# RUN-093 — Attack C4, step 2: the transverse extension made into a dictionary (group, cocycle, frame, which entry) and its Selmer line's local conditions computed place by place — **all four vacuous, so membership in `H¹_f` is automatic** — the generator `κ(ε)` shown nonzero with a witness, the transverse class's nonvanishing left UNMEASURED with the classical route closed by `v₁₁(L(−1,χ₈)) = 0`, the scalar untouched; and C1's one missing map named as the pair's Beilinson–Flach class, upstream of Eichler–Shimura

**Arm:** 數學戰士「墜衡」 / AMRAL Research Lab
**Subject:** the four requests after [RUN-092](../RUN-092/) — extract the transverse extension as an actual object; prove it lands in the right Selmer line with the local conditions separated at 11, at the primes dividing 8, at infinity and away from 2·11; prove it nonzero with an actual witness; and only then the comparison scalar — plus the shortest actual chain for C1
**Tools:** `src95_c4_transverse_extension_selmer.py`, `src11_gate_drill.py`
**Logs:** `src95-c4-transverse-extension-selmer.json`, `src11-gate-drill.json`
**Deliverable:** [`handoff/ATTACK-C4-TRANSVERSE-EXTENSION/`](../../packages/ATTACK-C4-TRANSVERSE-EXTENSION/) — [REPORT.md](../../packages/ATTACK-C4-TRANSVERSE-EXTENSION/), result.json, INPUTS.json, MANIFEST.json, `evidence/`, `logs/`, `code/`

**Result. Of the four requests, the second is fully settled and it is the one that mattered. STEP 1 — the extension is now a dictionary rather than a phrase. With `ψ = χ₈χ_cyc` (RUN-091's second Jordan–Hölder character) and the ordering fixed once as `0 → ψ → V_Eis → 1 → 0`, the matrix `ρ(g) = (ψ(g), c(g); 0, 1)` has `ρ(gh) = ρ(g)ρ(h)` forcing `c(gh) = ψ(g)c(h) + c(g)`, so `c` is a 1-cocycle valued in `ψ` and `[c] ∈ Ext¹(1, ψ) = H¹(Q, Q₁₁(χ₈)(1))` — RUN-092's target line; the class is the **upper-right entry**, the frame is `e₁` spanning the sub with `e₂` any lift of the quotient, changing that lift conjugates by `(1, b; 0, 1)` and moves `c` by the coboundary `(ψ(g) − 1)b`, and the extension splits exactly when `[c] = 0`. All three identities are verified on explicit 2×2 matrices over `Z/11⁴`, 40 instances. The reverse ordering lands in `H¹(Q, Q₁₁(χ₈)(−1))` — a different group — so the convention is load-bearing and is fixed here once; a defect that flips it goes red. Nothing is lifted yet: what is pinned is the dictionary, not a class. STEP 2 — the local conditions, and the headline. For `V = Q₁₁(χ₈)(1)`: at 11, `χ₈` has conductor 8 hence is **unramified** there with `χ₈(11) = −1 ≠ 1`, so `H⁰ = 0` and `H² = H⁰(Q₁₁, Q₁₁(χ₈))^* = 0`, the local Euler characteristic at `p` gives `dim H¹(Q₁₁,V) = 1`, and Bloch–Kato gives `dim H¹_f = dim H⁰ + dim D_dR/Fil⁰ = 0 + 1 = 1` — **`H¹ = H¹_f`, so every class is automatically crystalline at 11**; at 2, `χ₈` is ramified so `I₂` acts nontrivially, `H⁰ = H² = 0` and the Euler characteristic at `ℓ ≠ p` gives `dim H¹(Q₂,V) = 0`, the whole local group vanishing; at infinity, `χ₈` is even so conjugation acts by `−1` and `V^c = 0`, and 11 is odd so `H¹(Gal(C/R),V) = 0`; away from `2·11`, `(χ₈χ_cyc)(Frob_ℓ) = χ₈(ℓ)ℓ` is never 1 (checked at twelve primes) so `H⁰ = 0` and `H¹(Q_ℓ,V) = 0`. **All four vacuous: `H¹_f(Q,V) = H¹(G_{Q,{2,11,∞}}, V)`, so membership in `H¹_f` — as opposed to merely `H¹` — costs nothing, and the requirement the round singled out as most important is discharged.** The global Euler characteristic cross-checks it: `χ = −1 + 0 = −1` with `H⁰ = 0` gives `dim H¹ = 1 + dim H²`, which agrees with RUN-092's `dim H¹_f = 1` exactly when `H² = 0`. STEP 3 — the target line's generator is genuinely nonzero (`ε` is the fundamental unit, so `ε = η¹¹` would force `1 = 11n`; witness `q = 23`, where `ε ↦ 6` and `6^{(q−1)/11} = 13 ≠ 1`), but **the transverse class's nonvanishing is UNMEASURED** and the gate refuses to claim it — every witness the request lists needs the overconvergent θ-critical eigenspace. One route is now closed by computation: `B_{2,χ₈} = 2`, so `L(−1,χ₈) = −1` and `v₁₁ = 0` — **11 does not divide the L-value, so there is no classical mod-11 Eisenstein congruence**, and the transverse class cannot come from a cuspidal congruence; if nonzero it must come from the θ-critical overconvergent direction. STEP 4 — the scalar is untouched by instruction, and guessing it from `log₁₁(ε)/(11√2) ≡ 5` is a planted defect. On "one scalar or two": of the four frames only Eichler–Shimura is non-canonical, so its normalisation and `c_ES/Kum` are not independent — they collapse to one exactly when the ES frame is pinned to the canonical quotient frame, and that pinning is itself UNMEASURED. AND FOR C1: the auxiliary transverse extension does **not** transport to the partners — it is partner-blind, while `B̂_i` needs the Euler system class of the **pair**. The one missing map is the **Rankin–Eisenstein / Beilinson–Flach class for `(f_i, g)` over the slope-1 Coleman family**, and it sits **upstream** of the Eichler–Shimura comparison, not downstream; neither of the two candidate chains as posed is right. Consequence: **closing C4 does not unblock C1** — different objects, either order.**

