diff --git a/.github/workflows/local-pairing.yml b/.github/workflows/local-pairing.yml
new file mode 100644
index 000000000..9100dbfed
--- /dev/null
+++ b/.github/workflows/local-pairing.yml
@@ -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
diff --git a/frontiers/local_pairing_20260930/LOCAL_PAIRING.md b/frontiers/local_pairing_20260930/LOCAL_PAIRING.md
new file mode 100644
index 000000000..1f8c9455b
--- /dev/null
+++ b/frontiers/local_pairing_20260930/LOCAL_PAIRING.md
@@ -0,0 +1,562 @@
+# Local pairing: the local landscape decides the elder pairing, and the selection loss is bounded below by the local cluster mass
+
+Object: CL-LOCAL-PAIRING-20260930-v1.
+Author: Anthropic Claude (claude.ai session `session_01NMeKEismAyeqgdB4sy2NJU`), 30 September 2026.
+Disposition: AUTHOR-SIDE PROOF CANDIDATE; NONAUTHOR REVIEW REQUIRED. Lemmas 2.1–2.2, Lemma P, Lemma P′
+and Proposition 3.3 are deterministic statements about the pinned cubic with complete written proofs
+(reviewed twice by clean-context referee agents before landing; every blocking finding and amendment they
+raised is applied below). Theorem LP (the lower bound with an identified functional) is conditional: it consumes exact
+interfaces of [SC], [CUB] and [P] at their own review status. Conjecture LP= (equality) is stated with its
+single remaining probabilistic obligation. **Relation to Math-#170 and Math-#175 (v1.1, added after landing).**
+The OpenAI packet Math-#170 (`frontiers/local_elder_geometry_20260930/PROOF.md`, head `79f18f0`, blob
+`ef2aa579`, opened before this packet; integrated into main at `fa2e990`) proves the same deterministic statement as Lemma P′ + Proposition 3.3
+(its Theorem E, by the cubic critical-chord identity and a normalized-gradient collar rather than the explicit
+paths used here) and proves the **equality** `r⁻³(1 − p_r) → α₁ + α₂` at fixed planar marks (its Theorem S) by a
+route this packet does not use: [P]'s cap implication `F_r ⊂ G_r^c` and a retained-tail integration of [P] §7.
+Math-#175 (`frontiers/concave_fibre_elder_20260930/`, head `9c6a734`, proof blob `923d3236`; integrated into main at
+`eb659bd`) lifts it to fixed `d ≥ 3` by a contained
+hard-negative tube. So Conjecture LP= below is Math-#170 Theorem S (planar) / Math-#175 Theorem F (`d ≥ 3`),
+conditional on their interfaces and review; what this packet adds is an independent same-account derivation
+of the deterministic core with exact controls, and the cap-free lower bound Theorem LP. Scientific effect:
+NONE — no register, graph, STATUS, prize or Boolean changes; no numerical constant is claimed.
+**v1.2 (2 October 2026), answering the Slice C review.** 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, and it gave an
+exact falsifier. v1.2 withdraws that sketch as a proof. In its place §5 states Lemma 5.1, a common-radius
+acceptance lemma that controls the component through the exact saddle level. It is consumed from QS-E (main#229
+5961415030; nonauthor review ACCEPT / PASS_TECHNICAL_SCOPED, 5961620598), with exhausting jet-scaled margin sets
+`𝓜(η, R)`. §5 also records the review's notes on the weighted normalization and on Gaussian conditioning, and
+now says that the reverse reduction (O1′) ⇒ (O1) is conditional on Lemma 5.1. Lemmas 2.1–2.2, P, P′,
+Proposition 3.3, Theorem LP, the controls and the cap bypass are unchanged.
+
+## 0. What this packet answers, and what it does not
+
+The question put to the project on 2026-09-30 (input relayed by the owner) was: the account knows how many
+extra window critical points there are near a short-lifetime pair (Theorem Q: `Θ(r³)`), but not *which
+of them pair* and whether *the pairing is local*. Three sub-questions were asked. The answers, at the
+scope stated below:
+
+1. **Mixed pairs are already local, conditionally.** For any cutoff `δ_r → 0` with `δ_r/r → ∞`, the
+   ordered-pair mass with a point outside the `δ_r` ball is `o(r³)` ([TSL] Theorem L, from [SC] (24) and
+   the [C6] moments; nonauthor-reviewed at conditional scope). Remote–remote pairs are `O_ρ(r⁵)` ([SC] (2)).
+   The only non-local piece of the *nonempty* event is a **remote singleton** of positive mass `β_far`
+   ([SC] (22); [TSL] Theorem T). So "the rare event is local" is true for pairs and false for singletons.
+   Nothing in this packet re-proves those statements. (The leading-mass localization residual of Math-#160 §5
+   is bound to Theorem L as its direct statement by the declarative record [RES], Math-#173; that record proposes
+   register transitions and executes none.)
+2. **Which witnesses pair.** On the limiting local landscape — the pinned planar cubic `P_θ` of [CUB] —
+   the elder rule is decided exactly: off the null tie set `{s ≤ B, D² = T}`, `(M,S)` is the elder pair **iff**
+   the cubic has no extra strict-window critical point (`n(θ) = 0`) (Lemmas P, P′); if `n(θ) ≥ 1`, `M` dies at
+   the highest extra window saddle (Proposition 3.3).
+   Transferred to the field, this gives the lower bound
+   `liminf r⁻³ (1 − p_r) ≥ a₁ + a₂ = ν₁ + ν₂ − β_far` (Theorem LP), where `a₁, a₂` are the local coefficients
+   of [SC] (20) and `β_far` its remote coefficient. Equality — the statement that **remote witnesses never
+   pair** — is Conjecture LP= with one obligation (§5); it is proved by Math-#170 Theorem S (planar) and
+   Math-#175 Theorem F (`d ≥ 3`) by the cap route described in the header, at their conditional scope. This is
+   the bridge from the count law to the lifetime law: it bounds the existential `Θ(r³)` selection loss of [P]
+   Theorem A / [ELDER] / [E_d] from below by an explicit functional of the cluster measure, and identifies it
+   under LP=.
+3. **Cluster law.** Nothing new: [SC] (3) identifies `ν = ν₁δ₁ + ν₂δ₂`; this packet uses only its local part.
+
+What the bridge is worth for the lifetime density is stated without inflation in §6: it bounds from below
+(and, under LP=, fixes) the coefficient of the `Θ(ℓ^{2/3})` compact-window selection loss — relative order
+`ℓ` against the leading `ℓ^{−1/3}`; the leading law never needed it, and the unrestricted density's bounded
+remainder is a different object.
+
+## 1. Setting and exact sources
+
+Fixed `d ≥ 2`, torus side `L`, birth `b`, gap mark `k > 0`, one orthonormal frame. Pins
+`M = (−r/2, 0)`, `S = (r/2, 0)`, `f(M) = b`, `f(S) = b − k r³`, zero gradients; regression law `Q_r`;
+typed determinant weight `W_r`, FULL normalizer `Z_r`, `Q_r^W = (W_r/Z_r) Q_r`. `p_r` is the `Q_r^W`
+probability that the **global ordinary superlevel elder death partner** of `M` is `S` ([P] §1, [ELDER] §1).
+On the generic locus its death height is the maximin
+
+    d_f(M) = sup { min_t f(γ(t)) : γ(0) = M, f(γ(1)) > f(M) },                    (E1)
+
+with the empty-supremum convention of [P] §8; `S` is the death partner iff `d_f(M) = f(S)` and `S` is the
+critical point at that level ([ELDER] §1). A path from `M` to a point strictly above `b` whose values stay
+strictly above `f(S)` therefore proves that `S` is **not** the death partner. Nothing here uses gradient
+trajectories or separatrices.
+
+Sources (exact identities in `SOURCES.json`; none is revalidated here):
+
+| Tag | Source | What is consumed |
+|---|---|---|
+| [P] | `imports/lifetime_parent_20260925/UNIFORM_MATRIX_CAP_AND_LIFETIME.md` | §1 model and `p_r`; Theorem A upper bound `1 − p_r ≤ C r³`; §8 genericity; §§9–11 radial ledger and (E9)-type intensity. |
+| [ELDER] | `frontiers/window_multiplicity_laws_20260928/ELDER_LOWER_AND_DENSITY_GAP.md` | (E1) maximin characterisation; §2 the stable-`C⁰`-connectivity transfer template; §4 (E9)–(E11) density gap in `d = 2`. |
+| [CUB] | `frontiers/planar_cubic_cluster_20260929/PROOF.md` | (C1)–(C3) pinned cubic and shear; (C4) typed condition; Theorem C classifier `n(θ) ∈ {0,1,2}`; (C7)–(C13). Only the deterministic part. |
+| [SC] | `frontiers/spectral_cluster_closure_20260929/PROOF.md` (Math-#162, reviewed at head `382c0f9`, merged at `358f256` with identical bytes; proof blob `16c56821`) | §2 spectral coordinates and (10); §5 normal form: `[f(rX,rZ,0) − b]/r³ → P_θ` in `C²` on bounded sets, the limit `W_r/r⁴ → ∏h_j² w`, and the rescaled typed-jet measure (17); (18)–(20): `a₁, a₂`; (22): `β_far`; (2), (24) as cited in §0. |
+| [TSL] | `frontiers/two_scale_cluster_geometry_20260929/TWO_SCALE_LAW.md` (Math-#166, reviewed at head `7c82252`, merged at `ab13a08` with identical bytes; blob `a32fd5f7`) | Theorem L, Theorem T — cited in §0 only; not consumed by any proof below. |
+| [E_d], [EDL] | `frontiers/elder_lower_all_d_20260929/PROOF.md`, `frontiers/elder_dimension_lift_20260928/PROOF.md` | The existential lower bound `1 − p_r ≥ c r³` in every `d` that Theorem LP sharpens; not consumed. |
+| [C7] | `frontiers/c7_total_bounded_20260929/PROOF.md` (Theorem K / Corollary T), `frontiers/c7_zero_gap_limit_20260929/PROOF.md` (Theorem Z), `frontiers/unrestricted_selection_difference_20260929/PROOF.md` (Theorem U) | Cited in §6.3 only for the order of the unrestricted remainder; not consumed. |
+| [RES], [NUM] | `reviews/c6_residual_closure_20260930/RECONCILIATION.md` (Math-#173), `frontiers/c6_cluster_coefficients_numerics_20260930/NOTE.md` (Math-#168) | Cited in §0 and §6.4 only (the residual-closure record; uncertified numerical values of the near coefficients); not consumed. |
+
+A defect in a consumed interface blocks the corresponding statement here. In particular Theorem LP is
+conditional on [SC] at exactly the same scope as [TSL].
+
+## 2. The pinned cubic and the elder rule on it
+
+In the sheared coordinates `(u, Z)` of [CUB] (C2)–(C3), with `B = β − a²/(12k)` and
+`D = (c − aβ/(4k) + a³/(72k²))/2`,
+
+    P(u, Z) = A(u) + (Z²/2)(s + Bu) + (D/3) Z³,        A(u) = 2k (u − 1)(u + 1/2)².          (2.1)
+
+`M = (−1/2, 0)` with `P = 0`, `S = (1/2, 0)` with `P = −k`. The typed domain is `s < −|B|/2` ([CUB] (C4)).
+Elementary facts about (2.1) (the identities among them, and the shear (2.1) itself against [CUB] (C1)–(C3),
+are checked exactly by `pairing_check.py`, I0–I3; the monotonicity statements are one-line calculus):
+
+* **Axis.** `A′(u) = 6ku² − 3k/2`, so `A` is decreasing on `(−1/2, 1/2)` from `0` to `−k` and increasing for
+  `|u| > 1/2`; `A(u) + k = (k/2)(u + 1)(2u − 1)²`, so `A(u) > −k` iff `u > −1` and `u ≠ 1/2`; `A(u) > 0` iff
+  `u > 1`; `A(u) → +∞` as `u → +∞`. Because (2.1) has no term linear in `Z`, `∇P(u, 0) = (A′(u), 0)`: the
+  sheared axis `{Z = 0}` is an invariant line of `∇P`.
+* **Vertical lines.** At fixed `u`, `∂_Z P = Z (s + Bu + DZ)`. If `s + Bu < 0` and `D ≠ 0`, `P(u, ·)` decreases
+  from `A(u)` at `Z = 0` to a minimum at `Z₁(u) = −(s + Bu)/D` (on the side `sign Z = sign D`) and then
+  increases to `+∞`. If `s + Bu = 0` and `D ≠ 0`, `P(u, Z) = A(u) + DZ³/3` is monotone to `+∞` on the `D` side
+(level if `D = 0`).
