sqrt15/5) (m = 2) and arctan(1/sqrt7) (m = 4, from lam = 3/16) arise in intermediate terms and cancel"},"scientific_effect":"NONE"}},"vandermonde":{"final_line":"ValueError: an even p3-power occurred; J_k would carry an error function","kind":"guard"}},"results_bytes":16765,"results_sha256":"b878c1d3534a1ce72f01bb461a06b8777888bc0a5fdd9413fc1a4f8a565a2bb3","schema":1,"script":"closed_form_d4.py","script_git_blob":"e087f4c0007ae2473d28230771588da57f794aab"}
diff --git a/tools/side24_mutant_contracts/side24-d5-coefficient.json b/tools/side24_mutant_contracts/side24-d5-coefficient.json
new file mode 100644
index 000000000..a2781afb6
--- /dev/null
+++ b/tools/side24_mutant_contracts/side24-d5-coefficient.json
@@ -0,0 +1 @@
+{"directory":"coefficients/side24_d5_20260930","family":"side24-d5-coefficient","mutants":{"cumulative-half":{"kind":"report","report":{"certified":true,"checks":{"CLOSED_FORM_D5":true,"D2_EXACT":true,"D3_CONSISTENT":true,"FLOAT_INSIDE":true,"IMAGE_BOUND":true,"LIBRARY_EXACT":true,"MEHTA_Z4":false,"MONOTONE":true,"PINNED":false,"SIDE24_CONSISTENT":true,"TRANSFER_BOUND":true,"TRUNCATION_NESTING":false,"WIDTHS":true},"coefficients":{"c_2_ref":["0.0734069193060342710301359629577740500176642446843326718326939","0.0734069193060342710301359629577740500176642446843326718516467"],"c_3_ref":["0.0417759318405983433429366654285755564666815196610446423810796","0.0417759318405983433429366654285755564666815196610448333783896"],"c_4_ref":["0.0233216660029528350945211949528568851531013723630981034059901","0.0233216660029528350945211949528568851531013723633825654059788"],"c_5_24":["0.0131775012055573304295912516063076684815093177529785159329521","0.0131775012055573304295912516063076684815093213561308597531588"],"c_5_ref":["0.0131775012055573304295912516063076684815093177529785159329523","0.0131775012055573304295912516063076684815093213561308597531586"],"c_5_ref_closed":["0.0131775012055573304295912516063076684815093177529785159329602","0.0131775012055573304295912516063076684815093213561308597531509"]},"cone_moments":{"D_2":["2.38384359055015523513604925862744194136738585267665775580671","2.38384359055015523513604925862744194136738585267666865399451"],"D_2_exact_29/6-sqrt6":["2.38384359055015523513604925862744194136738585267666320490062","2.38384359055015523513604925862744194136738585267666320490067"],"D_3":["5.32318026888989689492319873968726063578438852940845576405060","5.32318026888989689492319873968726063578438852947338433618854"],"D_3_Math-199":["5.32318026888989689492319873968726063578438852940845576405060","5.32318026888989689492319873968726063578438852947338433618854"],"D_4":["14.1737836294718774152704475535282200540633255861937840925411","14.1737836294718774152704475535282200540633294617617623178672"],"D_4_coarse":["14.0415821891942423615411885601824364879419337106452527815946","14.0415821934054391841392635978156143834928345951187006831042"]},"constants":{"(3/2)^(1/3)":["1.14471424255333186780804221193967700891590692078793107209905","1.14471424255333186780804221193967700891590692078793107209906"],"1536pi":["4825.48631591392241427862023671731643012685219744016254101693","4825.48631591392241427862023671731643012685219744016254101804"],"Gamma(7/6)":["0.927719333630039200708349482534621018566466519145475576700351","0.927719333630039200708349482534621018566466519145475576939062"],"Z_4":["4825.48631591392241427862023671731643012685219744016254101437","4825.48631591392241427862023671731643012685219744016254102065"],"Z_4_quadrature":["4817.77788037263695367750483678554498490612010501985079663825","4817.77788037263695367750483678554498490612010502162815701116"],"pi":["3.14159265358979323846264338327950288419716939937510582097457","3.14159265358979323846264338327950288419716939937510582097528"],"|S^4|":["26.3189450695716229835586426663364030275031984193087750037589","26.3189450695716229835586426663364030275031984193087750037782"]},"exact_relations":{"D_4":"sqrt(3/7) . 24/Z_4 . (1/6) I_4","D_m":"sqrt(3/(m+3)) Z_m^-1 int_{mu<0} prod mu_i^2 |Delta(mu)| exp(-|mu|^2/4 + (sum mu)^2/(4(m+3))) dmu","L_a":"L_a(T) = int_0^T e^{-v^2/24} sum_k e_{a,k}(v) J_k(v) dv, accumulated cell by cell","Z_4":"4! sqrt pi (1/6) int dT e^{-T^2/48} int_0^T dv e^{-v^2/24} int_0^v V e^{-p3^2/8} dp3 = 1536 pi","c_d_ref":"Gamma(7/6) (3/2)^(1/3) |S^{d-1}| D_{d-1} / (sqrt3 sqrt pi (2 pi)^d); d = 5: Gamma(7/6) (3/2)^(1/3) D_4 / (12 sqrt3 pi^(7/2))","decoupling":"alpha = 3m/(4(m+3)), beta = 3T/(2(m+3)); erfc branch exp(-T^2/(4m(m-1)) - |u|^2/4), polynomial branch exp(-T^2/((m-1)(m+3)) - |u|^2/4), u = t - (T/(m-1)) 1","erfc argument":"beta/(2 sqrt alpha) = T sqrt21/28; erfc(x) <= e^{-x^2} in the tail","m=4":"p1 = (T - 2 p2 - p3)/3, v = 2 p2 + p3, p2 = (v - p3)/2; |u|^2/4 = v^2/24 + p3^2/8; dp1 dp2 dp3 = dT dv dp3/6; region 0 < p3 < v < T","transfer":"E_5 = 60500 (76 24^6 + 15) e^-288; eps = 126 E_5; c_{5,24}/c_{5,ref} in [1/(1+delta), 1+delta], delta = (1+eps)^10/(1-eps)^10 - 1"},"float_reference_10sig":{"D_2_orthant":"2.383704698","D_3_orthant":"5.309848377","D_4_a_integrated_3D":"14.52840747","D_4_orthant":"13.51071422"},"mutant":"cumulative-half","object":"CL-SIDE24-D5-COEFFICIENT-20260930-v1","passed":false,"quadrature":{"K":40,"cells":"width 1/2 on [0, 30], width 1 on [30, 60]","rho":"6"},"scientific_effect":"NONE","transfer":{"E_5":["7.36250545404696408692679395342008992909499098015444669312995E-111","7.36250545404696408692679395342008992909499098015444669313000E-111"],"delta":["1.85535137441983494990555207626186266213193772699892056666874E-107","1.85535137441983494990555207626186266213193772699892056666882E-107"],"epsilon":["9.27675687209917474952776038130931331065968863499460283334373E-109","9.27675687209917474952776038130931331065968863499460283334380E-109"],"exponents":{"a":"29/3 (<= 10 used)","b":10},"n_jet":21}}},"erfc-branch":{"kind":"report","report":{"certified":true,"checks":{"CLOSED_FORM_D5":true,"D2_EXACT":true,"D3_CONSISTENT":true,"FLOAT_INSIDE":false,"IMAGE_BOUND":true,"LIBRARY_EXACT":true,"MEHTA_Z4":true,"MONOTONE":true,"PINNED":false,"SIDE24_CONSISTENT":true,"TRANSFER_BOUND":true,"TRUNCATION_NESTING":true,"WIDTHS":true},"coefficients":{"c_2_ref":["0.0734069193060342710301359629577740500176642446843326718326939","0.0734069193060342710301359629577740500176642446843326718516467"],"c_3_ref":["0.0417759318405983433429366654285755564666815196610446423810796","0.0417759318405983433429366654285755564666815196610448333783896"],"c_4_ref":["0.0233216660029528350945211949528568851531013723630981034059901","0.0233216660029528350945211949528568851531013723633825654059788"],"c_5_24":["0.0123696512269971112168949035124426635679638004412583341383702","0.0123696512269971112168949035124426635679638022451742760899064"],"c_5_ref":["0.0123696512269971112168949035124426635679638004412583341383704","0.0123696512269971112168949035124426635679638022451742760899062"],"c_5_ref_closed":["0.0123696512269971112168949035124426635679638004412583341383778","0.0123696512269971112168949035124426635679638022451742760898989"]},"cone_moments":{"D_2":["2.38384359055015523513604925862744194136738585267665775580671","2.38384359055015523513604925862744194136738585267666865399451"],"D_2_exact_29/6-sqrt6":["2.38384359055015523513604925862744194136738585267666320490062","2.38384359055015523513604925862744194136738585267666320490067"],"D_3":["5.32318026888989689492319873968726063578438852940845576405060","5.32318026888989689492319873968726063578438852947338433618854"],"D_3_Math-199":["5.32318026888989689492319873968726063578438852940845576405060","5.32318026888989689492319873968726063578438852947338433618854"],"D_4":["13.3048563099010678908348111059004219391008126245600525991741","13.3048563099010678908348111059004219391008145648607096120266"],"D_4_coarse":["13.3048563088270919155690268903110838246964785939160741565373","13.3048563109750438661009000137781072877567229653148984555572"]},"constants":{"(3/2)^(1/3)":["1.14471424255333186780804221193967700891590692078793107209905","1.14471424255333186780804221193967700891590692078793107209906"],"1536pi":["4825.48631591392241427862023671731643012685219744016254101693","4825.48631591392241427862023671731643012685219744016254101804"],"Gamma(7/6)":["0.927719333630039200708349482534621018566466519145475576700351","0.927719333630039200708349482534621018566466519145475576939062"],"Z_4":["4825.48631591392241427862023671731643012685219744016254101437","4825.48631591392241427862023671731643012685219744016254102065"],"Z_4_quadrature":["4825.48631591392241427862023671731643012685219743927374007126","4825.48631591392241427862023671731643012685219744105134196381"],"pi":["3.14159265358979323846264338327950288419716939937510582097457","3.14159265358979323846264338327950288419716939937510582097528"],"|S^4|":["26.3189450695716229835586426663364030275031984193087750037589","26.3189450695716229835586426663364030275031984193087750037782"]},"exact_relations":{"D_4":"sqrt(3/7) . 24/Z_4 . (1/6) I_4","D_m":"sqrt(3/(m+3)) Z_m^-1 int_{mu<0} prod mu_i^2 |Delta(mu)| exp(-|mu|^2/4 + (sum mu)^2/(4(m+3))) dmu","L_a":"L_a(T) = int_0^T e^{-v^2/24} sum_k e_{a,k}(v) J_k(v) dv, accumulated cell by cell","Z_4":"4! sqrt pi (1/6) int dT e^{-T^2/48} int_0^T dv e^{-v^2/24} int_0^v V e^{-p3^2/8} dp3 = 1536 pi","c_d_ref":"Gamma(7/6) (3/2)^(1/3) |S^{d-1}| D_{d-1} / (sqrt3 sqrt pi (2 pi)^d); d = 5: Gamma(7/6) (3/2)^(1/3) D_4 / (12 sqrt3 pi^(7/2))","decoupling":"alpha = 3m/(4(m+3)), beta = 3T/(2(m+3)); erfc branch exp(-T^2/(4m(m-1)) - |u|^2/4), polynomial branch exp(-T^2/((m-1)(m+3)) - |u|^2/4), u = t - (T/(m-1)) 1","erfc argument":"beta/(2 sqrt alpha) = T sqrt21/28; erfc(x) <= e^{-x^2} in the tail","m=4":"p1 = (T - 2 p2 - p3)/3, v = 2 p2 + p3, p2 = (v - p3)/2; |u|^2/4 = v^2/24 + p3^2/8; dp1 dp2 dp3 = dT dv dp3/6; region 0 < p3 < v < T","transfer":"E_5 = 60500 (76 24^6 + 15) e^-288; eps = 126 E_5; c_{5,24}/c_{5,ref} in [1/(1+delta), 1+delta], delta = (1+eps)^10/(1-eps)^10 - 1"},"float_reference_10sig":{"D_2_orthant":"2.383704698","D_3_orthant":"5.309848377","D_4_a_integrated_3D":"14.52840747","D_4_orthant":"13.51071422"},"mutant":"erfc-branch","object":"CL-SIDE24-D5-COEFFICIENT-20260930-v1","passed":false,"quadrature":{"K":40,"cells":"width 1/2 on [0, 30], width 1 on [30, 60]","rho":"6"},"scientific_effect":"NONE","transfer":{"E_5":["7.36250545404696408692679395342008992909499098015444669312995E-111","7.36250545404696408692679395342008992909499098015444669313000E-111"],"delta":["1.85535137441983494990555207626186266213193772699892056666874E-107","1.85535137441983494990555207626186266213193772699892056666882E-107"],"epsilon":["9.27675687209917474952776038130931331065968863499460283334373E-109","9.27675687209917474952776038130931331065968863499460283334380E-109"],"exponents":{"a":"29/3 (<= 10 used)","b":10},"n_jet":21}}},"image-shells":{"kind":"report","report":{"certified":true,"checks":{"CLOSED_FORM_D5":true,"D2_EXACT":true,"D3_CONSISTENT":true,"FLOAT_INSIDE":true,"IMAGE_BOUND":false,"LIBRARY_EXACT":true,"MEHTA_Z4":true,"MONOTONE":true,"PINNED":true,"SIDE24_CONSISTENT":true,"TRANSFER_BOUND":true,"TRUNCATION_NESTING":true,"WIDTHS":true},"coefficients":{"c_2_ref":["0.0734069193060342710301359629577740500176642446843326718326939","0.0734069193060342710301359629577740500176642446843326718516467"],"c_3_ref":["0.0417759318405983433429366654285755564666815196610446423810796","0.0417759318405983433429366654285755564666815196610448333783896"],"c_4_ref":["0.0233216660029528350945211949528568851531013723630981034059901","0.0233216660029528350945211949528568851531013723633825654059788"],"c_5_24":["0.0132193193800849680760334875146413965210528883241234179643775","0.0132193193800849680760334875146413965210528919272763929401089"],"c_5_ref":["0.0132193193800849680760334875146413965210528883241234179643777","0.0132193193800849680760334875146413965210528919272763929401087"],"c_5_ref_closed":["0.0132193193800849680760334875146413965210528883241234179643856","0.0132193193800849680760334875146413965210528919272763929401010"]},"cone_moments":{"D_2":["2.38384359055015523513604925862744194136738585267665775580671","2.38384359055015523513604925862744194136738585267666865399451"],"D_2_exact_29/6-sqrt6":["2.38384359055015523513604925862744194136738585267666320490062","2.38384359055015523513604925862744194136738585267666320490067"],"D_3":["5.32318026888989689492319873968726063578438852940845576405060","5.32318026888989689492319873968726063578438852947338433618854"],"D_3_Math-199":["5.32318026888989689492319873968726063578438852940845576405060","5.32318026888989689492319873968726063578438852947338433618854"],"D_4":["14.2187634589773576013064656778740783173406003782976930345926","14.2187634589773576013064656778740783173406042538663501338228"],"D_4_coarse":["14.2187634568717473865521252219639301878256781952949559611488","14.2187634610829678160609861730738485181580582236181611512260"]},"constants":{"(3/2)^(1/3)":["1.14471424255333186780804221193967700891590692078793107209905","1.14471424255333186780804221193967700891590692078793107209906"],"1536pi":["4825.48631591392241427862023671731643012685219744016254101693","4825.48631591392241427862023671731643012685219744016254101804"],"Gamma(7/6)":["0.927719333630039200708349482534621018566466519145475576700351","0.927719333630039200708349482534621018566466519145475576939062"],"Z_4":["4825.48631591392241427862023671731643012685219744016254101437","4825.48631591392241427862023671731643012685219744016254102065"],"Z_4_quadrature":["4825.48631591392241427862023671731643012685219743927374007126","4825.48631591392241427862023671731643012685219744105134196381"],"pi":["3.14159265358979323846264338327950288419716939937510582097457","3.14159265358979323846264338327950288419716939937510582097528"],"|S^4|":["26.3189450695716229835586426663364030275031984193087750037589","26.3189450695716229835586426663364030275031984193087750037782"]},"exact_relations":{"D_4":"sqrt(3/7) . 24/Z_4 . (1/6) I_4","D_m":"sqrt(3/(m+3)) Z_m^-1 int_{mu<0} prod mu_i^2 |Delta(mu)| exp(-|mu|^2/4 + (sum mu)^2/(4(m+3))) dmu","L_a":"L_a(T) = int_0^T e^{-v^2/24} sum_k e_{a,k}(v) J_k(v) dv, accumulated cell by cell","Z_4":"4! sqrt pi (1/6) int dT e^{-T^2/48} int_0^T dv e^{-v^2/24} int_0^v V e^{-p3^2/8} dp3 = 1536 pi","c_d_ref":"Gamma(7/6) (3/2)^(1/3) |S^{d-1}| D_{d-1} / (sqrt3 sqrt pi (2 pi)^d); d = 5: Gamma(7/6) (3/2)^(1/3) D_4 / (12 sqrt3 pi^(7/2))","decoupling":"alpha = 3m/(4(m+3)), beta = 3T/(2(m+3)); erfc branch exp(-T^2/(4m(m-1)) - |u|^2/4), polynomial branch exp(-T^2/((m-1)(m+3)) - |u|^2/4), u = t - (T/(m-1)) 1","erfc argument":"beta/(2 sqrt alpha) = T sqrt21/28; erfc(x) <= e^{-x^2} in the tail","m=4":"p1 = (T - 2 p2 - p3)/3, v = 2 p2 + p3, p2 = (v - p3)/2; |u|^2/4 = v^2/24 + p3^2/8; dp1 dp2 dp3 = dT dv dp3/6; region 0 < p3 < v < T","transfer":"E_5 = 60500 (76 24^6 + 15) e^-288; eps = 126 E_5; c_{5,24}/c_{5,ref} in [1/(1+delta), 1+delta], delta = (1+eps)^10/(1-eps)^10 - 1"},"float_reference_10sig":{"D_2_orthant":"2.383704698","D_3_orthant":"5.309848377","D_4_a_integrated_3D":"14.52840747","D_4_orthant":"13.51071422"},"mutant":"image-shells","object":"CL-SIDE24-D5-COEFFICIENT-20260930-v1","passed":false,"quadrature":{"K":40,"cells":"width 1/2 on [0, 30], width 1 on [30, 60]","rho":"6"},"scientific_effect":"NONE","transfer":{"E_5":["3.68125272702348204346339697671004496454749549007722334656497E-111","3.68125272702348204346339697671004496454749549007722334656500E-111"],"delta":["9.27675687209917474952776038130931331065968863499460283334372E-108","9.27675687209917474952776038130931331065968863499460283334394E-108"],"epsilon":["4.63837843604958737476388019065465665532984431749730141667186E-109","4.63837843604958737476388019065465665532984431749730141667190E-109"],"exponents":{"a":"29/3 (<= 10 used)","b":10},"n_jet":21}}},"jacobian":{"kind":"report","report":{"certified":true,"checks":{"CLOSED_FORM_D5":true,"D2_EXACT":true,"D3_CONSISTENT":true,"FLOAT_INSIDE":false,"IMAGE_BOUND":true,"LIBRARY_EXACT":true,"MEHTA_Z4":false,"MONOTONE":false,"PINNED":false,"SIDE24_CONSISTENT":true,"TRANSFER_BOUND":true,"TRUNCATION_NESTING":true,"WIDTHS":true},"coefficients":{"c_2_ref":["0.0734069193060342710301359629577740500176642446843326718326939","0.0734069193060342710301359629577740500176642446843326718516467"],"c_3_ref":["0.0417759318405983433429366654285755564666815196610446423810796","0.0417759318405983433429366654285755564666815196610448333783896"],"c_4_ref":["0.0233216660029528350945211949528568851531013723630981034059901","0.0233216660029528350945211949528568851531013723633825654059788"],"c_5_24":["0.0396579581402549042281004625439241895631586649723702538931399","0.0396579581402549042281004625439241895631586757818291788203167"],"c_5_ref":["0.0396579581402549042281004625439241895631586649723702538931403","0.0396579581402549042281004625439241895631586757818291788203163"],"c_5_ref_closed":["0.0396579581402549042281004625439241895631586649723702538931642","0.0396579581402549042281004625439241895631586757818291788202929"]},"cone_moments":{"D_2":["2.38384359055015523513604925862744194136738585267665775580671","2.38384359055015523513604925862744194136738585267666865399451"],"D_2_exact_29/6-sqrt6":["2.38384359055015523513604925862744194136738585267666320490062","2.38384359055015523513604925862744194136738585267666320490067"],"D_3":["5.32318026888989689492319873968726063578438852940845576405060","5.32318026888989689492319873968726063578438852947338433618854"],"D_3_Math-199":["5.32318026888989689492319873968726063578438852940845576405060","5.32318026888989689492319873968726063578438852947338433618854"],"D_4":["42.6562903769320728039193970336222349520218011348930791037856","42.6562903769320728039193970336222349520218127615990504014579"],"D_4_coarse":["42.6562903706152421596563756658917905634770345858848678834521","42.6562903832489034481829585192215455544741746708544834536723"]},"constants":{"(3/2)^(1/3)":["1.14471424255333186780804221193967700891590692078793107209905","1.14471424255333186780804221193967700891590692078793107209906"],"1536pi":["4825.48631591392241427862023671731643012685219744016254101693","4825.48631591392241427862023671731643012685219744016254101804"],"Gamma(7/6)":["0.927719333630039200708349482534621018566466519145475576700351","0.927719333630039200708349482534621018566466519145475576939062"],"Z_4":["4825.48631591392241427862023671731643012685219744016254101437","4825.48631591392241427862023671731643012685219744016254102065"],"Z_4_quadrature":["14476.4589477417672428358607101519492903805565923178212202112","14476.4589477417672428358607101519492903805565923231540258951"],"pi":["3.14159265358979323846264338327950288419716939937510582097457","3.14159265358979323846264338327950288419716939937510582097528"],"|S^4|":["26.3189450695716229835586426663364030275031984193087750037589","26.3189450695716229835586426663364030275031984193087750037782"]},"exact_relations":{"D_4":"sqrt(3/7) . 