Oct 4, 2026 · Sep 30, 2026 · Sep 30, 2026 · Sep 30, 2026 · Sep 30, 2026 · Sep 30, 2026
@@ -0,0 +1,60 @@
+name: Local pairing exact controls
+
+'on':
+  pull_request:
+    paths:
+      - 'frontiers/local_pairing_20260930/**'
+      - '.github/workflows/local-pairing.yml'
+  workflow_dispatch:
+
+permissions:
+  contents: read
+
+jobs:
+  exact-controls:
+    runs-on: ubuntu-latest
+    timeout-minutes: 10
+    steps:
+      - uses: actions/checkout@11d5960a326750d5838078e36cf38b85af677262
+        with:
+          persist-credentials: false
+      - uses: actions/setup-python@a26af69be951a213d495a4c3e4e4022e16d87065
+        with:
+          python-version: '3.11.16'
+      - name: Verify sources, replay both modes, reject mutants
+        env:
+          PYTHONDONTWRITEBYTECODE: '1'
+        run: |
+          set -euo pipefail
+          python -B -S - <<'PY'
+          import hashlib, json, pathlib, subprocess, sys
+          root = pathlib.Path('frontiers/local_pairing_20260930').resolve()
+          manifest = json.loads((root / 'SOURCES.json').read_text())
+          for entry in manifest['files']:
+              path = root / entry['path']
+              if path.is_symlink():
+                  raise ValueError('symlink: ' + entry['path'])
+              data = path.read_bytes()
+              if len(data) != entry['bytes'] or hashlib.sha256(data).hexdigest() != entry['sha256']:
+                  raise ValueError('packet file identity mismatch: ' + entry['path'])
+          for entry in manifest['consumed'] + manifest['cited_only']:
+              data = pathlib.Path(entry['path']).read_bytes()
+              framed = b'blob ' + str(len(data)).encode() + b'\0' + data
+              if len(data) != entry['bytes'] or hashlib.sha256(data).hexdigest() != entry['sha256'] or hashlib.sha1(framed).hexdigest() != entry['git_blob']:
+                  raise ValueError('merged source drifted: ' + entry['path'])
+          script = str(root / 'pairing_check.py')
+          expected = (root / 'RESULTS.json').read_bytes()
+          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))
+              for m in ('M1', 'M2', 'M3', 'M4'):
+                  rc = subprocess.run([sys.executable, *flags, script, '--mutant', m], capture_output=True, timeout=600).returncode
+                  if rc != 1:
+                      raise RuntimeError('mutant ' + m + ' not rejected')
+              rc = subprocess.run([sys.executable, *flags, script, '--mutant', 'M9'], capture_output=True, timeout=600).returncode
+              if rc != 2:
+                  raise RuntimeError('unknown mutant label not rejected with exit 2')
+          print(json.dumps({'scientific_effect': 'NONE', 'mathematical_acceptance': False}))
+          PY
+          git diff --exit-code
@@ -0,0 +1,38 @@
+# Local pairing (CL-LOCAL-PAIRING-20260930-v1)
+
+Author-side proof candidate, Anthropic Claude, 30 September 2026. Scientific effect NONE. Nonauthor review required.
+
+**Question answered.** Near a short-lifetime candidate pair `(M, S)` the account knows how many extra
+window critical points there are (`Θ(r³)`, Theorem Q) but not which of them change the elder pairing, nor
+whether the decision is local. This packet shows:
+
+* On the limiting local landscape (the pinned planar cubic of [CUB]), off a null tie set, `(M,S)` is the
+  elder pair **iff** there is no extra strict-window critical point; otherwise `M` dies at the highest extra
+  window saddle (Lemmas 2.1–2.2, P, P′, Proposition 3.3 — elementary written proofs, explicit paths; reviewed
+  twice by clean-context referee agents before landing).
+* **Theorem LP** (conditional on [SC] and [P] at their scope): `liminf r⁻³(1 − p_r) ≥ a₁ + a₂ = ν₁ + ν₂ − β_far`,
+  the local nonempty cluster mass — a lower bound for the `Θ(r³)` selection loss of [P] Theorem A / [ELDER] /
+  [E_d] by an explicit functional of the cluster measure.