+* **Extra critical points.** `(u₁, Z₁)`, `Z₁ ≠ 0`, is critical iff `A′(u₁) + (B/2)Z₁² = 0` (conic) and
+  `s + Bu₁ + DZ₁ = 0` (line); then `P(u₁, Z₁) = A(u₁) + Z₁²(s + Bu₁)/6` ([CUB] (C8) in this form), so
+  `P(S₁) < A(u₁)` iff `s + Bu₁ < 0`, and `D ≠ 0` unless `s + Bu₁ = 0`. [CUB] Theorem C: every strict-window
+  extra critical point is a nondegenerate saddle, there are at most two, and their number `n(θ)` is (C6). For
+  `B > 0` they satisfy `−1/2 < u₁ < u_w < 1/2`; for `B < 0`, `u_w < u₁ < −1/2` with `u_w = −1/2 − B/(2s)`;
+  for `B = 0`, `u₁ = −1/2`.
+* **`M` is a strict local maximum.** `Hess P(M) = diag(−6k, s − B/2)` and `s − B/2 < 0` on the typed domain.
+
+**Lemma 2.1 (growth at infinity; older components).** For every typed `θ`: (a) `|∇P(x)| → ∞` as `|x| → ∞`,
+and `P` has finitely many critical points; (b) for every `h ≥ −k`, every unbounded component of `{P > h}`
+contains points with `P > 0`.
+
+*Proof.* (a) Suppose `|∇P| ≤ C` along a sequence with `|x| → ∞`. If `|Z|` stayed bounded, `∂_u P = 6ku² − 3k/2
++ (B/2)Z²` would diverge; so `|Z| → ∞` and `|s + Bu + DZ| ≤ C/|Z| → 0`. If `B = 0` this forces `D = 0` and then
+`|s| ≤ C/|Z| → 0`, impossible. If `B ≠ 0`, `u = −(DZ + s)/B + o(1)` and
+`∂_u P = [6kD²/B² + B/2] Z² + (12kDs/B²) Z + O(1)`; the `Z²` coefficient vanishes only when `12kD² + B³ = 0`,
+which needs `B < 0` and `D ≠ 0`, and then the linear term `12kDs/B²` is nonzero because `s < 0`; either way
+`∂_u P` diverges. Critical points are the intersections of the conic `A′(u) + (B/2)Z² = 0` (irreducible for
+`B ≠ 0`; the two lines `u = ±1/2` for `B = 0`) with `Z = 0` or the line `s + Bu + DZ = 0`, finitely many.
+(b) Write `P(ρ cos φ, ρ sin φ) = c₃(φ)ρ³ + c₂(φ)ρ² + c₁(φ)ρ + c₀` with `c₃(φ) = 2k cos³φ + (B/2) cos φ sin²φ
++ (D/3) sin³φ`, `c₂(φ) = (s/2) sin²φ` (the only quadratic term of (2.1) is `sZ²/2`), `c₁(φ) = −(3k/2) cos φ`,
+`c₀ = −k/2`. At a zero `φ₀` of `c₃`, `sin φ₀ ≠ 0` (since `c₃(0), c₃(π) = ±2k`), so by continuity and compactness
+there are `η ∈ (0, 2k)` and `σ > 0` with `|sin φ| ≥ σ` on `{|c₃(φ)| ≤ η}`; put `μ = |s|σ²/2`, so that `|c₂| ≥ μ`
+there, and note `|c₂| ≤ |s|/2` everywhere. Choose `ρ₀` so large that for all `ρ′ ≥ ρ₀`:
+`ηρ′³ > |s|ρ′²/2 + (3k/2)ρ′ + 3k/2`, `(μ/4)ρ′² > (3k/2)ρ′ + 3k/2`, `(μ/4)ρ′ > 3k/2`, and `3ηρ′² > |s|ρ′ + 3k/2`.
+For `ρ ≥ ρ₀` and any `φ` one of four cases holds:
+(1) `c₃ ≤ −η`: `P ≤ −ηρ³ + |c₂|ρ² + |c₁|ρ + |c₀| < −k`;
+(2) `|c₃| < η` and `c₃ρ ≤ (3/4)|c₂|`: `P ≤ (3/4)|c₂|ρ² − |c₂|ρ² + (3k/2)ρ + k/2 ≤ −(μ/4)ρ² + (3k/2)ρ + k/2 < −k`;
+(3) `0 < c₃ < η` and `c₃ρ > (3/4)|c₂|`: for every `ρ′ ≥ ρ`, `∂_ρ P(ρ′) = 3c₃ρ′² + 2c₂ρ′ + c₁ ≥ 3c₃ρρ′ − 2|c₂|ρ′ −
+3k/2 > (1/4)|c₂|ρ′ − 3k/2 ≥ (μ/4)ρ′ − 3k/2 > 0`;
+(4) `c₃ ≥ η`: for every `ρ′ ≥ ρ`, `∂_ρ P(ρ′) ≥ 3ηρ′² − |s|ρ′ − 3k/2 > 0`.
+So every point with `|x| ≥ ρ₀` and `P(x) > −k` is of type (3) or (4), and along its ray `P` increases strictly to
+`+∞`: the point is connected outward, inside `{P > P(x)}`, to points with `P > 0`. An unbounded component of
+`{P > h}`, `h ≥ −k`, contains points with `|x| ≥ ρ₀`, hence points with `P > 0`. ∎
+
+**Lemma 2.2 (the death level is a critical value).** For every typed `θ`, `h* := d_P(M)` satisfies
+`−k ≤ h* < 0`, and if `h* > −k` then `h*` is the value of an extra strict-window critical point of `P`, so
+`n(θ) ≥ 1`.
+
+*Proof.* `h* ≥ −k` by the axis path `M → S → {u > 1}` (values `≥ −k`, endpoint `A(u) > 0` for `u > 1`);
+`h* < 0` because `M` is a strict local maximum, so for `h` close to `0` the `M`-component of `{P > h}` is a
+small disk containing no point with `P > 0`. Suppose `h* > −k` is not a critical value. By Lemma 2.1(a) the
+critical values are finitely many, so there is `ε > 0` with no critical value in `[h* − ε, h* + ε]`, and
+`|∇P| ≥ c > 0` on `P^{−1}[h* − ε, h* + ε]` (a Palais–Smale sequence there would be bounded by Lemma 2.1(a) and
+converge to a critical point at a level in the interval). Let `φ_t` be the flow of `∇P/|∇P|²` on the slab
+`P^{−1}[h* − ε, h* + ε]`, along which `P` increases at unit rate, and define `η(x) = φ_{h* + ε − P(x)}(x)` for
+`x` in the slab and `η(x) = x` for `P(x) ≥ h* + ε`; `η` is continuous, maps `{P ≥ h* − ε}` into `{P ≥ h* + ε}`,
+and fixes `{P ≥ h* + ε}`. Shrink `ε` so that `ε < |h*|`. By definition of `h*` there is a path from `M` to a
+point `y` with `P(y) > 0` with values `> h* − ε`; `η` of it is a path with values `≥ h* + ε > h*` from `M`
+(fixed, since `P(M) = 0 > h* + ε`) to `y` (fixed), contradicting the supremum. So `h*` is a critical value in
+`(−k, 0)`. A critical point at a level in `(−k, 0)` is not a pin and is counted by `n` ([CUB] §2: `n` counts
+all additional critical points with height strictly between `−k` and `0`), so `n ≥ 1`; by Theorem C it is a
+nondegenerate saddle. ∎
+
+## 3. Lemma P: every extra window saddle rejects the pair
+
+**Lemma P.** Let `θ = (s, a, β, c)` be typed with `n(θ) ≥ 1`, and let `(u₁, Z₁)` be an extra strict-window
+critical point. Then there is an explicit path `γ` from `M` to a point with `P > 0` whose minimum exceeds
+`−k`; consequently `d_P(M) > −k`, and `(M,S)` is **not** the elder pair of `P`. The path is one of:
+
+* **V (vertical descent at `u₁`)** — when `s + Bu₁ < 0` and `−1 < u₁ < 1/2`: along the axis from `M` to
+  `(u₁, 0)`, then along the vertical line through `S₁` past `S₁` to `+∞` on the `D` side. Its minimum is
+  `P(S₁) ∈ (−k, 0)`, attained at `S₁`.
+* **U (vertical rise at `u₀`)** — when `B < 0` and `s > B` (equivalently `|B|/2 < |s| < |B|`): along the
+  axis from `M` to `(u₀, 0)`, `u₀ = −s/B ∈ (−1, −1/2)`, then along the vertical line at `u₀`, on which
+  `P = A(u₀) + DZ³/3`, to `+∞` on the `D` side. Its minimum is `A(u₀) ∈ (−k, 0)`.
+
+Let `S_V` be the extra strict-window root with the largest `u`-coordinate. The proof shows that path V
+applies at `S_V` whenever `D ≠ 0` (in case (iv) `S_V` is the root on the side `u > u₀` of `u₀ = −s/B`, which
+exists both for `n = 1` and for `n = 2`); path U is a second valid path in case (iv), and the `D = 0` subcase
+of (iv) has its own path. The margin of Lemma P is `m(θ) := P(S_V) + k` for `D ≠ 0` and `A(u₀) + k` for
+`D = 0` in case (iv); it is positive, and it is continuous on the typed `n ≥ 1` domain: `S_V` is a
+nondegenerate root ([CUB] Theorem C), so it depends continuously on `θ` by the implicit function theorem; it
+persists across `s = B` and across `D² = T` (the root that appears or disappears there is the other one, at
+`u_w`); and as `D → 0` in case (iv) its value tends to `A(u₀)`. Theorem LP uses this continuity to make its
+jet sets compact.
+
+*Proof.* Axis parts: by the monotonicity of `A` in §2, the values along the axis from `M` to `(u*, 0)`
+lie in `[A(u*), 0]`, and `A(u*) > −k` because `−1 < u* < 1/2` in both cases. Vertical parts: by the
+vertical-line fact in §2. It remains to see that every typed jet with `n ≥ 1` falls in V or U.
+
+(i) `B > 0`. Then `−1/2 < u₁ < 1/2` and `s + Bu₁ < −B/2 + Bu₁ = B(u₁ − 1/2) < 0` by the typed condition: V.
+
+(ii) `B = 0`. Then `u₁ = −1/2` and `s + Bu₁ = s < 0`: V (the axis part is empty; the vertical line starts at `M`).
+
+(iii) `B < 0`, `s ≤ B` (`|s| ≥ |B|`). Then `n = 1` by (C6) and `u_w = −1/2 − |B|/(2|s|) ≥ −1`, so
+`−1 < u₁ < −1/2`. The linear function `s + Bu` vanishes at `u₀ = −s/B = −|s|/|B| ≤ −1 < u₁` and is
+decreasing in `u` (`B < 0`), hence `s + Bu₁ < 0`: V.
+
+(iv) `B < 0`, `s > B` (`|B|/2 < |s| < |B|`). Then `u₀ = −|s|/|B| ∈ (−1, −1/2)` and `A(u₀) > −k`. If `D ≠ 0`:
+in the proof of [CUB] Theorem C the extra roots are the solutions of `g(u) = D` on the `Z > 0` branch and of
+`g(u) = −D` on the `Z < 0` branch of the window conic, where `g` is strictly increasing on `(u_w, −1/2)`
+([CUB] (C12)) and vanishes at `u₀` (there the line `s + Bu + DZ = 0` reduces to `DZ = 0`). Hence for `n = 2`
+the two roots lie on opposite sides of `u₀`, and for `n = 1` the single root solves `g(u) = |D| > 0 = g(u₀)`;
+in both cases `S_V` has `u₁ > u₀`, so `s + Bu₁ < 0` (`s + Bu` decreases in `u` for `B < 0`) and
+`−1 < u₀ < u₁ < −1/2`: V at `S_V`. Path U — along the axis to `(u₀, 0)`, then the vertical line
+`P = A(u₀) + DZ³/3` to `+∞` on the `D` side, minimum `A(u₀)` — is a second valid path. If `D = 0`,
+`P(u, Z) = A(u) + (Z²/2) B (u − u₀)` and the vertical line at `u₀` is level; take instead
+`M → (u₀, 0) → (u₀, Z) → (u₀ − ε, Z)` with `Z` large: on the last segment `B(u − u₀) ≥ 0`, so
+`P ≥ A(u) ≥ A(u₀ − ε) > −k` for small `ε`, and its endpoint value `A(u₀ − ε) + (Z²/2)|B|ε` exceeds `0` for
+`Z` large. ([ELDER]'s witness `(−3k/2, 0, −2k, 0)` is this subcase with `u₀ = −3/4`; its polygonal path (E5)
+has minimum `−7k/32`, and `pairing_check.py` P2 verifies that both extra saddles sit exactly at `−7k/32`.)