24/Z_4 . (1/6) I_4","D_m":"sqrt(3/(m+3)) Z_m^-1 int_{mu<0} prod mu_i^2 |Delta(mu)| exp(-|mu|^2/4 + (sum mu)^2/(4(m+3))) dmu","L_a":"L_a(T) = int_0^T e^{-v^2/24} sum_k e_{a,k}(v) J_k(v) dv, accumulated cell by cell","Z_4":"4! sqrt pi (1/6) int dT e^{-T^2/48} int_0^T dv e^{-v^2/24} int_0^v V e^{-p3^2/8} dp3 = 1536 pi","c_d_ref":"Gamma(7/6) (3/2)^(1/3) |S^{d-1}| D_{d-1} / (sqrt3 sqrt pi (2 pi)^d); d = 5: Gamma(7/6) (3/2)^(1/3) D_4 / (12 sqrt3 pi^(7/2))","decoupling":"alpha = 3m/(4(m+3)), beta = 3T/(2(m+3)); erfc branch exp(-T^2/(4m(m-1)) - |u|^2/4), polynomial branch exp(-T^2/((m-1)(m+3)) - |u|^2/4), u = t - (T/(m-1)) 1","erfc argument":"beta/(2 sqrt alpha) = T sqrt21/28; erfc(x) <= e^{-x^2} in the tail","m=4":"p1 = (T - 2 p2 - p3)/3, v = 2 p2 + p3, p2 = (v - p3)/2; |u|^2/4 = v^2/24 + p3^2/8; dp1 dp2 dp3 = dT dv dp3/6; region 0 < p3 < v < T","transfer":"E_5 = 60500 (76 24^6 + 15) e^-288; eps = 126 E_5; c_{5,24}/c_{5,ref} in [1/(1+delta), 1+delta], delta = (1+eps)^10/(1-eps)^10 - 1"},"float_reference_10sig":{"D_2_orthant":"2.383704698","D_3_orthant":"5.309848377","D_4_a_integrated_3D":"14.52840747","D_4_orthant":"13.51071422"},"mutant":"jacobian","object":"CL-SIDE24-D5-COEFFICIENT-20260930-v1","passed":false,"quadrature":{"K":40,"cells":"width 1/2 on [0, 30], width 1 on [30, 60]","rho":"6"},"scientific_effect":"NONE","transfer":{"E_5":["7.36250545404696408692679395342008992909499098015444669312995E-111","7.36250545404696408692679395342008992909499098015444669313000E-111"],"delta":["1.85535137441983494990555207626186266213193772699892056666874E-107","1.85535137441983494990555207626186266213193772699892056666882E-107"],"epsilon":["9.27675687209917474952776038130931331065968863499460283334373E-109","9.27675687209917474952776038130931331065968863499460283334380E-109"],"exponents":{"a":"29/3 (<= 10 used)","b":10},"n_jet":21}}},"mehta":{"kind":"report","report":{"certified":true,"checks":{"CLOSED_FORM_D5":true,"D2_EXACT":true,"D3_CONSISTENT":true,"FLOAT_INSIDE":false,"IMAGE_BOUND":true,"LIBRARY_EXACT":true,"MEHTA_Z4":false,"MONOTONE":false,"PINNED":false,"SIDE24_CONSISTENT":true,"TRANSFER_BOUND":true,"TRUNCATION_NESTING":true,"WIDTHS":true},"coefficients":{"c_2_ref":["0.0734069193060342710301359629577740500176642446843326718326939","0.0734069193060342710301359629577740500176642446843326718516467"],"c_3_ref":["0.0417759318405983433429366654285755564666815196610446423810796","0.0417759318405983433429366654285755564666815196610448333783896"],"c_4_ref":["0.0233216660029528350945211949528568851531013723630981034059901","0.0233216660029528350945211949528568851531013723633825654059788"],"c_5_24":["0.000248606743194497398100059172018518446971493225831870143262679","0.000248606743194497398100059172018518446971493293593916023609947"],"c_5_ref":["0.000248606743194497398100059172018518446971493225831870143262682","0.000248606743194497398100059172018518446971493293593916023609944"],"c_5_ref_closed":["0.000248606743194497398100059172018518446971493225831870143262831","0.000248606743194497398100059172018518446971493293593916023609798"]},"cone_moments":{"D_2":["2.38384359055015523513604925862744194136738585267665775580671","2.38384359055015523513604925862744194136738585267666865399451"],"D_2_exact_29/6-sqrt6":["2.38384359055015523513604925862744194136738585267666320490062","2.38384359055015523513604925862744194136738585267666320490067"],"D_3":["5.32318026888989689492319873968726063578438852940845576405060","5.32318026888989689492319873968726063578438852947338433618854"],"D_3_Math-199":["5.32318026888989689492319873968726063578438852940845576405060","5.32318026888989689492319873968726063578438852947338433618854"],"D_4":["0.267402607816149668960181444861623130944715129939809026692857","0.267402607816149668960181444861623130944715202824991248677955"],"D_4_coarse":["0.267402607776550890620976344119831062052275159538058163954836","0.267402607855748447299389931479809670180107167602791152497369"]},"constants":{"(3/2)^(1/3)":["1.14471424255333186780804221193967700891590692078793107209905","1.14471424255333186780804221193967700891590692078793107209906"],"1536pi":["4825.48631591392241427862023671731643012685219744016254101693","4825.48631591392241427862023671731643012685219744016254101804"],"Gamma(7/6)":["0.927719333630039200708349482534621018566466519145475576700351","0.927719333630039200708349482534621018566466519145475576939062"],"Z_4":["256588.554094005097510724412616035949148787811904251879407305","256588.554094005097510724412616035949148787811904251879407752"],"Z_4_quadrature":["4825.48631591392241427862023671731643012685219743927374007126","4825.48631591392241427862023671731643012685219744105134196381"],"pi":["3.14159265358979323846264338327950288419716939937510582097457","3.14159265358979323846264338327950288419716939937510582097528"],"|S^4|":["26.3189450695716229835586426663364030275031984193087750037589","26.3189450695716229835586426663364030275031984193087750037782"]},"exact_relations":{"D_4":"sqrt(3/7) . 24/Z_4 . (1/6) I_4","D_m":"sqrt(3/(m+3)) Z_m^-1 int_{mu<0} prod mu_i^2 |Delta(mu)| exp(-|mu|^2/4 + (sum mu)^2/(4(m+3))) dmu","L_a":"L_a(T) = int_0^T e^{-v^2/24} sum_k e_{a,k}(v) J_k(v) dv, accumulated cell by cell","Z_4":"4! sqrt pi (1/6) int dT e^{-T^2/48} int_0^T dv e^{-v^2/24} int_0^v V e^{-p3^2/8} dp3 = 1536 pi","c_d_ref":"Gamma(7/6) (3/2)^(1/3) |S^{d-1}| D_{d-1} / (sqrt3 sqrt pi (2 pi)^d); d = 5: Gamma(7/6) (3/2)^(1/3) D_4 / (12 sqrt3 pi^(7/2))","decoupling":"alpha = 3m/(4(m+3)), beta = 3T/(2(m+3)); erfc branch exp(-T^2/(4m(m-1)) - |u|^2/4), polynomial branch exp(-T^2/((m-1)(m+3)) - |u|^2/4), u = t - (T/(m-1)) 1","erfc argument":"beta/(2 sqrt alpha) = T sqrt21/28; erfc(x) <= e^{-x^2} in the tail","m=4":"p1 = (T - 2 p2 - p3)/3, v = 2 p2 + p3, p2 = (v - p3)/2; |u|^2/4 = v^2/24 + p3^2/8; dp1 dp2 dp3 = dT dv dp3/6; region 0 < p3 < v < T","transfer":"E_5 = 60500 (76 24^6 + 15) e^-288; eps = 126 E_5; c_{5,24}/c_{5,ref} in [1/(1+delta), 1+delta], delta = (1+eps)^10/(1-eps)^10 - 1"},"float_reference_10sig":{"D_2_orthant":"2.383704698","D_3_orthant":"5.309848377","D_4_a_integrated_3D":"14.52840747","D_4_orthant":"13.51071422"},"mutant":"mehta","object":"CL-SIDE24-D5-COEFFICIENT-20260930-v1","passed":false,"quadrature":{"K":40,"cells":"width 1/2 on [0, 30], width 1 on [30, 60]","rho":"6"},"scientific_effect":"NONE","transfer":{"E_5":["7.36250545404696408692679395342008992909499098015444669312995E-111","7.36250545404696408692679395342008992909499098015444669313000E-111"],"delta":["1.85535137441983494990555207626186266213193772699892056666874E-107","1.85535137441983494990555207626186266213193772699892056666882E-107"],"epsilon":["9.27675687209917474952776038130931331065968863499460283334373E-109","9.27675687209917474952776038130931331065968863499460283334380E-109"],"exponents":{"a":"29/3 (<= 10 used)","b":10},"n_jet":21}}},"recurrence":{"kind":"report","report":{"certified":true,"checks":{"CLOSED_FORM_D5":true,"D2_EXACT":false,"D3_CONSISTENT":false,"FLOAT_INSIDE":false,"IMAGE_BOUND":true,"LIBRARY_EXACT":true,"MEHTA_Z4":true,"MONOTONE":true,"PINNED":false,"SIDE24_CONSISTENT":false,"TRANSFER_BOUND":true,"TRUNCATION_NESTING":true,"WIDTHS":true},"coefficients":{"c_2_ref":["0.0734069193060342710301359629577740500176642446843326718326939","0.0734069193060342710301359629577740500176642446843326718516467"],"c_3_ref":["0.0407892691572937282017742657472035223510267804246933518457154","0.0407892691572937282017742657472035223510267804246935368519180"],"c_4_ref":["0.0212167922784642336278468368688454193619790585194223888318101","0.0212167922784642336278468368688454193619790585196949706532499"],"c_5_24":["0.0104105530242710195877724030734441393030274113699977020069447","0.0104105530242710195877724030734441393030274581140009710569624"],"c_5_ref":["0.0104105530242710195877724030734441393030274113699977020069449","0.0104105530242710195877724030734441393030274581140009710569622"],"c_5_ref_closed":["0.0104105530242710195877724030734441393030274113699977020069511","0.0104105530242710195877724030734441393030274581140009710569561"]},"cone_moments":{"D_2":["2.32754204537803833215472815132730519441816387234869976218835","2.32754204537803833215472815132730519441816387234871031852251"],"D_2_exact_29/6-sqrt6":["2.38384359055015523513604925862744194136738585267666320490062","2.38384359055015523513604925862744194136738585267666320490067"],"D_3":["4.84274193839996289334197430567817435779451992391719768679301","4.84274193839996289334197430567817435779451992397941460320561"],"D_3_Math-199":["5.32318026888989689492319873968726063578438852940845576405060","5.32318026888989689492319873968726063578438852947338433618854"],"D_4":["11.1976408673696445281346559818449780467563111222876764258800","11.1976408673696445281346559818449780467563614003622503360618"],"D_4_coarse":["11.1976408653178809467299045098391088077032503127603207277847","11.1976408694214081095396076508696012828715501662351196082526"]},"constants":{"(3/2)^(1/3)":["1.14471424255333186780804221193967700891590692078793107209905","1.14471424255333186780804221193967700891590692078793107209906"],"1536pi":["4825.48631591392241427862023671731643012685219744016254101693","4825.48631591392241427862023671731643012685219744016254101804"],"Gamma(7/6)":["0.927719333630039200708349482534621018566466519145475576700351","0.927719333630039200708349482534621018566466519145475576939062"],"Z_4":["4825.48631591392241427862023671731643012685219744016254101437","4825.48631591392241427862023671731643012685219744016254102065"],"Z_4_quadrature":["4825.48631591392241427862023671731643012685219743927374007126","4825.48631591392241427862023671731643012685219744105134196381"],"pi":["3.14159265358979323846264338327950288419716939937510582097457","3.14159265358979323846264338327950288419716939937510582097528"],"|S^4|":["26.3189450695716229835586426663364030275031984193087750037589","26.3189450695716229835586426663364030275031984193087750037782"]},"exact_relations":{"D_4":"sqrt(3/7) . 24/Z_4 . (1/6) I_4","D_m":"sqrt(3/(m+3)) Z_m^-1 int_{mu<0} prod mu_i^2 |Delta(mu)| exp(-|mu|^2/4 + (sum mu)^2/(4(m+3))) dmu","L_a":"L_a(T) = int_0^T e^{-v^2/24} sum_k e_{a,k}(v) J_k(v) dv, accumulated cell by cell","Z_4":"4! sqrt pi (1/6) int dT e^{-T^2/48} int_0^T dv e^{-v^2/24} int_0^v V e^{-p3^2/8} dp3 = 1536 pi","c_d_ref":"Gamma(7/6) (3/2)^(1/3) |S^{d-1}| D_{d-1} / (sqrt3 sqrt pi (2 pi)^d); d = 5: Gamma(7/6) (3/2)^(1/3) D_4 / (12 sqrt3 pi^(7/2))","decoupling":"alpha = 3m/(4(m+3)), beta = 3T/(2(m+3)); erfc branch exp(-T^2/(4m(m-1)) - |u|^2/4), polynomial branch exp(-T^2/((m-1)(m+3)) - |u|^2/4), u = t - (T/(m-1)) 1","erfc argument":"beta/(2 sqrt alpha) = T sqrt21/28; erfc(x) <= e^{-x^2} in the tail","m=4":"p1 = (T - 2 p2 - p3)/3, v = 2 p2 + p3, p2 = (v - p3)/2; |u|^2/4 = v^2/24 + p3^2/8; dp1 dp2 dp3 = dT dv dp3/6; region 0 < p3 < v < T","transfer":"E_5 = 60500 (76 24^6 + 15) e^-288; eps = 126 E_5; c_{5,24}/c_{5,ref} in [1/(1+delta), 1+delta], delta = (1+eps)^10/(1-eps)^10 - 1"},"float_reference_10sig":{"D_2_orthant":"2.383704698","D_3_orthant":"5.309848377","D_4_a_integrated_3D":"14.52840747","D_4_orthant":"13.51071422"},"mutant":"recurrence","object":"CL-SIDE24-D5-COEFFICIENT-20260930-v1","passed":false,"quadrature":{"K":40,"cells":"width 1/2 on [0, 30], width 1 on [30, 60]","rho":"6"},"scientific_effect":"NONE","transfer":{"E_5":["7.36250545404696408692679395342008992909499098015444669312995E-111","7.36250545404696408692679395342008992909499098015444669313000E-111"],"delta":["1.85535137441983494990555207626186266213193772699892056666874E-107","1.85535137441983494990555207626186266213193772699892056666882E-107"],"epsilon":["9.27675687209917474952776038130931331065968863499460283334373E-109","9.27675687209917474952776038130931331065968863499460283334380E-109"],"exponents":{"a":"29/3 (<= 10 used)","b":10},"n_jet":21}}},"remainder-dropped":{"kind":"report","report":{"certified":true,"checks":{"CLOSED_FORM_D5":true,"D2_EXACT":true,"D3_CONSISTENT":true,"FLOAT_INSIDE":true,"IMAGE_BOUND":true,"LIBRARY_EXACT":true,"MEHTA_Z4":true,"MONOTONE":true,"PINNED":false,"SIDE24_CONSISTENT":true,"TRANSFER_BOUND":true,"TRUNCATION_NESTING":false,"WIDTHS":true},"coefficients":{"c_2_ref":["0.0734069193060342710301359629577740500176642446843326718326939","0.0734069193060342710301359629577740500176642446843326718516467"],"c_3_ref":["0.0417759318405983433429366654285755564666815196610446423810796","0.0417759318405983433429366654285755564666815196610448333783896"],"c_4_ref":["0.0233216660029528350945211949528568851531013723630981034059901","0.0233216660029528350945211949528568851531013723633825654059788"],"c_5_24":["0.0132193194631583022360189082361133306203200702895853292477555","0.0132193194631583022360189082361133306203200702895853292512897"],"c_5_ref":["0.0132193194631583022360189082361133306203200702895853292477557","0.0132193194631583022360189082361133306203200702895853292512895"],"c_5_ref_closed":["0.0132193194631583022360189082361133306203200702895853292477636","0.0132193194631583022360189082361133306203200702895853292512817"]},"cone_moments":{"D_2":["2.38384359055015523513604925862744194136738585267665775580671","2.38384359055015523513604925862744194136738585267666865399451"],"D_2_exact_29/6-sqrt6":["2.38384359055015523513604925862744194136738585267666320490062","2.38384359055015523513604925862744194136738585267666320490067"],"D_3":["5.32318026888989689492319873968726063578438852940845576405060","5.32318026888989689492319873968726063578438852947338433618854"],"D_3_Math-199":["5.32318026888989689492319873968726063578438852940845576405060","5.32318026888989689492319873968726063578438852947338433618854"],"D_4":["14.2187635483314350849706765750027959830732262497797390841046","14.2187635483314350849706765750027959830732262497797390842175"],"D_4_coarse":["14.2187634589773576013065556975188893529918682094565585561295","14.2187634589773576013065556975188893529918682094565585562446"]},"constants":{"(3/2)^(1/3)":["1.14471424255333186780804221193967700891590692078793107209905","1.14471424255333186780804221193967700891590692078793107209906"],"1536pi":["4825.48631591392241427862023671731643012685219744016254101693","4825.48631591392241427862023671731643012685219744016254101804"],"Gamma(7/6)":["0.927719333630039200708349482534621018566466519145475576700351","0.927719333630039200708349482534621018566466519145475576939062"],"Z_4":["4825.48631591392241427862023671731643012685219744016254101437","4825.48631591392241427862023671731643012685219744016254102065"],"Z_4_quadrature":["4825.48631591392241427862023671731643012685219743927374007126","4825.48631591392241427862023671731643012685219744105134196381"],"pi":["3.14159265358979323846264338327950288419716939937510582097457","3.14159265358979323846264338327950288419716939937510582097528"],"|S^4|":["26.3189450695716229835586426663364030275031984193087750037589","26.3189450695716229835586426663364030275031984193087750037782"]},"exact_relations":{"D_4":"sqrt(3/7) . 24/Z_4 . (1/6) I_4","D_m":"sqrt(3/(m+3)) Z_m^-1 int_{mu<0} prod mu_i^2 |Delta(mu)| exp(-|mu|^2/4 + (sum mu)^2/(4(m+3))) dmu","L_a":"L_a(T) = int_0^T e^{-v^2/24} sum_k e_{a,k}(v) J_k(v) dv, accumulated cell by cell","Z_4":"4! sqrt pi (1/6) int dT e^{-T^2/48} int_0^T dv e^{-v^2/24} int_0^v V e^{-p3^2/8} dp3 = 1536 pi","c_d_ref":"Gamma(7/6) (3/2)^(1/3) |S^{d-1}| D_{d-1} / (sqrt3 sqrt pi (2 pi)^d); d = 5: Gamma(7/6) (3/2)^(1/3) D_4 / (12 sqrt3 pi^(7/2))","decoupling":"alpha = 3m/(4(m+3)), beta = 3T/(2(m+3)); erfc branch exp(-T^2/(4m(m-1)) - |u|^2/4), polynomial branch exp(-T^2/((m-1)(m+3)) - |u|^2/4), u = t - (T/(m-1)) 1","erfc argument":"beta/(2 sqrt alpha) = T sqrt21/28; erfc(x) <= e^{-x^2} in the tail","m=4":"p1 = (T - 2 p2 - p3)/3, v = 2 p2 + p3, p2 = (v - p3)/2; |u|^2/4 = v^2/24 + p3^2/8; dp1 dp2 dp3 = dT dv dp3/6; region 0 < p3 < v < T","transfer":"E_5 = 60500 (76 24^6 + 15) e^-288; eps = 126 E_5; c_{5,24}/c_{5,ref} in [1/(1+delta), 1+delta], delta = (1+eps)^10/(1-eps)^10 - 1"},"float_reference_10sig":{"D_2_orthant":"2.383704698","D_3_orthant":"5.309848377","D_4_a_integrated_3D":"14.52840747","D_4_orthant":"13.51071422"},"mutant":"remainder-dropped","object":"CL-SIDE24-D5-COEFFICIENT-20260930-v1","passed":false,"quadrature":{"K":40,"cells":"width 1/2 on [0, 30], width 1 on [30, 60]","rho":"6"},"scientific_effect":"NONE","transfer":{"E_5":["7.36250545404696408692679395342008992909499098015444669312995E-111","7.36250545404696408692679395342008992909499098015444669313000E-111"],"delta":["1.85535137441983494990555207626186266213193772699892056666874E-107","1.85535137441983494990555207626186266213193772699892056666882E-107"],"epsilon":["9.27675687209917474952776038130931331065968863499460283334373E-109","9.27675687209917474952776038130931331065968863499460283334380E-109"],"exponents":{"a":"29/3 (<= 10 used)","b":10},"n_jet":21}}},"sandwich-exponent":{"kind":"report","report":{"certified":true,"checks":{"CLOSED_FORM_D5":true,"D2_EXACT":true,"D3_CONSISTENT":true,"FLOAT_INSIDE":true,"IMAGE_BOUND":true,"LIBRARY_EXACT":true,"MEHTA_Z4":true,"MONOTONE":true,"PINNED":true,"SIDE24_CONSISTENT":true,"TRANSFER_BOUND":false,"TRUNCATION_NESTING":true,"WIDTHS":true},"coefficients":{"c_2_ref":["0.0734069193060342710301359629577740500176642446843326718326939","0.0734069193060342710301359629577740500176642446843326718516467"],"c_3_ref":["0.0417759318405983433429366654285755564666815196610446423810796","0.0417759318405983433429366654285755564666815196610448333783896"],"c_4_ref":["0.0233216660029528350945211949528568851531013723630981034059901","0.0233216660029528350945211949528568851531013723633825654059788"],"c_5_24":["0.0132193193800849680760334875146413965210528883241234179643775","0.0132193193800849680760334875146413965210528919272763929401089"],"c_5_ref":["0.0132193193800849680760334875146413965210528883241234179643777","0.0132193193800849680760334875146413965210528919272763929401087"],"c_5_ref_closed":["0.0132193193800849680760334875146413965210528883241234179643856","0.0132193193800849680760334875146413965210528919272763929401010"]},"cone_moments":{"D_2":["2.38384359055015523513604925862744194136738585267665775580671","2.38384359055015523513604925862744194136738585267666865399451"],"D_2_exact_29/6-sqrt6":["2.38384359055015523513604925862744194136738585267666320490062","2.38384359055015523513604925862744194136738585267666320490067"],"D_3":["5.32318026888989689492319873968726063578438852940845576405060","5.32318026888989689492319873968726063578438852947338433618854"],"D_3_Math-199":["5.32318026888989689492319873968726063578438852940845576405060","5.32318026888989689492319873968726063578438852947338433618854"],"D_4":["14.2187634589773576013064656778740783173406003782976930345926","14.2187634589773576013064656778740783173406042538663501338228"],"D_4_coarse":["14.2187634568717473865521252219639301878256781952949559611488","14.2187634610829678160609861730738485181580582236181611512260"]},"constants":{"(3/2)^(1/3)":["1.14471424255333186780804221193967700891590692078793107209905","1.14471424255333186780804221193967700891590692078793107209906"],"1536pi":["4825.48631591392241427862023671731643012685219744016254101693","4825.48631591392241427862023671731643012685219744016254101804"],"Gamma(7/6)":["0.927719333630039200708349482534621018566466519145475576700351","0.927719333630039200708349482534621018566466519145475576939062"],"Z_4":["4825.48631591392241427862023671731643012685219744016254101437","4825.48631591392241427862023671731643012685219744016254102065"],"Z_4_quadrature":["4825.48631591392241427862023671731643012685219743927374007126","4825.48631591392241427862023671731643012685219744105134196381"],"pi":["3.14159265358979323846264338327950288419716939937510582097457","3.14159265358979323846264338327950288419716939937510582097528"],"|S^4|":["26.3189450695716229835586426663364030275031984193087750037589","26.3189450695716229835586426663364030275031984193087750037782"]},"exact_relations":{"D_4":"sqrt(3/7) . 