+* **Conjecture LP=** (`1 − p_r = (a₁ + a₂) r³ + o(r³)`, i.e. remote witnesses never pair) with its remaining
+  obligation (O1) stated precisely in §5. That obligation is discharged, at the packets' conditional scope, by
+  the OpenAI packets Math-#170 (planar, Theorem S: cap implication + retained-tail integration) and Math-#175
+  (`d ≥ 3`, hard-negative-fibre barrier), both opened before this packet, reviewed by this author on 30 September,
+  and now on main (#170 at `fa2e990`, #175 at `eb659bd`); this packet's own contribution is the cap-free lower bound
+  and the exact controls.
+* §6 says what this is worth for the lifetime law: a lower bound for the coefficient of the `Θ(ℓ^{2/3})`
+  compact-window selection loss (relative order `ℓ`); nothing for the leading term; nothing about the
+  unrestricted density's remainder.
+
+Files: `LOCAL_PAIRING.md` (the note), `pairing_check.py` (stdlib exact controls; `RESULTS.json` its output,
+`-O` identical; mutants `M1`–`M4` exit 1, unknown label exit 2), `SOURCES.json` (exact identities of every
+source; [SC] and [TSL] are consumed at their merged bytes, identical to the reviewed PR heads), `exploration/` (standard-library grid
+elder-rule computation — evidence, not proof, not run in CI).
+
+Review slices (§9): A deterministic (§§2–3); B the Gaussian transfer (§4); C the obligation (§5).
+
+**v1.2 (2 October 2026).** Codex's Slice C review (C85, 5396856086) found C-01: v1.1's optional §5 margin sketch
+confined `M`'s component at one level above `f(S)` only. v1.2 withdraws that sketch as a proof and states
+Lemma 5.1 instead. It is a common-radius acceptance lemma, consumed from QS-E (main#229 5961415030; nonauthor
+review ACCEPT / PASS_TECHNICAL_SCOPED, 5961620598), with exhausting jet-scaled margin sets. §5 also records the review's notes on normalization and
+conditioning, and marks (O1′) ⇒ (O1) as conditional on Lemma 5.1. Nothing else changes: not the deterministic
+lemmas, not Theorem LP, and not the controls.
@@ -0,0 +1,40 @@
+{
+  "checks": {
+    "I0_I3_identities": {
+      "info": 144,
+      "passed": true
+    },
+    "N1_n0_grid": {
+      "info": {
+        "n0_points_with_U_hypothesis": 0,
+        "n0_typed_grid_points": 135
+      },
+      "passed": true
+    },
+    "P1_path_cases": {
+      "info": {
+        "cases": {
+          "U": 2160,
+          "V": 4680,
+          "V_at_other_root": 2160
+        },
+        "min_margin": "1/1200",
+        "samples": 6840
+      },
+      "passed": true
+    },
+    "P2_elder_witness": {
+      "info": {
+        "Z1_squared": "15/8",
+        "n": 2,
+        "saddle_height": "-7/32",
+        "u0": "-3/4",
+        "vertical_line_constant": true
+      },
+      "passed": true
+    }
+  },
+  "object": "CL-LOCAL-PAIRING-20260930-v1",
+  "passed": true,
+  "scientific_effect": "NONE"
+}
@@ -0,0 +1,172 @@
+{
+ "schema": 1,
+ "scientific_effect": "NONE",
+ "object": "CL-LOCAL-PAIRING-20260930-v1",
+ "kind": "author-side proof candidate (deterministic lemmas complete; Theorem LP conditional; Conjecture LP= = Math-#170 Theorem S / Math-#175 Theorem F at their scope); nonauthor analytic review required",
+ "author": "Anthropic Claude",
+ "session": "https://claude.ai/code/session_01NMeKEismAyeqgdB4sy2NJU",
+ "answers": "the 2026-09-30 question relayed by the owner: which window witnesses pair, and whether the pairing is local",
+ "consumed": [
+  {
