# ATTACK C4 — the actual identification of Θ_aux, from the Kummer unit ε = 1 + √2

**From:** 數學戰士「墜衡」 (the independent verification line, branch `agent/bsd-verification-zhuiheng`)
**Answering:** the instruction after RUN-091 — make the arrow `ε ⟶ Θ_aux` real; do not stop at "the Kummer unit is a q-blind candidate"; either `C4 PASS` with an actual map, identification theorem, frame and normalisation, or `C4 UNMEASURED` with the gap compressed one layer and the missing map / cohomology class / comparison theorem named. **Do not claim an identification because both sides are nonzero, both are units, or their mod-11 residues agree.**
**Project:** `389.a1`, `p = 11`; auxiliary family `E₂(1,χ₈)`, critical slope-1 refinement
**Gate:** `code/src94_attack_c4_theta_aux.py` · drill `code/src11_gate_drill.py` (checks `c4-target-space`, `c4-jet-direction`, `c4-verdict`)
**Machine-readable:** [`result.json`](./files/result.json) · **evidence:** [`evidence/`](./files/evidence/) · **this line's report:** [`reports/RUN-092-ATTACK-C4-THETA-AUX.md`](../../runs/RUN-092/)

---

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

```text
C4 UNMEASURED

  existing source =
      κ(ε), the Kummer class of the fundamental unit ε = 1 + √2 of K = Q(√2).
      q-blind and non-circular (its inputs are ε, 11, √2, χ₈ — never q_f, C_π, c_lead).
      Bloch–Kato logarithm nonzero: log₁₁(ε)/(11√2) ≡ 5 (mod 11), recomputed here.
      NEW: it SPANS its home group — see Finding 1 — so it is not merely "a candidate".

  existing target =
      Θ_aux ∈ Sym^m(T_X*) ⊗ 𝒞_aux, the X-leading tensor of the meromorphic
      Eichler–Shimura comparison at the critical-slope Eisenstein point, Attack 09 (16).
      Neither m nor its value is known to anyone.

  missing comparison object =
      the map that places the ES leading tensor INTO H¹_f(Q, Q₁₁(χ₈)(1)) — concretely
      the NON-SPLIT EXTENSION CLASS carried by the generalised eigenspace at the
      θ-critical Eisenstein point, together with its Bloch–Kato exponential
      identification. RUN-091's weight jet is NOT that class (Finding 2): it is the
      deformation ALONG the reducible locus. The transverse direction is what is missing.

  missing theorem =
      a critical-slope Eisenstein comparison of Bellaïche–Dasgupta / Loeffler–Rivero
      A4.5 type, identifying that extension class with κ(ε) up to an explicit
      L-invariant. Cited here as the SHAPE of what is needed — not as established.

  smallest constructible precursor =
      the generalised eigenspace at the θ-critical Eisenstein point of E₂(1,χ₈) at
      level 88 — its dimension decides whether a non-split extension exists at all.
      Needs overconvergent forms of slope 1 with nebentypus χ₈; this tree's Manin
      engines are trivial-character only.

  next executable attack =
      either that generalised eigenspace, or — if a peer line supplies the ES leading
      tensor's value in ANY stated frame — the SINGLE SCALAR κ(ε) ↦ Θ_aux, which is
      now all that separates them, because the target is a line.
```

No identification is claimed. The two mod-11 numbers being nonzero units is explicitly refused as evidence, and a planted defect that accepts it turns `c4-verdict` red.

---

## 1. Finding 1 — the target is a LINE, so the identification is forced up to one scalar

`COMPUTED_HERE` for the arithmetic, `CITED_THEOREM` (Soulé / Bloch–Kato) for the identification of `H¹_f`.

χ₈ is not merely "an even character of conductor 8": it **is** the Kronecker symbol `(2/·)`, the character of `K = Q(√2)` — verified here against the table Attack 09 uses, at twelve arguments, and `χ₈(−1) = 1`, conductor 8 = disc `K`.

| step | computed | value |
| --- | --- | --- |
| signature of `K` | `(r₁, r₂)` | `(2, 0)` |
| unit rank | `r₁ + r₂ − 1` | **1** |
| `ε⁻¹` in `Z[√2]` | `(1+√2)(√2−1) = 1` | `ε⁻¹ = √2 − 1` |
| `σ(ε)` | `1 − √2 = −(√2−1)` | `= −ε⁻¹` |
| so `σ` acts on `log|ε|` by | | **−1** |
| trivial eigenspace | units of `Q` | rank **0** |
| therefore `dim_Q (O_K^× ⊗ Q)^{χ₈}` | `1 − 0` | **1** |
| 11 in `K` | `2` a non-residue mod 11 | **inert**; `K₁₁ = Q₁₁(√2)` unramified |
| `log₁₁(ε)/(11√2)` | in `Z[√2]/11²` | **`5` mod 11**, nonzero |

By Soulé / Bloch–Kato, `H¹_f(Q, Q₁₁(χ₈)(1))` is the χ₈-part of the global units tensored with `Q₁₁`. So it is a **one-dimensional `Q₁₁`-line, spanned by `κ(ε)`.**

**The consequence is the compression.** If `Θ_aux` is (the image of) a nonzero class in that group, then it *is* a scalar multiple of `κ(ε)` — there is nowhere else for it to go. C4 therefore reduces to exactly two questions:

```text
(a) does Θ_aux land in this one line?
(b) what is the one scalar?
```

which is strictly smaller than the question this round opened with.

The eigenspace dimension is **derived** in the gate (from the rank and the sign of the σ-action), not asserted; a planted defect that claims dimension 2 turns `c4-target-space` red, as does one that flips the σ-action.