---

## Step 1 — the extension, as a dictionary

```text
          ( ψ(g)   c(g) )
  ρ(g) =  (   0      1  )          0 → ψ → V_Eis → 1 → 0,   ψ = χ₈·χ_cyc
```

| asked | answered |
| --- | --- |
| extension group | `Ext¹(1, ψ) = H¹(Q, Q₁₁(χ₈)(1))` — the target line |
| extension cocycle | `c(gh) = ψ(g)c(h) + c(g)`, forced by `ρ(gh) = ρ(g)ρ(h)` |
| basis / frame | `e₁` spans the sub `ψ`; `e₂` is any lift of the trivial quotient |
| which off-diagonal entry | the **upper-right** `c(g)` |

Changing the lift `e₂ ↦ e₂ + b e₁` conjugates by `(1, b; 0, 1)` and moves `c` by `(ψ(g) − 1)b`; split ⟺ `[c] = 0`. Verified on explicit matrices over `Z/11⁴`, 40 instances (matrix law, coboundary law, split criterion). **Reverse ordering** (sub trivial): `Ext¹(ψ, 1) = H¹(Q, Q₁₁(χ₈)(−1))`, not the target — the convention is fixed once, and flipping it is a planted defect.

**Lifting status:** the identities hold integrally over `Z/11ⁿ` for every `n` tested; no class has been exhibited, so no lifting obstruction arises yet.

## Step 2 — the local conditions

| place | computed inputs | `H⁰` | `H²` | `H¹` | `H¹_f` | verdict |
| --- | --- | --- | --- | --- | --- | --- |
| **11** | conductor 8 ⇒ unramified at 11; `χ₈(11) = −1` | 0 | 0 | **1** | **1** | `H¹ = H¹_f` — vacuous |
| **2** | `χ₈` ramified at 2, `I₂` nontrivial | 0 | 0 | **0** | 0 | vacuous |
| **∞** | `χ₈(−1) = 1`, `c` acts by `−1`, `V^c = 0`; 11 odd | — | — | **0** | 0 | vacuous |
| **away** | `χ₈(ℓ)ℓ ≠ 1` at 12 primes | 0 | 0 | **0** | 0 | vacuous |

```text
⇒  H¹_f(Q, V) = H¹(G_{Q,{2,11,∞}}, V)      MEMBERSHIP IS AUTOMATIC
```

Cross-check (global Euler characteristic): `χ = −dim V + dim V^c = −1`, `H⁰ = 0` ⇒ `dim H¹ = 1 + dim H²`; RUN-092 gives `dim H¹_f = 1`; consistent iff `H² = 0`.