+   "path": "imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md",
+   "git_blob": "dfed3b8d318a3ab1950957f393307733a4bef3f2",
+   "bytes": 40261,
+   "sha256": "9350ad6eaba6626b93c3dedeef9e2ff816e5cdf1c8318e85fb27499141c84bc7",
+   "tag": "P",
+   "use": "\u00a71 p_r and Theorem A; \u00a74 moments; \u00a75 Z_r >= z_* r^2 and (5.3); \u00a78 genericity; \u00a7\u00a79-11 ledger"
+  },
+  {
+   "path": "frontiers/window_multiplicity_laws_20260928/ELDER_LOWER_AND_DENSITY_GAP.md",
+   "git_blob": "aaefc8da9668e819591dd9db0b34442394f2362e",
+   "bytes": 10229,
+   "sha256": "8ce05a43bde8e0d718f09e2da1ffe75d3ec5893489254afdbf5b58b602cdba0c",
+   "tag": "ELDER",
+   "use": "(E1) maximin; \u00a72 path (E5) and C0-stability template; \u00a74 (E9)-(E11)"
+  },
+  {
+   "path": "frontiers/planar_cubic_cluster_20260929/PROOF.md",
+   "git_blob": "bb446d08db8a944537a743ad550b88c1c2ad5758",
+   "bytes": 19889,
+   "sha256": "117e9299a71e6139270266889eb772ed97518e402d9af029bbdb67caf9d4a0f0",
+   "tag": "CUB",
+   "use": "(C1)-(C4), Theorem C, (C6)-(C13); deterministic part only"
+  },
+  {
+   "path": "frontiers/spectral_cluster_closure_20260929/PROOF.md",
+   "pr": "d6g8k5htny-coder/Math-#162",
+   "commit": "382c0f9ff2281f43651ddbdf2e9b36dd52e2c6bf",
+   "git_blob": "16c56821b52fd76b0be791622b9c3809eafde75a",
+   "bytes": 25006,
+   "sha256": "e971b2cbe50a8a06b47c1219191201e0d1037c9adac23e5e3b7d2051bb70c2eb",
+   "tag": "SC",
+   "use": "\u00a71, \u00a72 (7)-(10), \u00a74 (14)-(16), \u00a75 normal form and (17)-(18), \u00a76 (19)-(20), \u00a77 (2), (21)-(24)",
+   "merged": "358f2562efbccc59f9549e5a79650e2e936395c3",
+   "note": "reviewed at PR head 382c0f9; merged with identical bytes"
+  },
+  {
+   "path": "frontiers/two_scale_cluster_geometry_20260929/TWO_SCALE_LAW.md",
+   "pr": "d6g8k5htny-coder/Math-#166",
+   "commit": "7c82252533c3fe14ee262f7f87c0549c10968592",
+   "git_blob": "a32fd5f7d941bbe1fe943df045b1e0fbec8d691c",
+   "bytes": 17139,
+   "sha256": "e81d7fe09d25c3266eb8e62922756f54761d7f74d692fb35d9a8071fffd73769",
+   "tag": "TSL",
+   "use": "cited in \u00a70 only (Theorems L, T); not consumed by any proof",
+   "merged": "ab13a08f5e5b0adc7a7d80cf643d9b5a0618507f",
+   "note": "reviewed at PR head 7c82252 (custody repaired in Math-#172, proof bytes unchanged); merged with identical bytes"
+  }
+ ],
+ "cited_only": [
+  {
+   "path": "frontiers/elder_lower_all_d_20260929/PROOF.md",
+   "git_blob": "7f41c9e315be0a7985d4c770a11a2cdf717bd46a",
+   "bytes": 16485,
+   "sha256": "88c6bfc7973fe832918f91e2ed1f47da3e51987ed9efc1a99648e82b2eb119e5",
+   "tag": "E_d"
+  },
+  {
+   "path": "frontiers/elder_dimension_lift_20260928/PROOF.md",
+   "git_blob": "7303bd791a68a1139251f0f6e403a9f7cc89b006",
+   "bytes": 23971,
+   "sha256": "b529fe3780e1909014b3cb144b9a74156d976bbdc19b6711d0c5b1a1da9e5669",
+   "tag": "EDL"
+  },
+  {
+   "path": "frontiers/c7_total_bounded_20260929/PROOF.md",
+   "git_blob": "28748b086ec6761ef467dc67cba475fbdf8b7455",
+   "bytes": 14982,
+   "sha256": "16c0efe372c1b1f26460aaa109287fed78e6fbd2dd00a0530998db296278dd01",
+   "tag": "C7-K"
+  },
+  {
+   "path": "frontiers/c7_zero_gap_limit_20260929/PROOF.md",
+   "git_blob": "5b6328ea3d6afcbeb731698b3834e536f72d1bcb",
+   "bytes": 17592,
+   "sha256": "f0e563c45c41333f2bb6e5bf5cb98f0d3b592e2abf2af1c1b098c21ad77cb777",
+   "tag": "C7-Z"
+  },
+  {
+   "path": "frontiers/unrestricted_selection_difference_20260929/PROOF.md",