24/Z_4 . (1/6) I_4","D_m":"sqrt(3/(m+3)) Z_m^-1 int_{mu<0} prod mu_i^2 |Delta(mu)| exp(-|mu|^2/4 + (sum mu)^2/(4(m+3))) dmu","L_a":"L_a(T) = int_0^T e^{-v^2/24} sum_k e_{a,k}(v) J_k(v) dv, accumulated cell by cell","Z_4":"4! sqrt pi (1/6) int dT e^{-T^2/48} int_0^T dv e^{-v^2/24} int_0^v V e^{-p3^2/8} dp3 = 1536 pi","c_d_ref":"Gamma(7/6) (3/2)^(1/3) |S^{d-1}| D_{d-1} / (sqrt3 sqrt pi (2 pi)^d); d = 5: Gamma(7/6) (3/2)^(1/3) D_4 / (12 sqrt3 pi^(7/2))","decoupling":"alpha = 3m/(4(m+3)), beta = 3T/(2(m+3)); erfc branch exp(-T^2/(4m(m-1)) - |u|^2/4), polynomial branch exp(-T^2/((m-1)(m+3)) - |u|^2/4), u = t - (T/(m-1)) 1","erfc argument":"beta/(2 sqrt alpha) = T sqrt21/28; erfc(x) <= e^{-x^2} in the tail","m=4":"p1 = (T - 2 p2 - p3)/3, v = 2 p2 + p3, p2 = (v - p3)/2; |u|^2/4 = v^2/24 + p3^2/8; dp1 dp2 dp3 = dT dv dp3/6; region 0 < p3 < v < T","transfer":"E_5 = 60500 (76 24^6 + 15) e^-288; eps = 126 E_5; c_{5,24}/c_{5,ref} in [1/(1+delta), 1+delta], delta = (1+eps)^10/(1-eps)^10 - 1"},"float_reference_10sig":{"D_2_orthant":"2.383704698","D_3_orthant":"5.309848377","D_4_a_integrated_3D":"14.52840747","D_4_orthant":"13.51071422"},"mutant":"sandwich-exponent","object":"CL-SIDE24-D5-COEFFICIENT-20260930-v1","passed":false,"quadrature":{"K":40,"cells":"width 1/2 on [0, 30], width 1 on [30, 60]","rho":"6"},"scientific_effect":"NONE","transfer":{"E_5":["7.36250545404696408692679395342008992909499098015444669312995E-111","7.36250545404696408692679395342008992909499098015444669313000E-111"],"delta":["1.66981623697785145491499686863567639591874395429902851000187E-107","1.66981623697785145491499686863567639591874395429902851000200E-107"],"epsilon":["9.27675687209917474952776038130931331065968863499460283334373E-109","9.27675687209917474952776038130931331065968863499460283334380E-109"],"exponents":{"a":"29/3 (<= 10 used)","b":10},"n_jet":21}}},"shift-variance":{"kind":"report","report":{"certified":true,"checks":{"CLOSED_FORM_D5":true,"D2_EXACT":true,"D3_CONSISTENT":true,"FLOAT_INSIDE":false,"IMAGE_BOUND":true,"LIBRARY_EXACT":true,"MEHTA_Z4":true,"MONOTONE":false,"PINNED":false,"SIDE24_CONSISTENT":true,"TRANSFER_BOUND":true,"TRUNCATION_NESTING":true,"WIDTHS":true},"coefficients":{"c_2_ref":["0.0734069193060342710301359629577740500176642446843326718326939","0.0734069193060342710301359629577740500176642446843326718516467"],"c_3_ref":["0.0417759318405983433429366654285755564666815196610446423810796","0.0417759318405983433429366654285755564666815196610448333783896"],"c_4_ref":["0.0233216660029528350945211949528568851531013723630981034059901","0.0233216660029528350945211949528568851531013723633825654059788"],"c_5_24":["0.0591759708792297785593923857978417427883205929715612918603759","0.0591759708792297785593923857978417427883205960349088493701797"],"c_5_ref":["0.0591759708792297785593923857978417427883205929715612918603765","0.0591759708792297785593923857978417427883205960349088493701791"],"c_5_ref_closed":["0.0591759708792297785593923857978417427883205929715612918604120","0.0591759708792297785593923857978417427883205960349088493701442"]},"cone_moments":{"D_2":["2.38384359055015523513604925862744194136738585267665775580671","2.38384359055015523513604925862744194136738585267666865399451"],"D_2_exact_29/6-sqrt6":["2.38384359055015523513604925862744194136738585267666320490062","2.38384359055015523513604925862744194136738585267666320490067"],"D_3":["5.32318026888989689492319873968726063578438852940845576405060","5.32318026888989689492319873968726063578438852947338433618854"],"D_3_Math-199":["5.32318026888989689492319873968726063578438852940845576405060","5.32318026888989689492319873968726063578438852947338433618854"],"D_4":["63.6499586850660065518550616607172288346109617869654449437689","63.6499586850660065518550616607172288346109650819168006292017"],"D_4_coarse":["63.6499586832265656481704168204544592951254238539077761189810","63.6499586869054474555394385599157501334231002970601621449954"]},"constants":{"(3/2)^(1/3)":["1.14471424255333186780804221193967700891590692078793107209905","1.14471424255333186780804221193967700891590692078793107209906"],"1536pi":["4825.48631591392241427862023671731643012685219744016254101693","4825.48631591392241427862023671731643012685219744016254101804"],"Gamma(7/6)":["0.927719333630039200708349482534621018566466519145475576700351","0.927719333630039200708349482534621018566466519145475576939062"],"Z_4":["4825.48631591392241427862023671731643012685219744016254101437","4825.48631591392241427862023671731643012685219744016254102065"],"Z_4_quadrature":["4825.48631591392241427862023671731643012685219743927374007126","4825.48631591392241427862023671731643012685219744105134196381"],"pi":["3.14159265358979323846264338327950288419716939937510582097457","3.14159265358979323846264338327950288419716939937510582097528"],"|S^4|":["26.3189450695716229835586426663364030275031984193087750037589","26.3189450695716229835586426663364030275031984193087750037782"]},"exact_relations":{"D_4":"sqrt(3/7) . 24/Z_4 . (1/6) I_4","D_m":"sqrt(3/(m+3)) Z_m^-1 int_{mu<0} prod mu_i^2 |Delta(mu)| exp(-|mu|^2/4 + (sum mu)^2/(4(m+3))) dmu","L_a":"L_a(T) = int_0^T e^{-v^2/24} sum_k e_{a,k}(v) J_k(v) dv, accumulated cell by cell","Z_4":"4! sqrt pi (1/6) int dT e^{-T^2/48} int_0^T dv e^{-v^2/24} int_0^v V e^{-p3^2/8} dp3 = 1536 pi","c_d_ref":"Gamma(7/6) (3/2)^(1/3) |S^{d-1}| D_{d-1} / (sqrt3 sqrt pi (2 pi)^d); d = 5: Gamma(7/6) (3/2)^(1/3) D_4 / (12 sqrt3 pi^(7/2))","decoupling":"alpha = 3m/(4(m+3)), beta = 3T/(2(m+3)); erfc branch exp(-T^2/(4m(m-1)) - |u|^2/4), polynomial branch exp(-T^2/((m-1)(m+3)) - |u|^2/4), u = t - (T/(m-1)) 1","erfc argument":"beta/(2 sqrt alpha) = T sqrt21/28; erfc(x) <= e^{-x^2} in the tail","m=4":"p1 = (T - 2 p2 - p3)/3, v = 2 p2 + p3, p2 = (v - p3)/2; |u|^2/4 = v^2/24 + p3^2/8; dp1 dp2 dp3 = dT dv dp3/6; region 0 < p3 < v < T","transfer":"E_5 = 60500 (76 24^6 + 15) e^-288; eps = 126 E_5; c_{5,24}/c_{5,ref} in [1/(1+delta), 1+delta], delta = (1+eps)^10/(1-eps)^10 - 1"},"float_reference_10sig":{"D_2_orthant":"2.383704698","D_3_orthant":"5.309848377","D_4_a_integrated_3D":"14.52840747","D_4_orthant":"13.51071422"},"mutant":"shift-variance","object":"CL-SIDE24-D5-COEFFICIENT-20260930-v1","passed":false,"quadrature":{"K":40,"cells":"width 1/2 on [0, 30], width 1 on [30, 60]","rho":"6"},"scientific_effect":"NONE","transfer":{"E_5":["7.36250545404696408692679395342008992909499098015444669312995E-111","7.36250545404696408692679395342008992909499098015444669313000E-111"],"delta":["1.85535137441983494990555207626186266213193772699892056666874E-107","1.85535137441983494990555207626186266213193772699892056666882E-107"],"epsilon":["9.27675687209917474952776038130931331065968863499460283334373E-109","9.27675687209917474952776038130931331065968863499460283334380E-109"],"exponents":{"a":"29/3 (<= 10 used)","b":10},"n_jet":21}}},"sphere-area":{"kind":"report","report":{"certified":true,"checks":{"CLOSED_FORM_D5":false,"D2_EXACT":true,"D3_CONSISTENT":true,"FLOAT_INSIDE":true,"IMAGE_BOUND":true,"LIBRARY_EXACT":true,"MEHTA_Z4":true,"MONOTONE":true,"PINNED":false,"SIDE24_CONSISTENT":true,"TRANSFER_BOUND":true,"TRUNCATION_NESTING":true,"WIDTHS":true},"coefficients":{"c_2_ref":["0.0734069193060342710301359629577740500176642446843326718326939","0.0734069193060342710301359629577740500176642446843326718516467"],"c_3_ref":["0.0417759318405983433429366654285755564666815196610446423810796","0.0417759318405983433429366654285755564666815196610448333783896"],"c_4_ref":["0.0233216660029528350945211949528568851531013723630981034059901","0.0233216660029528350945211949528568851531013723633825654059788"],"c_5_24":["0.0198289790701274521140502312719620947815793324861851269465691","0.0198289790701274521140502312719620947815793378909145894101606"],"c_5_ref":["0.0198289790701274521140502312719620947815793324861851269465693","0.0198289790701274521140502312719620947815793378909145894101604"],"c_5_ref_closed":["0.0132193193800849680760334875146413965210528883241234179643856","0.0132193193800849680760334875146413965210528919272763929401010"]},"cone_moments":{"D_2":["2.38384359055015523513604925862744194136738585267665775580671","2.38384359055015523513604925862744194136738585267666865399451"],"D_2_exact_29/6-sqrt6":["2.38384359055015523513604925862744194136738585267666320490062","2.38384359055015523513604925862744194136738585267666320490067"],"D_3":["5.32318026888989689492319873968726063578438852940845576405060","5.32318026888989689492319873968726063578438852947338433618854"],"D_3_Math-199":["5.32318026888989689492319873968726063578438852940845576405060","5.32318026888989689492319873968726063578438852947338433618854"],"D_4":["14.2187634589773576013064656778740783173406003782976930345926","14.2187634589773576013064656778740783173406042538663501338228"],"D_4_coarse":["14.2187634568717473865521252219639301878256781952949559611488","14.2187634610829678160609861730738485181580582236181611512260"]},"constants":{"(3/2)^(1/3)":["1.14471424255333186780804221193967700891590692078793107209905","1.14471424255333186780804221193967700891590692078793107209906"],"1536pi":["4825.48631591392241427862023671731643012685219744016254101693","4825.48631591392241427862023671731643012685219744016254101804"],"Gamma(7/6)":["0.927719333630039200708349482534621018566466519145475576700351","0.927719333630039200708349482534621018566466519145475576939062"],"Z_4":["4825.48631591392241427862023671731643012685219744016254101437","4825.48631591392241427862023671731643012685219744016254102065"],"Z_4_quadrature":["4825.48631591392241427862023671731643012685219743927374007126","4825.48631591392241427862023671731643012685219744105134196381"],"pi":["3.14159265358979323846264338327950288419716939937510582097457","3.14159265358979323846264338327950288419716939937510582097528"],"|S^4|":["39.4784176043574344753379639995046045412547976289631625056437","39.4784176043574344753379639995046045412547976289631625056621"]},"exact_relations":{"D_4":"sqrt(3/7) . 24/Z_4 . (1/6) I_4","D_m":"sqrt(3/(m+3)) Z_m^-1 int_{mu<0} prod mu_i^2 |Delta(mu)| exp(-|mu|^2/4 + (sum mu)^2/(4(m+3))) dmu","L_a":"L_a(T) = int_0^T e^{-v^2/24} sum_k e_{a,k}(v) J_k(v) dv, accumulated cell by cell","Z_4":"4! sqrt pi (1/6) int dT e^{-T^2/48} int_0^T dv e^{-v^2/24} int_0^v V e^{-p3^2/8} dp3 = 1536 pi","c_d_ref":"Gamma(7/6) (3/2)^(1/3) |S^{d-1}| D_{d-1} / (sqrt3 sqrt pi (2 pi)^d); d = 5: Gamma(7/6) (3/2)^(1/3) D_4 / (12 sqrt3 pi^(7/2))","decoupling":"alpha = 3m/(4(m+3)), beta = 3T/(2(m+3)); erfc branch exp(-T^2/(4m(m-1)) - |u|^2/4), polynomial branch exp(-T^2/((m-1)(m+3)) - |u|^2/4), u = t - (T/(m-1)) 1","erfc argument":"beta/(2 sqrt alpha) = T sqrt21/28; erfc(x) <= e^{-x^2} in the tail","m=4":"p1 = (T - 2 p2 - p3)/3, v = 2 p2 + p3, p2 = (v - p3)/2; |u|^2/4 = v^2/24 + p3^2/8; dp1 dp2 dp3 = dT dv dp3/6; region 0 < p3 < v < T","transfer":"E_5 = 60500 (76 24^6 + 15) e^-288; eps = 126 E_5; c_{5,24}/c_{5,ref} in [1/(1+delta), 1+delta], delta = (1+eps)^10/(1-eps)^10 - 1"},"float_reference_10sig":{"D_2_orthant":"2.383704698","D_3_orthant":"5.309848377","D_4_a_integrated_3D":"14.52840747","D_4_orthant":"13.51071422"},"mutant":"sphere-area","object":"CL-SIDE24-D5-COEFFICIENT-20260930-v1","passed":false,"quadrature":{"K":40,"cells":"width 1/2 on [0, 30], width 1 on [30, 60]","rho":"6"},"scientific_effect":"NONE","transfer":{"E_5":["7.36250545404696408692679395342008992909499098015444669312995E-111","7.36250545404696408692679395342008992909499098015444669313000E-111"],"delta":["1.85535137441983494990555207626186266213193772699892056666874E-107","1.85535137441983494990555207626186266213193772699892056666882E-107"],"epsilon":["9.27675687209917474952776038130931331065968863499460283334373E-109","9.27675687209917474952776038130931331065968863499460283334380E-109"],"exponents":{"a":"29/3 (<= 10 used)","b":10},"n_jet":21}}},"vandermonde":{"kind":"report","report":{"certified":true,"checks":{"CLOSED_FORM_D5":true,"D2_EXACT":false,"D3_CONSISTENT":false,"FLOAT_INSIDE":false,"IMAGE_BOUND":true,"LIBRARY_EXACT":true,"MEHTA_Z4":true,"MONOTONE":false,"PINNED":false,"SIDE24_CONSISTENT":false,"TRANSFER_BOUND":true,"TRUNCATION_NESTING":true,"WIDTHS":true},"coefficients":{"c_2_ref":["0.0734069193060342710301359629577740500176642446843326718326939","0.0734069193060342710301359629577740500176642446843326718516467"],"c_3_ref":["0.0345227141991587609278963003134510556863572492000787955216016","0.0345227141991587609278963003134510556863572492000788102367989"],"c_4_ref":["0.0100891982783390193484136921479201246570727833514406500722959","0.0100891982783390193484136921479201246570727833514407991499130"],"c_5_24":["0.00158639184717738646105150854498227141350832362117998133018051","0.00158639184717738646105150854498227141350832362123036573105399"],"c_5_ref":["0.00158639184717738646105150854498227141350832362117998133018053","0.00158639184717738646105150854498227141350832362123036573105397"],"c_5_ref_closed":["0.00158639184717738646105150854498227141350832362117998133018148","0.00158639184717738646105150854498227141350832362123036573105303"]},"cone_moments":{"D_2":["1.96995608107734641188773612549525733323255983498115354468098","1.96995608107734641188773612549525733323255983498115438385889"],"D_2_exact_29/6-sqrt6":["2.38384359055015523513604925862744194136738585267666320490062","2.38384359055015523513604925862744194136738585267666320490067"],"D_3":["2.30286383474371893847418311751945948921381753752185851032203","2.30286383474371893847418311751945948921381753752189253675590"],"D_3_Math-199":["5.32318026888989689492319873968726063578438852940845576405060","5.32318026888989689492319873968726063578438852947338433618854"],"D_4":["1.70633069522830694711471002108715891112109476847653695537290","1.70633069522830694711471002108715891112109476853073065866372"],"D_4_coarse":["1.70633069522830538717927262895521519835651746821332873917830","1.70633069522830850705014368149018105038936156657368676653244"]},"constants":{"(3/2)^(1/3)":["1.14471424255333186780804221193967700891590692078793107209905","1.14471424255333186780804221193967700891590692078793107209906"],"1536pi":["4825.48631591392241427862023671731643012685219744016254101693","4825.48631591392241427862023671731643012685219744016254101804"],"Gamma(7/6)":["0.927719333630039200708349482534621018566466519145475576700351","0.927719333630039200708349482534621018566466519145475576939062"],"Z_4":["4825.48631591392241427862023671731643012685219744016254101437","4825.48631591392241427862023671731643012685219744016254102065"],"Z_4_quadrature":["4825.48631591392241427862023671731643012685219743927374007126","4825.48631591392241427862023671731643012685219744105134196381"],"pi":["3.14159265358979323846264338327950288419716939937510582097457","3.14159265358979323846264338327950288419716939937510582097528"],"|S^4|":["26.3189450695716229835586426663364030275031984193087750037589","26.3189450695716229835586426663364030275031984193087750037782"]},"exact_relations":{"D_4":"sqrt(3/7) . 24/Z_4 . (1/6) I_4","D_m":"sqrt(3/(m+3)) Z_m^-1 int_{mu<0} prod mu_i^2 |Delta(mu)| exp(-|mu|^2/4 + (sum mu)^2/(4(m+3))) dmu","L_a":"L_a(T) = int_0^T e^{-v^2/24} sum_k e_{a,k}(v) J_k(v) dv, accumulated cell by cell","Z_4":"4! sqrt pi (1/6) int dT e^{-T^2/48} int_0^T dv e^{-v^2/24} int_0^v V e^{-p3^2/8} dp3 = 1536 pi","c_d_ref":"Gamma(7/6) (3/2)^(1/3) |S^{d-1}| D_{d-1} / (sqrt3 sqrt pi (2 pi)^d); d = 5: Gamma(7/6) (3/2)^(1/3) D_4 / (12 sqrt3 pi^(7/2))","decoupling":"alpha = 3m/(4(m+3)), beta = 3T/(2(m+3)); erfc branch exp(-T^2/(4m(m-1)) - |u|^2/4), polynomial branch exp(-T^2/((m-1)(m+3)) - |u|^2/4), u = t - (T/(m-1)) 1","erfc argument":"beta/(2 sqrt alpha) = T sqrt21/28; erfc(x) <= e^{-x^2} in the tail","m=4":"p1 = (T - 2 p2 - p3)/3, v = 2 p2 + p3, p2 = (v - p3)/2; |u|^2/4 = v^2/24 + p3^2/8; dp1 dp2 dp3 = dT dv dp3/6; region 0 < p3 < v < T","transfer":"E_5 = 60500 (76 24^6 + 15) e^-288; eps = 126 E_5; c_{5,24}/c_{5,ref} in [1/(1+delta), 1+delta], delta = (1+eps)^10/(1-eps)^10 - 1"},"float_reference_10sig":{"D_2_orthant":"2.383704698","D_3_orthant":"5.309848377","D_4_a_integrated_3D":"14.52840747","D_4_orthant":"13.51071422"},"mutant":"vandermonde","object":"CL-SIDE24-D5-COEFFICIENT-20260930-v1","passed":false,"quadrature":{"K":40,"cells":"width 1/2 on [0, 30], width 1 on [30, 60]","rho":"6"},"scientific_effect":"NONE","transfer":{"E_5":["7.36250545404696408692679395342008992909499098015444669312995E-111","7.36250545404696408692679395342008992909499098015444669313000E-111"],"delta":["1.85535137441983494990555207626186266213193772699892056666874E-107","1.85535137441983494990555207626186266213193772699892056666882E-107"],"epsilon":["9.27675687209917474952776038130931331065968863499460283334373E-109","9.27675687209917474952776038130931331065968863499460283334380E-109"],"exponents":{"a":"29/3 (<= 10 used)","b":10},"n_jet":21}}}},"results_bytes":5435,"results_sha256":"81c4bc072b7eab4d2f1725b3d873076dac6bc31ab1a06779d3299d9e81b3cbaf","schema":1,"script":"cone_moment_d5.py","script_git_blob":"9d918805865031c8ef9ae5d4d0c3266e1a396a98"}
diff --git a/tools/side24_mutant_contracts/side24-d6-coefficient.json b/tools/side24_mutant_contracts/side24-d6-coefficient.json
new file mode 100644
index 000000000..11e48b6e4
--- /dev/null
+++ b/tools/side24_mutant_contracts/side24-d6-coefficient.json
@@ -0,0 +1 @@
+{"directory":"coefficients/side24_d6_20260930","family":"side24-d6-coefficient","mutants":{"cumulative-half":{"kind":"report","report":{"certified":true,"checks":{"CLOSED_FORM_D6":true,"D1_EXACT":true,"D2_EXACT":true,"D3_EXACT":true,"D4_CERTIFIED_PATH":false,"D4_EXACT":true,"FLOAT_INSIDE":false,"IMAGE_BOUND":true,"JK_EXACT":true,"LAYER_EXACT":true,"LIBRARY_EXACT":true,"MEHTA_Z5":false,"MONOTONE":true,"PINNED":false,"ROUTE_CONSISTENT":false,"SIDE24_CONSISTENT":true,"STRUCTURE":true,"TRANSFER_BOUND":true,"TRUNCATION_NESTING":false,"WIDTHS":true},"coefficients":{"c_2_ref":["0.0734069193060342710301359629577740500176642446843326718326939","0.0734069193060342710301359629577740500176642446843326718516467"],"c_3_ref":["0.0417759318405983433429366654285755564666815196610447378743313","0.0417759318405983433429366654285755564666815196610447378851390"],"c_4_ref":["0.0233216660029528350945211949528568851531013723632403344029580","0.0233216660029528350945211949528568851531013723632403344090136"],"c_5_ref":["0.0132193193800849680760334875146413965210528901256999054505061","0.0132193193800849680760334875146413965210528901256999054539805"],"c_6_24":["0.00763709418399938871104495130339622303698812062399916308102812","0.00763709418399938871104495130339622303705680401001444588997040"],"c_6_ref":["0.00763709418399938871104495130339622303698812062399916308102820","0.00763709418399938871104495130339622303705680401001444588997032"],"c_6_ref_closed":["0.00763709418399938871104495130339622303698812062399916308103336","0.00763709418399938871104495130339622303705680401001444588996511"]},"cone_moments":{"D_1":"4/3","D_2":"29/6  +  -1 sqrt(6)","D_3":"50/3  +  -76/3 pi^(-1)  +  -200/9 pi^(-1) arctan(1/2)","D_4":"6695/54  +  -405/32 sqrt(6)  +  -1375/72 sqrt(21) pi^(-1)  +  -405/16 sqrt(6) pi^(-1) arctan(1/4 sqrt(14))  +  -6695/27 pi^(-1) arctan(1/7 sqrt(21))  +  405/16 sqrt(6) pi^(-1) arctan(1/7 sqrt(14))","D_4_certified_path":["14.1737836294718774152704475535282200540633273165861324655780","14.1737836294718774152704475535282200540633277313694139448277"],"D_4_interval":["14.2187634589773576013064656778740783173406023160820215841833","14.2187634589773576013064656778740783173406023160820215842323"],"D_5":["43.8106411155825739436051577890449899572541794073024782701302","43.8106411155825739436051577890449899576481857011652565589590"],"D_5_coarse":["42.9075503075969408225584232661620597585391386672186484524695","42.9075503076421612824132098037866812799919754930900093063304"],"D_5_route_certified_e":["43.7580506247069198811496859614374179649968168679260068326997","43.7580506247069198811501907272235636803028956920150100321680"]},"constants":{"(3/2)^(1/3)":["1.14471424255333186780804221193967700891590692078793107209905","1.14471424255333186780804221193967700891590692078793107209906"],"46080 pi^(3/2)":["256588.554094005097510724412616035949148787811904251879407479","256588.554094005097510724412616035949148787811904251879407574"],"Gamma(7/6)":["0.927719333630039200708349482534621018566466519145475576700351","0.927719333630039200708349482534621018566466519145475576939062"],"Z_5":["256588.554094005097510724412616035949148787811904251879407480","256588.554094005097510724412616035949148787811904251879407574"],"Z_5_layered":["256065.551243145019165366845502644555448476458681860747582421","256065.551243145019165366845502644555448533388129704404119059"],"pi":["3.14159265358979323846264338327950288419716939937510582097457","3.14159265358979323846264338327950288419716939937510582097528"],"|S^5|":["31.0062766802998201754763150671013952022252885658851076941334","31.0062766802998201754763150671013952022252885658851076941545"]},"exact_relations":{"D_m":"sqrt(3/(m+3)) Z_m^-1 int_{mu<0} prod mu_i^2 |Delta(mu)| exp(-|mu|^2/4 + (sum mu)^2/(4(m+3))) dmu","Z_5":"5! (1/24) 2 sqrt(pi/5) int_{0<q<v<w<T} V e^{-T^2/80 - w^2/48 - v^2/24 - q^2/8} = 2^(15/2) (2 pi)^(5/2) prod_{j<=5} Gamma(1+j/2)/Gamma(3/2)^5 = 46080 pi^(3/2)","c_d_ref":"Gamma(7/6) (3/2)^(1/3) |S^{d-1}| D_{d-1} / (sqrt3 sqrt pi (2 pi)^d); d = 6: |S^5| = pi^3, c_{6,ref} = Gamma(7/6) (3/2)^(1/3) D_5 / (64 sqrt3 pi^(7/2))","exponents":"erfc branch T^2/80 + w^2/48 + v^2/24 + q^2/8 with erfc(sqrt30 T/40); polynomial branch T^2/32 + w^2/48 + v^2/24 + q^2/8","layers":"q: J_k (odd k, no erf); v: incomplete Gaussian moments G_n(mu, w), mu in {1/24, 1/6}; w: exact for the polynomial and e^{-mu w^2} parts (G_n(1/48 + mu, T)), certified cumulative Omega_{a,mu}(T) = int_0^T Q(w) e^{-w^2/48} erf(sqrt(mu) w) dw for the erf parts; T: exact two-erf moments plus the certified Taylor quadrature","m=5 coordinates":"T = 4p1 + 3p2 + 2p3 + p4, w = 3p2 + 2p3 + p4, v = 2p3 + p4, q = p4; dp = dT dw dv dq/24; 0 < q < v < w < T","transfer":"E_6 = 314496 (76 24^6 + 15) e^-288; eps = 168 E_6; c_{6,24}/c_{6,ref} in [1/(1+delta), 