# ATTACK C4, step 2 — the transverse extension, its Selmer line, and the local conditions

**From:** 數學戰士「墜衡」 (the independent verification line, branch `agent/bsd-verification-zhuiheng`)
**Answering:** the four requests after RUN-092 — (1) extract the transverse extension as an actual object; (2) prove it lands in the right Selmer line, local conditions separated; (3) prove it is nonzero with an actual witness; (4) only then the comparison scalar. Plus: the shortest actual chain for C1.
**Gate:** `code/src95_c4_transverse_extension_selmer.py` · drill `code/src11_gate_drill.py` (checks `c4-extension`, `c4-local-conditions`, `c4-nonvanishing`, `c4-transverse`)
**Machine-readable:** [`result.json`](./files/result.json) · **evidence:** [`evidence/`](./files/evidence/) · **this line's report:** [`reports/RUN-093-C4-TRANSVERSE-EXTENSION-SELMER.md`](../../runs/RUN-093/)

---

## 0. The answer, in the format asked for

```text
C4 STATUS:

target Selmer line:
  H^1_f(Q, Q_11(chi8)(1))
  dimension = 1                       (RUN-092: the chi_8-part of the units of Q(sqrt2))
  generator  = kappa(epsilon), epsilon = 1 + sqrt2, NONZERO (fundamental unit; witness q = 23)

transverse extension:
  constructed     = NO
  extension group = Ext^1(1, psi) = H^1(Q, Q_11(chi8)(1))     [psi = chi_8 * chi_cyc]
  cocycle         = c(gh) = psi(g) c(h) + c(g)
  basis/frame     = e_1 spans the sub psi ; e_2 any lift of the trivial quotient
  which entry     = the UPPER-RIGHT entry of  rho(g) = ( psi(g)  c(g) ; 0  1 )
  ordering fixed  = 0 -> psi -> V_Eis -> 1 -> 0      (the reverse gives H^1(Q, Q_11(chi8)(-1)), NOT the target)
  local conditions:
      at 11              VACUOUS   H^1(Q_11,V) = H^1_f(Q_11,V), both dimension 1
      at 2 (| 8)         VACUOUS   H^1(Q_2,V) = 0
      at infinity        VACUOUS   11 odd, so H^1(Gal(C/R),V) = 0
      away from 2*11     VACUOUS   H^1(Q_l,V) = 0
      ==> MEMBERSHIP IN H^1_f IS AUTOMATIC
  nonzero witness = NONE

comparison:
  Theta_aux = c_ES/Kum * kappa(epsilon)
  c_ES/Kum still UNMEASURED because the class-level identification is not done:
      membership is settled, nonvanishing is not, and the round's instruction is
      class first, scalar last. The value log_11(eps)/(11 sqrt2) = 5 is explicitly
      NOT used to suggest a scalar.

C4 verdict = UNMEASURED
  split:  C4-membership    CLOSED
          C4-nonvanishing  OPEN
          C4-exact-scalar  OPEN
```

```text
minimal map still missing for C1 =
  the Rankin-Eisenstein / Beilinson-Flach class for the PAIR (f_i, g) over the
  slope-1 Coleman family. It is UPSTREAM of the Eichler-Shimura comparison, not
  downstream of it, and the auxiliary transverse extension is NOT on its path.
```

---

## 1. Step 1 — the extension as an actual object, convention fixed once

Write `ψ = χ₈·χ_cyc`, the second Jordan–Hölder character identified in RUN-091. In the basis `(e₁, e₂)` with `e₁` spanning the sub and `e₂` any lift of the quotient:

```text
          ( ψ(g)   c(g) )
  ρ(g) =  (   0      1  )          0 → ψ → V_Eis → 1 → 0
```

`ρ(gh) = ρ(g)ρ(h)` **forces** `c(gh) = ψ(g)c(h) + c(g)` — so `c` is a 1-cocycle valued in `ψ`, and

```text
  [c] ∈ Ext¹(1, ψ) = H¹(Q, Q₁₁(χ₈)(1))
```

which is exactly RUN-092's target line. Changing the lift `e₂ ↦ e₂ + b·e₁` conjugates `ρ` by `(1 b; 0 1)` and moves `c` by the coboundary `(ψ(g) − 1)·b`; the extension splits iff `[c] = 0`. **The class is the upper-right entry.**

All three identities — the matrix product forcing the cocycle law, the lift-change being a coboundary, and split ⟺ coboundary — are verified in the gate on explicit 2×2 matrices over `Z/11⁴`, on 40 instances.