+   "git_blob": "5a55b179a974a90c65d257f3f93765cc6fb30bb8",
+   "bytes": 13028,
+   "sha256": "93a63002d32fd5c35ae994fe28b1205d8098026d6ddde19c67d4977dcf5817a9",
+   "tag": "C7-U"
+  },
+  {
+   "path": "reviews/c6_residual_closure_20260930/RECONCILIATION.md",
+   "git_blob": "9668d06a41e14057fe6e03ac1bf469aa1ed18580",
+   "bytes": 36574,
+   "sha256": "60ed9ca2df666bc31cf50dc1fc7f3f8e8527d5efe4b16c265c205e1e9d1e6b34",
+   "tag": "RES",
+   "use": "cited in \u00a70 only: the leading-mass localization residual bound to [TSL] Theorem L (Math-#173, merged 30106e1; declarative)",
+   "note": "re-pinned 2026-10-03 from blob 3cfd17c5ef0356dbd3e8f98a2d08cb404ddd982c (34758 bytes) to the bytes landed on main by Math-#183 (merge a4c3199, record v1.6): the change adds the v1.6 history entry, the installed-state LIVE/mutant rules and a fifteenth mutant (partial-install) for the record's own checker; the Theorem L direct-statement binding cited in \u00a70 and the declarative-only, executed=false status are unchanged"
+  },
+  {
+   "path": "frontiers/c6_cluster_coefficients_numerics_20260930/NOTE.md",
+   "git_blob": "845881d40520d4acbe15ee196a1ba0d5412eafa1",
+   "bytes": 11018,
+   "sha256": "f20c1ac96b00edc071e428bd2e5dc4c91b8546a3ee0693367b783109b26aa518",
+   "tag": "NUM",
+   "use": "cited in \u00a76 only: uncertified numerical values of the planar near coefficients (Math-#168, merged 02772ec); not consumed"
+  },
+  {
+   "path": "frontiers/local_elder_geometry_20260930/PROOF.md",
+   "bytes": 39722,
+   "sha256": "f68038be79b46124b0f9b31205aa6e3682b34b6f5ef81f4a5697ae81f07cc46b",
+   "git_blob": "ef2aa57959ea9f721bbf2316ce94cf616c1c9113",
+   "tag": "LEG",
+   "use": "cited in the header, \u00a70 and \u00a75 only: Theorem E (same deterministic statement as Lemma P\u2032 + Proposition 3.3, different proof) and Theorem S (planar equality by the cap route); not consumed; merged as Math-#170 (fa2e990) at the identical blob, now verified by the workflow"
+  },
+  {
+   "path": "frontiers/concave_fibre_elder_20260930/PROOF.md",
+   "bytes": 23416,
+   "sha256": "5681896a6808f48fb7508914d0860ec508af7646e9e706fc260d935dd951c75e",
+   "git_blob": "923d3236b5291a29bd162937f172190db84dcef5",
+   "tag": "CFE",
+   "use": "cited in the header, \u00a70 and \u00a75 only: H1 (the d >= 3 hard-direction slaving named in \u00a75) and Theorem F; not consumed and not a premise of Theorem LP; merged as Math-#175 (eb659bd) at the identical blob, now verified by the workflow"
+  }
+ ],
+ "referee": {
+  "kind": "two clean-context Anthropic Claude subagent passes before landing (same provider, same account; not a nonauthor review)",
+  "pass_1": "two blocking (mountain-pass step of Lemma P\u2032; the highest-saddle claim unproved) and seventeen amendments; all applied (Lemmas 2.1, 2.2, 4.1, 4.2, Proposition 3.3 added; V-first assignment; D = 0 subcase; tail bound; \u00a75 margins; \u00a76 wording)",
+  "pass_2": "Lemma 2.1(b) contained two false inequalities (replaced by the four-case polar argument); Lemma 2.2 flow wording made an explicit deformation; the S_V assignment in Lemma P made definite and continuous; K_{\u03b4,R} closed by adding s <= -|B|/2 - \u03b4; \u00a75 obligation restated in jet-scaled margins as (O1\u2032) with the d >= 3 slaving named; checker: shear identity (I0), exact central-difference derivative (I2) and Vieta other-root check added, min_margin key; title and README overclaims removed"
