# RUN-092 — Attack C4, the arrow `ε = 1+√2 ⟶ Θ_aux`: the target turns out to be a **line** — `χ₈` is the character of `Q(√2)`, whose unit rank is 1 and on whose fundamental unit `σ` acts by `−1`, so `H¹_f(Q, Q₁₁(χ₈)(1))` is one-dimensional and spanned by `κ(ε)`, which forces any identification up to a single scalar — and the Eisenstein weight jet is **not** in it: divided by `char₂` its `X`-direction is `log₁₁(ℓ)` for `χ₈` and `χ₅` alike, so it is the cyclotomic class, blind to the field. C4 UNMEASURED, the naive arrow refuted on type grounds, the missing object named as the extension class at the θ-critical Eisenstein point

**Arm:** 數學戰士「墜衡」 / AMRAL Research Lab
**Subject:** the instruction after [RUN-091](../RUN-091/) — attack C4 directly, the actual identification of the q-blind auxiliary leading tensor, without stopping at the Kummer unit as a q-blind candidate; with the explicit prohibition on declaring an identification merely because both sides are nonzero, both are units, or the mod-11 residues agree
**Tools:** `src94_attack_c4_theta_aux.py`, `src11_gate_drill.py`
**Logs:** `src94-attack-c4-theta-aux.json`, `src11-gate-drill.json`
**Deliverable:** [`handoff/ATTACK-C4-THETA-AUX/`](../../packages/ATTACK-C4-THETA-AUX/) — [REPORT.md](../../packages/ATTACK-C4-THETA-AUX/), result.json, INPUTS.json, MANIFEST.json, `evidence/`, `logs/`, `code/`

**Result. C4 is UNMEASURED, and the round's value is in two structural findings that make that verdict informative rather than empty. FINDING 1 — the target is a line. `χ₈` is not merely an even character of conductor 8: it *is* the Kronecker symbol `(2/·)`, the character of `K = Q(√2)`, checked here at twelve arguments against the table Attack 09 uses. `K` is real quadratic, so its unit rank is `r₁ + r₂ − 1 = 1`; in `Z[√2]`, `(1+√2)(√2−1) = 1`, so `ε⁻¹ = √2 − 1` and `σ(ε) = 1 − √2 = −ε⁻¹` — the nontrivial automorphism acts by `−1` on `log|ε|`. The trivial eigenspace is the unit group of `Q`, of rank 0, so the whole rank sits in the `χ₈`-part: `dim_Q (O_K^× ⊗ Q)^{χ₈} = 1`, derived in the gate from the rank and the sign rather than asserted. By Soulé / Bloch–Kato (cited) `H¹_f(Q, Q₁₁(χ₈)(1))` is that `χ₈`-part tensored up, hence a **one-dimensional `Q₁₁`-line spanned by the Kummer class `κ(ε)`**; 11 is inert in `K` (2 is a non-residue mod 11) so the localisation sits in the unramified `Q₁₁(√2)`, and the Bloch–Kato logarithm is nonzero, `log₁₁(ε)/(11√2) ≡ 5 (mod 11)`, recomputed here in `Z[√2]/11²`. **The consequence is the compression: if `Θ_aux` is a nonzero class in that group then it *is* a scalar multiple of `κ(ε)` — there is nowhere else for it to go — so C4 collapses to "does `Θ_aux` land in this one line, and what is the one scalar".** FINDING 2 — the object RUN-091 built is not that class, and the reason is a type mismatch rather than a numerical one. Divide the trace jet's `X`-coefficient by `char₂(ℓ) = χ₈(ℓ)ℓ` and what is left is `log₁₁(ℓ)`, the cyclotomic class at `Frob_ℓ`. Run the identical construction with `χ₅`, the character of `Q(√5)` (which passes Attack 09's own criterion, RUN-077), and the quotient is the **same `log₁₁(ℓ)` at every prime tested**. So the weight direction of an Eisenstein family carries no information about which real quadratic field is in play: it is `char₂ ⊗ (log₁₁ ∘ χ_cyc)`, the `X`-derivative of the character `χ₈χ_cyc^{1+X}` — a deformation **within** the reducible locus, living in `H¹(Q, Q₁₁) ≅ Hom(G_Q, Q₁₁)` — whereas `κ(ε)` lives in `H¹_f(Q, Q₁₁(χ₈)(1))` and is specific to `Q(√2)`. Different groups; the naive arrow "trace jet ↔ Kummer class" is refuted on type grounds, exactly the refusal the round demanded. **What `Θ_aux` would have to be instead is named: not the deformation along the reducible locus but the one transverse to it — the non-split extension class carried by the generalised eigenspace at the θ-critical Eisenstein point, which does live on the line `κ(ε)` spans** — and the theorem that would identify it with `κ(ε)` up to an L-invariant is cited only as the *shape* of what is needed (Bellaïche–Dasgupta / Loeffler–Rivero A4.5 type), never as established. The required checks all carried: q-blindness under `q_f ↦ u q_f` on 40 instances with the construction's inputs limited to `ε, 11, √2, χ₈`; non-circularity (none of `q_f`, `C_π`, `c_lead`); the four frames tabulated with the Eichler–Shimura one marked as the single non-canonical frame needing a comparison scalar; and the round's required mutation `Θ'_aux = q_f·Θ_aux` turning `c4-verdict` red. A planted defect that claims the identification because both sides are nonzero units also goes red.**