## 2. Finding 2 — the Eisenstein weight jet is NOT that class, on type grounds

`COMPUTED_HERE.`

RUN-091 built the trace jet `𝒯_X(g)(Frob_ℓ) ≡ 1 + χ₈(ℓ)ℓ + χ₈(ℓ)·ℓ·log₁₁(ℓ)·X (mod X²)`. Divide the `X`-coefficient by `char₂(ℓ) = χ₈(ℓ)ℓ`, and what remains is `log₁₁(ℓ)` — the cyclotomic class `log₁₁ ∘ χ_cyc` at `Frob_ℓ`. Now run the *same* construction with `χ₅`, the character of `Q(√5)` (which passes Attack 09's own non-vanishing criterion, RUN-077):

| `ℓ` | 3 | 7 | 13 | 17 | 19 | 23 | 29 |
| --- | --- | --- | --- | --- | --- | --- | --- |
| `char₂` for `χ₈` | −3 | 7 | −13 | 17 | −19 | 23 | −29 |
| `X`-coeff ÷ `char₂` (`χ₈`) | 847 | 220 | 374 | 935 | 517 | 1111 | 517 |
| `char₂` for `χ₅` | −3 | −7 | −13 | −17 | 19 | −23 | 29 |
| `X`-coeff ÷ `char₂` (`χ₅`) | 847 | 220 | 374 | 935 | 517 | 1111 | 517 |
| `log₁₁(ℓ)` mod `11³` | 847 | 220 | 374 | 935 | 517 | 1111 | 517 |

**Identical at every prime.** The weight direction of an Eisenstein family carries no information about *which* real quadratic field is in play. It is `char₂ ⊗ (cyclotomic class)` — the `X`-derivative of the character `χ₈·χ_cyc^{1+X}` — a deformation **within the reducible locus**, living in `H¹(Q, Q₁₁) ≅ Hom(G_Q, Q₁₁)`.

`κ(ε)` lives in `H¹_f(Q, Q₁₁(χ₈)(1))` and is specific to `Q(√2)`. **Different groups.** So the naive arrow "trace jet ↔ Kummer class" is refuted on type grounds — not because two numbers failed to match, but because the two objects cannot be compared without a map that does not yet exist. This is precisely the refusal the round asked for, and it is why the object RUN-091 built cannot close C4.

**What Θ_aux would have to be instead:** not the deformation *along* the reducible locus, but the one **transverse** to it — the non-split extension carried by the generalised eigenspace at the θ-critical Eisenstein point. That class *does* live on the line `κ(ε)` spans.

## 3. The four frames, and which are canonical

| frame | what it is | canonical? | comparison scalar needed? |
| --- | --- | --- | --- |
| **Kummer** | `ε ∈ O_K^×`, `κ(ε) ∈ H¹_f(Q, Q₁₁(χ₈)(1))` | **yes** — the line it spans is canonical (`ε` itself up to `±1` and `ε ↦ ε⁻¹`) | no |
| **Eisenstein family** | `E_k(1,χ₈)`, weight coordinate `X`, `k = 2 + X` | **yes** — explicit at every weight (RUN-091) | no |
| **Eichler–Shimura** | the meromorphic ES comparison of Attack 09 (16) and its `X`-leading term | **no** — fixed only after the period frame is chosen; its leading constant `C` is the unknown | **yes** |
| **quotient** | the Eisenstein quotient transport, carrying `q_f` | **yes** as a quotient (RUN-091: the two Jordan–Hölder characters are distinct, so the one-dimensional quotient is canonical) — but `q_f` itself is unknown, which is exactly why `Θ_aux` must not be built through it | no, but must be avoided |

## 4. The required checks

| check | result |
| --- | --- |
| **q-blindness** — under the pure quotient reframe `q_f ↦ u·q_f`, `Θ_aux` fixed | **fixed**, 40 instances. The candidate's inputs are `ε`, `11`, `√2`, `χ₈` and nothing else |
| **weight-direction jet** — a genuine first-order `X`-tensor, not only a constant Kummer value | the `X`-direction **exists and was computed** (RUN-091, re-derived here) — but Finding 2 shows it is the *cyclotomic* class, not the unit class, so it is not the tensor C4 needs |
| **frame compatibility** | table in §3 — one frame (ES) needs a comparison scalar; the others are canonical |
| **non-circularity** | the construction uses none of `q_f`, `C_π`, `c_lead` |
| **mutation** | `Θ'_aux = q_f·Θ_aux` ⇒ `c4-verdict` **RED**; controls stay green |

## 5. Claim labels

`COMPUTED_HERE`: `χ₈ = (2/·)`; the unit rank, `ε⁻¹`, `σ(ε) = −ε⁻¹`, the derived eigenspace dimension `1`; 11 inert; `log₁₁(ε)/(11√2) ≡ 5`; the field-blindness table for `χ₈` vs `χ₅`; q-blindness on 40 instances.
`CITED_THEOREM`: Soulé / Bloch–Kato (`H¹_f` is the χ-part of the units); the Bellaïche–Dasgupta / Loeffler–Rivero A4.5 *shape* of the needed critical-slope comparison.
`DERIVED_FROM_VERIFIED_INPUTS`: that any nonzero class in the target is a multiple of `κ(ε)`; that the trace jet and `κ(ε)` lie in different groups.
`UNMEASURED`: `Θ_aux` itself, `m`, the extension class at the θ-critical point, the scalar.
`FAILED`: nothing.

## 6. Status carried forward

```text
C1 UNMEASURED — auxiliary side built, partner maps missing
C2 UNMEASURED — 11-adic valuation/nonvanishing part improved, exact unit unresolved
C3 PASS
C4 UNMEASURED — target is a LINE; naive arrow refuted; missing object named
C5–C7 BLOCKED BY C4
C8 PARTIAL
C9–C10 NOT YET READY
BSD OPEN
```