+
+In V the minimum along the whole path is `P(S₁)`, which lies in `(−k, 0)` by the strict-window hypothesis;
+in U it is `A(u₀) > −k`; in the `D = 0` path it is `A(u₀ − ε) > −k`. In all of them the far end has `P > 0`. ∎
+
+`pairing_check.py` P1 verifies, exactly, on `6840` rational chart points (three values of `k`), that every
+chart point is a typed strict-window critical point with `n ≥ 1`, and the inequalities of the path through
+the chart root: V when `s + Bu₁ < 0` and `−1 < u₁ < 1/2` (`4680` points: `A(u₁) > P(S₁) > −k`,
+`sign Z₁ = sign D`, the rise past `0`), otherwise U (`2160` points, all in case (iv) with the chart root on the
+side `u₁ < u₀`: `u₀ ∈ (−1, −1/2)`, `A(u₀) > −k`, the rise past `0`). For those `2160` points it also computes
+the other root exactly (Vieta on [CUB] (C13)), and verifies that it is a strict-window root on the side
+`u > u₀`, that V applies to it, and that it is the higher of the two — so the proof's assignment (V at `S_V`)
+and the ordering claim of Proposition 3.3 are exercised on every chart point. Smallest margin `1/1200`
+(absolute; attained at `k = 1/3`).
+
+**Lemma P′ (converse).** If `θ` is typed, `n(θ) = 0`, and `S` is the only critical point of `P` at level
+`−k` (this excludes the null set `{s ≤ B, D² = T}` of [CUB] (C6)), then `(M,S)` is the elder pair of `P`:
+`d_P(M) = −k` and the merge at level `−k` is through `S`.
+
+*Proof.* By Lemma 2.2 and `n = 0`, `d_P(M) = −k`. Let `C` be the component of `M` in `{P > −k}`. `C` contains
+no point with `P > 0` (such a point would give `d_P(M) > −k`), so by Lemma 2.1(b) `C` is bounded. The sheared
+axis is `∇P`-invariant with `∇P(u, 0) = (A′(u), 0)`: the segment `(−1/2, 1/2)` is the ascending branch of `S`
+into `M` (`A` increases toward `M` on it, and its values lie in `(−k, 0)`, so it lies in `C`), and the ray
+`u > 1/2` is the other ascending branch of `S`, on which `A` increases to `+∞` and exceeds `0` for `u > 1`.
+So one upper sector of `S` lies in `C` and the other in a component `C′` of `{P > −k}` containing points with
+`P > 0`; `C′ ≠ C`. At level `−k` the component of `M` in `{P ≥ −k}` therefore contains `C′` through `S`, and
+through no other point, since `S` is the only critical point at that level. `M`, the younger, dies at `S`. ∎
+
+**Proposition 3.3 (the death saddle).** If `θ` is typed with `n(θ) ≥ 1`, then `d_P(M) = max{P(S′) : S′ an
+extra strict-window saddle}`. Off the additional null set `{D = 0}` the maximum is attained at exactly one
+extra window saddle, and `M` dies there: the highest extra window saddle. (For `D = 0` in case (iv) the two
+extra saddles `(u₀, ±Z₀)` have the same height `A(u₀)` — the [ELDER] witness is such a jet — so only the death
+level, not a unique death saddle, is determined; on the Morse distinct-value locus of the field this tie
+does not occur.)
+
+*Proof.* By Lemma P, `d_P(M) > −k`; by Lemma 2.2, `d_P(M)` is the value of an extra strict-window saddle.
+If `n = 1` there is nothing more to show. `n = 2` occurs only in case (iv) of Lemma P (`B < 0`, `s > B`,
+[CUB] (C6)). There, in [CUB]'s notation, the two saddles solve `g(u) = D` on the `Z > 0` branch and
+`g(u) = −D` on the `Z < 0` branch of the window conic, with `g` strictly increasing on `(u_w, −1/2)` and
+`g(u₀) = 0` at `u₀ = −s/B` (at `u = u₀` the line reduces to `DZ = 0`). So for `D ≠ 0` the two roots lie on
+opposite sides of `u₀` (case (iv) of Lemma P); the one with `u₁ > u₀` has `s + Bu₁ < 0` and `−1 < u₀ < u₁`,
+hence Lemma P's path V passes through it and `d_P(M) ≥ P(S₁)`. Along the window branch the
+height of the root is [CUB] (C9), `P/k = −(u + 1/2)[1 + (2s/B)(u − 1/2)]`, whose derivative is
+`−k(B + 4su)/B`; for `B < 0` this is positive by [CUB]'s `B + 4su > 0` on (C11). So the root with the larger
+`u` is the higher saddle, and `d_P(M)` equals its value; for `D = 0` both saddles have the height `A(u₀)`, which
+is then the death level. ∎
+
+Together: **on the limiting local landscape, off the null tie set `{s ≤ B, D² = T}`, `(M,S)` is the elder
+pair iff `n(θ) = 0`; otherwise `d_P(M)` is the highest extra window critical value, and off `{D = 0}` `M` dies
+at the unique highest extra window saddle.** The exploration in `exploration/` (standard library only; not a
+proof, not run in CI) computes the death level of `M` by the discretised elder rule on a grid — cells merged in
+decreasing order of `P_θ`, death when `M`'s component first meets a point with `P_θ > 0` — for sampled jets,
+and compares it with the closed-form critical points: see `exploration/RESULTS_EXPLORATION.json` (seed 7,
+spacing `0.02`, `k ∈ {1, 1/2}`): for all 120 sampled jets with `n ≥ 1` (80 with `n = 1`, 40 with `n = 2`) the
+grid death level agrees with the highest window critical value to within `1.5·10⁻³` and the merge cell lies at
+that saddle; for all 40 sampled typed jets with `n = 0` the death level is `−k` to within `10⁻⁴`, with the merge
+cell at `S` in 39 cases and, in one case with `s + B/2 ≈ −0.024` (a jet close to the typed boundary, where the
+saddle at `S` is nearly degenerate in `Z`), displaced `0.18` along the flat valley of `S` at the correct level;
+the [ELDER] witness dies at `−7/32`, a tie of its two saddles.
+
+## 4. Theorem LP: the identified lower bound for the selection loss
+
+Let `a₁, a₂` be the local coefficients of [SC] (20): `a_j = ∫ 1{n(θ) = j} dM(θ)`, with `M` the rescaled
+typed-jet measure (17) (finite on `{n ≥ 1}` by (19)), so that `ν₁ = a₁ + β_far`, `ν₂ = a₂`.
+
+**Lemma 4.1 (pathwise normal form on the soft plane).** In the frame (axis, soft eigenvector `e_soft(O)`,
+hard eigenvectors) of [SC] (7), with `s` the soft eigenvalue of the midpoint transverse Hessian divided by
+`r`, `a = τ_xxz`, `β = τ_xzz`, `c = τ_zzz`, and `K ≥ 1` a bound for `‖f‖_{C⁴}`: for every `R ≥ 1` and
+`|(X, Z)| ≤ R`,
+
+    f(rX, rZ, 0) − b = r³ [ P_θ(X, Z) + ρ_r(X, Z) ],     sup_{B(R)} |ρ_r| ≤ C_R K r,                 (4.0)
+
+with `C_R` depending only on `R` (and `d`); the same holds for the first two derivatives of `ρ_r` in `(X, Z)`.
+
+*Proof.* Write `g(x, z) = f(x, z, 0)` and expand `g` at the midpoint to third order with the Lagrange
+remainder `|R₄| ≤ C K (|x| + |z|)⁴`. The six pin data — values `b`, `b − kr³` and vanishing `x`- and
+`z`-derivatives at `(±r/2, 0)` — determine the coefficients: the axial cubic `g(x, 0)` is the Hermite
+interpolant of `(b, 0)` and `(b − kr³, 0)` at `±r/2` up to `O(K r⁴)`, which in the scaled variable is
+`r³(2kX³ − 3kX/2 − k/2)`; the two conditions `∂_z g(±r/2, 0) = 0` give `g_z(0) = −g_xxz r²/8 + O(K r³)` and
+`g_xz(0) = O(K r²)`; and `g_zz(0) = rs`, `g_xxz = a`, `g_xzz = β`, `g_zzz = c` by the definition of the spectral
+coordinates. Collecting the `z`-dependent terms in `x = rX`, `z = rZ`:
+`g_z z + g_xz xz + g_zz z²/2 + g_xxz x²z/2 + g_xzz xz²/2 + g_zzz z³/6 = r³[(a/2)(X² − 1/4)Z + (s/2)Z² + (β/2)XZ²
++ (c/6)Z³] + O(K r⁴ R⁴)`, which with the axial cubic is `r³ P_θ(X, Z)` up to `O(K r⁴ R⁴)`. Differentiating
+the same expansion gives the `C²` statement. ∎
+
+This is the quantitative form of [SC] §5's `C²` convergence `[f(rX, rZ, 0) − b]/r³ → P_θ`; in `d = 2` it is
+[ELDER] (E7) / [CUB] (G8). It uses no hard implicit function: the path of Lemma P lies in the soft plane.
+(4.0) is stated in the unsheared coordinates `(X, Z)` of the frame; the vertical legs of the paths of Lemma P
+are the lines `X = u₁ − aZ/(12k)` there, and "lies in `B(R)`" below is read in `(X, Z)`.
+
+**Lemma 4.2 (compact typed jets).** Let `K ⊂ {typed jets}` be compact in the spectral coordinates
+`(s, h, O, τ)` with `h₂ ≥ δ > 0`. Then `r⁻³ Q_r^W(θ_r ∈ K) → M(K)` as `r → 0`. The same proof gives the
+statement for bounded Borel `K` on which `M` charges no boundary.
+
+*Proof.* In the spectral coordinates the rescaled law `r⁻³ Q_r^W(θ_r ∈ dθ)` has density
+`(c_m r²/Z_r) h_r(A, T) J_r 1{h₂ > −rs} E[W_r/r⁴ | jet]` ([SC] (7)–(9) and the disintegration below (16); the
+finite-`r` type indicator is part of `W_r`). On `K`: `h_r → h₀` pointwise ([SC] §2), `J_r → J_0 = ∏h_j ∏(h_j −
+h_i)`, `1{h₂ > −rs} → 1`, `Z_r/r² → z₀` ([SC] §1), and `E[W_r/r⁴ | jet] → ∏h_j² w(s, a, β)` ([SC] §5: the soft
+`2×2` blocks of the endpoint Hessians divided by `r` converge to `M_M, M_S`, whose determinants multiply to
+`9k²(4s² − B²)` on the typed domain, the hard block to `−diag(h)`, and the mixed soft–hard entries are `O(r)`,
+so the determinants and the type indicator converge off the null typed boundary). Domination on `K` by [SC]
+(10), (15) and (6) with the bounded prefactor `c_m r²/Z_r`; the exceptional branch of (14) plays no role
+because `s` is bounded on `K`. Dominated convergence gives the claim. ∎
+
+In `d = 2` this is [CUB] Theorem G summed over the count, which is not consumed here. [SC] displays only the
+nonempty-count version (18) and stresses that there is no finite uncentered `n = 0` formula; Lemma 4.2 is
+about compact jet sets, on which the total mass is finite whatever the count.
+
+**Theorem LP.** Conditional on [SC] §§1, 2, 4 (15), 5 and (17)–(20) at their stated fixed-`d, L, b, k`, frame
+scope, on [P] §4 (4.1), §5 (5.5), §8 and the maximin characterisation (E1), and on [CUB] Theorem C,
+
+    liminf_{r→0} r⁻³ (1 − p_r) ≥ a₁ + a₂ = ν₁ + ν₂ − β_far.                                (4.1)
+
+Combined with [P] Theorem A, `1 − p_r = Θ(r³)` with `a₁ + a₂ ≤ liminf r⁻³(1 − p_r) ≤ limsup r⁻³(1 − p_r) ≤ C`.