1+delta], delta = (1+eps)^14/(1-eps)^14 - 1"},"float_reference_10sig":{"D_5_layered_trapezoid":"44.12971862"},"layers":{"Z_5":{"certified_part":["24112.7849833655005660416093418999767297120299868652734943920","24112.7849833655005660416093418999767297192120139310225083511"],"exact_part":"183296/81 sqrt(5) pi^(1)  +  596992/6561 sqrt(15) pi^(1)  +  -3191680/729  +  -366592/81 sqrt(5) arctan(1/5 sqrt(5))  +  -1193984/6561 sqrt(15) arctan(1/15 sqrt(15))","kmax":7},"atoms":["1","arctan(1/5 sqrt(15))","arctan(1/2 sqrt(1))","arctan(1/4 sqrt(14))","arctan(1/7 sqrt(21))","arctan(1/7 sqrt(14))","arctan(1/7 sqrt(7))","arctan(1/5 sqrt(5))","arctan(1/15 sqrt(15))","arctan(1/3 sqrt(7))"],"m=5":{"G_0_constant":"2/15 sqrt(30) pi^(1/2)","T_powers_erfc":[0,1,2,3,4,5,6,7,8,9,10,11,12,14],"T_powers_poly":[0,1,2,3,4,5,6,7,8,9,10,11,13],"alpha":"15/32","beta":"3/16 T","certified_functions_(erfc, poly)":[28,26],"certified_part_erfc_branch":["2156637.79720088429227158860901630206847432936199316862563037","2156637.79720088429227158860901630206849983759689706357824189"],"certified_part_poly_branch":["2961802.38104603922216459318558888507341114660353320436329868","2961802.38104603922216459318558888507341114661056743170324294"],"exact_part_erfc_branch":"37325849600/19683 sqrt(5) pi^(1)  +  -2526199047775232/215233605  +  1342173730389322381312/420378134765625 sqrt(5)  +  -307725678950598344704/252226880859375 sqrt(15)  +  6553487824267442728475648/148371201531123046875 sqrt(105)  +  -149303398400/19683 sqrt(5) arctan(1/5 sqrt(5))  +  -46870014525440/129140163 sqrt(15) arctan(1/15 sqrt(15))  +  46870014525440/129140163 sqrt(15) arctan(1/3 sqrt(7))","exact_part_poly_branch":"931333683623590068224/756680642578125 sqrt(2) pi^(1/2)  +  -694064242779548254208/420378134765625 sqrt(6) pi^(1/2)  +  309876730511277843629146112/445113604593369140625 sqrt(14) pi^(1/2)","kappa":"1/40 sqrt(30)","kappa^2":"3/160","lambda_erfc":"1/80","lambda_poly":"1/32","prefactor":"1/36864 sqrt(6) pi^(-3/2)","q_powers":[1,3,5,7,9,11]}},"mutant":"cumulative-half","object":"CL-SIDE24-D6-COEFFICIENT-20260930-v1","passed":false,"quadrature":{"K":40,"cells":"width 1/2 on [0, 30], width 1 on [30, 90]","rho":"6"},"scientific_effect":"NONE","transfer":{"E_6":["3.82723721533215540079690742342942909477794757569364110281421E-110","3.82723721533215540079690742342942909477794757569364110281424E-110"],"delta":["1.8003323860922459005348652519812034461835465396062887747638E-106","1.80033238609224590053486525198120344618354653960628877476392E-106"],"epsilon":["6.42975852175802107333880447136144087922695192716531705272787E-108","6.42975852175802107333880447136144087922695192716531705272793E-108"],"exponents":{"a":"79/6 (<= 14 used)","b":"27/2 (<= 14 used)"},"n_jet":28}}},"erf-inside":{"kind":"report","report":{"certified":true,"checks":{"CLOSED_FORM_D6":true,"D1_EXACT":true,"D2_EXACT":true,"D3_EXACT":true,"D4_CERTIFIED_PATH":true,"D4_EXACT":true,"FLOAT_INSIDE":false,"IMAGE_BOUND":true,"JK_EXACT":true,"LAYER_EXACT":true,"LIBRARY_EXACT":true,"MEHTA_Z5":false,"MONOTONE":true,"PINNED":false,"ROUTE_CONSISTENT":true,"SIDE24_CONSISTENT":true,"STRUCTURE":true,"TRANSFER_BOUND":true,"TRUNCATION_NESTING":true,"WIDTHS":true},"coefficients":{"c_2_ref":["0.0734069193060342710301359629577740500176642446843326718326939","0.0734069193060342710301359629577740500176642446843326718516467"],"c_3_ref":["0.0417759318405983433429366654285755564666815196610447378743313","0.0417759318405983433429366654285755564666815196610447378851390"],"c_4_ref":["0.0233216660029528350945211949528568851531013723632403344029580","0.0233216660029528350945211949528568851531013723632403344090136"],"c_5_ref":["0.0132193193800849680760334875146413965210528901256999054505061","0.0132193193800849680760334875146413965210528901256999054539805"],"c_6_24":["0.00551844245899755077490531129712183432066244379716027032805899","0.00551844245899755077490531129712183432073114552562385981000424"],"c_6_ref":["0.00551844245899755077490531129712183432066244379716027032805905","0.00551844245899755077490531129712183432073114552562385981000418"],"c_6_ref_closed":["0.00551844245899755077490531129712183432066244379716027032806279","0.00551844245899755077490531129712183432073114552562385981000044"]},"cone_moments":{"D_1":"4/3","D_2":"29/6  +  -1 sqrt(6)","D_3":"50/3  +  -76/3 pi^(-1)  +  -200/9 pi^(-1) arctan(1/2)","D_4":"6695/54  +  -405/32 sqrt(6)  +  -1375/72 sqrt(21) pi^(-1)  +  -405/16 sqrt(6) pi^(-1) arctan(1/4 sqrt(14))  +  -6695/27 pi^(-1) arctan(1/7 sqrt(21))  +  405/16 sqrt(6) pi^(-1) arctan(1/7 sqrt(14))","D_4_certified_path":["14.2187634589773576013064656778740783173406021086903800669014","14.2187634589773576013064656778740783173406025234736631015112"],"D_4_interval":["14.2187634589773576013064656778740783173406023160820215841833","14.2187634589773576013064656778740783173406023160820215842323"],"D_5":["31.6568705666435253621578073399149552593273181894941632569136","31.6568705666435253621578073399149552597214297058932340043611"],"D_5_coarse":["31.6568705666209042166450430388015261771940293475175155172060","31.6568705666661465076706897000006787887918996728579591236626"],"D_5_route_certified_e":["31.6568705666435253621575548356580787955274985813103586083243","31.6568705666435253621580598441718314417723792570749157041283"]},"constants":{"(3/2)^(1/3)":["1.14471424255333186780804221193967700891590692078793107209905","1.14471424255333186780804221193967700891590692078793107209906"],"46080 pi^(3/2)":["256588.554094005097510724412616035949148787811904251879407479","256588.554094005097510724412616035949148787811904251879407574"],"Gamma(7/6)":["0.927719333630039200708349482534621018566466519145475576700351","0.927719333630039200708349482534621018566466519145475576939062"],"Z_5":["256588.554094005097510724412616035949148787811904251879407480","256588.554094005097510724412616035949148787811904251879407574"],"Z_5_layered":["96401.2738289793057389442748118743758676656840601648260851529","96401.2738289793057389442748118743758677226135049625663594019"],"pi":["3.14159265358979323846264338327950288419716939937510582097457","3.14159265358979323846264338327950288419716939937510582097528"],"|S^5|":["31.0062766802998201754763150671013952022252885658851076941334","31.0062766802998201754763150671013952022252885658851076941545"]},"exact_relations":{"D_m":"sqrt(3/(m+3)) Z_m^-1 int_{mu<0} prod mu_i^2 |Delta(mu)| exp(-|mu|^2/4 + (sum mu)^2/(4(m+3))) dmu","Z_5":"5! (1/24) 2 sqrt(pi/5) int_{0<q<v<w<T} V e^{-T^2/80 - w^2/48 - v^2/24 - q^2/8} = 2^(15/2) (2 pi)^(5/2) prod_{j<=5} Gamma(1+j/2)/Gamma(3/2)^5 = 46080 pi^(3/2)","c_d_ref":"Gamma(7/6) (3/2)^(1/3) |S^{d-1}| D_{d-1} / (sqrt3 sqrt pi (2 pi)^d); d = 6: |S^5| = pi^3, c_{6,ref} = Gamma(7/6) (3/2)^(1/3) D_5 / (64 sqrt3 pi^(7/2))","exponents":"erfc branch T^2/80 + w^2/48 + v^2/24 + q^2/8 with erfc(sqrt30 T/40); polynomial branch T^2/32 + w^2/48 + v^2/24 + q^2/8","layers":"q: J_k (odd k, no erf); v: incomplete Gaussian moments G_n(mu, w), mu in {1/24, 1/6}; w: exact for the polynomial and e^{-mu w^2} parts (G_n(1/48 + mu, T)), certified cumulative Omega_{a,mu}(T) = int_0^T Q(w) e^{-w^2/48} erf(sqrt(mu) w) dw for the erf parts; T: exact two-erf moments plus the certified Taylor quadrature","m=5 coordinates":"T = 4p1 + 3p2 + 2p3 + p4, w = 3p2 + 2p3 + p4, v = 2p3 + p4, q = p4; dp = dT dw dv dq/24; 0 < q < v < w < T","transfer":"E_6 = 314496 (76 24^6 + 15) e^-288; eps = 168 E_6; c_{6,24}/c_{6,ref} in [1/(1+delta), 1+delta], delta = (1+eps)^14/(1-eps)^14 - 1"},"float_reference_10sig":{"D_5_layered_trapezoid":"44.12971862"},"layers":{"Z_5":{"certified_part":["3970.07843593897983706387490244346252878448878912787686364293","3970.07843593897983706387490244346252879167081580936335855565"],"exact_part":"183296/81 sqrt(5) pi^(1)  +  596992/6561 sqrt(15) pi^(1)  +  -3191680/729  +  -366592/81 sqrt(5) arctan(1/5 sqrt(5))  +  -1193984/6561 sqrt(15) arctan(1/15 sqrt(15))","kmax":7},"atoms":["1","arctan(1/5 sqrt(15))","arctan(1/2 sqrt(1))","arctan(1/4 sqrt(14))","arctan(1/7 sqrt(21))","arctan(1/7 sqrt(14))","arctan(1/7 sqrt(7))","arctan(1/5 sqrt(5))","arctan(1/15 sqrt(15))","arctan(1/3 sqrt(7))"],"m=5":{"G_0_constant":"2/15 sqrt(30) pi^(1/2)","T_powers_erfc":[0,1,2,3,4,5,6,7,8,9,10,11,12,14],"T_powers_poly":[0,1,2,3,4,5,6,7,8,9,10,11,13],"alpha":"15/32","beta":"3/16 T","certified_functions_(erfc, poly)":[28,26],"certified_part_erfc_branch":["1518348.88997068001249079355344432122313471801620669474113798","1518348.88997068001249079355344432122316023306329031292808928"],"certified_part_poly_branch":["2769510.72066488039983521914098007360876944068179699406360613","2769510.72066488039983521914098007360876944068883141478035164"],"exact_part_erfc_branch":"37325849600/19683 sqrt(5) pi^(1)  +  -2526199047775232/215233605  +  1342173730389322381312/420378134765625 sqrt(5)  +  -307725678950598344704/252226880859375 sqrt(15)  +  6553487824267442728475648/148371201531123046875 sqrt(105)  +  -149303398400/19683 sqrt(5) arctan(1/5 sqrt(5))  +  -46870014525440/129140163 sqrt(15) arctan(1/15 sqrt(15))  +  46870014525440/129140163 sqrt(15) arctan(1/3 sqrt(7))","exact_part_poly_branch":"931333683623590068224/756680642578125 sqrt(2) pi^(1/2)  +  -694064242779548254208/420378134765625 sqrt(6) pi^(1/2)  +  309876730511277843629146112/445113604593369140625 sqrt(14) pi^(1/2)","kappa":"1/40 sqrt(30)","kappa^2":"3/160","lambda_erfc":"1/80","lambda_poly":"1/32","prefactor":"1/36864 sqrt(6) pi^(-3/2)","q_powers":[1,3,5,7,9,11]}},"mutant":"erf-inside","object":"CL-SIDE24-D6-COEFFICIENT-20260930-v1","passed":false,"quadrature":{"K":40,"cells":"width 1/2 on [0, 30], width 1 on [30, 90]","rho":"6"},"scientific_effect":"NONE","transfer":{"E_6":["3.82723721533215540079690742342942909477794757569364110281421E-110","3.82723721533215540079690742342942909477794757569364110281424E-110"],"delta":["1.8003323860922459005348652519812034461835465396062887747638E-106","1.80033238609224590053486525198120344618354653960628877476392E-106"],"epsilon":["6.42975852175802107333880447136144087922695192716531705272787E-108","6.42975852175802107333880447136144087922695192716531705272793E-108"],"exponents":{"a":"79/6 (<= 14 used)","b":"27/2 (<= 14 used)"},"n_jet":28}}},"erfc-branch":{"kind":"report","report":{"certified":true,"checks":{"CLOSED_FORM_D6":true,"D1_EXACT":true,"D2_EXACT":true,"D3_EXACT":true,"D4_CERTIFIED_PATH":true,"D4_EXACT":true,"FLOAT_INSIDE":true,"IMAGE_BOUND":true,"JK_EXACT":true,"LAYER_EXACT":true,"LIBRARY_EXACT":true,"MEHTA_Z5":true,"MONOTONE":true,"PINNED":false,"ROUTE_CONSISTENT":true,"SIDE24_CONSISTENT":true,"STRUCTURE":true,"TRANSFER_BOUND":true,"TRUNCATION_NESTING":true,"WIDTHS":true},"coefficients":{"c_2_ref":["0.0734069193060342710301359629577740500176642446843326718326939","0.0734069193060342710301359629577740500176642446843326718516467"],"c_3_ref":["0.0417759318405983433429366654285755564666815196610447378743313","0.0417759318405983433429366654285755564666815196610447378851390"],"c_4_ref":["0.0233216660029528350945211949528568851531013723632403344029580","0.0233216660029528350945211949528568851531013723632403344090136"],"c_5_ref":["0.0132193193800849680760334875146413965210528901256999054505061","0.0132193193800849680760334875146413965210528901256999054539805"],"c_6_24":["0.00749041612833451712081601486949792882465439196073749659724045","0.00749041612833451712081601486949792882468873366109791202812739"],"c_6_ref":["0.00749041612833451712081601486949792882465439196073749659724053","0.00749041612833451712081601486949792882468873366109791202812731"],"c_6_ref_closed":["0.00749041612833451712081601486949792882465439196073749659724560","0.00749041612833451712081601486949792882468873366109791202812224"]},"cone_moments":{"D_1":"4/3","D_2":"29/6  +  -1 sqrt(6)","D_3":"50/3  +  -76/3 pi^(-1)  +  -200/9 pi^(-1) arctan(1/2)","D_4":"6695/54  +  -405/32 sqrt(6)  +  -1375/72 sqrt(21) pi^(-1)  +  -405/16 sqrt(6) pi^(-1) arctan(1/4 sqrt(14))  +  -6695/27 pi^(-1) arctan(1/7 sqrt(21))  +  405/16 sqrt(6) pi^(-1) arctan(1/7 sqrt(14))","D_4_certified_path":["14.2187634589773576013064656778740783173406021086903800669014","14.2187634589773576013064656778740783173406025234736631015112"],"D_4_interval":["14.2187634589773576013064656778740783173406023160820215841833","14.2187634589773576013064656778740783173406023160820215842323"],"D_5":["42.9692138002394576879326139693212960097121191357063248372013","42.9692138002394576879326139693212960099091223248173357425236"],"D_5_coarse":["42.9692138002279563536232730087460336024460225554250931260448","42.9692138002509590222419946359479677102061768916521662765223"],"D_5_route_certified_e":["42.9692138002394576879324855586427287896992850892957139986315","42.9692138002394576879327423799998632442906281317198005916579"]},"constants":{"(3/2)^(1/3)":["1.14471424255333186780804221193967700891590692078793107209905","1.14471424255333186780804221193967700891590692078793107209906"],"46080 pi^(3/2)":["256588.554094005097510724412616035949148787811904251879407479","256588.554094005097510724412616035949148787811904251879407574"],"Gamma(7/6)":["0.927719333630039200708349482534621018566466519145475576700351","0.927719333630039200708349482534621018566466519145475576939062"],"Z_5":["256588.554094005097510724412616035949148787811904251879407480","256588.554094005097510724412616035949148787811904251879407574"],"Z_5_layered":["256588.554094005097510724412616035949148757223338670240957447","256588.554094005097510724412616035949148814152786723333504992"],"pi":["3.14159265358979323846264338327950288419716939937510582097457","3.14159265358979323846264338327950288419716939937510582097528"],"|S^5|":["31.0062766802998201754763150671013952022252885658851076941334","31.0062766802998201754763150671013952022252885658851076941545"]},"exact_relations":{"D_m":"sqrt(3/(m+3)) Z_m^-1 int_{mu<0} prod mu_i^2 |Delta(mu)| exp(-|mu|^2/4 + (sum mu)^2/(4(m+3))) dmu","Z_5":"5! (1/24) 2 sqrt(pi/5) int_{0<q<v<w<T} V e^{-T^2/80 - w^2/48 - v^2/24 - q^2/8} = 2^(15/2) (2 pi)^(5/2) prod_{j<=5} Gamma(1+j/2)/Gamma(3/2)^5 = 46080 pi^(3/2)","c_d_ref":"Gamma(7/6) (3/2)^(1/3) |S^{d-1}| D_{d-1} / (sqrt3 sqrt pi (2 pi)^d); d = 6: |S^5| = pi^3, c_{6,ref} = Gamma(7/6) (3/2)^(1/3) D_5 / (64 sqrt3 pi^(7/2))","exponents":"erfc branch T^2/80 + w^2/48 + v^2/24 + q^2/8 with erfc(sqrt30 T/40); polynomial branch T^2/32 + w^2/48 + v^2/24 + q^2/8","layers":"q: J_k (odd k, no erf); v: incomplete Gaussian moments G_n(mu, w), mu in {1/24, 1/6}; w: exact for the polynomial and e^{-mu w^2} parts (G_n(1/48 + mu, T)), certified cumulative Omega_{a,mu}(T) = int_0^T Q(w) e^{-w^2/48} erf(sqrt(mu) w) dw for the erf parts; T: exact two-erf moments plus the certified Taylor quadrature","m=5 coordinates":"T = 4p1 + 3p2 + 2p3 + p4, w = 3p2 + 2p3 + p4, v = 2p3 + p4, q = p4; dp = dT dw dv dq/24; 0 < q < v < w < T","transfer":"E_6 = 314496 (76 24^6 + 15) e^-288; eps = 168 E_6; c_{6,24}/c_{6,ref} in [1/(1+delta), 1+delta], delta = (1+eps)^14/(1-eps)^14 - 1"},"float_reference_10sig":{"D_5_layered_trapezoid":"42.96851267"},"layers":{"Z_5":{"certified_part":["24178.7652584712167214089274033053575141507099675457961027994","24178.7652584712167214089274033053575141578919946379668572785"],"exact_part":"183296/81 sqrt(5) pi^(1)  +  596992/6561 sqrt(15) pi^(1)  +  -3191680/729  +  -366592/81 sqrt(5) arctan(1/5 sqrt(5))  +  -1193984/6561 sqrt(15) arctan(1/15 sqrt(15))","kmax":7},"atoms":["1","arctan(1/5 sqrt(15))","arctan(1/2 sqrt(1))","arctan(1/4 sqrt(14))","arctan(1/7 sqrt(21))","arctan(1/7 sqrt(14))","arctan(1/7 sqrt(7))","arctan(1/5 sqrt(5))","arctan(1/15 sqrt(15))","arctan(1/3 sqrt(7))"],"m=5":{"G_0_constant":"1/15 sqrt(30) pi^(1/2)","T_powers_erfc":[0,1,2,3,4,5,6,7,8,9,10,11,12,14],"T_powers_poly":[0,1,2,3,4,5,6,7,8,9,10,11,13],"alpha":"15/32","beta":"3/16 T","certified_functions_(erfc, poly)":[28,26],"certified_part_erfc_branch":["2171340.67575158928149811730903868726756164117400995326826133","2171340.67575158928149811730903868726758714940894103808208401"],"certified_part_poly_branch":["2969588.09339018881243789877924110799593795308619456695381198","2969588.09339018881243789877924110799593795309322879745417960"],"exact_part_erfc_branch":"37325849600/19683 sqrt(5) pi^(1)  +  -2526199047775232/215233605  +  1342173730389322381312/420378134765625 sqrt(5)  +  -307725678950598344704/252226880859375 sqrt(15)  +  6553487824267442728475648/148371201531123046875 sqrt(105)  +  -149303398400/19683 sqrt(5) arctan(1/5 sqrt(5))  +  -46870014525440/129140163 sqrt(15) arctan(1/15 sqrt(15))  +  46870014525440/129140163 sqrt(15) arctan(1/3 sqrt(7))","exact_part_poly_branch":"931333683623590068224/756680642578125 sqrt(2) pi^(1/2)  +  -694064242779548254208/420378134765625 sqrt(6) pi^(1/2)  +  309876730511277843629146112/445113604593369140625 sqrt(14) pi^(1/2)","kappa":"1/40 sqrt(30)","kappa^2":"3/160","lambda_erfc":"1/80","lambda_poly":"1/32","prefactor":"1/36864 sqrt(6) pi^(-3/2)","q_powers":[1,3,5,7,9,11]}},"mutant":"erfc-branch","object":"CL-SIDE24-D6-COEFFICIENT-20260930-v1","passed":false,"quadrature":{"K":40,"cells":"width 1/2 on [0, 30], width 1 on [30, 90]","rho":"6"},"scientific_effect":"NONE","transfer":{"E_6":["3.82723721533215540079690742342942909477794757569364110281421E-110","3.82723721533215540079690742342942909477794757569364110281424E-110"],"delta":["1.8003323860922459005348652519812034461835465396062887747638E-106","1.80033238609224590053486525198120344618354653960628877476392E-106"],"epsilon":["6.42975852175802107333880447136144087922695192716531705272787E-108","6.42975852175802107333880447136144087922695192716531705272793E-108"],"exponents":{"a":"79/6 (<= 14 used)","b":"27/2 (<= 14 used)"},"n_jet":28}}},"image-shells":{"kind":"report","report":{"certified":true,"checks":{"CLOSED_FORM_D6":true,"D1_EXACT":true,"D2_EXACT":true,"D3_EXACT":true,"D4_CERTIFIED_PATH":true,"D4_EXACT":true,"FLOAT_INSIDE":true,"IMAGE_BOUND":false,"JK_EXACT":true,"LAYER_EXACT":true,"LIBRARY_EXACT":true,"MEHTA_Z5":true,"MONOTONE":true,"PINNED":true,"ROUTE_CONSISTENT":true,"SIDE24_CONSISTENT":true,"STRUCTURE":true,"TRANSFER_BOUND":true,"TRUNCATION_NESTING":true,"WIDTHS":true},"coefficients":{"c_2_ref":["0.0734069193060342710301359629577740500176642446843326718326939","0.0734069193060342710301359629577740500176642446843326718516467"],"c_3_ref":["0.0417759318405983433429366654285755564666815196610447378743313","0.0417759318405983433429366654285755564666815196610447378851390"],"c_4_ref":["0.0233216660029528350945211949528568851531013723632403344029580","0.0233216660029528350945211949528568851531013723632403344090136"],"c_5_ref":["0.0132193193800849680760334875146413965210528901256999054505061","0.0132193193800849680760334875146413965210528901256999054539805"],"c_6_24":["0.00769287862923684827866663128122739008112489691080122434668364","0.00769287862923684827866663128122739008119358029688972504148796"],"c_6_ref":["0.00769287862923684827866663128122739008112489691080122434668372","0.00769287862923684827866663128122739008119358029688972504148788"],"c_6_ref_closed":["0.00769287862923684827866663128122739008112489691080122434668894","0.00769287862923684827866663128122739008119358029688972504148264"]},"cone_moments":{"D_1":"4/3","D_2":"29/6  +  -1 sqrt(6)","D_3":"50/3  +  -76/3 pi^(-1)  +  -200/9 pi^(-1) arctan(1/2)","D_4":"6695/54  +  -405/32 sqrt(6)  +  -1375/72 sqrt(21) pi^(-1)  +  -405/16 sqrt(6) pi^(-1) arctan(1/4 sqrt(14))  +  -6695/27 pi^(-1) arctan(1/7 sqrt(21))  +  405/16 sqrt(6) pi^(-1) arctan(1/7 sqrt(14))","D_4_certified_path":["14.2187634589773576013064656778740783173406021086903800669014","14.2187634589773576013064656778740783173406025234736631015112"],"D_4_interval":["14.2187634589773576013064656778740783173406023160820215841833","14.2187634589773576013064656778740783173406023160820215842323"],"D_5":["44.1306518750741327103620248954026331485852349159356435082422","44.1306518750741327103620248954026331489792412102184405166029"],"D_5_coarse":["44.1306518750515223664261051710832611996349172943016710320497","44.1306518750967430542978265663853391541885558936237109467677"],"D_5_route_certified_e":["44.1306518750741327103617725110811619399972170664905125471480","44.1306518750741327103622772797241043578020958769281829486597"]},"constants":{"(3/2)^(1/3)":["1.14471424255333186780804221193967700891590692078793107209905","1.14471424255333186780804221193967700891590692078793107209906"],"46080 