**The opposite ordering is a different group.** With the sub trivial and the quotient `ψ`, the class lies in `Ext¹(ψ, 1) = H¹(Q, Q₁₁(χ₈)(−1))` — **not** the target. So the convention is not cosmetic, and it is fixed here once: **the sub is `ψ`**. A planted defect that flips it turns `c4-extension` red.

**Lifting status:** nothing is being lifted yet. The identities hold integrally over `Z/11ⁿ` for every `n` tested; what is fixed here is the dictionary (group, cocycle, entry, frame), not a class. No class has been exhibited, so no lifting obstruction arises.

## 2. Step 2 — the local conditions, all four, all vacuous

`V = Q₁₁(χ₈)(1)`. Inputs computed; formulas cited (local Euler characteristics, local Tate duality, Bloch–Kato, global Euler characteristic).

| place | inputs computed | `H⁰` | `H²` | `H¹` | `H¹_f` | verdict |
| --- | --- | --- | --- | --- | --- | --- |
| **11** (= p) | `χ₈` has conductor 8 ⇒ **unramified at 11**; `χ₈(11) = −1 ≠ 1` | 0 | 0 | **1** | **1** | `H¹ = H¹_f` — **vacuous** |
| **2** (∣ 8) | `χ₈` **ramified at 2**, `I₂` acts nontrivially | 0 | 0 | **0** | 0 | whole group vanishes — **vacuous** |
| **∞** | `χ₈(−1) = 1` (even) ⇒ `c` acts by `−1`, `V^c = 0`; 11 odd | — | — | **0** | 0 | **vacuous** |
| **away from 2·11** | `(χ₈χ_cyc)(Frob_ℓ) = χ₈(ℓ)·ℓ`, never `1` (checked at 12 primes) | 0 | 0 | **0** | 0 | **vacuous** |

Working at 11: `H⁰(Q₁₁,V) = 0` because `χ₈χ_cyc` is nontrivial (`χ_cyc` is infinitely ramified); `H²(Q₁₁,V) = H⁰(Q₁₁, V*(1))^* = H⁰(Q₁₁, Q₁₁(χ₈))^* = 0` because `χ₈` is unramified there with `Frob₁₁ ↦ −1 ≠ 1`; the local Euler characteristic at `p` then gives `dim H¹ = 1`; and Bloch–Kato gives `dim H¹_f = dim H⁰ + dim D_dR/Fil⁰ = 0 + 1 = 1` (the Tate twist puts `Fil⁰ = 0`). **The two coincide: every class is automatically crystalline at 11.**

```text
  ⇒  H¹_f(Q, V) = H¹(G_{Q,{2,11,∞}}, V)
  ⇒  MEMBERSHIP IN H¹_f IS AUTOMATIC for any class unramified outside 2·11,
     which the extension is by construction.
```

**Cross-check.** The global Euler characteristic gives `χ = −dim V + dim V^c = −1 + 0 = −1`, and `H⁰ = 0`, so `dim H¹ = 1 + dim H²`. RUN-092's unit computation gives `dim H¹_f = 1`. The two agree exactly when `H² = 0` — consistent, and an independent confirmation of the dimension.

**This settles step 2 of the request.** The `[Θ_aux] ∈ H¹_f` requirement, as opposed to merely `∈ H¹`, costs nothing here: there is no condition to impose.

## 3. Step 3 — nonvanishing: the generator yes, the transverse class no

**The target line's generator is genuinely nonzero.** `ε = 1 + √2` is the *fundamental* unit of `Q(√2)`: every unit is `±εⁿ`, so `ε = η¹¹` would force `1 = 11n`. Computational witness: `q = 23` splits in `K` (`√2 ≡ 5`), `ε ↦ 6` in `F_q`, and `6^{(q−1)/11} = 13 ≠ 1`, so `ε` is not an 11th power. Hence `κ(ε) ≠ 0`. `COMPUTED_HERE`.