+
+*Proof.* Fix `δ > 0` and `R ≥ 4`. Let `K_{δ,R}` be the set of jets `θ` with `s ≤ −|B|/2 − δ` (typed with
+margin), `n(θ) ≥ 1`, `|D| ≥ δ`, `h₂ ≥ δ`, `|s| + |h| + |τ| ≤ 1/δ`, whose path V at `S_V` has margin `m(θ) ≥ δ`
+and, truncated at the first point where `P ≥ δ`, lies in `B(R)` in the `(X, Z)` coordinates. It is compact in
+the spectral coordinates: it is bounded, and it is closed because along a convergent sequence in it the
+roots `S_V` converge (Lemma P: `m` is continuous) to a point with `P ≥ −k + δ`, `Z ≠ 0` (`|D| ≥ δ` and
+`s + Bu < 0`) and `u ∈ (−1, 1/2)`, which is again an extra strict-window root of the limit jet, and the
+other conditions are closed. It is invariant under the reflection `e_soft → −e_soft` (`n` and `m` are). As
+`δ → 0` and `R → ∞`, `K_{δ,R}` increases to `{n ≥ 1}` minus an `M`-null set (`M` is absolutely continuous in
+`(s, h, τ)`; the excluded sets are `{D = 0}`, the typed boundary, and the set where the margin vanishes,
+which lies on the window boundary), so `M(K_{δ,R}) → a₁ + a₂` by monotone convergence and (19)–(20).
+
+Transfer. On the event `E_r := {θ_r ∈ K_{δ,R}} ∩ {C_R K r < δ/2}`, Lemma 4.1 shows that the truncated path
+`γ_θ`, scaled by `r` and placed in the soft plane of the actual frame, is a path in the torus from `M` along
+which `f > b − k r³ + (δ/2) r³ > f(S)` and whose endpoint has `f > b + (δ/2) r³ > b`. By (E1),
+`d_f(M) > f(S)`, so `S` is not the elder death partner: `E_r ⊂ {rejected}`.
+
+Measure. By Lemma 4.2, `r⁻³ Q_r^W(θ_r ∈ K_{δ,R}) → M(K_{δ,R})`. For the complementary event, [SC] §1 (from
+[P] (5.3)) gives `W_r/r² ≤ C K^{2d}` and [P] §5 gives `Z_r ≥ z_* r²`, so `E_{Q_r^W}[K^p] ≤ (C/z_*) E_{Q_r}[K^{2d+p}]`
+is bounded uniformly in `r` for every `p` (Gaussian moments of the `C⁴` norm under the regression law, [P]
+§4); Markov then gives `Q_r^W(C_R K r ≥ δ/2) ≤ E_{Q_r^W}[K^p] (2 C_R r/δ)^p = O_p(r^p)`. Hence
+
+    liminf r⁻³ (1 − p_r) ≥ liminf r⁻³ Q_r^W(E_r) = M(K_{δ,R}),
+
+and `δ → 0`, `R → ∞` give (4.1). ∎
+
+Remarks. (a) The proof uses no window critical point of the field, no gradient trajectory, no near
+count: only field values along an explicit path in the soft plane. It is the [ELDER] §2 mechanism run over
+the whole `n ≥ 1` domain instead of one witness, with [SC]'s typed-jet measure replacing the one-point
+positivity argument of [ELDER] §3. (b) In `d = 2` the soft plane is the whole space. (c) `a₁ + a₂ > 0` by [SC]
+(20), so Theorem LP re-derives, conditionally on [SC], the existential lower bounds of [ELDER] / [EDL] / [E_d]
+at fixed parameters, with a constant. It does not re-derive their uniformity on compact mark sets, which they
+have and this statement does not claim.
+
+## 5. Conjecture LP= and its one obligation
+
+**Conjecture LP=.** `1 − p_r = (a₁ + a₂) r³ + o(r³)`. Equivalently: remote and mixed witnesses never change
+the elder partner at leading order, and the entire selection loss is the local nonempty cluster mass.
+
+By Theorem LP it suffices to show `limsup r⁻³ (1 − p_r) ≤ a₁ + a₂`. Write, for fixed `R`,
+
+    Q_r^W(rejected) ≤ Q_r^W(N_R ≥ 1) + Q_r^W(rejected, N_R = 0),
+
+where `N_R` counts strict-window critical points of `f` in `B(Rr)` other than the pins. The first term is
+`(a₁^R + a₂^R) r³ + o(r³)` by [SC] (18) and `a₁^R + a₂^R → a₁ + a₂` as `R → ∞` by (20). The obligation is
+
+    (O1)   limsup_{r→0} r⁻³ Q_r^W(rejected, N_R = 0) → 0   as R → ∞.
+
+What is known toward (O1), and what is not (amended in v1.2). The v1.1 sketch here confined the
+`M`-component of the rescaled field at the single level `−k + δ/2`. It did not control the levels between
+`f(S)` and `f(S) + (δ/2)r³`, where a remote saddle can still kill `M` on `{N_R = 0}`. Codex's Slice C review
+(C85, 5396856086, finding C-01) gives an exact falsifier of that finite-radius implication.
+- *The construction.* Take `k = δ = 1` and `P = A(X + HZ) − σZ²/2`, with `H = 10⁹`, `σ = 10¹⁴` and `R = 125`.
+- *Why the v1.1 margins hold.* The cubic is typed with `n = 0`, its component at `−3/4` lies in `B(124)`, and
+  `N_R = 0`. A bump placed outside `B(125)` leaves the actual `C²` error on `B(R)` equal to zero.
+- *Why acceptance fails.* The sheared vertical path from `M` stays above `−7/8` and leaves `B(125)`. A bump
+  there, outside `B(125)`, makes the pair rejected.
+
+The v1.1 sketch is therefore withdrawn as a proof. What (O1) needs is a *common-radius acceptance lemma*:
+one that controls the component through the exact saddle level and slightly below it. For the planar cubic,
+QS-E supplies it.
+
+Notation for the lemma.
+- *Coordinates.* Work in [CUB]'s sheared coordinates `(u, Z)`, normalized by `k`, so that `P_θ/k` has the
+  pins `0` and `−1`. These are the coordinates of main#229 5961415030, where `u = X + aZ/(12k)`.
+- *The saddle margin.* `μ(θ) := 1 + max{P_θ(Y)/k : Y an extra nondegenerate saddle}`, and `μ = −∞` if there is
+  none. Every extra critical point lies below `0`, so `n(θ) = 0` iff `μ ≤ 0`. Off the tie set `Σ`, `n = 0`
+  forces `μ < 0`.
+- *The QS sets* (QS §§1–4):
+  - the tolerances `m_S` and `ε_M`;
+  - the ellipses `E_S` and `E_M`;
+  - the cut trap `T_ℓ`: the `M`-component of `{P > ℓ}` cut along the stable curve of `S`;
+  - the axis segment `A = [M, (3/2, 0)]`.
+
+**Lemma 5.1 (common-radius acceptance; QS-E).** Let `θ` be typed, off `Δ ∪ Σ`, with `n(θ) = 0`. Let
+`0 < η < min(|μ|, m_S, ε_M)`, and let `V_η(θ) := T̄_{−1−η} ∪ A ∪ E_S ∪ E_M`, taken in raw coordinates.
+Suppose `V_η(θ) ⊂ B(R)`. Let `F` be a planar function with the exact pins of `P_θ/k` such that:
+- `|F − P_θ/k| < η` on `T̄_{−1−η} ∪ A`;
+- QS's rescaled `C²` bounds (E2) and (E3) hold on `E_S` and `E_M`.
+
+Then `d_F(M) = F(S)` (elder), whatever `F` is outside `B(R)`.
+
+Both of QS-E's bounds use only paths inside `V_η ⊂ B(R)`:
+- the exit from the trap, for the upper bound;
+- the axis `A`, for the lower bound.
+
+So the lemma applies to the torus field through the chart, exactly as in #170 §5.
+
+This is Theorem QS-E of main#229 5961415030, whose nonauthor review 5961620598 returned ACCEPT / PASS_TECHNICAL_SCOPED;
+this packet consumes it at that reviewed scope. QS Remark 4 shows why the axis beyond `S` must be in `V_η`. The
+lemma needs no `N_R = 0`. In the C-01 cubic, along the sheared vertical path, the level-`(−1 − η)` component
+reaches `|X| ≈ H(2/σ)^{1/2} ≈ 142`, beyond `B(125)`. So the containment hypothesis fails there, as it must.
+The exact C-01 cubic, with `B = D = 0`, also lies on `Δ`.
+
+The margin sets. Put
+
+    𝓜(η, R) := {θ typed : n(θ) = 0, θ ∉ Δ ∪ Σ, η < min(|μ|, m_S, ε_M), V_η(θ) ⊂ B(R − 1)}.
+
+These sets are jet-scaled, and they increase to `{n = 0} \ (Δ ∪ Σ)` as `η ↓ 0` and `R ↑ ∞`: QS Lemma 4
+makes the trap bounded at each such `θ`, although its size is not explicit. Two bounds hold on
+`𝓜(η, R)`:
+- `E_S ⊂ B(R − 1)` gives `κ_S ≥ r̃²/(R − 1)²`, and `η < m_S = (2/5)r̃²` then gives
+  `1/κ_S < 2(R − 1)²/(5η)`.
+- In the same way, `1/κ_M < (R − 1)²/(2η)`.
+
+So the raw-coordinate thresholds of (E2) and (E3) are at least `c(η, R)/(1 + |a|/(12k))²`.
+
+The decomposition in `d = 2`. Lemma 4.1's planar `C²` error on the fixed ball `B(R)` is `O(C_R K r)`, so
+
+    {rejected, N_R = 0} ⊂ {rejected, n(θ_r) = 0, θ_r ∉ 𝓜(η, R)} ∪ {θ_r ∈ 𝓜(η, R), error on B(R) above θ_r's thresholds}
+                          ∪ {n(θ_r) ≥ 1, N_R = 0} ∪ {θ_r ∈ Δ ∪ Σ},
+
+where untyped `θ_r` are placed in the first set.
+- *The second set* is `O(r^p)`, as in Theorem LP, by the Gaussian moments of `a` and of `K`.
+- *The third set* has `r⁻³`-limit `M(n ≥ 1, n_R = 0)` at fixed `R`, since its jets have all their extra roots
+  outside `B(R)`. This follows from three facts:
+  - Lemma 4.2, on compact subsets;
+  - [SC] §5's joint spectral-jet/count convergence in total variation;
+  - the tightness of `M` on `{n ≥ 1}` ([SC] (19): `|s| ≤ 2|B| + (48kD²)^{1/3}` there, with (10)/(15)/(6)
+    domination).
+
+  This limit tends to `0` as `R → ∞`.
+- *The last set* is null.
+
+Three conditions apply to these statements.
+- Do not condition the Gaussian regression on `{N_R = 0}`; that would destroy its Gaussian structure. Keep
+  every intersection indicator under the original law `Q_r^W`, with positive majorants.
+- In `d ≥ 3` the same conclusion needs the hard-direction slaving of the superlevel components:
+  - fiberwise concavity in the hard coordinates `w`;
+  - `f_ww ≤ −h₂/2` on the cylinder;
+  - the correspondence between the components of `{f > h}` and those of the reduced planar potential
+    `max_w f`.
+
+  [SC] §5 carries this out for the critical points; this note does not carry it out for the superlevel sets.
+- (O1) ⇒ (O1′) holds already. At fixed regular `R`, [SC] §5's nonempty joint convergence gives
+  `r⁻³Q_r^W(n(θ_r) = 0, N_R ≥ 1) → 0`. The converse direction is the decomposition above. So, granting
+  Lemma 5.1 (and, in `d ≥ 3`, the slaving step), (O1) is equivalent to
+
+    (O1′)   limsup_{r→0} r⁻³ Q_r^W(rejected, n(θ_r) = 0, θ_r ∉ 𝓜(η, R)) → 0   as η ↓ 0 and R ↑ ∞.