pi^(3/2)":["256588.554094005097510724412616035949148787811904251879407479","256588.554094005097510724412616035949148787811904251879407574"],"Gamma(7/6)":["0.927719333630039200708349482534621018566466519145475576700351","0.927719333630039200708349482534621018566466519145475576939062"],"Z_5":["256588.554094005097510724412616035949148787811904251879407480","256588.554094005097510724412616035949148787811904251879407574"],"Z_5_layered":["256588.554094005097510724412616035949148757223338670240957447","256588.554094005097510724412616035949148814152786723333504992"],"pi":["3.14159265358979323846264338327950288419716939937510582097457","3.14159265358979323846264338327950288419716939937510582097528"],"|S^5|":["31.0062766802998201754763150671013952022252885658851076941334","31.0062766802998201754763150671013952022252885658851076941545"]},"exact_relations":{"D_m":"sqrt(3/(m+3)) Z_m^-1 int_{mu<0} prod mu_i^2 |Delta(mu)| exp(-|mu|^2/4 + (sum mu)^2/(4(m+3))) dmu","Z_5":"5! (1/24) 2 sqrt(pi/5) int_{0<q<v<w<T} V e^{-T^2/80 - w^2/48 - v^2/24 - q^2/8} = 2^(15/2) (2 pi)^(5/2) prod_{j<=5} Gamma(1+j/2)/Gamma(3/2)^5 = 46080 pi^(3/2)","c_d_ref":"Gamma(7/6) (3/2)^(1/3) |S^{d-1}| D_{d-1} / (sqrt3 sqrt pi (2 pi)^d); d = 6: |S^5| = pi^3, c_{6,ref} = Gamma(7/6) (3/2)^(1/3) D_5 / (64 sqrt3 pi^(7/2))","exponents":"erfc branch T^2/80 + w^2/48 + v^2/24 + q^2/8 with erfc(sqrt30 T/40); polynomial branch T^2/32 + w^2/48 + v^2/24 + q^2/8","layers":"q: J_k (odd k, no erf); v: incomplete Gaussian moments G_n(mu, w), mu in {1/24, 1/6}; w: exact for the polynomial and e^{-mu w^2} parts (G_n(1/48 + mu, T)), certified cumulative Omega_{a,mu}(T) = int_0^T Q(w) e^{-w^2/48} erf(sqrt(mu) w) dw for the erf parts; T: exact two-erf moments plus the certified Taylor quadrature","m=5 coordinates":"T = 4p1 + 3p2 + 2p3 + p4, w = 3p2 + 2p3 + p4, v = 2p3 + p4, q = p4; dp = dT dw dv dq/24; 0 < q < v < w < T","transfer":"E_6 = 314496 (76 24^6 + 15) e^-288; eps = 168 E_6; c_{6,24}/c_{6,ref} in [1/(1+delta), 1+delta], delta = (1+eps)^14/(1-eps)^14 - 1"},"float_reference_10sig":{"D_5_layered_trapezoid":"44.12971862"},"layers":{"Z_5":{"certified_part":["24178.7652584712167214089274033053575141507099675457961027994","24178.7652584712167214089274033053575141578919946379668572785"],"exact_part":"183296/81 sqrt(5) pi^(1)  +  596992/6561 sqrt(15) pi^(1)  +  -3191680/729  +  -366592/81 sqrt(5) arctan(1/5 sqrt(5))  +  -1193984/6561 sqrt(15) arctan(1/15 sqrt(15))","kmax":7},"atoms":["1","arctan(1/5 sqrt(15))","arctan(1/2 sqrt(1))","arctan(1/4 sqrt(14))","arctan(1/7 sqrt(21))","arctan(1/7 sqrt(14))","arctan(1/7 sqrt(7))","arctan(1/5 sqrt(5))","arctan(1/15 sqrt(15))","arctan(1/3 sqrt(7))"],"m=5":{"G_0_constant":"2/15 sqrt(30) pi^(1/2)","T_powers_erfc":[0,1,2,3,4,5,6,7,8,9,10,11,12,14],"T_powers_poly":[0,1,2,3,4,5,6,7,8,9,10,11,13],"alpha":"15/32","beta":"3/16 T","certified_functions_(erfc, poly)":[28,26],"certified_part_erfc_branch":["2171340.67575158928149811730903868726756164117400995326826133","2171340.67575158928149811730903868726758714940894103808208401"],"certified_part_poly_branch":["2969588.09339018881243789877924110799593795308619456695381198","2969588.09339018881243789877924110799593795309322879745417960"],"exact_part_erfc_branch":"37325849600/19683 sqrt(5) pi^(1)  +  -2526199047775232/215233605  +  1342173730389322381312/420378134765625 sqrt(5)  +  -307725678950598344704/252226880859375 sqrt(15)  +  6553487824267442728475648/148371201531123046875 sqrt(105)  +  -149303398400/19683 sqrt(5) arctan(1/5 sqrt(5))  +  -46870014525440/129140163 sqrt(15) arctan(1/15 sqrt(15))  +  46870014525440/129140163 sqrt(15) arctan(1/3 sqrt(7))","exact_part_poly_branch":"931333683623590068224/756680642578125 sqrt(2) pi^(1/2)  +  -694064242779548254208/420378134765625 sqrt(6) pi^(1/2)  +  309876730511277843629146112/445113604593369140625 sqrt(14) pi^(1/2)","kappa":"1/40 sqrt(30)","kappa^2":"3/160","lambda_erfc":"1/80","lambda_poly":"1/32","prefactor":"1/36864 sqrt(6) pi^(-3/2)","q_powers":[1,3,5,7,9,11]}},"mutant":"image-shells","object":"CL-SIDE24-D6-COEFFICIENT-20260930-v1","passed":false,"quadrature":{"K":40,"cells":"width 1/2 on [0, 30], width 1 on [30, 90]","rho":"6"},"scientific_effect":"NONE","transfer":{"E_6":["1.91361860766607770039845371171471454738897378784682055140710E-110","1.91361860766607770039845371171471454738897378784682055140712E-110"],"delta":["9.00166193046122950267432625990601723091773269803144387381897E-107","9.00166193046122950267432625990601723091773269803144387381931E-107"],"epsilon":["3.21487926087901053666940223568072043961347596358265852636392E-108","3.21487926087901053666940223568072043961347596358265852636397E-108"],"exponents":{"a":"79/6 (<= 14 used)","b":"27/2 (<= 14 used)"},"n_jet":28}}},"jacobian":{"kind":"report","report":{"certified":true,"checks":{"CLOSED_FORM_D6":true,"D1_EXACT":true,"D2_EXACT":true,"D3_EXACT":true,"D4_CERTIFIED_PATH":true,"D4_EXACT":true,"FLOAT_INSIDE":true,"IMAGE_BOUND":true,"JK_EXACT":true,"LAYER_EXACT":true,"LIBRARY_EXACT":true,"MEHTA_Z5":false,"MONOTONE":false,"PINNED":false,"ROUTE_CONSISTENT":true,"SIDE24_CONSISTENT":true,"STRUCTURE":true,"TRANSFER_BOUND":true,"TRUNCATION_NESTING":true,"WIDTHS":true},"coefficients":{"c_2_ref":["0.0734069193060342710301359629577740500176642446843326718326939","0.0734069193060342710301359629577740500176642446843326718516467"],"c_3_ref":["0.0417759318405983433429366654285755564666815196610447378743313","0.0417759318405983433429366654285755564666815196610447378851390"],"c_4_ref":["0.0233216660029528350945211949528568851531013723632403344029580","0.0233216660029528350945211949528568851531013723632403344090136"],"c_5_ref":["0.0132193193800849680760334875146413965210528901256999054505061","0.0132193193800849680760334875146413965210528901256999054539805"],"c_6_24":["0.0307715145169473931146665251249095603244995876432048973867344","0.0307715145169473931146665251249095603247743211875589001659519"],"c_6_ref":["0.0307715145169473931146665251249095603244995876432048973867348","0.0307715145169473931146665251249095603247743211875589001659515"],"c_6_ref_closed":["0.0307715145169473931146665251249095603244995876432048973867555","0.0307715145169473931146665251249095603247743211875589001659306"]},"cone_moments":{"D_1":"4/3","D_2":"29/6  +  -1 sqrt(6)","D_3":"50/3  +  -76/3 pi^(-1)  +  -200/9 pi^(-1) arctan(1/2)","D_4":"6695/54  +  -405/32 sqrt(6)  +  -1375/72 sqrt(21) pi^(-1)  +  -405/16 sqrt(6) pi^(-1) arctan(1/4 sqrt(14))  +  -6695/27 pi^(-1) arctan(1/7 sqrt(21))  +  405/16 sqrt(6) pi^(-1) arctan(1/7 sqrt(14))","D_4_certified_path":["14.2187634589773576013064656778740783173406021086903800669014","14.2187634589773576013064656778740783173406025234736631015112"],"D_4_interval":["14.2187634589773576013064656778740783173406023160820215841833","14.2187634589773576013064656778740783173406023160820215842323"],"D_5":["176.522607500296530841448099581610532594340939663742574032968","176.522607500296530841448099581610532595916964840873762066412"],"D_5_coarse":["176.522607500206089465704420684333044798539669177206684128198","176.522607500386972217191306265541356616754223574494843787071"],"D_5_route_certified_e":["176.522607500296530841447090044324647759988868265962050188591","176.522607500296530841449109118896417431208383507712731794639"]},"constants":{"(3/2)^(1/3)":["1.14471424255333186780804221193967700891590692078793107209905","1.14471424255333186780804221193967700891590692078793107209906"],"46080 pi^(3/2)":["256588.554094005097510724412616035949148787811904251879407479","256588.554094005097510724412616035949148787811904251879407574"],"Gamma(7/6)":["0.927719333630039200708349482534621018566466519145475576700351","0.927719333630039200708349482534621018566466519145475576939062"],"Z_5":["256588.554094005097510724412616035949148787811904251879407480","256588.554094005097510724412616035949148787811904251879407574"],"Z_5_layered":["1026354.21637602039004289765046414379659502889335468096382979","1026354.21637602039004289765046414379659525661114689333401998"],"pi":["3.14159265358979323846264338327950288419716939937510582097457","3.14159265358979323846264338327950288419716939937510582097528"],"|S^5|":["31.0062766802998201754763150671013952022252885658851076941334","31.0062766802998201754763150671013952022252885658851076941545"]},"exact_relations":{"D_m":"sqrt(3/(m+3)) Z_m^-1 int_{mu<0} prod mu_i^2 |Delta(mu)| exp(-|mu|^2/4 + (sum mu)^2/(4(m+3))) dmu","Z_5":"5! (1/24) 2 sqrt(pi/5) int_{0<q<v<w<T} V e^{-T^2/80 - w^2/48 - v^2/24 - q^2/8} = 2^(15/2) (2 pi)^(5/2) prod_{j<=5} Gamma(1+j/2)/Gamma(3/2)^5 = 46080 pi^(3/2)","c_d_ref":"Gamma(7/6) (3/2)^(1/3) |S^{d-1}| D_{d-1} / (sqrt3 sqrt pi (2 pi)^d); d = 6: |S^5| = pi^3, c_{6,ref} = Gamma(7/6) (3/2)^(1/3) D_5 / (64 sqrt3 pi^(7/2))","exponents":"erfc branch T^2/80 + w^2/48 + v^2/24 + q^2/8 with erfc(sqrt30 T/40); polynomial branch T^2/32 + w^2/48 + v^2/24 + q^2/8","layers":"q: J_k (odd k, no erf); v: incomplete Gaussian moments G_n(mu, w), mu in {1/24, 1/6}; w: exact for the polynomial and e^{-mu w^2} parts (G_n(1/48 + mu, T)), certified cumulative Omega_{a,mu}(T) = int_0^T Q(w) e^{-w^2/48} erf(sqrt(mu) w) dw for the erf parts; T: exact two-erf moments plus the certified Taylor quadrature","m=5 coordinates":"T = 4p1 + 3p2 + 2p3 + p4, w = 3p2 + 2p3 + p4, v = 2p3 + p4, q = p4; dp = dT dw dv dq/24; 0 < q < v < w < T","transfer":"E_6 = 314496 (76 24^6 + 15) e^-288; eps = 168 E_6; c_{6,24}/c_{6,ref} in [1/(1+delta), 1+delta], delta = (1+eps)^14/(1-eps)^14 - 1"},"float_reference_10sig":{"D_5_layered_trapezoid":"176.5188745"},"layers":{"Z_5":{"certified_part":["24178.7652584712167214089274033053575141507099675457961027994","24178.7652584712167214089274033053575141578919946379668572785"],"exact_part":"183296/81 sqrt(5) pi^(1)  +  596992/6561 sqrt(15) pi^(1)  +  -3191680/729  +  -366592/81 sqrt(5) arctan(1/5 sqrt(5))  +  -1193984/6561 sqrt(15) arctan(1/15 sqrt(15))","kmax":7},"atoms":["1","arctan(1/5 sqrt(15))","arctan(1/2 sqrt(1))","arctan(1/4 sqrt(14))","arctan(1/7 sqrt(21))","arctan(1/7 sqrt(14))","arctan(1/7 sqrt(7))","arctan(1/5 sqrt(5))","arctan(1/15 sqrt(15))","arctan(1/3 sqrt(7))"],"m=5":{"G_0_constant":"2/15 sqrt(30) pi^(1/2)","T_powers_erfc":[0,1,2,3,4,5,6,7,8,9,10,11,12,14],"T_powers_poly":[0,1,2,3,4,5,6,7,8,9,10,11,13],"alpha":"15/32","beta":"3/16 T","certified_functions_(erfc, poly)":[28,26],"certified_part_erfc_branch":["2171340.67575158928149811730903868726756164117400995326826133","2171340.67575158928149811730903868726758714940894103808208401"],"certified_part_poly_branch":["2969588.09339018881243789877924110799593795308619456695381198","2969588.09339018881243789877924110799593795309322879745417960"],"exact_part_erfc_branch":"37325849600/19683 sqrt(5) pi^(1)  +  -2526199047775232/215233605  +  1342173730389322381312/420378134765625 sqrt(5)  +  -307725678950598344704/252226880859375 sqrt(15)  +  6553487824267442728475648/148371201531123046875 sqrt(105)  +  -149303398400/19683 sqrt(5) arctan(1/5 sqrt(5))  +  -46870014525440/129140163 sqrt(15) arctan(1/15 sqrt(15))  +  46870014525440/129140163 sqrt(15) arctan(1/3 sqrt(7))","exact_part_poly_branch":"931333683623590068224/756680642578125 sqrt(2) pi^(1/2)  +  -694064242779548254208/420378134765625 sqrt(6) pi^(1/2)  +  309876730511277843629146112/445113604593369140625 sqrt(14) pi^(1/2)","kappa":"1/40 sqrt(30)","kappa^2":"3/160","lambda_erfc":"1/80","lambda_poly":"1/32","prefactor":"1/9216 sqrt(6) pi^(-3/2)","q_powers":[1,3,5,7,9,11]}},"mutant":"jacobian","object":"CL-SIDE24-D6-COEFFICIENT-20260930-v1","passed":false,"quadrature":{"K":40,"cells":"width 1/2 on [0, 30], width 1 on [30, 90]","rho":"6"},"scientific_effect":"NONE","transfer":{"E_6":["3.82723721533215540079690742342942909477794757569364110281421E-110","3.82723721533215540079690742342942909477794757569364110281424E-110"],"delta":["1.8003323860922459005348652519812034461835465396062887747638E-106","1.80033238609224590053486525198120344618354653960628877476392E-106"],"epsilon":["6.42975852175802107333880447136144087922695192716531705272787E-108","6.42975852175802107333880447136144087922695192716531705272793E-108"],"exponents":{"a":"79/6 (<= 14 used)","b":"27/2 (<= 14 used)"},"n_jet":28}}},"layer-init":{"kind":"report","report":{"certified":true,"checks":{"CLOSED_FORM_D6":true,"D1_EXACT":true,"D2_EXACT":true,"D3_EXACT":true,"D4_CERTIFIED_PATH":false,"D4_EXACT":false,"FLOAT_INSIDE":true,"IMAGE_BOUND":true,"JK_EXACT":true,"LAYER_EXACT":false,"LIBRARY_EXACT":true,"MEHTA_Z5":false,"MONOTONE":false,"PINNED":false,"ROUTE_CONSISTENT":false,"SIDE24_CONSISTENT":false,"STRUCTURE":true,"TRANSFER_BOUND":true,"TRUNCATION_NESTING":true,"WIDTHS":true},"coefficients":{"c_2_ref":["0.0734069193060342710301359629577740500176642446843326718326939","0.0734069193060342710301359629577740500176642446843326718516467"],"c_3_ref":["0.0417759318405983433429366654285755564666815196610447378743313","0.0417759318405983433429366654285755564666815196610447378851390"],"c_4_ref":["0.0233216660029528350945211949528568851531013723632403344029580","0.0233216660029528350945211949528568851531013723632403344090136"],"c_5_ref":["0.00312020630084981533114554892362482561674890340507555551653036","0.00312020630084981533114554892362482561674890340507555551738232"],"c_6_24":["0.0202679116469422708943590873368263950636306652806431696862809","0.0202679116469422708943590873368263950636993486667316703843496"],"c_6_ref":["0.0202679116469422708943590873368263950636306652806431696862812","0.0202679116469422708943590873368263950636993486667316703843493"],"c_6_ref_closed":["0.0202679116469422708943590873368263950636306652806431696862949","0.0202679116469422708943590873368263950636993486667316703843356"]},"cone_moments":{"D_1":"4/3","D_2":"29/6  +  -1 sqrt(6)","D_3":"50/3  +  -76/3 pi^(-1)  +  -200/9 pi^(-1) arctan(1/2)","D_4":"376507/4374  +  2514176/177147 sqrt(3)  +  -405/32 sqrt(6)  +  -2457770783/106288200 sqrt(21) pi^(-1)  +  -405/16 sqrt(6) pi^(-1) arctan(1/4 sqrt(14))  +  -376507/2187 pi^(-1) arctan(1/7 sqrt(21))  +  405/16 sqrt(6) pi^(-1) arctan(1/7 sqrt(14))  +  -5028352/177147 sqrt(3) pi^(-1) arctan(1/7 sqrt(7))","D_4_certified_path":["14.2187634589773576013064656778740783173406021086903800669014","14.2187634589773576013064656778740783173406025234736631015112"],"D_4_interval":["3.35610889330893099329120372165535199237327016202855818846111","3.35610889330893099329120372165535199237327016202855818850699"],"D_5":["116.268070280825418177121647330509250276387637918916711376116","116.268070280825418177121647330509250276781644213199508384477"],"D_5_coarse":["116.268070280802807833185727606189878327437320297282738899924","116.268070280848028521057449001491956281990958896604778814642"],"D_5_route_certified_e":["59.7830430969280753312457575504351982258397577326159287093447","59.7830430969280753312462626350585626905747180932658464320717"]},"constants":{"(3/2)^(1/3)":["1.14471424255333186780804221193967700891590692078793107209905","1.14471424255333186780804221193967700891590692078793107209906"],"46080 pi^(3/2)":["256588.554094005097510724412616035949148787811904251879407479","256588.554094005097510724412616035949148787811904251879407574"],"Gamma(7/6)":["0.927719333630039200708349482534621018566466519145475576700351","0.927719333630039200708349482534621018566466519145475576939062"],"Z_5":["256588.554094005097510724412616035949148787811904251879407480","256588.554094005097510724412616035949148787811904251879407574"],"Z_5_layered":["286577.348318386087279989074316181888641618369899945004795250","286577.348318386087279989074316181888641675299347998097342782"],"pi":["3.14159265358979323846264338327950288419716939937510582097457","3.14159265358979323846264338327950288419716939937510582097528"],"|S^5|":["31.0062766802998201754763150671013952022252885658851076941334","31.0062766802998201754763150671013952022252885658851076941545"]},"exact_relations":{"D_m":"sqrt(3/(m+3)) Z_m^-1 int_{mu<0} prod mu_i^2 |Delta(mu)| exp(-|mu|^2/4 + (sum mu)^2/(4(m+3))) dmu","Z_5":"5! (1/24) 2 sqrt(pi/5) int_{0<q<v<w<T} V e^{-T^2/80 - w^2/48 - v^2/24 - q^2/8} = 2^(15/2) (2 pi)^(5/2) prod_{j<=5} Gamma(1+j/2)/Gamma(3/2)^5 = 46080 pi^(3/2)","c_d_ref":"Gamma(7/6) (3/2)^(1/3) |S^{d-1}| D_{d-1} / (sqrt3 sqrt pi (2 pi)^d); d = 6: |S^5| = pi^3, c_{6,ref} = Gamma(7/6) (3/2)^(1/3) D_5 / (64 sqrt3 pi^(7/2))","exponents":"erfc branch T^2/80 + w^2/48 + v^2/24 + q^2/8 with erfc(sqrt30 T/40); polynomial branch T^2/32 + w^2/48 + v^2/24 + q^2/8","layers":"q: J_k (odd k, no erf); v: incomplete Gaussian moments G_n(mu, w), mu in {1/24, 1/6}; w: exact for the polynomial and e^{-mu w^2} parts (G_n(1/48 + mu, T)), certified cumulative Omega_{a,mu}(T) = int_0^T Q(w) e^{-w^2/48} erf(sqrt(mu) w) dw for the erf parts; T: exact two-erf moments plus the certified Taylor quadrature","m=5 coordinates":"T = 4p1 + 3p2 + 2p3 + p4, w = 3p2 + 2p3 + p4, v = 2p3 + p4, q = p4; dp = dT dw dv dq/24; 0 < q < v < w < T","transfer":"E_6 = 314496 (76 24^6 + 15) e^-288; eps = 168 E_6; c_{6,24}/c_{6,ref} in [1/(1+delta), 1+delta], delta = (1+eps)^14/(1-eps)^14 - 1"},"float_reference_10sig":{"D_5_layered_trapezoid":"116.267137"},"layers":{"Z_5":{"certified_part":["24178.7652584712167214089274033053575141507099675457961027994","24178.7652584712167214089274033053575141578919946379668572785"],"exact_part":"106496/81 sqrt(5) pi^(1)  +  965632/6561 sqrt(15) pi^(1)  +  24333760/6561  +  -212992/81 sqrt(5) arctan(1/5 sqrt(5))  +  -1931264/6561 sqrt(15) arctan(1/15 sqrt(15))","kmax":7},"atoms":["1","arctan(1/5 sqrt(15))","arctan(1/2 sqrt(1))","arctan(1/4 sqrt(14))","arctan(1/7 sqrt(21))","arctan(1/7 sqrt(14))","arctan(1/7 sqrt(7))","arctan(1/5 sqrt(5))","arctan(1/15 sqrt(15))","arctan(1/3 sqrt(7))"],"m=5":{"G_0_constant":"2/15 sqrt(30) pi^(1/2)","T_powers_erfc":[0,1,2,3,4,5,6,7,8,9,10,11,12,14],"T_powers_poly":[0,1,2,3,4,5,6,7,8,9,10,11,13],"alpha":"15/32","beta":"3/16 T","certified_functions_(erfc, poly)":[28,26],"certified_part_erfc_branch":["2171340.67575158928149811730903868726756164117400995326826133","2171340.67575158928149811730903868726758714940894103808208401"],"certified_part_poly_branch":["2969588.09339018881243789877924110799593795308619456695381198","2969588.09339018881243789877924110799593795309322879745417960"],"exact_part_erfc_branch":"28925465600/19683 sqrt(5) pi^(1)  +  -9268337768199389753/2066242608000  +  1332123018364889608192/420378134765625 sqrt(5)  +  -421758106855146029056/252226880859375 sqrt(15)  +  -1740790427386340926436111881/18991513795983750000000 sqrt(105)  +  -115701862400/19683 sqrt(5) arctan(1/5 sqrt(5))  +  -85292942950400/129140163 sqrt(15) arctan(1/15 sqrt(15))  +  85292942950400/129140163 sqrt(15) arctan(1/3 sqrt(7))","exact_part_poly_branch":"910769088170876174336/756680642578125 sqrt(2) pi^(1/2)  +  -8196571416593072128/420378134765625 sqrt(6) pi^(1/2)  +  -33593277943383081522495488/445113604593369140625 sqrt(14) pi^(1/2)","kappa":"1/40 sqrt(30)","kappa^2":"3/160","lambda_erfc":"1/80","lambda_poly":"1/32","prefactor":"1/36864 sqrt(6) pi^(-3/2)","q_powers":[1,3,5,7,9,11]}},"mutant":"layer-init","object":"CL-SIDE24-D6-COEFFICIENT-20260930-v1","passed":false,"quadrature":{"K":40,"cells":"width 1/2 on [0, 30], width 1 on [30, 90]","rho":"6"},"scientific_effect":"NONE","transfer":{"E_6":["3.82723721533215540079690742342942909477794757569364110281421E-110","3.82723721533215540079690742342942909477794757569364110281424E-110"],"delta":["1.8003323860922459005348652519812034461835465396062887747638E-106","1.80033238609224590053486525198120344618354653960628877476392E-106"],"epsilon":["6.42975852175802107333880447136144087922695192716531705272787E-108","6.42975852175802107333880447136144087922695192716531705272793E-108"],"exponents":{"a":"79/6 (<= 14 used)","b":"27/2 (<= 14 