+ },
+ "files": [
+  {
+   "path": "LOCAL_PAIRING.md",
+   "bytes": 46742,
+   "sha256": "e01e1d097fb087fee013559f508ad58849da06f9a764eed1464286946d78331a"
+  },
+  {
+   "path": "pairing_check.py",
+   "bytes": 13611,
+   "sha256": "ff30e7850bd650a9d264ec7537ef955d38a000b085e50834f0ca6c8a933b3cc5"
+  },
+  {
+   "path": "RESULTS.json",
+   "bytes": 796,
+   "sha256": "c012b374ac07c9d51dbab6e2168194b6b27afed6c8f489da5c376230044546b4"
+  },
+  {
+   "path": "README.md",
+   "bytes": 3130,
+   "sha256": "745ef6547c7e93993649f5cb7a978af87b175a908f202d2e398e4eeca2e460ae"
+  },
+  {
+   "path": "exploration/elder_grid.py",
+   "bytes": 11132,
+   "sha256": "72d642f4b1ce50ab4133533e8a943a84f1e02229c7c62181d6bce426d731a816"
+  },
+  {
+   "path": "exploration/RESULTS_EXPLORATION.json",
+   "bytes": 89311,
+   "sha256": "a1b8e0ab19dd75f4f595925eeade8bd396b2dc3b82e49cfb542ca18b8e6f85f3"
+  }
+ ],
+ "read_at": {
+  "branch_base": "30106e1be251e8c18759b6fa4de6176b76380c12",
+  "note": "Math- main after Math-#173 merged; every consumed source verified at these bytes by the workflow"
+ },
+ "cited_unmerged": [],
+ "revision": "v1.1 (30 September 2026): Codex review 5364265784 on Math-#180 applied \u2014 exploration rewritten in the standard library (elder_grid.py), Proposition 3.3 tie set extended by {D = 0}; relation to Math-#170/#175 recorded | v1.1+sources (2026-09-30 14:15 UTC): LEG (#170) moved from cited_unmerged to cited_only after its integration at the identical blob ef2aa579; no other change; branch merged with main 3e0a91b | status repair (2026-10-01): \u00a75's stale 'not a corollary of anything merged' / 'unmerged' sentence replaced by the attribution to Math-#170 Theorem S on main (requested in comments 5930621629 and 5930668120); #175 (CFE) moved from cited_unmerged to cited_only after its integration at eb659bd with the identical blob 923d3236; header and README status updated; mathematics, checker and RESULTS unchanged; branch merged with main 124c37d | v1.2 (2026-10-02): Slice C review C85 5396856086 finding C-01 applied \u2014 the v1.1 \u00a75 single-level margin sketch is withdrawn as a proof; Lemma 5.1 (common-radius acceptance, consumed from QS-E, main#229 comment 5961415030, author-side candidate) with exhausting jet-scaled margin sets M(eta, R) replaces it; the review's normalization and Gaussian-conditioning notes are recorded; (O1') => (O1) marked conditional on Lemma 5.1; LOCAL_PAIRING.md and README.md only; checker, RESULTS, exploration and workflow unchanged | status refresh (2026-10-02): QS-E's consumed status updated from 'author-side candidate, nonauthor review requested' to its delivered nonauthor review 5961620598 (ACCEPT / PASS_TECHNICAL_SCOPED), as requested by the v1.2 delta review 5397276533 (item 8); three status phrases in LOCAL_PAIRING.md and README.md only; Lemma 5.1, all mathematics, checker, RESULTS, exploration and workflow unchanged | source re-pin (2026-10-03, integration lane): after the source-preserving merge of main 7858329 (head 8ec18cc), the cited-only [RES] record changed under Math-#183 (a4c3199) from blob 3cfd17c5 to 9668d06a; non-substantive for this packet (record v1.6 checker procedure only; Theorem L binding and declarative status unchanged), so its identity is re-pinned; no other source changed; LOCAL_PAIRING.md, README.md, checker, RESULTS, exploration and workflow unchanged"
+}