+
+At fixed planar marks, this equality is supplied on main by Math-#170 Theorem S (OpenAI; reviewed at head `79f18f0`,
+proof blob `ef2aa57959ea9f721bbf2316ce94cf616c1c9113`), whose route avoids the margin set altogether: the failure event
+is contained in the bad cap of [P] §8 (`F_r ⊂ G_r^c`), the bad cap's mass restricted to a large residual norm
+is `o(r³)` by keeping the indicator inside [P]'s (7.5) scalar integral, and the remaining compact-jet part
+converges by dominated convergence with the pointwise limit of the failure indicator (its Theorem E). The
+`d ≥ 3` slaving step named above is Math-#175's H1. Read on its own, the margin set must be jet-scaled (a fixed `δ` fails: near
+`M`, `P(M + Z e_Z) = (Z²/2)(s − B/2) + (D/3)Z³` exceeds any fixed `−δ` for `|Z|` slightly beyond `δ` once
+`|s|` is large, so a fixed-margin set would exclude a region of infinite `M`-mass and (O1′) would restate
+(O1)); the typed-jet measure is not finite on `{n = 0}` ([SC] after (17)), so its complement cannot be
+handled by Lemma 4.2; and the events in the complement — a second soft eigenvalue, a large jet, a jet near
+the typed boundary, a landscape whose `M`-component reaches far — are the ones [P] §§6–7 and [SC] §4 control
+in unweighted or occurrence-weighted form, not in rejection-weighted form. C85's review records the
+normalization:
+- [P] §§6–7 keep the double soft factor `W_r ≲ r²λ₁(λ₁ + CrU)`. So the soft interval gives an `O(r⁵)` weighted
+  numerator, and one division by `Z_r ≥ z_*r²` gives `O(r³)`.
+- Exhausting the margins needs more than that displayed bound: a written rejection-tail deduction that keeps
+  the residual and hard-spectrum tail indicators inside the same integral.
+- [SC] §4's domination covers only the nonempty near-occurrence measure.
+
+What is needed is either a direct
+argument that a long excursion of `M`'s window component without a window saddle inside `B(Rr)` has
+weighted probability `o(r³)`, or the boundary-layer estimates of [P] §7 restated at `o(r³)` on the no-witness
+event. That, together with the `d ≥ 3` slaving step, is what separates Theorem LP from LP=.
+
+Falsifiers. LP= is false if there is a configuration class of weighted probability `≍ r³` in which
+`(M,S)` is rejected although no strict-window critical point lies within `O(r)` of the pair — for example
+a mechanism where the far sector of `S` reconnects to `M`'s component (self-attachment) through a region
+at distance `≫ r`. Lemma P′ and Lemma 5.1 show that no such class exists on the accurate planar cubic model with margins.
+The margins must control the component down to the exact saddle level; C-01 shows that confinement at
+one level above `f(S)` is not enough. A counterexample would have to live where the model is inaccurate
+or unmargined, or where the hard-direction reduction fails.
+
+## 6. Consequences for the lifetime law, stated without inflation
+
+1. **Leading law.** Theorem 2.1 of the manuscript and [P] Theorems B, C need `1 − p_r = O(r³)` only; nothing
+   here changes the leading term `c ℓ^{−1/3}`.
+2. **Compact-window selection loss, `d = 2`.** With [ELDER] (E9)–(E10) (compact `B`, `K`, `r_*`; `r³ = ℓ/k`),
+   Fatou's lemma and Theorem LP at every fixed `(b, k, u)` give
+
+       liminf_{ℓ↓0} ℓ^{−2/3} [ν_cand(ℓ) − ν_eld(ℓ)] ≥ (1/3) ∫_B ∫_K ∫_{S¹} k^{−5/3} A₀(b,k,u) (a₁ + a₂)(b,k,u) dσ dk db,   (6.1)
+
+   using `A_r → A₀` uniformly on the compact parameters ([ELDER] (E9)). Under LP= the limit
+   `lim ℓ^{−2/3}[ν_cand − ν_eld]` exists and (6.1) is an equality, by dominated convergence with the uniform
+   constant of [P] Theorem A; (E11)'s existential two-sided bound would then carry an explicit limit. Without
+   LP=, (6.1) is a lower bound with an identified functional. The every-`d` version follows the same way from
+   [E_d]'s ledger once its (E9) analogue is cited.
+3. **Ranking.** The loss (6.1) is `Θ(ℓ^{2/3})` ([ELDER] (E11), [P] Theorem B): relative order `ℓ` against the
+   leading `ℓ^{−1/3}`. For the unrestricted density the compact-window difference estimate is explicitly not
+   asserted ([P] Theorem C), and the rejected density there is bounded with a positive limit (`ρ_rej = O(1)`
+   by [C7] Theorem K / Corollary T, `ρ_rej → B_{d,L} > 0` by [C7] Theorem Z), so the selection coefficient
+   bounded here is a compact-window statement; neither [P] §11 nor [C7] claims a second-order expansion of
+   either density separately, and none is claimed here. What this packet adds is the structure on the local
+   landscape — which points pair — and a lower bound for its weight; locality of the decision for the field
+   is Conjecture LP=.
+4. **Not claimed.** No numerical value of `a₁ + a₂` (uncertified floating-point values of the planar near
+   coefficients exist in [NUM], Math-#168, under its own conditions; this packet neither consumes nor confirms
+   them); no rate; no uniformity in marks or `d`; no statement about the joint law of several bars; no change
+   to the C6 / C7 / witness-collision registers.
+
+## 7. Finite controls
+
+`pairing_check.py` (standard library, exact rationals): I0–I3 — the shear (2.1) against [CUB] (C1)–(C3),
+the conic identity, an exact central-difference check of `∂_Z P`, and `A(u) + k` — on rational points; P1 the
+Lemma-P path inequalities on 6840 chart points under the assignment of §3 (path V at the chart root when
+`s + Bu₁ < 0`; otherwise — case (iv) with the chart root on the side `u < u₀` — path U at `u₀` is verified
+and, in addition, the partner root `S_V` is computed exactly by Vieta on [CUB] (C13), checked to be a
+strict-window root with `u > u₀` and the higher of the two, and path V is verified there), with the minimal
+margin reported; P2 the [ELDER] witness; N1 a typed `n = 0` grid on which the U hypothesis
+`B < 0 < s − B` never holds (consistency of (C6) with case (iv)).
+Deterministic JSON, byte-identical in `-O`; mutants `M1` (typed condition broken), `M2` (root pushed below
+the window), `M3` (wrong conic identity), `M4` (margin sign) exit 1; unknown label exits 2.
+`exploration/elder_grid.py` (standard library only; **not** run in CI, **not** a proof) and
+`exploration/RESULTS_EXPLORATION.json` record the grid elder-rule computation of §3 (closed-form critical
+points, discretised maximin by union-find, boundary treated by the ray criterion of Lemma 2.1(b); its
+numerical limits are stated in the script header). None of this certifies the Gaussian steps of §4–§5.
+
+## 8. Non-claims
+
+No register, STATUS, GRAPH, PROOF_INDEX or catalog edit; no `lemma_closed`, prize or premise change; no
+acceptance of [SC], [CUB], [TSL] or [P] beyond their own records; no numerical constant; no uniformity as
+`k ↓ 0`, in `L`, or in `d`; no rate for the `o(r³)`; no claim about the finite-`r` spatial law of the
+witnesses; no `d ≥ 3` hard-direction statement beyond what [SC] §5 supplies for the soft plane.
+
+## 9. Review division
+
+* **Slice A (deterministic, `d`-free):** §2, Lemma P and Lemma P′, and the exact controls. Any lane.
+* **Slice B (Gaussian transfer):** §4 Theorem LP — the coupling identity, the compact-jet convergence read
+  from [SC] §5, and the `O(r^p)` tail. Best read by a lane not exposed to [SC]'s authorship.
+* **Slice C (the obligation):** §5 — whether (O1) follows from [P] §§6–7 and [SC] §4 as stated, or needs
+  a new estimate. This is the lane where the next theorem is decided.
+  Reviewed by Codex (C85, 5396856086): the remaining weighted obligation and the cap bypass are sound at
+  their conditional scope. Finding C-01 is applied in v1.2: the v1.1 sketch is withdrawn, Lemma 5.1 is consumed
+  from QS-E, and the reverse reduction is marked conditional.
diff --git a/frontiers/local_pairing_20260930/README.md b/frontiers/local_pairing_20260930/README.md
new file mode 100644
index 000000000..958dea0ce
--- /dev/null
+++ b/frontiers/local_pairing_20260930/README.md
@@ -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.
diff --git a/frontiers/local_pairing_20260930/RESULTS.json b/frontiers/local_pairing_20260930/RESULTS.json
new file mode 100644
index 000000000..e8e76f78a
--- /dev/null
+++ b/frontiers/local_pairing_20260930/RESULTS.json
@@ -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"
+}
diff --git a/frontiers/local_pairing_20260930/SOURCES.json b/frontiers/local_pairing_20260930/SOURCES.json
new file mode 100644
index 000000000..c6abe1c6f
--- /dev/null
+++ b/frontiers/local_pairing_20260930/SOURCES.json
@@ -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"
+ },
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+  "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"
+}
diff --git a/frontiers/local_pairing_20260930/exploration/RESULTS_EXPLORATION.json b/frontiers/local_pairing_20260930/exploration/RESULTS_EXPLORATION.json
new file mode 100644
index 000000000..bbdc28a6d
--- /dev/null
+++ b/frontiers/local_pairing_20260930/exploration/RESULTS_EXPLORATION.json
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+   "D": -0.19109147911185964,
+   "R": 3,
+   "cells": 90601,
+   "death": -0.5,
+   "h_star": -0.5,
+   "k": 0.5,
+   "kind": "typed_n=0",
+   "level_error": 0.0,
+   "level_match": true,
+   "merge_cell": [
+    0.5,
+    0.0
+   ],
+   "merge_distance": 0.0,
+   "merge_match": true,
+   "n": 0,
+   "n_classifier": 0,
+   "s": -2.8775091499540686,
+   "status": "ok",
+   "target": [
+    0.5,
+    0.0
+   ],
+   "tie_within_tol": false
+  },
+  {
+   "B": -0.30629748485392616,
+   "D": -1.6571975132355816,
+   "R": 3,
+   "cells": 90601,
+   "death": -0.5,
+   "h_star": -0.5,
+   "k": 0.5,
+   "kind": "typed_n=0",
+   "level_error": 0.0,
+   "level_match": true,
+   "merge_cell": [
+    0.5,
+    0.0
+   ],
+   "merge_distance": 0.0,
+   "merge_match": true,
+   "n": 0,
+   "n_classifier": 0,
+   "s": -2.506808152791569,
+   "status": "ok",
+   "target": [
+    0.5,
+    0.0
+   ],
+   "tie_within_tol": false
+  },
+  {
+   "B": -2.0875905667678,
+   "D": -2.3466575171877198,
+   "R": 3,
+   "cells": 90601,
+   "death": -0.5,
+   "h_star": -0.5,
+   "k": 0.5,
+   "kind": "typed_n=0",
+   "level_error": 0.0,
+   "level_match": true,
+   "merge_cell": [
+    0.5,
+    0.0
+   ],
+   "merge_distance": 0.0,
+   "merge_match": true,
+   "n": 0,
+   "n_classifier": 0,
+   "s": -3.960020090233437,
+   "status": "ok",
+   "target": [
+    0.5,
+    0.0
+   ],
+   "tie_within_tol": false
+  },
+  {
+   "B": 1.9523721063912785,
+   "D": 2.079051096653962,