used)"},"n_jet":28}}},"mehta":{"kind":"report","report":{"certified":true,"checks":{"CLOSED_FORM_D6":true,"D1_EXACT":true,"D2_EXACT":true,"D3_EXACT":true,"D4_CERTIFIED_PATH":true,"D4_EXACT":true,"FLOAT_INSIDE":true,"IMAGE_BOUND":true,"JK_EXACT":true,"LAYER_EXACT":true,"LIBRARY_EXACT":true,"MEHTA_Z5":false,"MONOTONE":false,"PINNED":false,"ROUTE_CONSISTENT":true,"SIDE24_CONSISTENT":true,"STRUCTURE":true,"TRANSFER_BOUND":true,"TRUNCATION_NESTING":true,"WIDTHS":true},"coefficients":{"c_2_ref":["0.0734069193060342710301359629577740500176642446843326718326939","0.0734069193060342710301359629577740500176642446843326718516467"],"c_3_ref":["0.0417759318405983433429366654285755564666815196610447378743313","0.0417759318405983433429366654285755564666815196610447378851390"],"c_4_ref":["0.0233216660029528350945211949528568851531013723632403344029580","0.0233216660029528350945211949528568851531013723632403344090136"],"c_5_ref":["0.0132193193800849680760334875146413965210528901256999054505061","0.0132193193800849680760334875146413965210528901256999054539805"],"c_6_24":["0.0000566634025581088305907671635439374744270075540007462839014672","0.0000566634025581088305907671635439374744275134549180174570583592"],"c_6_ref":["0.0000566634025581088305907671635439374744270075540007462839014678","0.0000566634025581088305907671635439374744275134549180174570583586"],"c_6_ref_closed":["0.0000566634025581088305907671635439374744270075540007462839015065","0.0000566634025581088305907671635439374744275134549180174570583203"]},"cone_moments":{"D_1":"4/3","D_2":"29/6  +  -1 sqrt(6)","D_3":"50/3  +  -76/3 pi^(-1)  +  -200/9 pi^(-1) arctan(1/2)","D_4":"6695/54  +  -405/32 sqrt(6)  +  -1375/72 sqrt(21) pi^(-1)  +  -405/16 sqrt(6) pi^(-1) arctan(1/4 sqrt(14))  +  -6695/27 pi^(-1) arctan(1/7 sqrt(21))  +  405/16 sqrt(6) pi^(-1) arctan(1/7 sqrt(14))","D_4_certified_path":["14.2187634589773576013064656778740783173406021086903800669014","14.2187634589773576013064656778740783173406025234736631015112"],"D_4_interval":["14.2187634589773576013064656778740783173406023160820215841833","14.2187634589773576013064656778740783173406023160820215842323"],"D_5":["0.325052949990080708069598505647194931183199828091573152077811","0.325052949990080708069598505647194931186101958534464279109636"],"D_5_coarse":["0.325052949989914167157910495540684764694757517380213716436950","0.325052949990247248981285646208758241556744998656272913186090"],"D_5_route_certified_e":["0.325052949990080708069596646661099332671124610409037559914623","0.325052949990080708069600364633290529699906912737417296995187"]},"constants":{"(3/2)^(1/3)":["1.14471424255333186780804221193967700891590692078793107209905","1.14471424255333186780804221193967700891590692078793107209906"],"46080 pi^(3/2)":["256588.554094005097510724412616035949148787811904251879407479","256588.554094005097510724412616035949148787811904251879407574"],"Gamma(7/6)":["0.927719333630039200708349482534621018566466519145475576700351","0.927719333630039200708349482534621018566466519145475576939062"],"Z_5":["34835617.2623466768872118224364023416948666781970895514970409","34835617.2623466768872118224364023416948666781970895514970539"],"Z_5_layered":["256588.554094005097510724412616035949148757223338670240957447","256588.554094005097510724412616035949148814152786723333504992"],"pi":["3.14159265358979323846264338327950288419716939937510582097457","3.14159265358979323846264338327950288419716939937510582097528"],"|S^5|":["31.0062766802998201754763150671013952022252885658851076941334","31.0062766802998201754763150671013952022252885658851076941545"]},"exact_relations":{"D_m":"sqrt(3/(m+3)) Z_m^-1 int_{mu<0} prod mu_i^2 |Delta(mu)| exp(-|mu|^2/4 + (sum mu)^2/(4(m+3))) dmu","Z_5":"5! (1/24) 2 sqrt(pi/5) int_{0<q<v<w<T} V e^{-T^2/80 - w^2/48 - v^2/24 - q^2/8} = 2^(15/2) (2 pi)^(5/2) prod_{j<=5} Gamma(1+j/2)/Gamma(3/2)^5 = 46080 pi^(3/2)","c_d_ref":"Gamma(7/6) (3/2)^(1/3) |S^{d-1}| D_{d-1} / (sqrt3 sqrt pi (2 pi)^d); d = 6: |S^5| = pi^3, c_{6,ref} = Gamma(7/6) (3/2)^(1/3) D_5 / (64 sqrt3 pi^(7/2))","exponents":"erfc branch T^2/80 + w^2/48 + v^2/24 + q^2/8 with erfc(sqrt30 T/40); polynomial branch T^2/32 + w^2/48 + v^2/24 + q^2/8","layers":"q: J_k (odd k, no erf); v: incomplete Gaussian moments G_n(mu, w), mu in {1/24, 1/6}; w: exact for the polynomial and e^{-mu w^2} parts (G_n(1/48 + mu, T)), certified cumulative Omega_{a,mu}(T) = int_0^T Q(w) e^{-w^2/48} erf(sqrt(mu) w) dw for the erf parts; T: exact two-erf moments plus the certified Taylor quadrature","m=5 coordinates":"T = 4p1 + 3p2 + 2p3 + p4, w = 3p2 + 2p3 + p4, v = 2p3 + p4, q = p4; dp = dT dw dv dq/24; 0 < q < v < w < T","transfer":"E_6 = 314496 (76 24^6 + 15) e^-288; eps = 168 E_6; c_{6,24}/c_{6,ref} in [1/(1+delta), 1+delta], delta = (1+eps)^14/(1-eps)^14 - 1"},"float_reference_10sig":{"D_5_layered_trapezoid":"0.3250460759"},"layers":{"Z_5":{"certified_part":["24178.7652584712167214089274033053575141507099675457961027994","24178.7652584712167214089274033053575141578919946379668572785"],"exact_part":"183296/81 sqrt(5) pi^(1)  +  596992/6561 sqrt(15) pi^(1)  +  -3191680/729  +  -366592/81 sqrt(5) arctan(1/5 sqrt(5))  +  -1193984/6561 sqrt(15) arctan(1/15 sqrt(15))","kmax":7},"atoms":["1","arctan(1/5 sqrt(15))","arctan(1/2 sqrt(1))","arctan(1/4 sqrt(14))","arctan(1/7 sqrt(21))","arctan(1/7 sqrt(14))","arctan(1/7 sqrt(7))","arctan(1/5 sqrt(5))","arctan(1/15 sqrt(15))","arctan(1/3 sqrt(7))"],"m=5":{"G_0_constant":"2/15 sqrt(30) pi^(1/2)","T_powers_erfc":[0,1,2,3,4,5,6,7,8,9,10,11,12,14],"T_powers_poly":[0,1,2,3,4,5,6,7,8,9,10,11,13],"alpha":"15/32","beta":"3/16 T","certified_functions_(erfc, poly)":[28,26],"certified_part_erfc_branch":["2171340.67575158928149811730903868726756164117400995326826133","2171340.67575158928149811730903868726758714940894103808208401"],"certified_part_poly_branch":["2969588.09339018881243789877924110799593795308619456695381198","2969588.09339018881243789877924110799593795309322879745417960"],"exact_part_erfc_branch":"37325849600/19683 sqrt(5) pi^(1)  +  -2526199047775232/215233605  +  1342173730389322381312/420378134765625 sqrt(5)  +  -307725678950598344704/252226880859375 sqrt(15)  +  6553487824267442728475648/148371201531123046875 sqrt(105)  +  -149303398400/19683 sqrt(5) arctan(1/5 sqrt(5))  +  -46870014525440/129140163 sqrt(15) arctan(1/15 sqrt(15))  +  46870014525440/129140163 sqrt(15) arctan(1/3 sqrt(7))","exact_part_poly_branch":"931333683623590068224/756680642578125 sqrt(2) pi^(1/2)  +  -694064242779548254208/420378134765625 sqrt(6) pi^(1/2)  +  309876730511277843629146112/445113604593369140625 sqrt(14) pi^(1/2)","kappa":"1/40 sqrt(30)","kappa^2":"3/160","lambda_erfc":"1/80","lambda_poly":"1/32","prefactor":"1/3538944 sqrt(3) pi^(-3/2)","q_powers":[1,3,5,7,9,11]}},"mutant":"mehta","object":"CL-SIDE24-D6-COEFFICIENT-20260930-v1","passed":false,"quadrature":{"K":40,"cells":"width 1/2 on [0, 30], width 1 on [30, 90]","rho":"6"},"scientific_effect":"NONE","transfer":{"E_6":["3.82723721533215540079690742342942909477794757569364110281421E-110","3.82723721533215540079690742342942909477794757569364110281424E-110"],"delta":["1.8003323860922459005348652519812034461835465396062887747638E-106","1.80033238609224590053486525198120344618354653960628877476392E-106"],"epsilon":["6.42975852175802107333880447136144087922695192716531705272787E-108","6.42975852175802107333880447136144087922695192716531705272793E-108"],"exponents":{"a":"79/6 (<= 14 used)","b":"27/2 (<= 14 used)"},"n_jet":28}}},"recurrence":{"kind":"report","report":{"certified":true,"checks":{"CLOSED_FORM_D6":true,"D1_EXACT":true,"D2_EXACT":false,"D3_EXACT":false,"D4_CERTIFIED_PATH":true,"D4_EXACT":false,"FLOAT_INSIDE":true,"IMAGE_BOUND":true,"JK_EXACT":true,"LAYER_EXACT":true,"LIBRARY_EXACT":true,"MEHTA_Z5":true,"MONOTONE":true,"PINNED":false,"ROUTE_CONSISTENT":true,"SIDE24_CONSISTENT":false,"STRUCTURE":true,"TRANSFER_BOUND":true,"TRUNCATION_NESTING":true,"WIDTHS":true},"coefficients":{"c_2_ref":["0.0734069193060342710301359629577740500176642446843326718326939","0.0734069193060342710301359629577740500176642446843326718516467"],"c_3_ref":["0.0407892691572937282017742657472035223510267804246934443435423","0.0407892691572937282017742657472035223510267804246934443540948"],"c_4_ref":["0.0212167922784642336278468368688454193619790585195586797397750","0.0212167922784642336278468368688454193619790585195586797452834"],"c_5_ref":["0.0104105530242710195877724030734441393030274347419993365301535","0.0104105530242710195877724030734441393030274347419993365337101"],"c_6_24":["0.00456531676719494210934994849269202330250761728721639630051241","0.00456531676719494210934994849269202330328591997528417653550768"],"c_6_ref":["0.00456531676719494210934994849269202330250761728721639630051246","0.00456531676719494210934994849269202330328591997528417653550763"],"c_6_ref_closed":["0.00456531676719494210934994849269202330250761728721639630051555","0.00456531676719494210934994849269202330328591997528417653550455"]},"cone_moments":{"D_1":"4/3","D_2":"671/150  +  -11213/12800 sqrt(6)","D_3":"59/30  +  36434/1875 pi^(-1)  +  -1009/45 pi^(-1) arctan(1/2)","D_4":"-1265353/1350  +  -15328/25 sqrt(3)  +  1377446847/3276800 sqrt(6)  +  693103377557/4608000000 sqrt(21) pi^(-1)  +  1377446847/1638400 sqrt(6) pi^(-1) arctan(1/4 sqrt(14))  +  1265353/675 pi^(-1) arctan(1/7 sqrt(21))  +  -1377446847/1638400 sqrt(6) pi^(-1) arctan(1/7 sqrt(14))  +  30656/25 sqrt(3) pi^(-1) arctan(1/7 sqrt(7))","D_4_certified_path":["11.1976408673696445281346559818449780467563127857466993394169","11.1976408673696445281346559818449780467563597369032274225292"],"D_4_interval":["11.197640867369644528134655981844978046756336261324963380487","11.197640867369644528134655981844978046756336261324963381408"],"D_5":["26.1892088335865490699708021367647339298594246170856331530694","26.1892088335865490699708021367647339343242038806005350605182"],"D_5_coarse":["26.1892088334295514405124336959685260362490164325805419387135","26.1892088337435466994290402994765776523547607562406873755278"],"D_5_route_certified_e":["26.1892088335865490699690573289097711009803067549651239788668","26.1892088335865490699725469446196967583379665765186768611540"]},"constants":{"(3/2)^(1/3)":["1.14471424255333186780804221193967700891590692078793107209905","1.14471424255333186780804221193967700891590692078793107209906"],"46080 pi^(3/2)":["256588.554094005097510724412616035949148787811904251879407479","256588.554094005097510724412616035949148787811904251879407574"],"Gamma(7/6)":["0.927719333630039200708349482534621018566466519145475576700351","0.927719333630039200708349482534621018566466519145475576939062"],"Z_5":["256588.554094005097510724412616035949148787811904251879407480","256588.554094005097510724412616035949148787811904251879407574"],"Z_5_layered":["256588.554094005097510724412616035949148757223338670240957447","256588.554094005097510724412616035949148814152786723333504992"],"pi":["3.14159265358979323846264338327950288419716939937510582097457","3.14159265358979323846264338327950288419716939937510582097528"],"|S^5|":["31.0062766802998201754763150671013952022252885658851076941334","31.0062766802998201754763150671013952022252885658851076941545"]},"exact_relations":{"D_m":"sqrt(3/(m+3)) Z_m^-1 int_{mu<0} prod mu_i^2 |Delta(mu)| exp(-|mu|^2/4 + (sum mu)^2/(4(m+3))) dmu","Z_5":"5! (1/24) 2 sqrt(pi/5) int_{0<q<v<w<T} V e^{-T^2/80 - w^2/48 - v^2/24 - q^2/8} = 2^(15/2) (2 pi)^(5/2) prod_{j<=5} Gamma(1+j/2)/Gamma(3/2)^5 = 46080 pi^(3/2)","c_d_ref":"Gamma(7/6) (3/2)^(1/3) |S^{d-1}| D_{d-1} / (sqrt3 sqrt pi (2 pi)^d); d = 6: |S^5| = pi^3, c_{6,ref} = Gamma(7/6) (3/2)^(1/3) D_5 / (64 sqrt3 pi^(7/2))","exponents":"erfc branch T^2/80 + w^2/48 + v^2/24 + q^2/8 with erfc(sqrt30 T/40); polynomial branch T^2/32 + w^2/48 + v^2/24 + q^2/8","layers":"q: J_k (odd k, no erf); v: incomplete Gaussian moments G_n(mu, w), mu in {1/24, 1/6}; w: exact for the polynomial and e^{-mu w^2} parts (G_n(1/48 + mu, T)), certified cumulative Omega_{a,mu}(T) = int_0^T Q(w) e^{-w^2/48} erf(sqrt(mu) w) dw for the erf parts; T: exact two-erf moments plus the certified Taylor quadrature","m=5 coordinates":"T = 4p1 + 3p2 + 2p3 + p4, w = 3p2 + 2p3 + p4, v = 2p3 + p4, q = p4; dp = dT dw dv dq/24; 0 < q < v < w < T","transfer":"E_6 = 314496 (76 24^6 + 15) e^-288; eps = 168 E_6; c_{6,24}/c_{6,ref} in [1/(1+delta), 1+delta], delta = (1+eps)^14/(1-eps)^14 - 1"},"float_reference_10sig":{"D_5_layered_trapezoid":"26.18830807"},"layers":{"Z_5":{"certified_part":["24178.7652584712167214089274033053575141507099675457961027994","24178.7652584712167214089274033053575141578919946379668572785"],"exact_part":"183296/81 sqrt(5) pi^(1)  +  596992/6561 sqrt(15) pi^(1)  +  -3191680/729  +  -366592/81 sqrt(5) arctan(1/5 sqrt(5))  +  -1193984/6561 sqrt(15) arctan(1/15 sqrt(15))","kmax":7},"atoms":["1","arctan(1/5 sqrt(15))","arctan(1/2 sqrt(1))","arctan(1/4 sqrt(14))","arctan(1/7 sqrt(21))","arctan(1/7 sqrt(14))","arctan(1/7 sqrt(7))","arctan(1/5 sqrt(5))","arctan(1/15 sqrt(15))","arctan(1/3 sqrt(7))"],"m=5":{"G_0_constant":"2/15 sqrt(30) pi^(1/2)","T_powers_erfc":[0,1,2,3,4,5,6,7,8,9,10,11,12,14],"T_powers_poly":[0,1,2,3,4,5,6,7,8,9,10,11,13],"alpha":"15/32","beta":"3/16 T","certified_functions_(erfc, poly)":[28,26],"certified_part_erfc_branch":["1573360.16760866583088141863891182297416827400739139057609018","1573360.16760866583088141863891182297445732690361026474386970"],"certified_part_poly_branch":["2969588.09339018881243789877924110799593795308619456695381198","2969588.09339018881243789877924110799593795309322879745417960"],"exact_part_erfc_branch":"-57496231242752/98415 sqrt(5) pi^(1)  +  -455802706912310272/215233605  +  258469996108070010927104/140126044921875 sqrt(5)  +  7235228408715065655296/252226880859375 sqrt(15)  +  -8898811216730332409661924352/148371201531123046875 sqrt(105)  +  229984924971008/98415 sqrt(5) arctan(1/5 sqrt(5))  +  -28419517033152512/129140163 sqrt(15) arctan(1/15 sqrt(15))  +  28419517033152512/129140163 sqrt(15) arctan(1/3 sqrt(7))","exact_part_poly_branch":"931333683623590068224/756680642578125 sqrt(2) pi^(1/2)  +  -694064242779548254208/420378134765625 sqrt(6) pi^(1/2)  +  309876730511277843629146112/445113604593369140625 sqrt(14) pi^(1/2)","kappa":"1/40 sqrt(30)","kappa^2":"3/160","lambda_erfc":"1/80","lambda_poly":"1/32","prefactor":"1/36864 sqrt(6) pi^(-3/2)","q_powers":[1,3,5,7,9,11]}},"mutant":"recurrence","object":"CL-SIDE24-D6-COEFFICIENT-20260930-v1","passed":false,"quadrature":{"K":40,"cells":"width 1/2 on [0, 30], width 1 on [30, 90]","rho":"6"},"scientific_effect":"NONE","transfer":{"E_6":["3.82723721533215540079690742342942909477794757569364110281421E-110","3.82723721533215540079690742342942909477794757569364110281424E-110"],"delta":["1.8003323860922459005348652519812034461835465396062887747638E-106","1.80033238609224590053486525198120344618354653960628877476392E-106"],"epsilon":["6.42975852175802107333880447136144087922695192716531705272787E-108","6.42975852175802107333880447136144087922695192716531705272793E-108"],"exponents":{"a":"79/6 (<= 14 used)","b":"27/2 (<= 14 used)"},"n_jet":28}}},"remainder-dropped":{"kind":"report","report":{"certified":true,"checks":{"CLOSED_FORM_D6":true,"D1_EXACT":true,"D2_EXACT":true,"D3_EXACT":true,"D4_CERTIFIED_PATH":false,"D4_EXACT":true,"FLOAT_INSIDE":true,"IMAGE_BOUND":true,"JK_EXACT":true,"LAYER_EXACT":true,"LIBRARY_EXACT":true,"MEHTA_Z5":false,"MONOTONE":true,"PINNED":false,"ROUTE_CONSISTENT":false,"SIDE24_CONSISTENT":true,"STRUCTURE":true,"TRANSFER_BOUND":true,"TRUNCATION_NESTING":true,"WIDTHS":true},"coefficients":{"c_2_ref":["0.0734069193060342710301359629577740500176642446843326718326939","0.0734069193060342710301359629577740500176642446843326718516467"],"c_3_ref":["0.0417759318405983433429366654285755564666815196610447378743313","0.0417759318405983433429366654285755564666815196610447378851390"],"c_4_ref":["0.0233216660029528350945211949528568851531013723632403344029580","0.0233216660029528350945211949528568851531013723632403344090136"],"c_5_ref":["0.0132193193800849680760334875146413965210528901256999054505061","0.0132193193800849680760334875146413965210528901256999054539805"],"c_6_24":["0.00769287862923720407581665066838559861864667603757395255024483","0.00769287862923720407581665066838559861864667603757395255253746"],"c_6_ref":["0.00769287862923720407581665066838559861864667603757395255024491","0.00769287862923720407581665066838559861864667603757395255253738"],"c_6_ref_closed":["0.00769287862923720407581665066838559861864667603757395255025015","0.00769287862923720407581665066838559861864667603757395255253216"]},"cone_moments":{"D_1":"4/3","D_2":"29/6  +  -1 sqrt(6)","D_3":"50/3  +  -76/3 pi^(-1)  +  -200/9 pi^(-1) arctan(1/2)","D_4":"6695/54  +  -405/32 sqrt(6)  +  -1375/72 sqrt(21) pi^(-1)  +  -405/16 sqrt(6) pi^(-1) arctan(1/4 sqrt(14))  +  -6695/27 pi^(-1) arctan(1/7 sqrt(21))  +  405/16 sqrt(6) pi^(-1) arctan(1/7 sqrt(14))","D_4_certified_path":["14.2187634590030051962357745457558418785526599100297543648948","14.2187634590030051962357745457558418785526599100297543649946"],"D_4_interval":["14.2187634589773576013064656778740783173406023160820215841833","14.2187634589773576013064656778740783173406023160820215842323"],"D_5":["44.1306518750761737616930539840837607165071463287755006633878","44.1306518750761737616930539840837607165071463287755006650843"],"D_5_coarse":["44.1306518750761737616930539840837607165071463287755006633878","44.1306518750761737616930539840837607165071463287755006650843"],"D_5_route_certified_e":["44.1306518750741327103617725110811619399972170664905125471480","44.1306518750741327103622772797241043578020958769281829486597"]},"constants":{"(3/2)^(1/3)":["1.14471424255333186780804221193967700891590692078793107209905","1.14471424255333186780804221193967700891590692078793107209906"],"46080 pi^(3/2)":["256588.554094005097510724412616035949148787811904251879407479","256588.554094005097510724412616035949148787811904251879407574"],"Gamma(7/6)":["0.927719333630039200708349482534621018566466519145475576700351","0.927719333630039200708349482534621018566466519145475576939062"],"Z_5":["256588.554094005097510724412616035949148787811904251879407480","256588.554094005097510724412616035949148787811904251879407574"],"Z_5_layered":["256588.554093971052053468012788488833217990578232184374773041","256588.554093971052053468012788488833218047507279826776093607"],"pi":["3.14159265358979323846264338327950288419716939937510582097457","3.14159265358979323846264338327950288419716939937510582097528"],"|S^5|":["31.0062766802998201754763150671013952022252885658851076941334","31.0062766802998201754763150671013952022252885658851076941545"]},"exact_relations":{"D_m":"sqrt(3/(m+3)) Z_m^-1 int_{mu<0} prod mu_i^2 |Delta(mu)| exp(-|mu|^2/4 + (sum mu)^2/(4(m+3))) dmu","Z_5":"5! (1/24) 2 sqrt(pi/5) int_{0<q<v<w<T} V e^{-T^2/80 - w^2/48 - v^2/24 - q^2/8} = 2^(15/2) (2 pi)^(5/2) prod_{j<=5} Gamma(1+j/2)/Gamma(3/2)^5 = 46080 pi^(3/2)","c_d_ref":"Gamma(7/6) (3/2)^(1/3) |S^{d-1}| D_{d-1} / (sqrt3 sqrt pi (2 pi)^d); d = 6: |S^5| = pi^3, c_{6,ref} = Gamma(7/6) (3/2)^(1/3) D_5 / (64 sqrt3 pi^(7/2))","exponents":"erfc branch T^2/80 + w^2/48 + v^2/24 + q^2/8 with erfc(sqrt30 T/40); polynomial branch T^2/32 + w^2/48 + v^2/24 + q^2/8","layers":"q: J_k (odd k, no erf); v: incomplete Gaussian moments G_n(mu, w), mu in {1/24, 1/6}; w: exact for the polynomial and e^{-mu w^2} parts (G_n(1/48 + mu, T)), certified cumulative Omega_{a,mu}(T) = int_0^T Q(w) e^{-w^2/48} erf(sqrt(mu) w) dw for the erf parts; T: exact two-erf moments plus the certified Taylor quadrature","m=5 coordinates":"T = 4p1 + 3p2 + 2p3 + p4, w = 3p2 + 2p3 + p4, v = 2p3 + p4, q = p4; dp = dT dw dv dq/24; 0 < q < v < w < T","transfer":"E_6 = 314496 (76 24^6 + 15) e^-288; eps = 168 E_6; c_{6,24}/c_{6,ref} in [1/(1+delta), 1+delta], delta = (1+eps)^14/(1-eps)^14 - 1"},"float_reference_10sig":{"D_5_layered_trapezoid":"44.12971862"},"layers":{"Z_5":{"certified_part":["24178.7652584669216613873939205263472651731197292245595315273","24178.7652584669216613873939205263472651803017058022684260383"],"exact_part":"183296/81 sqrt(5) pi^(1)  +  596992/6561 sqrt(15) pi^(1)  +  -3191680/729  +  -366592/81 sqrt(5) arctan(1/5 sqrt(5))  +  -1193984/6561 sqrt(15) arctan(1/15 sqrt(15))","kmax":7},"atoms":["1","arctan(1/5 sqrt(15))","arctan(1/2 sqrt(1))","arctan(1/4 sqrt(14))","arctan(1/7 sqrt(21))","arctan(1/7 sqrt(14))","arctan(1/7 sqrt(7))","arctan(1/5 sqrt(5))","arctan(1/15 sqrt(15))","arctan(1/3 sqrt(7))"],"m=5":{"G_0_constant":"2/15 sqrt(30) pi^(1/2)","T_powers_erfc":[0,1,2,3,4,5,6,7,8,9,10,11,12,14],"T_powers_poly":[0,1,2,3,4,5,6,7,8,9,10,11,13],"alpha":"15/32","beta":"3/16 