**The transverse class's nonvanishing is UNMEASURED**, and the gate refuses to claim it. Every witness the request lists — non-semisimplicity of the generalised eigenspace, a nonzero off-diagonal cocycle, a reducibility ideal, a congruence module, a local restriction, an explicit matrix coefficient — needs the **overconvergent** θ-critical eigenspace, which needs slope-1 forms with nebentypus `χ₈`. Not a stdlib computation.

**But one route is now closed, by computation.** The Eisenstein congruence criterion:

```text
  B_{2,χ₈} = 8 · Σ_{a=1}^{8} χ₈(a)·B₂(a/8) = 2          (RUN-077's number, recomputed)
  L(−1, χ₈) = −B_{2,χ₈}/2 = −1
  v₁₁(L(−1,χ₈)) = 0
```

**11 does not divide the L-value, so there is no classical mod-11 Eisenstein congruence at this point.** The transverse class therefore cannot be produced by a cuspidal congruence; if it is nonzero it must come from the θ-critical *overconvergent* direction. That is a real narrowing of where the witness can come from — and a warning: the architecture cannot anchor on this class until someone shows it is not zero.

## 4. Step 4 — the scalar, and how many there really are

Not attempted, by instruction: class-level identification first. `c_ES/Kum` is `UNMEASURED`, and the gate treats guessing it from `log₁₁(ε)/(11√2) ≡ 5` as a defect (`c4-nonvanishing` goes red).

**On the "one scalar or two" question.** Of the four frames (RUN-092), only the Eichler–Shimura frame is non-canonical. So the ES frame's own normalisation and `c_ES/Kum` are *not independent*: fixing the ES frame fixes the comparison, and the two collapse to one. They collapse **exactly when** the ES frame is pinned to the quotient frame, which RUN-091 showed is canonical. So the answer is: **one scalar, provided the ES normalisation is pinned to the canonical quotient — and that pinning is itself UNMEASURED.** It is not two independent scalars, and it is not automatically one.

## 5. The C1 question — the shortest actual chain

**Finding: the auxiliary transverse extension does not transport to the partners at all.** It is a global object attached to `g` alone and is partner-blind; `B̂_i` needs the Euler system class of the **pair** `(E_i, g)`. C4's object and C1's object sit on *different paths* and meet only at the normalisation.

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

**Neither of the two candidate routes as stated.** It does not run `auxiliary generalised eigenspace → ES comparison → partner BF class`, because the auxiliary eigenspace is partner-blind. And it does not need `pseudocharacter → Galois rep → local quotient → ES map → BF class` either, because the pseudocharacter and the canonical quotient are already built (RUN-091). **The pair's BF class is the single input, and it sits upstream of the ES comparison, not downstream.**

**Consequence for sequencing:** closing C4 does **not** unblock C1. They need different objects — C4 needs the θ-critical extension, C1 needs the pair's BF class. Either can be attacked first, and neither waits on the other.

## 6. Claim labels

`COMPUTED_HERE`: the cocycle/coboundary dictionary on explicit matrices over `Z/11⁴`; every local input (conductor, `χ₈(11) = −1`, evenness, `χ₈(ℓ)ℓ ≠ 1` at 12 primes); the derived dimensions at all four places; `B_{2,χ₈} = 2`, `L(−1,χ₈) = −1`, `v₁₁ = 0`; `κ(ε) ≠ 0` with the `q = 23` witness.
`CITED_THEOREM`: local Euler characteristic formulas; local Tate duality; Bloch–Kato `dim H¹_f(Q_p,V) = dim H⁰ + dim D_dR/Fil⁰`; the global Euler characteristic formula; Soulé/Bloch–Kato for `H¹_f` as the unit part (RUN-092).
`DERIVED_FROM_VERIFIED_INPUTS`: membership in `H¹_f` is automatic; the extension's Ext group under the fixed ordering; that closing C4 does not unblock C1.
`UNMEASURED`: the transverse class itself, its nonvanishing, `c_ES/Kum`, the ES-frame pinning.
`FAILED`: nothing.

## 7. Status carried forward

```text
C1 UNMEASURED — auxiliary side built; the pair's BF class is the one missing map
C2 UNMEASURED — 11-adic valuation/nonvanishing improved, exact unit unresolved
C3 PASS
C4 UNMEASURED — membership CLOSED, nonvanishing OPEN, scalar OPEN
C5–C7 BLOCKED BY C4
C8 PARTIAL
C9–C10 NOT YET READY
BSD OPEN
```