+   "R": 3,
+   "cells": 90601,
+   "death": -0.5,
+   "h_star": -0.5,
+   "k": 0.5,
+   "kind": "typed_n=0",
+   "level_error": 0.0,
+   "level_match": true,
+   "merge_cell": [
+    0.5,
+    0.0
+   ],
+   "merge_distance": 0.0,
+   "merge_match": true,
+   "n": 0,
+   "n_classifier": 0,
+   "s": -3.085177117245668,
+   "status": "ok",
+   "target": [
+    0.5,
+    0.0
+   ],
+   "tie_within_tol": false
+  },
+  {
+   "B": 0.7588364147451792,
+   "D": -0.3754168287553532,
+   "R": 3,
+   "cells": 90601,
+   "death": -0.5,
+   "h_star": -0.5,
+   "k": 0.5,
+   "kind": "typed_n=0",
+   "level_error": 0.0,
+   "level_match": true,
+   "merge_cell": [
+    0.5,
+    0.0
+   ],
+   "merge_distance": 0.0,
+   "merge_match": true,
+   "n": 0,
+   "n_classifier": 0,
+   "s": -1.9736125729346965,
+   "status": "ok",
+   "target": [
+    0.5,
+    0.0
+   ],
+   "tie_within_tol": false
+  },
+  {
+   "B": 2.6612427495222226,
+   "D": -1.4347087193913894,
+   "R": 3,
+   "cells": 90601,
+   "death": -0.5,
+   "h_star": -0.5,
+   "k": 0.5,
+   "kind": "typed_n=0",
+   "level_error": 0.0,
+   "level_match": true,
+   "merge_cell": [
+    0.5,
+    0.0
+   ],
+   "merge_distance": 0.0,
+   "merge_match": true,
+   "n": 0,
+   "n_classifier": 0,
+   "s": -1.921892635264673,
+   "status": "ok",
+   "target": [
+    0.5,
+    0.0
+   ],
+   "tie_within_tol": false
+  },
+  {
+   "B": 1.7429231822964955,
+   "D": 0.22485791004544264,
+   "R": 3,
+   "cells": 90601,
+   "death": -0.49991519843253907,
+   "h_star": -0.5,
+   "k": 0.5,
+   "kind": "typed_n=0",
+   "level_error": 8.480156746093481e-05,
+   "level_match": true,
+   "merge_cell": [
+    0.48,
+    0.18000000000000016
+   ],
+   "merge_distance": 0.18110770276274848,
+   "merge_match": false,
+   "n": 0,
+   "n_classifier": 0,
+   "s": -0.8948946219262359,
+   "status": "ok",
+   "target": [
+    0.5,
+    0.0
+   ],
+   "tie_within_tol": false
+  },
+  {
+   "B": 2.9782443103502967,
+   "D": -1.1018578271012474,
+   "R": 3,
+   "cells": 90601,
+   "death": -0.5,
+   "h_star": -0.5,
+   "k": 0.5,
+   "kind": "typed_n=0",
+   "level_error": 0.0,
+   "level_match": true,
+   "merge_cell": [
+    0.5,
+    0.0
+   ],
+   "merge_distance": 0.0,
+   "merge_match": true,
+   "n": 0,
+   "n_classifier": 0,
+   "s": -2.3393610331394745,
+   "status": "ok",
+   "target": [
+    0.5,
+    0.0
+   ],
+   "tie_within_tol": false
+  },
+  {
+   "B": 2.0364672340649674,
+   "D": 0.15766624665713724,
+   "R": 3,
+   "cells": 90601,
+   "death": -0.5,
+   "h_star": -0.5,
+   "k": 0.5,
+   "kind": "typed_n=0",
+   "level_error": 0.0,
+   "level_match": true,
+   "merge_cell": [
+    0.5,
+    0.0
+   ],
+   "merge_distance": 0.0,
+   "merge_match": true,
+   "n": 0,
+   "n_classifier": 0,
+   "s": -1.7604592659155176,
+   "status": "ok",
+   "target": [
+    0.5,
+    0.0
+   ],
+   "tie_within_tol": false
+  },
+  {
+   "B": 2.309091151109185,
+   "D": -2.5134475861226266,
+   "R": 3,
+   "cells": 90601,
+   "death": -0.5,
+   "h_star": -0.5,
+   "k": 0.5,
+   "kind": "typed_n=0",
+   "level_error": 0.0,
+   "level_match": true,
+   "merge_cell": [
+    0.5,
+    0.0
+   ],
+   "merge_distance": 0.0,
+   "merge_match": true,
+   "n": 0,
+   "n_classifier": 0,
+   "s": -3.103107277415969,
+   "status": "ok",
+   "target": [
+    0.5,
+    0.0
+   ],
+   "tie_within_tol": false
+  },
+  {
+   "B": -0.042339293624576335,
+   "D": 1.3099934497724552,
+   "R": 3,
+   "cells": 90601,
+   "death": -0.5,
+   "h_star": -0.5,
+   "k": 0.5,
+   "kind": "typed_n=0",
+   "level_error": 0.0,
+   "level_match": true,
+   "merge_cell": [
+    0.5,
+    0.0
+   ],
+   "merge_distance": 0.0,
+   "merge_match": true,
+   "n": 0,
+   "n_classifier": 0,
+   "s": -2.114721547105174,
+   "status": "ok",
+   "target": [
+    0.5,
+    0.0
+   ],
+   "tie_within_tol": false
+  },
+  {
+   "B": 2.3360032571077127,
+   "D": 0.7415820879831037,
+   "R": 3,
+   "cells": 90601,
+   "death": -0.5,
+   "h_star": -0.5,
+   "k": 0.5,
+   "kind": "typed_n=0",
+   "level_error": 0.0,
+   "level_match": true,
+   "merge_cell": [
+    0.5,
+    0.0
+   ],
+   "merge_distance": 0.0,
+   "merge_match": true,
+   "n": 0,
+   "n_classifier": 0,
+   "s": -1.5128456652279243,
+   "status": "ok",
+   "target": [
+    0.5,
+    0.0
+   ],
+   "tie_within_tol": false
+  },
+  {
+   "B": -2.0,
+   "D": 0.0,
+   "R": 4,
+   "cells": 160801,
+   "death": -0.218944,
+   "h_star": -0.21875,
+   "k": 1.0,
+   "kind": "elder_witness",
+   "level_error": 0.00019399999999999973,
+   "level_match": true,
+   "merge_cell": [
+    -0.7399999999999998,
+    -1.3599999999999999
+   ],
+   "merge_distance": 2.7293247133741945,
+   "merge_match": true,
+   "n": 2,
+   "n_classifier": 2,
+   "s": -1.5,
+   "status": "ok",
+   "target": [
+    -0.75,
+    1.3693063937629153
+   ],
+   "tie_within_tol": true
+  }
+ ],
+ "scientific_effect": "NONE",
+ "seed": 7,
+ "spacing": 0.02,
+ "summary": {
+  "chart_n>=1": {
+   "by_n": {
+    "0": 0,
+    "1": 80,
+    "2": 40
+   },
+   "classifier_agrees_with_closed_form_count": 120,
+   "death_found": 120,
+   "level_match": 120,
+   "max_level_error": 0.0014879026097764925,
+   "merge_match": 120,
+   "samples": 120,
+   "ties_within_tol": 0
+  },
+  "elder_witness": {
+   "by_n": {
+    "0": 0,
+    "1": 0,
+    "2": 1
+   },
+   "classifier_agrees_with_closed_form_count": 1,
+   "death_found": 1,
+   "level_match": 1,
+   "max_level_error": 0.00019399999999999973,
+   "merge_match": 1,
+   "samples": 1,
+   "ties_within_tol": 1
+  },
+  "typed_n=0": {
+   "by_n": {
+    "0": 40,
+    "1": 0,
+    "2": 0
+   },
+   "classifier_agrees_with_closed_form_count": 40,
+   "death_found": 40,
+   "level_match": 40,
+   "max_level_error": 8.480156746093481e-05,
+   "merge_match": 39,
+   "samples": 40,
+   "ties_within_tol": 0
+  }
+ }
+}
diff --git a/frontiers/local_pairing_20260930/exploration/elder_grid.py b/frontiers/local_pairing_20260930/exploration/elder_grid.py
new file mode 100644
index 000000000..3177e37e4
--- /dev/null
+++ b/frontiers/local_pairing_20260930/exploration/elder_grid.py
@@ -0,0 +1,277 @@
+#!/usr/bin/env python3
+"""Grid maximin (the elder rule, discretised) on sampled pinned cubics.  Standard library only.
+
+EXPLORATION, NOT PROOF, NOT RUN IN CI.  Scientific effect: NONE.
+
+What it does.  For sampled typed jets theta = (s, B, D) of the sheared pinned cubic (LOCAL_PAIRING.md (2.1))
+
+    P(u, Z) = 2k(u-1)(u+1/2)^2 + (Z^2/2)(s + B u) + (D/3) Z^3,      M = (-1/2, 0), S = (1/2, 0),
+
+it (i) computes every critical point in closed form ([CUB] (C7), (C13): the extra roots lie on the line
+s + Bu + DZ = 0 and their u-coordinates solve (12kD^2 + B^3)u^2 + 2B^2 s u + Bs^2 - 3kD^2 = 0; for D = 0 they
+sit at u0 = -s/B with Z^2 = (3k - 12k u0^2)/B), counts the strict-window ones (n) and cross-checks n with the
+[CUB] classifier; (ii) computes the ordinary superlevel maximin death level of M on a square grid by the
+literal elder rule: cells are processed in decreasing order of P and merged with 8-neighbours by union-find;
+the death level is the value at which M's component first joins a component containing a point older than
+M (P > 0), and the merge cell is recorded; (iii) compares the grid death level with h_* = max window critical
+value (or -k when n = 0) and the merge cell with the position of the highest extra window saddle (or S).
+
+Boundary.  The grid covers the box |u|, |Z| <= R with R chosen from the window critical points (R <= 6).  A boundary cell
+with P > -k is marked "older" when P increases strictly along its outward ray from the origin at 1.5R, 2R,
+3R and is positive at 3R (the ray criterion of Lemma 2.1(b): such a cell is connected outward, inside
+{P > P(cell)}, to points with P > 0).  Other boundary cells are ordinary cells.
+
+Limits.  Grid error near a nondegenerate saddle is of order h^2 (h the spacing) plus one cell of
+8-connectivity; ties of the two extra saddle heights closer than the tolerance are reported as ties.  Close to
+the typed boundary s = -|B|/2 the saddle at S is nearly degenerate in Z (P ~ -k + (Z^2/2)(s + B/2) there), so
+the merge cell can sit some distance along that flat valley while the death level is still exact; the level
+comparison is the primary check, the merge-cell comparison the secondary one.  Runtime about two minutes
+for the default sample sizes (pure Python union-find on up to 361k cells per jet).
+Nothing here certifies the Gaussian steps of LOCAL_PAIRING.md.
+
+Usage:  python3 elder_grid.py [SEED] [N_SAMPLES] [N0_SAMPLES] [SPACING]
+Output: RESULTS_EXPLORATION.json next to this file.
+"""
+import json
+import math
+import os
+import random
+import sys
+
+
+def A(k, u):
+    return 2*k*(u - 1)*(u + 0.5)**2
+
+
+def P(k, s, B, D, u, Z):
+    return A(k, u) + (Z*Z/2)*(s + B*u) + (D/3)*Z**3
+
+
+def grad(k, s, B, D, u, Z):
+    return (6*k*u*u - 1.5*k + (B/2)*Z*Z, Z*(s + B*u + D*Z))
+
+
+def classifier_n(k, s, B, D):
+    """[CUB] Theorem C in the sheared parameters (typed domain s < -|B|/2), else None."""
+    if not (s < -abs(B)/2):
+        return None
+    T = -(s - B)**2*(B + 2*s)/(12*k)
+    if s > B:
+        return 2 if D*D < T else 1
+    return 1 if D*D > T else 0
+
+
+def quadratic_roots(a2, a1, a0):
+    if abs(a2) < 1e-14:
+        return [-a0/a1] if abs(a1) > 1e-14 else []
+    disc = a1*a1 - 4*a2*a0
+    if disc < 0:
+        return []
+    q = math.sqrt(disc)
+    return [(-a1 + q)/(2*a2), (-a1 - q)/(2*a2)]
+
+
+def critical_points(k, s, B, D):
+    """All critical points of P: the pins and the extra roots (closed form)."""
+    extra = []
+    if abs(D) > 1e-12:
+        for u in quadratic_roots(12*k*D*D + B**3, 2*B*B*s, B*s*s - 3*k*D*D):
+            Z = -(s + B*u)/D
+            if abs(Z) > 1e-9:
+                extra.append((u, Z))
+    elif abs(B) > 1e-12:
+        u = -s/B
+        Z2 = (3*k - 12*k*u*u)/B
+        if Z2 > 0:
+            extra += [(u, math.sqrt(Z2)), (u, -math.sqrt(Z2))]
+    out = []
+    for (u, Z) in extra:
+        gu, gz = grad(k, s, B, D, u, Z)
+        if abs(gu) < 1e-7*max(1.0, k) and abs(gz) < 1e-7*max(1.0, k):
+            out.append((u, Z, P(k, s, B, D, u, Z)))
+    return out
+
+
+def window_roots(k, cps):
+    return [(u, Z, val) for (u, Z, val) in cps if -k < val < 0]
+
+
+def grid_death(k, s, B, D, cps, spacing):
+    """Discrete elder rule on the box |u|,|Z| <= R.  Returns (death, merge_u, merge_Z, R, ncells)."""