T","certified_functions_(erfc, poly)":[28,26],"certified_part_erfc_branch":["2171340.67575158928149811730903868726756164117400995326826133","2171340.67575158928149811730903868726758714940894103808208401"],"certified_part_poly_branch":["2969588.09339018881243789877924110799593795308619456695381198","2969588.09339018881243789877924110799593795309322879745417960"],"exact_part_erfc_branch":"37325849600/19683 sqrt(5) pi^(1)  +  -2526199047775232/215233605  +  1342173730389322381312/420378134765625 sqrt(5)  +  -307725678950598344704/252226880859375 sqrt(15)  +  6553487824267442728475648/148371201531123046875 sqrt(105)  +  -149303398400/19683 sqrt(5) arctan(1/5 sqrt(5))  +  -46870014525440/129140163 sqrt(15) arctan(1/15 sqrt(15))  +  46870014525440/129140163 sqrt(15) arctan(1/3 sqrt(7))","exact_part_poly_branch":"931333683623590068224/756680642578125 sqrt(2) pi^(1/2)  +  -694064242779548254208/420378134765625 sqrt(6) pi^(1/2)  +  309876730511277843629146112/445113604593369140625 sqrt(14) pi^(1/2)","kappa":"1/40 sqrt(30)","kappa^2":"3/160","lambda_erfc":"1/80","lambda_poly":"1/32","prefactor":"1/36864 sqrt(6) pi^(-3/2)","q_powers":[1,3,5,7,9,11]}},"mutant":"remainder-dropped","object":"CL-SIDE24-D6-COEFFICIENT-20260930-v1","passed":false,"quadrature":{"K":40,"cells":"width 1/2 on [0, 30], width 1 on [30, 90]","rho":"6"},"scientific_effect":"NONE","transfer":{"E_6":["3.82723721533215540079690742342942909477794757569364110281421E-110","3.82723721533215540079690742342942909477794757569364110281424E-110"],"delta":["1.8003323860922459005348652519812034461835465396062887747638E-106","1.80033238609224590053486525198120344618354653960628877476392E-106"],"epsilon":["6.42975852175802107333880447136144087922695192716531705272787E-108","6.42975852175802107333880447136144087922695192716531705272793E-108"],"exponents":{"a":"79/6 (<= 14 used)","b":"27/2 (<= 14 used)"},"n_jet":28}}},"sandwich-exponent":{"kind":"report","report":{"certified":true,"checks":{"CLOSED_FORM_D6":true,"D1_EXACT":true,"D2_EXACT":true,"D3_EXACT":true,"D4_CERTIFIED_PATH":true,"D4_EXACT":true,"FLOAT_INSIDE":true,"IMAGE_BOUND":true,"JK_EXACT":true,"LAYER_EXACT":true,"LIBRARY_EXACT":true,"MEHTA_Z5":true,"MONOTONE":true,"PINNED":true,"ROUTE_CONSISTENT":true,"SIDE24_CONSISTENT":true,"STRUCTURE":true,"TRANSFER_BOUND":false,"TRUNCATION_NESTING":true,"WIDTHS":true},"coefficients":{"c_2_ref":["0.0734069193060342710301359629577740500176642446843326718326939","0.0734069193060342710301359629577740500176642446843326718516467"],"c_3_ref":["0.0417759318405983433429366654285755564666815196610447378743313","0.0417759318405983433429366654285755564666815196610447378851390"],"c_4_ref":["0.0233216660029528350945211949528568851531013723632403344029580","0.0233216660029528350945211949528568851531013723632403344090136"],"c_5_ref":["0.0132193193800849680760334875146413965210528901256999054505061","0.0132193193800849680760334875146413965210528901256999054539805"],"c_6_24":["0.00769287862923684827866663128122739008112489691080122434668364","0.00769287862923684827866663128122739008119358029688972504148796"],"c_6_ref":["0.00769287862923684827866663128122739008112489691080122434668372","0.00769287862923684827866663128122739008119358029688972504148788"],"c_6_ref_closed":["0.00769287862923684827866663128122739008112489691080122434668894","0.00769287862923684827866663128122739008119358029688972504148264"]},"cone_moments":{"D_1":"4/3","D_2":"29/6  +  -1 sqrt(6)","D_3":"50/3  +  -76/3 pi^(-1)  +  -200/9 pi^(-1) arctan(1/2)","D_4":"6695/54  +  -405/32 sqrt(6)  +  -1375/72 sqrt(21) pi^(-1)  +  -405/16 sqrt(6) pi^(-1) arctan(1/4 sqrt(14))  +  -6695/27 pi^(-1) arctan(1/7 sqrt(21))  +  405/16 sqrt(6) pi^(-1) arctan(1/7 sqrt(14))","D_4_certified_path":["14.2187634589773576013064656778740783173406021086903800669014","14.2187634589773576013064656778740783173406025234736631015112"],"D_4_interval":["14.2187634589773576013064656778740783173406023160820215841833","14.2187634589773576013064656778740783173406023160820215842323"],"D_5":["44.1306518750741327103620248954026331485852349159356435082422","44.1306518750741327103620248954026331489792412102184405166029"],"D_5_coarse":["44.1306518750515223664261051710832611996349172943016710320497","44.1306518750967430542978265663853391541885558936237109467677"],"D_5_route_certified_e":["44.1306518750741327103617725110811619399972170664905125471480","44.1306518750741327103622772797241043578020958769281829486597"]},"constants":{"(3/2)^(1/3)":["1.14471424255333186780804221193967700891590692078793107209905","1.14471424255333186780804221193967700891590692078793107209906"],"46080 pi^(3/2)":["256588.554094005097510724412616035949148787811904251879407479","256588.554094005097510724412616035949148787811904251879407574"],"Gamma(7/6)":["0.927719333630039200708349482534621018566466519145475576700351","0.927719333630039200708349482534621018566466519145475576939062"],"Z_5":["256588.554094005097510724412616035949148787811904251879407480","256588.554094005097510724412616035949148787811904251879407574"],"Z_5_layered":["256588.554094005097510724412616035949148757223338670240957447","256588.554094005097510724412616035949148814152786723333504992"],"pi":["3.14159265358979323846264338327950288419716939937510582097457","3.14159265358979323846264338327950288419716939937510582097528"],"|S^5|":["31.0062766802998201754763150671013952022252885658851076941334","31.0062766802998201754763150671013952022252885658851076941545"]},"exact_relations":{"D_m":"sqrt(3/(m+3)) Z_m^-1 int_{mu<0} prod mu_i^2 |Delta(mu)| exp(-|mu|^2/4 + (sum mu)^2/(4(m+3))) dmu","Z_5":"5! (1/24) 2 sqrt(pi/5) int_{0<q<v<w<T} V e^{-T^2/80 - w^2/48 - v^2/24 - q^2/8} = 2^(15/2) (2 pi)^(5/2) prod_{j<=5} Gamma(1+j/2)/Gamma(3/2)^5 = 46080 pi^(3/2)","c_d_ref":"Gamma(7/6) (3/2)^(1/3) |S^{d-1}| D_{d-1} / (sqrt3 sqrt pi (2 pi)^d); d = 6: |S^5| = pi^3, c_{6,ref} = Gamma(7/6) (3/2)^(1/3) D_5 / (64 sqrt3 pi^(7/2))","exponents":"erfc branch T^2/80 + w^2/48 + v^2/24 + q^2/8 with erfc(sqrt30 T/40); polynomial branch T^2/32 + w^2/48 + v^2/24 + q^2/8","layers":"q: J_k (odd k, no erf); v: incomplete Gaussian moments G_n(mu, w), mu in {1/24, 1/6}; w: exact for the polynomial and e^{-mu w^2} parts (G_n(1/48 + mu, T)), certified cumulative Omega_{a,mu}(T) = int_0^T Q(w) e^{-w^2/48} erf(sqrt(mu) w) dw for the erf parts; T: exact two-erf moments plus the certified Taylor quadrature","m=5 coordinates":"T = 4p1 + 3p2 + 2p3 + p4, w = 3p2 + 2p3 + p4, v = 2p3 + p4, q = p4; dp = dT dw dv dq/24; 0 < q < v < w < T","transfer":"E_6 = 314496 (76 24^6 + 15) e^-288; eps = 168 E_6; c_{6,24}/c_{6,ref} in [1/(1+delta), 1+delta], delta = (1+eps)^14/(1-eps)^14 - 1"},"float_reference_10sig":{"D_5_layered_trapezoid":"44.12971862"},"layers":{"Z_5":{"certified_part":["24178.7652584712167214089274033053575141507099675457961027994","24178.7652584712167214089274033053575141578919946379668572785"],"exact_part":"183296/81 sqrt(5) pi^(1)  +  596992/6561 sqrt(15) pi^(1)  +  -3191680/729  +  -366592/81 sqrt(5) arctan(1/5 sqrt(5))  +  -1193984/6561 sqrt(15) arctan(1/15 sqrt(15))","kmax":7},"atoms":["1","arctan(1/5 sqrt(15))","arctan(1/2 sqrt(1))","arctan(1/4 sqrt(14))","arctan(1/7 sqrt(21))","arctan(1/7 sqrt(14))","arctan(1/7 sqrt(7))","arctan(1/5 sqrt(5))","arctan(1/15 sqrt(15))","arctan(1/3 sqrt(7))"],"m=5":{"G_0_constant":"2/15 sqrt(30) pi^(1/2)","T_powers_erfc":[0,1,2,3,4,5,6,7,8,9,10,11,12,14],"T_powers_poly":[0,1,2,3,4,5,6,7,8,9,10,11,13],"alpha":"15/32","beta":"3/16 T","certified_functions_(erfc, poly)":[28,26],"certified_part_erfc_branch":["2171340.67575158928149811730903868726756164117400995326826133","2171340.67575158928149811730903868726758714940894103808208401"],"certified_part_poly_branch":["2969588.09339018881243789877924110799593795308619456695381198","2969588.09339018881243789877924110799593795309322879745417960"],"exact_part_erfc_branch":"37325849600/19683 sqrt(5) pi^(1)  +  -2526199047775232/215233605  +  1342173730389322381312/420378134765625 sqrt(5)  +  -307725678950598344704/252226880859375 sqrt(15)  +  6553487824267442728475648/148371201531123046875 sqrt(105)  +  -149303398400/19683 sqrt(5) arctan(1/5 sqrt(5))  +  -46870014525440/129140163 sqrt(15) arctan(1/15 sqrt(15))  +  46870014525440/129140163 sqrt(15) arctan(1/3 sqrt(7))","exact_part_poly_branch":"931333683623590068224/756680642578125 sqrt(2) pi^(1/2)  +  -694064242779548254208/420378134765625 sqrt(6) pi^(1/2)  +  309876730511277843629146112/445113604593369140625 sqrt(14) pi^(1/2)","kappa":"1/40 sqrt(30)","kappa^2":"3/160","lambda_erfc":"1/80","lambda_poly":"1/32","prefactor":"1/36864 sqrt(6) pi^(-3/2)","q_powers":[1,3,5,7,9,11]}},"mutant":"sandwich-exponent","object":"CL-SIDE24-D6-COEFFICIENT-20260930-v1","passed":false,"quadrature":{"K":40,"cells":"width 1/2 on [0, 30], width 1 on [30, 90]","rho":"6"},"scientific_effect":"NONE","transfer":{"E_6":["3.82723721533215540079690742342942909477794757569364110281421E-110","3.82723721533215540079690742342942909477794757569364110281424E-110"],"delta":["1.67173721565708547906808916255397462859900750106298243370924E-106","1.67173721565708547906808916255397462859900750106298243370943E-106"],"epsilon":["6.42975852175802107333880447136144087922695192716531705272787E-108","6.42975852175802107333880447136144087922695192716531705272793E-108"],"exponents":{"a":"79/6 (<= 14 used)","b":"27/2 (<= 14 used)"},"n_jet":28}}},"shift-variance":{"kind":"report","report":{"certified":true,"checks":{"CLOSED_FORM_D6":true,"D1_EXACT":true,"D2_EXACT":true,"D3_EXACT":true,"D4_CERTIFIED_PATH":true,"D4_EXACT":true,"FLOAT_INSIDE":true,"IMAGE_BOUND":true,"JK_EXACT":true,"LAYER_EXACT":true,"LIBRARY_EXACT":true,"MEHTA_Z5":true,"MONOTONE":false,"PINNED":false,"ROUTE_CONSISTENT":true,"SIDE24_CONSISTENT":true,"STRUCTURE":false,"TRANSFER_BOUND":true,"TRUNCATION_NESTING":true,"WIDTHS":true},"coefficients":{"c_2_ref":["0.0734069193060342710301359629577740500176642446843326718326939","0.0734069193060342710301359629577740500176642446843326718516467"],"c_3_ref":["0.0417759318405983433429366654285755564666815196610447378743313","0.0417759318405983433429366654285755564666815196610447378851390"],"c_4_ref":["0.0233216660029528350945211949528568851531013723632403344029580","0.0233216660029528350945211949528568851531013723632403344090136"],"c_5_ref":["0.0132193193800849680760334875146413965210528901256999054505061","0.0132193193800849680760334875146413965210528901256999054539805"],"c_6_24":["0.0606894293887792690293623511108698287793249260133859735533930","0.0606894293887792690293623511108698287793822342058505598865649"],"c_6_ref":["0.0606894293887792690293623511108698287793249260133859735533937","0.0606894293887792690293623511108698287793822342058505598865642"],"c_6_ref_closed":["0.0606894293887792690293623511108698287793249260133859735534348","0.0606894293887792690293623511108698287793822342058505598865232"]},"cone_moments":{"D_1":"4/3","D_2":"29/6  +  -1 sqrt(6)","D_3":"50/3  +  -76/3 pi^(-1)  +  -200/9 pi^(-1) arctan(1/2)","D_4":"6695/54  +  -405/32 sqrt(6)  +  -1375/72 sqrt(21) pi^(-1)  +  -405/16 sqrt(6) pi^(-1) arctan(1/4 sqrt(14))  +  -6695/27 pi^(-1) arctan(1/7 sqrt(21))  +  405/16 sqrt(6) pi^(-1) arctan(1/7 sqrt(14))","D_4_certified_path":["14.2187634589773576013064656778740783173406021086903800669014","14.2187634589773576013064656778740783173406025234736631015112"],"D_4_interval":["14.2187634589773576013064656778740783173406023160820215841833","14.2187634589773576013064656778740783173406023160820215842323"],"D_5":["348.148490303011727652465385747662006869912896870460982231817","348.148490303011727652465385747662006870241648695770267395061"],"D_5_coarse":["348.148490302992583537222432071113810425241350071942341254098","348.148490303030871767708854646508638721600225922754862388590"],"D_5_route_certified_e":["348.148490303011727652465171893159398055426667325015602181223","348.148490303011727652465599602164615711620078892457847314972"]},"constants":{"(3/2)^(1/3)":["1.14471424255333186780804221193967700891590692078793107209905","1.14471424255333186780804221193967700891590692078793107209906"],"46080 pi^(3/2)":["256588.554094005097510724412616035949148787811904251879407479","256588.554094005097510724412616035949148787811904251879407574"],"Gamma(7/6)":["0.927719333630039200708349482534621018566466519145475576700351","0.927719333630039200708349482534621018566466519145475576939062"],"Z_5":["256588.554094005097510724412616035949148787811904251879407480","256588.554094005097510724412616035949148787811904251879407574"],"Z_5_layered":["256588.554094005097510724412616035949148757223338670240957447","256588.554094005097510724412616035949148814152786723333504992"],"pi":["3.14159265358979323846264338327950288419716939937510582097457","3.14159265358979323846264338327950288419716939937510582097528"],"|S^5|":["31.0062766802998201754763150671013952022252885658851076941334","31.0062766802998201754763150671013952022252885658851076941545"]},"exact_relations":{"D_m":"sqrt(3/(m+3)) Z_m^-1 int_{mu<0} prod mu_i^2 |Delta(mu)| exp(-|mu|^2/4 + (sum mu)^2/(4(m+3))) dmu","Z_5":"5! (1/24) 2 sqrt(pi/5) int_{0<q<v<w<T} V e^{-T^2/80 - w^2/48 - v^2/24 - q^2/8} = 2^(15/2) (2 pi)^(5/2) prod_{j<=5} Gamma(1+j/2)/Gamma(3/2)^5 = 46080 pi^(3/2)","c_d_ref":"Gamma(7/6) (3/2)^(1/3) |S^{d-1}| D_{d-1} / (sqrt3 sqrt pi (2 pi)^d); d = 6: |S^5| = pi^3, c_{6,ref} = Gamma(7/6) (3/2)^(1/3) D_5 / (64 sqrt3 pi^(7/2))","exponents":"erfc branch T^2/80 + w^2/48 + v^2/24 + q^2/8 with erfc(sqrt30 T/40); polynomial branch T^2/32 + w^2/48 + v^2/24 + q^2/8","layers":"q: J_k (odd k, no erf); v: incomplete Gaussian moments G_n(mu, w), mu in {1/24, 1/6}; w: exact for the polynomial and e^{-mu w^2} parts (G_n(1/48 + mu, T)), certified cumulative Omega_{a,mu}(T) = int_0^T Q(w) e^{-w^2/48} erf(sqrt(mu) w) dw for the erf parts; T: exact two-erf moments plus the certified Taylor quadrature","m=5 coordinates":"T = 4p1 + 3p2 + 2p3 + p4, w = 3p2 + 2p3 + p4, v = 2p3 + p4, q = p4; dp = dT dw dv dq/24; 0 < q < v < w < T","transfer":"E_6 = 314496 (76 24^6 + 15) e^-288; eps = 168 E_6; c_{6,24}/c_{6,ref} in [1/(1+delta), 1+delta], delta = (1+eps)^14/(1-eps)^14 - 1"},"float_reference_10sig":{"D_5_layered_trapezoid":"348.1452002"},"layers":{"Z_5":{"certified_part":["24178.7652584712167214089274033053575141507099675457961027994","24178.7652584712167214089274033053575141578919946379668572785"],"exact_part":"183296/81 sqrt(5) pi^(1)  +  596992/6561 sqrt(15) pi^(1)  +  -3191680/729  +  -366592/81 sqrt(5) arctan(1/5 sqrt(5))  +  -1193984/6561 sqrt(15) arctan(1/15 sqrt(15))","kmax":7},"atoms":["1","arctan(1/5 sqrt(15))","arctan(1/2 sqrt(1))","arctan(1/4 sqrt(14))","arctan(1/7 sqrt(21))","arctan(1/7 sqrt(14))","arctan(1/7 sqrt(7))","arctan(1/5 sqrt(5))","arctan(2/5 sqrt(5))","arctan(1/15 sqrt(15))"],"m=5":{"G_0_constant":"1/10 sqrt(70) pi^(1/2)","T_powers_erfc":[0,1,2,3,4,5,6,7,8,9,10,11,12,14],"T_powers_poly":[0,1,2,3,4,5,6,7,8,9,10,11,13],"alpha":"5/14","beta":"1/7 T","certified_functions_(erfc, poly)":[28,26],"certified_part_erfc_branch":["6072333.60743032607151865990249080262240566042508555596929399","6072333.60743032607151865990249080262242423820482870474005219"],"certified_part_poly_branch":["20062987.1449438950487978620019584927408393319721115611508269","20062987.1449438950487978620019584927408393321020348197572467"],"exact_part_erfc_branch":"12593853440/2187 sqrt(5) pi^(1)  +  1683100518400/14348907 sqrt(15) pi^(1)  +  -1464565764567087376909568/145964630126953125  +  666000658616731869184/1261134404296875 sqrt(15)  +  -45440103038551911328/28025208984375 sqrt(30)  +  -25187706880/2187 sqrt(5) arctan(1/5 sqrt(5))  +  -25187706880/2187 sqrt(5) arctan(2/5 sqrt(5))  +  -6732402073600/14348907 sqrt(15) arctan(1/15 sqrt(15))","exact_part_poly_branch":"281963028882967201952/252226880859375 sqrt(21) pi^(1/2)  +  917800073726634065408/1261134404296875 sqrt(42) pi^(1/2)  +  -1987191441627006426112/3003387451171875 sqrt(70) pi^(1/2)","kappa":"1/70 sqrt(70)","kappa^2":"1/70","lambda_erfc":"1/80","lambda_poly":"3/112","prefactor":"1/36864 sqrt(6) pi^(-3/2)","q_powers":[1,3,5,7,9,11]}},"mutant":"shift-variance","object":"CL-SIDE24-D6-COEFFICIENT-20260930-v1","passed":false,"quadrature":{"K":40,"cells":"width 1/2 on [0, 30], width 1 on [30, 90]","rho":"6"},"scientific_effect":"NONE","transfer":{"E_6":["3.82723721533215540079690742342942909477794757569364110281421E-110","3.82723721533215540079690742342942909477794757569364110281424E-110"],"delta":["1.8003323860922459005348652519812034461835465396062887747638E-106","1.80033238609224590053486525198120344618354653960628877476392E-106"],"epsilon":["6.42975852175802107333880447136144087922695192716531705272787E-108","6.42975852175802107333880447136144087922695192716531705272793E-108"],"exponents":{"a":"79/6 (<= 14 used)","b":"27/2 (<= 14 used)"},"n_jet":28}}},"tail-dropped":{"kind":"report","report":{"certified":true,"checks":{"CLOSED_FORM_D6":true,"D1_EXACT":true,"D2_EXACT":true,"D3_EXACT":true,"D4_CERTIFIED_PATH":false,"D4_EXACT":true,"FLOAT_INSIDE":true,"IMAGE_BOUND":true,"JK_EXACT":true,"LAYER_EXACT":true,"LIBRARY_EXACT":true,"MEHTA_Z5":false,"MONOTONE":true,"PINNED":false,"ROUTE_CONSISTENT":false,"SIDE24_CONSISTENT":true,"STRUCTURE":true,"TRANSFER_BOUND":true,"TRUNCATION_NESTING":false,"WIDTHS":true},"coefficients":{"c_2_ref":["0.0734069193060342710301359629577740500176642446843326718326939","0.0734069193060342710301359629577740500176642446843326718516467"],"c_3_ref":["0.0417759318405983433429366654285755564666815196610447378743313","0.0417759318405983433429366654285755564666815196610447378851390"],"c_4_ref":["0.0233216660029528350945211949528568851531013723632403344029580","0.0233216660029528350945211949528568851531013723632403344090136"],"c_5_ref":["0.0132193193800849680760334875146413965210528901256999054505061","0.0132193193800849680760334875146413965210528901256999054539805"],"c_6_24":["0.00769287857102547007039178001478644916727861308109043535111012","0.00769287857102547007039178001478644916727861308155889243048871"],"c_6_ref":["0.00769287857102547007039178001478644916727861308109043535111020","0.00769287857102547007039178001478644916727861308155889243048863"],"c_6_ref_closed":["0.00769287857102547007039178001478644916727861308109043535111540","0.00769287857102547007039178001478644916727861308155889243048340"]},"cone_moments":{"D_1":"4/3","D_2":"29/6  +  -1 sqrt(6)","D_3":"50/3  +  -76/3 pi^(-1)  +  -200/9 pi^(-1) arctan(1/2)","D_4":"6695/54  +  -405/32 sqrt(6)  +  -1375/72 sqrt(21) pi^(-1)  +  -405/16 sqrt(6) pi^(-1) arctan(1/4 sqrt(14))  +  -6695/27 pi^(-1) arctan(1/7 sqrt(21))  +  405/16 sqrt(6) pi^(-1) arctan(1/7 sqrt(14))","D_4_certified_path":["14.2187634589773446675284397327798994463294034911204779197287","14.2187634589773446675284397327798994463294034911660051095868"],"D_4_interval":["14.2187634589773576013064656778740783173406023160820215841833","14.2187634589773576013064656778740783173406023160820215842323"],"D_5":["44.1306515411411291251363966453332196777841944595710345071075","44.1306515411411291251363966453332196777841944622583661545584"],"D_5_coarse":["44.1306518750515223664261051710832611996349172943016710320500","44.1306518750967430542978265663853391541885558936237109467675"],"D_5_route_certified_e":["44.1306518750741327103617725110811619399972170664905125471481","44.1306518750741327103622772797241043578020958769281829486594"]},"constants":{"(3/2)^(1/3)":["1.14471424255333186780804221193967700891590692078793107209905","1.14471424255333186780804221193967700891590692078793107209906"],"46080 pi^(3/2)":["256588.554094005097510724412616035949148787811904251879407479","256588.554094005097510724412616035949148787811904251879407574"],"Gamma(7/6)":["0.927719333630039200708349482534621018566466519145475576700351","0.927719333630039200708349482534621018566466519145475576939062"],"Z_5":["256588.554094005097510724412616035949148787811904251879407480","256588.554094005097510724412616035949148787811904251879407574"],"Z_5_layered":["256494.458891708808438587453175109595087400488727706697322540","256494.458891708808438587453175109595087400488727707470294315"],"pi":["3.14159265358979323846264338327950288419716939937510582097457","3.14159265358979323846264338327950288419716939937510582097528"],"|S^5|":["31.0062766802998201754763150671013952022252885658851076941334","31.0062766802998201754763150671013952022252885658851076941545"]},"exact_relations":{"D_m":"sqrt(3/(m+3)) Z_m^-1 int_{mu<0} prod mu_i^2 |Delta(mu)| exp(-|mu|^2/4 + (sum mu)^2/(4(m+3))) dmu","Z_5":"5! (1/24) 