---

## Finding 1 — the target is a line

| step | computed | value |
| --- | --- | --- |
| `χ₈` vs Kronecker `(2/·)` at 12 arguments | identical | ✓, and `χ₈(−1) = 1`, conductor 8 = disc `K` |
| signature of `K = Q(√2)` | `(r₁, r₂)` | `(2, 0)` |
| unit rank | `r₁ + r₂ − 1` | **1** |
| `ε⁻¹` in `Z[√2]` | `(1+√2)(√2−1) = 1` | `√2 − 1` |
| `σ(ε)` | `1 − √2` | `= −ε⁻¹`, so `σ` acts by `−1` |
| trivial eigenspace | units of `Q` | rank **0** |
| **`dim_Q (O_K^× ⊗ Q)^{χ₈}`** | `1 − 0`, derived | **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 |

Soulé / Bloch–Kato (CITED): `H¹_f(Q, Q₁₁(χ₈)(1))` is the `χ₈`-part of the global units ⊗ `Q₁₁`. Hence it is a **line spanned by `κ(ε)`**, and any nonzero class in it is a scalar multiple of `κ(ε)`.

The dimension is *derived* in the gate — from the rank and the sign of the σ-action, with the plus part's rank 0 — not read back from the claim; defects that assert dimension 2, rank 2, or `σ = +1` each turn `c4-target-space` red.

## Finding 2 — the weight jet is blind to the field

| `ℓ` | 3 | 7 | 13 | 17 | 19 | 23 | 29 |
| --- | --- | --- | --- | --- | --- | --- | --- |
| `char₂` for `χ₈` = `χ₈(ℓ)ℓ` | −3 | 7 | −13 | 17 | −19 | 23 | −29 |
| `X`-coeff ÷ `char₂` (`χ₈`) | 847 | 220 | 374 | 935 | 517 | 1111 | 517 |
| `char₂` for `χ₅` = `χ₅(ℓ)ℓ` | −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, for two different quadratic fields. So:

```text
the jet's X-direction  =  char₂ ⊗ (log₁₁ ∘ χ_cyc)   ∈  H¹(Q, Q₁₁) = Hom(G_Q, Q₁₁)
                                                       — a deformation ALONG the reducible locus
κ(ε)                                                ∈  H¹_f(Q, Q₁₁(χ₈)(1))
                                                       — specific to Q(√2)
```

Different groups. The identification cannot be made by comparing these two, and no numerical agreement would license it.

## The frames

| frame | canonical? | comparison scalar? |
| --- | --- | --- |
| Kummer (`ε`, `κ(ε)`) | yes | no |
| Eisenstein family (`E_k(1,χ₈)`, `k = 2 + X`) | yes (RUN-091) | no |
| **Eichler–Shimura** (Attack 09 (16), its `X`-leading term) | **no** — fixed only after the period frame; `C` is the unknown | **yes** |
| quotient (the Eisenstein transport, carrying `q_f`) | yes as a quotient (RUN-091) | no — but must be avoided, `q_f` unknown |

## The required checks

* **q-blindness** — fixed under `q_f ↦ u·q_f`, 40 instances; inputs are `ε`, `11`, `√2`, `χ₈` only.
* **weight-direction jet** — exists and was computed (RUN-091, re-derived), but Finding 2 shows it is the wrong class.
* **non-circularity** — none of `q_f`, `C_π`, `c_lead` enters.
* **mutation** — `Θ'_aux = q_f·Θ_aux` ⇒ `c4-verdict` red; a claimed identification on a numerical match ⇒ red; `UNMEASURED → PASS` ⇒ red.

## The drill

**455 defects, 455 caught by the check named for each**, 0 uncaught, 0 caught by the wrong check, 119 controls undisturbed, over 134 checks.

The 7 planted for this gate, over its three checks `c4-target-space`, `c4-jet-direction`, `c4-verdict` — each caught by the check named for it (a defect in the target space or the jet also turns `c4-verdict` red, since the verdict rests on both):

| planted defect | named check | went red |
| --- | --- | --- |
| the candidate Theta_aux is built through the quotient, Theta' = q_f Theta | `c4-verdict` | `c4-verdict` |
| an identification is claimed because both sides are nonzero units | `c4-verdict` | `c4-verdict` |
| C4 is promoted from UNMEASURED to PASS with no comparison map | `c4-verdict` | `c4-verdict` |
| sigma is said to act by +1 on the fundamental unit of Q(sqrt 2) | `c4-target-space` | `c4-target-space`, `c4-verdict` |
| the unit rank of Q(sqrt 2) is taken as 2 | `c4-target-space` | `c4-target-space`, `c4-verdict` |
| the chi_8 eigenspace of the units is said to be 2-dimensional | `c4-target-space` | `c4-target-space`, `c4-verdict` |
| the Eisenstein weight jet is said to distinguish the quadratic field | `c4-jet-direction` | `c4-jet-direction`, `c4-verdict` |


## What this round does not claim

* **No identification.** `Θ_aux` is not exhibited, `m` is unknown, and the extension class at the θ-critical point is not constructed.
* **No L-invariant**, no `η_pdR`, no `κ_det`, no residual graph — none of them was touched, per the round's own exclusion list.
* The Bellaïche–Dasgupta / Loeffler–Rivero A4.5 reference is cited as **the shape of the needed theorem**, not as a result applied here.
* **BSD remains open.**