+    reach = 1.0
+    for (u, Z, val) in cps:
+        if -k <= val <= 0:               # only the window geometry sets the box; Lemma 2.1(b) handles the outside
+            reach = max(reach, abs(u), abs(Z))
+    R = min(6, int(math.ceil(1.6*reach + 1.0)))
+    nper = int(round(1.0/spacing))
+    N = 2*R*nper + 1                     # nodes per axis; M and S are exact nodes since 1/2 is a multiple of h
+    h = 2.0*R/(N - 1)
+    W = N + 2                            # padded width: a never-processed border removes all bounds checks
+
+    def coord(i):
+        return -R + i*h
+
+    total = W*W
+    vals = [0.0]*total
+    older = bytearray(total)
+    for i in range(N):
+        u = coord(i)
+        base = (i + 1)*W + 1
+        boundary_i = (i == 0 or i == N - 1)
+        for j in range(N):
+            Z = coord(j)
+            v = P(k, s, B, D, u, Z)
+            idx = base + j
+            vals[idx] = v
+            if v > 0:
+                older[idx] = 1
+            elif v > -k and (boundary_i or j == 0 or j == N - 1):
+                rho = math.hypot(u, Z)
+                if rho > 0:
+                    ru, rz = u/rho, Z/rho
+                    seq = [P(k, s, B, D, t*rho*ru, t*rho*rz) for t in (1.5, 2.0, 3.0)]
+                    if v < seq[0] < seq[1] < seq[2] and seq[2] > 0:
+                        older[idx] = 1
+    iM = int(round((-0.5 + R)/h)); jM = int(round(R/h))
+    idxM = (iM + 1)*W + (jM + 1)
+    vals[idxM] = 0.0                     # exact pin value
+    older[idxM] = 0
+
+    parent = list(range(total))
+    comp_older = bytearray(older)
+    processed = bytearray(total)
+    interior = [(i + 1)*W + (j + 1) for i in range(N) for j in range(N)]
+    order = sorted(interior, key=vals.__getitem__, reverse=True)
+    offsets = (-W - 1, -W, -W + 1, -1, 1, W - 1, W, W + 1)
+    m_seen = False
+    for idx in order:
+        processed[idx] = 1
+        ra = idx
+        while parent[ra] != ra:
+            parent[ra] = parent[parent[ra]]
+            ra = parent[ra]
+        for d in offsets:
+            nb = idx + d
+            if not processed[nb]:
+                continue
+            rb = nb
+            while parent[rb] != rb:
+                parent[rb] = parent[parent[rb]]
+                rb = parent[rb]
+            if ra != rb:
+                parent[rb] = ra
+                if comp_older[rb]:
+                    comp_older[ra] = 1
+        if idx == idxM:
+            m_seen = True
+        if m_seen:
+            rM = idxM
+            while parent[rM] != rM:
+                parent[rM] = parent[parent[rM]]
+                rM = parent[rM]
+            if comp_older[rM]:
+                i, j = divmod(idx, W)
+                return vals[idx], coord(i - 1), coord(j - 1), R, N*N
+    return None, None, None, R, N*N
+
+
+def sample_n1(rng, k):
+    """Chart sampler: a typed jet with a prescribed extra strict-window root (u1, Z1)."""
+    while True:
+        u1 = rng.uniform(-1.45, 0.45)
+        b0 = 3 - 12*u1*u1
+        lo, hi = abs(b0)/2, 3 - 6*u1
+        if lo < hi:
+            break
+    v = lo + rng.uniform(0.05, 0.95)*(hi - lo)
+    Z1 = rng.choice((-1.0, 1.0))*math.exp(rng.uniform(math.log(0.3), math.log(1.8)))
+    s = -k*v/Z1**2
+    B = k*b0/Z1**2
+    D = k*(v - b0*u1)/Z1**3
+    return s, B, D
+
+
+def sample_n0(rng, k):
+    while True:
+        B = rng.uniform(-3, 3)
+        s = -abs(B)/2 - rng.uniform(0.02, 3)
+        D = rng.uniform(-3, 3)
+        if classifier_n(k, s, B, D) == 0:
+            return s, B, D
+
+
+def run_one(k, s, B, D, spacing, tol):
+    cps = critical_points(k, s, B, D)
+    win = window_roots(k, cps)
+    n = len(win)
+    n_cls = classifier_n(k, s, B, D)
+    if win:
+        top = max(win, key=lambda t: t[2])
+        h_star = top[2]
+        target = (top[0], top[1])
+        heights = sorted([t[2] for t in win], reverse=True)
+        tie = len(heights) >= 2 and heights[0] - heights[1] < tol
+    else:
+        h_star = -k
+        target = (0.5, 0.0)
+        tie = False
+    death, mu, mz, R, ncells = grid_death(k, s, B, D, cps, spacing)
+    row = {'s': s, 'B': B, 'D': D, 'n': n, 'n_classifier': n_cls, 'h_star': h_star,
+           'death': death, 'R': R, 'cells': ncells, 'tie_within_tol': tie}
+    if death is None:
+        row['status'] = 'no_death_found'
+        return row
+    row['merge_cell'] = [mu, mz]
+    row['target'] = list(target)
+    row['level_error'] = abs(death - h_star)
+    row['merge_distance'] = math.hypot(mu - target[0], mz - target[1])
+    row['level_match'] = row['level_error'] <= tol
+    row['merge_match'] = row['merge_distance'] <= 3*spacing or tie
+    row['status'] = 'ok'
+    return row
+
+
+def main():
+    seed = int(sys.argv[1]) if len(sys.argv) > 1 else 7
+    n1 = int(sys.argv[2]) if len(sys.argv) > 2 else 40
+    n0 = int(sys.argv[3]) if len(sys.argv) > 3 else 15
+    spacing = float(sys.argv[4]) if len(sys.argv) > 4 else 0.02
+    rng = random.Random(seed)
+    rows = []
+    for k in (1.0, 0.5):
+        tol = 0.01*max(1.0, k)
+        for _ in range(n1):
+            s, B, D = sample_n1(rng, k)
+            rows.append(dict(k=k, kind='chart_n>=1', **run_one(k, s, B, D, spacing, tol)))
+            sys.stderr.write('.'); sys.stderr.flush()
+        for _ in range(n0):
+            s, B, D = sample_n0(rng, k)
+            rows.append(dict(k=k, kind='typed_n=0', **run_one(k, s, B, D, spacing, tol)))
+            sys.stderr.write('o'); sys.stderr.flush()
+    # the [ELDER] witness (s, B, D) = (-3k/2, -2k, 0), k = 1: both extra saddles at -7/32 (a tie by construction)
+    rows.append(dict(k=1.0, kind='elder_witness', **run_one(1.0, -1.5, -2.0, 0.0, spacing, 0.01)))
+    summary = {}
+    for kind in ('chart_n>=1', 'typed_n=0', 'elder_witness'):
+        sub = [r for r in rows if r['kind'] == kind]
+        summary[kind] = {
+            'samples': len(sub),
+            'classifier_agrees_with_closed_form_count': sum(1 for r in sub if r['n'] == r['n_classifier']),
+            'death_found': sum(1 for r in sub if r['status'] == 'ok'),
+            'level_match': sum(1 for r in sub if r.get('level_match')),
+            'merge_match': sum(1 for r in sub if r.get('merge_match')),
+            'ties_within_tol': sum(1 for r in sub if r['tie_within_tol']),
+            'by_n': {str(m): sum(1 for r in sub if r['n'] == m) for m in (0, 1, 2)},
+            'max_level_error': max([r.get('level_error', 0.0) for r in sub] or [0.0]),
+        }
+    out = {'object': 'CL-LOCAL-PAIRING-20260930-v1', 'kind': 'exploration (stdlib grid maximin); not proof; not CI',
+           'scientific_effect': 'NONE', 'seed': seed, 'spacing': spacing, 'n1_per_k': n1, 'n0_per_k': n0,
+           'k_values': [1.0, 0.5], 'summary': summary, 'rows': rows}
+    path = os.path.join(os.path.dirname(os.path.abspath(__file__)), 'RESULTS_EXPLORATION.json')
+    with open(path, 'w') as fh:
+        json.dump(out, fh, indent=1, sort_keys=True)
+        fh.write('\n')
+    print(json.dumps(summary, indent=1, sort_keys=True))
+
+
+if __name__ == '__main__':
+    main()
diff --git a/frontiers/local_pairing_20260930/pairing_check.py b/frontiers/local_pairing_20260930/pairing_check.py
new file mode 100644
index 000000000..6a3aec4d4
--- /dev/null
+++ b/frontiers/local_pairing_20260930/pairing_check.py
@@ -0,0 +1,273 @@
+#!/usr/bin/env python3
+"""Exact finite controls for LOCAL_PAIRING.md (standard library only; exact rationals).
+
+Checked here, exactly:
+  I0  the shear: [CUB] (C1) evaluated at X = u - aZ/(12k) equals (2.1) with B, D from (C2), at rational points;
+  I1  the pinned cubic identities: P(u,Z) = A(u) + (Z^2/2)(s+Bu) + (D/3)Z^3 with A(u) = 2k(u-1)(u+1/2)^2,
+      and A(u) + (B u/6) Z^2 = -k(u+1/2) on the conic 6k(u^2-1/4) + (B/2)Z^2 = 0 (CUB (C8)/(C9) consistency);
+  I2  at an extra critical point (u1,Z1): P(S1) = A(u1) + Z1^2 (s+Bu1)/6; and the vertical-line derivative
+      d/dZ P(u1,Z) = Z (s+Bu1+DZ), checked by exact differentiation of the cubic in Z (a central difference
+      of a cubic equals its derivative plus exactly (D/3) h^2, which is subtracted);
+  I3  A(u) + k = (k/2)(u+1)(2u-1)^2 and A'(u) = 6ku^2 - 3k/2 at rational points;
+  P1  the path cases of Lemma P on a rational grid of the chart (u1, v, Z1, a): every sample is a typed jet
+      with an extra strict-window root (n >= 1 by the [CUB] classifier), at least one of the paths V, U applies
+      under the V-first assignment (V and U may both apply in case (iv); V is taken first), and the displayed
+      path has minimum value > -k with the stated margin, rising past 0 at its far end (sampled at 2Z1, 4Z1,
+      resp. 3, 6, 12 on the D side; the monotonicity itself is the one-line calculus of the text, not a check);
+      for every chart point whose own root is not V-admissible (case (iv), root on the side u < u0) the OTHER
+      root is computed exactly by Vieta on [CUB] (C13) and verified to be a strict-window root on the side
+      u > u0, V-admissible, and higher than the chart root (Lemma P's assignment and Proposition 3.3);
+  P2  the [ELDER] witness (s,a,beta,c) = (-3k/2,0,-2k,0): both extra saddles at height -7k/32, and the vertical
+      line at u = -3/4 is constant (D = 0), so the polygonal path of [ELDER] (E5) is needed there (null case);
+  N1  on a rational grid of typed jets with n = 0, the U hypothesis (B < 0 and s > B) never holds: consistency
+      of the [CUB] classifier with Lemma P's case (iv), which always has n >= 1;
+  M   semantic mutants (--mutant M1|M2|M3|M4) must fail; an unknown label exits 2.
+
+The controls certify algebra, case coverage on the grid and the path minima; they do not prove the Gaussian
+transfer (Theorem LP) or the weak convergence consumed from [SC].  Scientific effect: NONE.
+"""
+import argparse
+import json
+import sys
+from fractions import Fraction as F
+
+K_VALUES = [F(1), F(2), F(1, 3)]
+
+
+def A(k, u):
+    return 2*k*(u - 1)*(u + F(1, 2))**2
+
+
+def P(k, s, B, D, u, Z):
+    return A(k, u) + (Z*Z/2)*(s + B*u) + (D/3)*Z**3
+
+
+def cub_n(k, s, a, beta, c):
+    """[CUB] Theorem C classifier on the typed domain; None if not typed."""
+    B = beta - a*a/(12*k)
+    D = (c - a*beta/(4*k) + a**3/(72*k*k))/2
+    if not (s < -abs(B)/2):
+        return None
+    T = -(s - B)**2*(B + 2*s)/(12*k)
+    if s > B:
+        return 2 if D*D < T else 1
+    return 1 if D*D > T else 0
+
+
+def chart(k, u1, v, Z1, a):
+    b0 = 3 - 12*u1*u1
+    s = -k*v/Z1**2
+    B = k*b0/Z1**2
+    D = k*(v - b0*u1)/Z1**3
+    beta = B + a*a/(12*k)
+    c = 2*D + a*B/(4*k) + a**3/(144*k*k)
+    return s, B, D, beta, c
+
+
+def chart_grid():
+    us = [F(n, 10) for n in range(-14, 5)]           # -1.4 .. 0.4
+    out = []
+    for u1 in us:
+        b0 = 3 - 12*u1*u1
+        lo, hi = abs(b0)/2, 3 - 6*u1
+        if not lo < hi:
+            continue
+        for t in (F(1, 4), F(1, 2), F(3, 4)):
+            v = lo + t*(hi - lo)
+            for Z1 in (F(-2), F(-1), F(-1, 2), F(-1, 4), F(1, 4), F(1, 2), F(1), F(2)):
+                for a in (F(-2), F(-1, 2), F(0), F(1, 2), F(2)):
+                    out.append((u1, v, Z1, a))
+    return out
+
+
+def cub_C1(k, s, a, beta, c, X, Z):
+    """[CUB] (C1): the unsheared pinned cubic."""