2 sqrt(pi/5) int_{0<q<v<w<T} V e^{-T^2/80 - w^2/48 - v^2/24 - q^2/8} = 2^(15/2) (2 pi)^(5/2) prod_{j<=5} Gamma(1+j/2)/Gamma(3/2)^5 = 46080 pi^(3/2)","c_d_ref":"Gamma(7/6) (3/2)^(1/3) |S^{d-1}| D_{d-1} / (sqrt3 sqrt pi (2 pi)^d); d = 6: |S^5| = pi^3, c_{6,ref} = Gamma(7/6) (3/2)^(1/3) D_5 / (64 sqrt3 pi^(7/2))","exponents":"erfc branch T^2/80 + w^2/48 + v^2/24 + q^2/8 with erfc(sqrt30 T/40); polynomial branch T^2/32 + w^2/48 + v^2/24 + q^2/8","layers":"q: J_k (odd k, no erf); v: incomplete Gaussian moments G_n(mu, w), mu in {1/24, 1/6}; w: exact for the polynomial and e^{-mu w^2} parts (G_n(1/48 + mu, T)), certified cumulative Omega_{a,mu}(T) = int_0^T Q(w) e^{-w^2/48} erf(sqrt(mu) w) dw for the erf parts; T: exact two-erf moments plus the certified Taylor quadrature","m=5 coordinates":"T = 4p1 + 3p2 + 2p3 + p4, w = 3p2 + 2p3 + p4, v = 2p3 + p4, q = p4; dp = dT dw dv dq/24; 0 < q < v < w < T","transfer":"E_6 = 314496 (76 24^6 + 15) e^-288; eps = 168 E_6; c_{6,24}/c_{6,ref} in [1/(1+delta), 1+delta], delta = (1+eps)^14/(1-eps)^14 - 1"},"float_reference_10sig":{"D_5_layered_trapezoid":"44.12971862"},"layers":{"Z_5":{"certified_part":["24166.8945252172250713269231590145170863765685690474529759184","24166.8945252172250713269231590145170863765685690475504914184"],"exact_part":"183296/81 sqrt(5) pi^(1)  +  596992/6561 sqrt(15) pi^(1)  +  -3191680/729  +  -366592/81 sqrt(5) arctan(1/5 sqrt(5))  +  -1193984/6561 sqrt(15) arctan(1/15 sqrt(15))","kmax":7},"atoms":["1","arctan(1/5 sqrt(15))","arctan(1/2 sqrt(1))","arctan(1/4 sqrt(14))","arctan(1/7 sqrt(21))","arctan(1/7 sqrt(14))","arctan(1/7 sqrt(7))","arctan(1/5 sqrt(5))","arctan(1/15 sqrt(15))","arctan(1/3 sqrt(7))"],"m=5":{"G_0_constant":"2/15 sqrt(30) pi^(1/2)","T_powers_erfc":[0,1,2,3,4,5,6,7,8,9,10,11,12,14],"T_powers_poly":[0,1,2,3,4,5,6,7,8,9,10,11,13],"alpha":"15/32","beta":"3/16 T","certified_functions_(erfc, poly)":[28,26],"certified_part_erfc_branch":["2171340.67575158928149811730903868726756164117400995326826134","2171340.67575158928149811730903868726758714940894103808208400"],"certified_part_poly_branch":["2969588.09339018881243789877924110799593795308619456695381199","2969588.09339018881243789877924110799593795309322879745417959"],"exact_part_erfc_branch":"37325849600/19683 sqrt(5) pi^(1)  +  -2526199047775232/215233605  +  1342173730389322381312/420378134765625 sqrt(5)  +  -307725678950598344704/252226880859375 sqrt(15)  +  6553487824267442728475648/148371201531123046875 sqrt(105)  +  -149303398400/19683 sqrt(5) arctan(1/5 sqrt(5))  +  -46870014525440/129140163 sqrt(15) arctan(1/15 sqrt(15))  +  46870014525440/129140163 sqrt(15) arctan(1/3 sqrt(7))","exact_part_poly_branch":"931333683623590068224/756680642578125 sqrt(2) pi^(1/2)  +  -694064242779548254208/420378134765625 sqrt(6) pi^(1/2)  +  309876730511277843629146112/445113604593369140625 sqrt(14) pi^(1/2)","kappa":"1/40 sqrt(30)","kappa^2":"3/160","lambda_erfc":"1/80","lambda_poly":"1/32","prefactor":"1/36864 sqrt(6) pi^(-3/2)","q_powers":[1,3,5,7,9,11]}},"mutant":"tail-dropped","object":"CL-SIDE24-D6-COEFFICIENT-20260930-v1","passed":false,"quadrature":{"K":40,"cells":"width 1/2 on [0, 30], width 1 on [30, 90]","rho":"6"},"scientific_effect":"NONE","transfer":{"E_6":["3.82723721533215540079690742342942909477794757569364110281421E-110","3.82723721533215540079690742342942909477794757569364110281424E-110"],"delta":["1.8003323860922459005348652519812034461835465396062887747638E-106","1.80033238609224590053486525198120344618354653960628877476392E-106"],"epsilon":["6.42975852175802107333880447136144087922695192716531705272787E-108","6.42975852175802107333880447136144087922695192716531705272793E-108"],"exponents":{"a":"79/6 (<= 14 used)","b":"27/2 (<= 14 used)"},"n_jet":28}}},"two-erf":{"kind":"report","report":{"certified":true,"checks":{"CLOSED_FORM_D6":true,"D1_EXACT":true,"D2_EXACT":true,"D3_EXACT":true,"D4_CERTIFIED_PATH":true,"D4_EXACT":true,"FLOAT_INSIDE":false,"IMAGE_BOUND":true,"JK_EXACT":true,"LAYER_EXACT":true,"LIBRARY_EXACT":true,"MEHTA_Z5":true,"MONOTONE":false,"PINNED":false,"ROUTE_CONSISTENT":false,"SIDE24_CONSISTENT":true,"STRUCTURE":true,"TRANSFER_BOUND":true,"TRUNCATION_NESTING":true,"WIDTHS":true},"coefficients":{"c_2_ref":["0.0734069193060342710301359629577740500176642446843326718326939","0.0734069193060342710301359629577740500176642446843326718516467"],"c_3_ref":["0.0417759318405983433429366654285755564666815196610447378743313","0.0417759318405983433429366654285755564666815196610447378851390"],"c_4_ref":["0.0233216660029528350945211949528568851531013723632403344029580","0.0233216660029528350945211949528568851531013723632403344090136"],"c_5_ref":["0.0132193193800849680760334875146413965210528901256999054505061","0.0132193193800849680760334875146413965210528901256999054539805"],"c_6_24":["-0.00578126801133068097780526580352291448910652989022098359732123","-0.00578126801133068097780526580352291448903784650413248290301505"],"c_6_ref":["-0.00578126801133068097780526580352291448910652989022098359732117","-0.00578126801133068097780526580352291448903784650413248290301511"],"c_6_ref_closed":["-0.00578126801133068097780526580352291448910652989022098359731725","-0.00578126801133068097780526580352291448903784650413248290301903"]},"cone_moments":{"D_1":"4/3","D_2":"29/6  +  -1 sqrt(6)","D_3":"50/3  +  -76/3 pi^(-1)  +  -200/9 pi^(-1) arctan(1/2)","D_4":"6695/54  +  -405/32 sqrt(6)  +  -1375/72 sqrt(21) pi^(-1)  +  -405/16 sqrt(6) pi^(-1) arctan(1/4 sqrt(14))  +  -6695/27 pi^(-1) arctan(1/7 sqrt(21))  +  405/16 sqrt(6) pi^(-1) arctan(1/7 sqrt(14))","D_4_certified_path":["14.2187634589773576013064656778740783173406021086903800669014","14.2187634589773576013064656778740783173406025234736631015112"],"D_4_interval":["14.2187634589773576013064656778740783173406023160820215841833","14.2187634589773576013064656778740783173406023160820215842323"],"D_5":["-33.1645848453800484328291478284176959755168974983986079265018","-33.1645848453800484328291478284176959751228912041158109181520"],"D_5_coarse":["-33.1645848454026587767650675527370679244672151200325804026943","-33.1645848453574380888933461574349899699135765207105404879871"],"D_5_route_certified_e":["44.1306518750741327103617725110811619399972170664905125471480","44.1306518750741327103622772797241043578020958769281829486597"]},"constants":{"(3/2)^(1/3)":["1.14471424255333186780804221193967700891590692078793107209905","1.14471424255333186780804221193967700891590692078793107209906"],"46080 pi^(3/2)":["256588.554094005097510724412616035949148787811904251879407479","256588.554094005097510724412616035949148787811904251879407574"],"Gamma(7/6)":["0.927719333630039200708349482534621018566466519145475576700351","0.927719333630039200708349482534621018566466519145475576939062"],"Z_5":["256588.554094005097510724412616035949148787811904251879407480","256588.554094005097510724412616035949148787811904251879407574"],"Z_5_layered":["256588.554094005097510724412616035949148757223338670240957447","256588.554094005097510724412616035949148814152786723333504992"],"pi":["3.14159265358979323846264338327950288419716939937510582097457","3.14159265358979323846264338327950288419716939937510582097528"],"|S^5|":["31.0062766802998201754763150671013952022252885658851076941334","31.0062766802998201754763150671013952022252885658851076941545"]},"exact_relations":{"D_m":"sqrt(3/(m+3)) Z_m^-1 int_{mu<0} prod mu_i^2 |Delta(mu)| exp(-|mu|^2/4 + (sum mu)^2/(4(m+3))) dmu","Z_5":"5! (1/24) 2 sqrt(pi/5) int_{0<q<v<w<T} V e^{-T^2/80 - w^2/48 - v^2/24 - q^2/8} = 2^(15/2) (2 pi)^(5/2) prod_{j<=5} Gamma(1+j/2)/Gamma(3/2)^5 = 46080 pi^(3/2)","c_d_ref":"Gamma(7/6) (3/2)^(1/3) |S^{d-1}| D_{d-1} / (sqrt3 sqrt pi (2 pi)^d); d = 6: |S^5| = pi^3, c_{6,ref} = Gamma(7/6) (3/2)^(1/3) D_5 / (64 sqrt3 pi^(7/2))","exponents":"erfc branch T^2/80 + w^2/48 + v^2/24 + q^2/8 with erfc(sqrt30 T/40); polynomial branch T^2/32 + w^2/48 + v^2/24 + q^2/8","layers":"q: J_k (odd k, no erf); v: incomplete Gaussian moments G_n(mu, w), mu in {1/24, 1/6}; w: exact for the polynomial and e^{-mu w^2} parts (G_n(1/48 + mu, T)), certified cumulative Omega_{a,mu}(T) = int_0^T Q(w) e^{-w^2/48} erf(sqrt(mu) w) dw for the erf parts; T: exact two-erf moments plus the certified Taylor quadrature","m=5 coordinates":"T = 4p1 + 3p2 + 2p3 + p4, w = 3p2 + 2p3 + p4, v = 2p3 + p4, q = p4; dp = dT dw dv dq/24; 0 < q < v < w < T","transfer":"E_6 = 314496 (76 24^6 + 15) e^-288; eps = 168 E_6; c_{6,24}/c_{6,ref} in [1/(1+delta), 1+delta], delta = (1+eps)^14/(1-eps)^14 - 1"},"float_reference_10sig":{"D_5_layered_trapezoid":"-33.1655181"},"layers":{"Z_5":{"certified_part":["24178.7652584712167214089274033053575141507099675457961027994","24178.7652584712167214089274033053575141578919946379668572785"],"exact_part":"183296/81 sqrt(5) pi^(1)  +  596992/6561 sqrt(15) pi^(1)  +  -3191680/729  +  -366592/81 sqrt(5) arctan(1/5 sqrt(5))  +  -1193984/6561 sqrt(15) arctan(1/15 sqrt(15))","kmax":7},"atoms":["1","arctan(1/5 sqrt(15))","arctan(1/2 sqrt(1))","arctan(1/4 sqrt(14))","arctan(1/7 sqrt(21))","arctan(1/7 sqrt(14))","arctan(1/7 sqrt(7))","arctan(1/5 sqrt(5))","arctan(1/15 sqrt(15))","arctan(1/3 sqrt(1))"],"m=5":{"G_0_constant":"2/15 sqrt(30) pi^(1/2)","T_powers_erfc":[0,1,2,3,4,5,6,7,8,9,10,11,12,14],"T_powers_poly":[0,1,2,3,4,5,6,7,8,9,10,11,13],"alpha":"15/32","beta":"3/16 T","certified_functions_(erfc, poly)":[28,26],"certified_part_erfc_branch":["2171340.67575158928149811730903868726756164117400995326826133","2171340.67575158928149811730903868726758714940894103808208401"],"certified_part_poly_branch":["2969588.09339018881243789877924110799593795308619456695381198","2969588.09339018881243789877924110799593795309322879745417960"],"exact_part_erfc_branch":"74651699200/59049 sqrt(5) pi^(1)  +  -2526199047775232/215233605  +  1342173730389322381312/420378134765625 sqrt(5)  +  -307725678950598344704/252226880859375 sqrt(15)  +  6553487824267442728475648/148371201531123046875 sqrt(105)  +  -149303398400/19683 sqrt(5) arctan(1/5 sqrt(5))  +  -46870014525440/129140163 sqrt(15) arctan(1/15 sqrt(15))  +  46870014525440/129140163 sqrt(15) arctan(1/3)","exact_part_poly_branch":"931333683623590068224/756680642578125 sqrt(2) pi^(1/2)  +  -694064242779548254208/420378134765625 sqrt(6) pi^(1/2)  +  309876730511277843629146112/445113604593369140625 sqrt(14) pi^(1/2)","kappa":"1/40 sqrt(30)","kappa^2":"3/160","lambda_erfc":"1/80","lambda_poly":"1/32","prefactor":"1/36864 sqrt(6) pi^(-3/2)","q_powers":[1,3,5,7,9,11]}},"mutant":"two-erf","object":"CL-SIDE24-D6-COEFFICIENT-20260930-v1","passed":false,"quadrature":{"K":40,"cells":"width 1/2 on [0, 30], width 1 on [30, 90]","rho":"6"},"scientific_effect":"NONE","transfer":{"E_6":["3.82723721533215540079690742342942909477794757569364110281421E-110","3.82723721533215540079690742342942909477794757569364110281424E-110"],"delta":["1.8003323860922459005348652519812034461835465396062887747638E-106","1.80033238609224590053486525198120344618354653960628877476392E-106"],"epsilon":["6.42975852175802107333880447136144087922695192716531705272787E-108","6.42975852175802107333880447136144087922695192716531705272793E-108"],"exponents":{"a":"79/6 (<= 14 used)","b":"27/2 (<= 14 used)"},"n_jet":28}}},"vandermonde":{"final_line":"ValueError: an even p3-power occurred; J_k would carry an error function","kind":"guard"}},"results_bytes":8319,"results_sha256":"82d1c19f9f503fd9211c6dcf356ec62aee9ec9eda44792bc38be4f3690637832","schema":1,"script":"cone_moment_d6.py","script_git_blob":"c5e56694d96c885373a7b70868259eeb9fdfa746"}
diff --git a/tools/side24_mutant_replay.py b/tools/side24_mutant_replay.py
new file mode 100644
index 000000000..d65db8b4f
--- /dev/null
+++ b/tools/side24_mutant_replay.py
@@ -0,0 +1,166 @@
+#!/usr/bin/env python3
+"""Authenticate the SIDE24 d = 5, 6 checkers' acceptance and rejection runs.
+
+Called AFTER each workflow's unchanged manifest and pin guards. Each checker is
+run from the repository root as `<script>` or `<script> --mutant LABEL`, as the
+workflows always ran it. It prints one JSON report on stdout and one timing line
+`seconds N.N` on stderr, then exits 0 if every check passed and 1 otherwise.
+
+For every child the protocol requires the exact exit status and a stderr that is
+exactly one timing line. The baseline's stdout must equal RESULTS.json byte for
+byte. A mutant's stdout must be one strict JSON report structurally equal to the
+report measured from the unchanged checker (tools/side24_mutant_contracts/, one
+file per family, each within the strict parser's size limit).
+One mutant, closed-form `vandermonde`, never reaches a report: the checker's own
+structural guard raises. Its reference is that guard: exit 1, empty stdout, and a
+traceback whose final line is the measured exception text. Any other crash,
+silent exit, extra diagnostic or different report fails. Parsing and comparison
+are those of tools/legacy_json_replay.py. This is an engineering protocol, not a
+sandbox or a mathematical acceptance test.
+"""
+from __future__ import annotations
+
+import concurrent.futures
+import hashlib
+import json
+from pathlib import Path
+import re
+import subprocess
+import sys
+from typing import Any
+
+from tools.legacy_json_replay import ProtocolError, _same, strict_json
+
+# Mutant order within each mode, as in each wrapper's original list.
+INVENTORIES = {
+    'side24-d5-closed-form': (
+        'shift-variance', 'vandermonde', 'erfc-sign', 'jacobian', 'mehta', 'by-parts',
+        'one-erf', 'two-erf', 'jk-init', 'gauss-half', 'boundary-term', 'atom-pi',
+        'cumulative-init', 'cumulative-recurrence'),
+    'side24-d5-coefficient': (
+        'shift-variance', 'vandermonde', 'erfc-branch', 'remainder-dropped', 'mehta',
+        'sphere-area', 'image-shells', 'sandwich-exponent', 'recurrence', 'jacobian',
+        'cumulative-half'),
+    'side24-d6-coefficient': (
+        'shift-variance', 'vandermonde', 'erfc-branch', 'jacobian', 'mehta', 'recurrence',
+        'layer-init', 'two-erf', 'cumulative-half', 'remainder-dropped', 'erf-inside',
+        'tail-dropped', 'image-shells', 'sandwich-exponent'),
+}
+MODES = (('normal', ('-B', '-S')), ('optimized', ('-B', '-O', '-S')))
+TIMING = re.compile(rb'\Aseconds [0-9]+\.[0-9]\n\Z')
+TRACEBACK = b'Traceback (most recent call last):\n'
+CONTRACTS = Path(__file__).with_name('side24_mutant_contracts')
+
+
+def _blob(data: bytes) -> str:
+    return hashlib.sha1(b'blob %d\0' % len(data) + data).hexdigest()
+
+
+def _contract(family: str, contracts: Path = CONTRACTS) -> dict[str, Any]:
+    if family not in INVENTORIES:
+        raise ProtocolError('unknown checker family')
+    entry = strict_json((Path(contracts) / (family + '.json')).read_bytes())
+    keys = {'schema', 'family', 'directory', 'script', 'script_git_blob', 'results_sha256',
+            'results_bytes', 'mutants'}
+    if (type(entry) is not dict or set(entry) != keys or type(entry['schema']) is not int
+            or entry['schema'] != 1 or entry['family'] != family):
+        raise ProtocolError('invalid family reference schema')
+    if (type(entry['script']) is not str or Path(entry['script']).name != entry['script']
+            or type(entry['directory']) is not str
+            or type(entry['results_bytes']) is not int or entry['results_bytes'] <= 0):
+        raise ProtocolError('invalid script or results metadata')
+    for key, size in (('script_git_blob', 40), ('results_sha256', 64)):
+        if type(entry[key]) is not str or not re.fullmatch('[0-9a-f]{%d}' % size, entry[key]):
+            raise ProtocolError('invalid reference digest')
+    mutants = entry['mutants']
+    if type(mutants) is not dict or set(mutants) != set(INVENTORIES[family]):
+        raise ProtocolError('missing, extra, or repeated mutant reference')
+    for label, ref in mutants.items():
+        if type(ref) is not dict or type(ref.get('kind')) is not str:
+            raise ProtocolError('invalid mutant reference')
+        if ref['kind'] == 'report':
+            report = ref.get('report')
+            if (set(ref) != {'kind', 'report'} or type(report) is not dict
+                    or report.get('mutant') != label or report.get('passed') is not False
+                    or type(report.get('checks')) is not dict
+                    or not any(v is False for v in report['checks'].values())):
+                raise ProtocolError('reference is not a rejection report')
+        elif ref['kind'] == 'guard':
+            line = ref.get('final_line')
+            if (set(ref) != {'kind', 'final_line'} or type(line) is not str
+                    or not line or '\n' in line or ': ' not in line):
+                raise ProtocolError('invalid guard reference')
+        else:
+            raise ProtocolError('unknown mutant reference kind')
+    return entry
+
+
+def _check(entry: dict[str, Any], label: str | None, results: bytes,
+           returncode: int, stdout: bytes, stderr: bytes) -> bool:
+    if label is None:
+        return returncode == 0 and stdout == results and bool(TIMING.match(stderr))
+    ref = entry['mutants'][label]
+    if returncode != 1:
+        return False
+    if ref['kind'] == 'guard':
+        tail = ref['final_line'].encode('utf-8') + b'\n'
+        return (not stdout and stderr.startswith(TRACEBACK) and stderr.endswith(b'\n' + tail)
+                and b'\nseconds ' not in stderr)
+    if not TIMING.match(stderr):
+        return False
+    try:
+        return _same(strict_json(stdout), ref['report'])
+    except ProtocolError:
+        return False
+
+
+def replay(family: str, repo: Path, *, timeout: float, workers: int = 4,
+           contracts: Path = CONTRACTS) -> list[dict[str, Any]]:
+    """Run the complete fixed inventory in both modes, or fail.
+
+    repo is the repository root, the children's working directory as before;
+    the packet directory inside it has already been source-verified. A timeout
+    or OSError is a failed run, never an intended rejection. No child input,
+    packet file or checker is changed.
+    """
+    entry = _contract(family, contracts)
+    repo = Path(repo).resolve()
+    root = repo / entry['directory']
+    script = root / entry['script']
+    if script.is_symlink() or not script.is_file():
+        raise ProtocolError('regular source checker required')
+    if _blob(script.read_bytes()) != entry['script_git_blob']:
+        raise ProtocolError('checker differs from the characterized source')
+    results = (root / 'RESULTS.json').read_bytes()
+    if (len(results) != entry['results_bytes']
+            or hashlib.sha256(results).hexdigest() != entry['results_sha256']):
+        raise ProtocolError('RESULTS.json differs from the characterized source')
+
+    def run(mode: str, flags: tuple[str, ...], label: str | None) -> dict[str, Any]:
+        command = [sys.executable, *flags, str(script)]
+        if label is not None:
+            command += ['--mutant', label]
+        try:
+            child = subprocess.run(command, cwd=repo, capture_output=True, timeout=timeout)
+            rc, out, err = child.returncode, child.stdout, child.stderr
+        except (subprocess.TimeoutExpired, OSError):
+            rc, out, err = None, b'', b''
+        ok = rc is not None and _check(entry, label, results, rc, out, err)
+        return {'family': family, 'mode': mode, 'label': label, 'returncode': rc,
+                'stdout_bytes': len(out), 'stderr_bytes': len(err),
+                'stdout_sha256': hashlib.sha256(out).hexdigest(),
+                'stderr_sha256': hashlib.sha256(err).hexdigest(), 'protocol_ok': ok}
+
+    records: list[dict[str, Any]] = []
+    for mode, flags in MODES:
+        labels = [None, *INVENTORIES[family]]
+        with concurrent.futures.ThreadPoolExecutor(max_workers=workers) as pool:
+            batch = list(pool.map(lambda label: run(mode, flags, label), labels))
+        for record in batch:
+            print(json.dumps(record, sort_keys=True), flush=True)
+        for record in batch:
+            if not record['protocol_ok']:
+                raise ProtocolError('SIDE24 protocol mismatch: %s/%s/%s' %
+                                    (family, mode, record['label'] or 'baseline'))
+        records.extend(batch)
+    return records