## Step 3 — nonvanishing

* **generator**: `κ(ε) ≠ 0`. `ε` is the fundamental unit of `Q(√2)`; witness `q = 23` (`√2 ≡ 5`, `ε ↦ 6`, `6^{(q−1)/11} = 13 ≠ 1`). `COMPUTED_HERE`.
* **transverse class**: `UNMEASURED`. Every listed witness needs the overconvergent θ-critical eigenspace (slope-1, nebentypus `χ₈`).
* **obstruction computed**: `B_{2,χ₈} = 2` ⇒ `L(−1,χ₈) = −1` ⇒ `v₁₁ = 0` ⇒ **no classical mod-11 Eisenstein congruence**. The transverse class cannot be produced by a cuspidal congruence. `COMPUTED_HERE`.

## Step 4 — the scalar, and how many

`c_ES/Kum` `UNMEASURED`; guessing it from `log₁₁(ε)/(11√2) ≡ 5` is a defect. **One scalar, not two — conditionally**: only the ES frame is non-canonical (RUN-092), so its normalisation and `c_ES/Kum` are dependent; they collapse to one exactly when the ES frame is pinned to the canonical quotient frame, and that pinning is itself `UNMEASURED`.

## C1 — the shortest actual chain, and the one missing map

```text
  1. auxiliary family g = E_k(1,χ₈), weight direction          BUILT   (RUN-091)
  2. partner f_i (389.a1 / 19.a1), ordinary at 11              BUILT   (RUN-091)
  3. Rankin–Eisenstein / Beilinson–Flach class for (f_i, g)    MISSING  ← the one map
  4. Perrin-Riou/Coleman image: R_i(X,t) = c_f(X)U_i(X,t)      follows from 3
  5. X-leading term                                            standard once 4 exists
  6. projection through the canonical Eisenstein quotient      BUILT   (RUN-091)
  7. B̂_i
```

Neither posed route is right: the auxiliary generalised eigenspace is **partner-blind**, and the pseudocharacter/quotient steps are already built. The pair's BF class is the single input and it is **upstream** of the ES comparison. **Closing C4 does not unblock C1.**

## The drill

**462 defects, 462 caught by the check named for each**, 0 uncaught, 0 caught by the wrong check, 123 controls undisturbed, over 138 checks.

The 7 planted for this gate, over its four checks `c4-extension`, `c4-local-conditions`, `c4-nonvanishing`, `c4-transverse` — each caught by the check named for it (the verdict check `c4-transverse` goes red alongside, since it rests on all three):

| planted defect | named check | went red |
| --- | --- | --- |
| the transverse extension is taken with the trivial character as the sub, landing in the wrong Ext group | `c4-extension` | `c4-extension`, `c4-transverse` |
| chi_8 is given conductor 11, so it is read as ramified at 11 | `c4-local-conditions` | `c4-local-conditions`, `c4-nonvanishing`, `c4-transverse` |
| the Bloch-Kato filtration dimension dim D_dR/Fil^0 is taken as 0 | `c4-local-conditions` | `c4-local-conditions`, `c4-transverse` |
| the Tate twist is dropped, so V is read as Q_11(chi_8) | `c4-local-conditions` | `c4-local-conditions`, `c4-transverse` |
| the transverse class is called nonzero with no witness | `c4-nonvanishing` | `c4-nonvanishing`, `c4-transverse` |
| the comparison scalar is guessed from log_11(eps)/(11 sqrt2) = 5 | `c4-nonvanishing` | `c4-nonvanishing`, `c4-transverse` |
| C4 is promoted from UNMEASURED to PASS with no nonvanishing witness | `c4-transverse` | `c4-transverse` |


## What this round does not claim

* **No transverse class is constructed**, and its nonvanishing is not asserted — only the dictionary, the local conditions, and one closed route.
* **No scalar.** `c_ES/Kum` is untouched; the mod-11 value of the Kummer logarithm is explicitly not used to suggest it.
* The Bellaïche–Dasgupta / Loeffler–Rivero A4.5 reference stays the **shape** of the needed theorem.
* Nothing on C5–C7, C9–C10; no residual graph; no BSD scalar. **BSD remains open.**