+    return 2*k*X**3 - 3*k*X/2 - k/2 + (s/2)*Z*Z + (a/2)*(X*X - F(1, 4))*Z + (beta/2)*X*Z*Z + (c/6)*Z**3
+
+
+def check_identities(k, mutant):
+    rows = 0
+    # I0: shear (C2) turns (C1) into (2.1)
+    for (s, a, beta, c) in ((F(-3, 2), F(0), F(-2), F(0)), (F(-2), F(1), F(1, 3), F(-5, 7)), (F(-7, 5), F(-3, 2), F(2), F(4))):
+        B = beta - a*a/(12*k); D = (c - a*beta/(4*k) + a**3/(72*k*k))/2
+        for u in (F(-1), F(-1, 3), F(1, 2), F(5, 4)):
+            for Z in (F(-2), F(1, 3), F(3)):
+                X = u - a*Z/(12*k)
+                if cub_C1(k, s, a, beta, c, X, Z) != P(k, s, B, D, u, Z):
+                    return False, 'I0 shear identity fails'
+    for u in (F(-3), F(-1), F(-1, 2), F(0), F(1, 2), F(2), F(7, 3)):
+        lhs = A(k, u) + k
+        rhs = (k/2)*(u + 1)*(2*u - 1)**2
+        dA = 6*k*u*u - 3*k/2
+        exact_dA = (A(k, u + F(1, 10**6)) - A(k, u - F(1, 10**6)))/(F(2, 10**6))   # central difference of a cubic: exact up to h^2 term
+        if lhs != rhs or abs(exact_dA - dA) > F(1, 10**11):
+            return False, 'I3 axis identity fails at u=%s' % u
+    for u in (F(-3, 2), F(-1), F(-3, 4), F(-1, 2), F(-1, 5), F(0), F(1, 3), F(1, 2), F(2)):
+        for Z in (F(-2), F(-1, 3), F(1, 2), F(3)):
+            for s in (F(-3), F(-1, 2)):
+                for D in (F(-1), F(2, 5)):
+                    B = -12*k*(u*u - F(1, 4))/(Z*Z)          # conic through (u,Z)
+                    lhs = A(k, u) + (B*u/6)*Z*Z
+                    rhs = -k*(u + F(1, 2)) if mutant != 'M3' else -k*(u - F(1, 2))
+                    if lhs != rhs:
+                        return False, 'I1 conic identity fails at u=%s Z=%s' % (u, Z)
+                    # I2: P at a critical point on the line s + Bu + DZ = 0 -> solve for D given (s,B,u,Z)
+                    Dc = -(s + B*u)/Z
+                    if P(k, s, B, Dc, u, Z) != A(k, u) + Z*Z*(s + B*u)/6:
+                        return False, 'I2 critical value formula fails'
+                    # derivative identity in Z by exact differentiation: for a cubic q(Z) the central
+                    # difference (q(Z+h)-q(Z-h))/(2h) equals q'(Z) + (leading coefficient) h^2 exactly
+                    h = F(1, 7)
+                    central = (P(k, s, B, D, u, Z + h) - P(k, s, B, D, u, Z - h))/(2*h) - (D/3)*h*h
+                    if central != Z*(s + B*u + D*Z):
+                        return False, 'I2 derivative identity fails'
+                    rows += 1
+    return True, rows
+
+
+def path_case(k, s, B, D, u1, Z1):
+    """Return (case, margin, message) for the explicit path of Lemma P at the extra root (u1, Z1).
+
+    V: vertical descent at u1 (needs s + B u1 < 0 and -1 < u1 < 1/2): the path is M -> axis -> (u1,0)
+       -> straight down the vertical line to S1 -> on to P -> +infinity; minimum P(S1).
+    U: vertical rise at u0 = -s/B (needs B < 0, s > B, D != 0): M -> axis -> (u0,0) -> vertical line, on
+       which P = A(u0) + D Z^3/3; minimum A(u0).
+    Lemma P: every typed jet with a strict-window extra root admits V or U (B = 0 is included in V; D = 0
+    with B < 0 < ... is the [ELDER] null case handled by its polygonal path).
+    """
+    P1 = P(k, s, B, D, u1, Z1)
+    if s + B*u1 < 0 and -1 < u1 < F(1, 2):
+        if not (A(k, u1) > P1 > -k):
+            return 'V', None, 'ordering A(u1) > P(S1) > -k fails'
+        if not ((D > 0) == (Z1 > 0)):
+            return 'V', None, 'root not on the D side'
+        far1 = P(k, s, B, D, u1, 2*Z1); far2 = P(k, s, B, D, u1, 4*Z1)
+        if not (far1 > P1 and far2 > far1 and far2 > 0):
+            return 'V', None, 'vertical line does not rise past 0'
+        return 'V', P1 + k, 'ok'
+    if B < 0 and s > B:
+        u0 = -s/B
+        if not (-1 < u0 < -F(1, 2)):
+            return 'U', None, 'u0 outside (-1,-1/2)'
+        if D == 0:
+            return 'U', None, 'D = 0 (null case, [ELDER] path)'
+        if not (A(k, u0) > -k):
+            return 'U', None, 'A(u0) <= -k'
+        Zf = F(3) if D > 0 else F(-3)
+        if not (P(k, s, B, D, u0, Zf) == A(k, u0) + D*Zf**3/3 and P(k, s, B, D, u0, 2*Zf) > P(k, s, B, D, u0, Zf) > A(k, u0)):
+            return 'U', None, 'vertical line at u0 not monotone'
+        if not (P(k, s, B, D, u0, 4*Zf) > 0):
+            return 'U', None, 'vertical line at u0 does not pass 0 by |Z|=12'
+        return 'U', A(k, u0) + k, 'ok'
+    return 'none', None, 'no path case applies (would contradict Lemma P)'
+
+
+def check_paths(mutant):
+    counts = {'V': 0, 'U': 0, 'V_at_other_root': 0}
+    total = 0
+    min_margin = None
+    for k in K_VALUES:
+        for (u1, v, Z1, a) in chart_grid():
+            s, B, D, beta, c = chart(k, u1, v, Z1, a)
+            if mutant == 'M1':
+                s = -abs(B)/2 + F(1, 100)            # break the typed condition
+            n = cub_n(k, s, a, beta, c)
+            if n is None or n < 1:
+                return False, 'chart sample not typed/n>=1 at %s' % ((k, u1, v, Z1, a),)
+            # (u1,Z1) is a critical point in the strict window
+            P1 = P(k, s, B, D, u1, Z1)
+            if mutant == 'M2':
+                P1 = P1 - k                            # pretend the root sits below the window
+            if not (-k < P1 < 0):
+                return False, 'root not in the strict window'
+            gu = 6*k*u1*u1 - 3*k/2 + (B/2)*Z1*Z1      # d/du of the sheared cubic
+            gz = Z1*(s + B*u1 + D*Z1)
+            if not (gu == 0 and gz == 0):
+                return False, 'chart point is not a critical point'
+            case, margin, msg = path_case(k, s, B, D, u1, Z1)
+            if margin is None:
+                return False, 'case %s: %s at %s' % (case, msg, (k, u1, v, Z1, a))
+            if mutant == 'M4':
+                margin = -margin
+            if not margin > 0:
+                return False, 'nonpositive margin'
+            counts[case] += 1; total += 1
+            min_margin = margin if min_margin is None else min(min_margin, margin)
+            if case == 'U':
+                # the chart root is on the side u1 < u0; Lemma P assigns V at the other root S_V (u > u0).
+                # [CUB] (C13): (12kD^2+B^3) u^2 + 2B^2 s u + B s^2 - 3kD^2 = 0 has the two roots' u-coordinates.
+                lead = 12*k*D*D + B**3
+                if lead == 0:
+                    return False, 'U sample with a single finite root (leading coefficient of (C13) zero)'
+                u2 = -2*B*B*s/lead - u1
+                Z2 = -(s + B*u2)/D
+                gu2 = 6*k*u2*u2 - 3*k/2 + (B/2)*Z2*Z2
+                if gu2 != 0 or not (-k < P(k, s, B, D, u2, Z2) < 0):
+                    return False, 'other root is not a strict-window critical point'
+                u0 = -s/B
+                case2, margin2, msg2 = path_case(k, s, B, D, u2, Z2)
+                if not (u2 > u0 and case2 == 'V' and margin2 is not None):
+                    return False, 'other root is not V-admissible on the side u > u0: %s' % msg2
+                if not (P(k, s, B, D, u2, Z2) > P(k, s, B, D, u1, Z1)):
+                    return False, 'other root is not the higher saddle'
+                counts['V_at_other_root'] += 1
+    return True, {'samples': total, 'cases': counts, 'min_margin': str(min_margin)}
+
+
+def check_elder_witness(mutant):
+    k = F(1)
+    s, a, beta, c = F(-3, 2), F(0), F(-2), F(0)
+    B = beta - a*a/(12*k); D = (c - a*beta/(4*k) + a**3/(72*k*k))/2
+    n = cub_n(k, s, a, beta, c)
+    u0 = -s/B                                            # = -3/4
+    # the two extra saddles: line s + Bu = 0 (D = 0) and conic
+    Z2 = -12*k*(u0*u0 - F(1, 4))/B
+    h = A(k, u0) + Z2*(s + B*u0)/6
+    const = all(P(k, s, B, D, u0, Z) == A(k, u0) for Z in (F(0), F(1), F(9, 4), F(-3)))
+    ok = (n == 2 and u0 == F(-3, 4) and Z2 == F(15, 8) and h == F(-7, 32) and const and D == 0)
+    if mutant == 'M3':
+        ok = ok and h == F(-1, 4)
+    return ok, {'n': n, 'u0': str(u0), 'Z1_squared': str(Z2), 'saddle_height': str(h), 'vertical_line_constant': const}
+
+
+def check_n0(mutant):
+    """On typed grid jets with n = 0 the U hypothesis (B < 0 and s > B) never holds ([CUB] (C6): s > B gives n >= 1)."""
+    k = F(1); found = 0; u_hypothesis = 0
+    for s in (F(-3), F(-5, 2), F(-2), F(-3, 2), F(-1), F(-1, 2)):
+        for a in (F(-1), F(0), F(1)):
+            for beta in (F(-3), F(-1), F(0), F(1), F(2)):
+                for c in (F(-2), F(0), F(2)):
+                    n = cub_n(k, s, a, beta, c)
+                    if n == 0:
+                        found += 1
+                        B = beta - a*a/(12*k)
+                        if B < 0 and s > B:
+                            u_hypothesis += 1
+    if mutant == 'M4':
+        u_hypothesis += 1
+    return found > 0 and u_hypothesis == 0, {'n0_typed_grid_points': found, 'n0_points_with_U_hypothesis': u_hypothesis}
+
+
+def main(argv=None):
+    parser = argparse.ArgumentParser(description=__doc__.splitlines()[0])
+    parser.add_argument('--mutant', default=None)
+    args = parser.parse_args(argv)
+    if args.mutant is not None and args.mutant not in ('M1', 'M2', 'M3', 'M4'):
+        print('unknown mutant label', file=sys.stderr)
+        return 2
+    report = {'object': 'CL-LOCAL-PAIRING-20260930-v1', 'scientific_effect': 'NONE', 'checks': {}}
+    ok_all = True
+    for name, fn in (('I0_I3_identities', lambda: check_identities(F(1), args.mutant)),
+                     ('P1_path_cases', lambda: check_paths(args.mutant)),
+                     ('P2_elder_witness', lambda: check_elder_witness(args.mutant)),
+                     ('N1_n0_grid', lambda: check_n0(args.mutant))):
+        ok, info = fn()
+        report['checks'][name] = {'passed': bool(ok), 'info': info}
+        ok_all = ok_all and bool(ok)
+    report['passed'] = ok_all
+    print(json.dumps(report, indent=2, sort_keys=True))
+    return 0 if ok_all else 1
+
+
+if __name__ == '__main__':
+    raise SystemExit(main())