# RUN-091 — Attack C1, the actual-object construction: the auxiliary side built in full — the Eisenstein family in the weight direction, its pseudocharacter, the trace jet `𝒯_X(g) mod X²` at eight Frobenius elements, and the canonical one-dimensional Eisenstein quotient that PC-001 §4 warns may not exist — the partner side named as the one remaining object, so C1 stays UNMEASURED; and Attack B computed: both congruence numbers are 11-units, so `Π_E/Π_0` is a unit and the partner ratio is nonzero

**Arm:** 數學戰士「墜衡」 / AMRAL Research Lab
**Subject:** the instruction after [RUN-090](../RUN-090/) — stop summarising and stop extending the symbolic framework; attack the C1 actual-object construction directly, turning the first UNMEASURED blocker into a measurable arithmetic object rather than re-verifying Round 030's algebra, ideally along `𝒯_X(g) mod X²` with a stated pseudocharacter identification; if stuck, drop to Attack B without idling
**Tools:** `src93_attack_c1_actual_object.py`, `src11_gate_drill.py`
**Logs:** `src93-attack-c1-actual-object.json`, `src11-gate-drill.json`
**Deliverable:** [`handoff/ATTACK-C1-ACTUAL-OBJECT/`](../../packages/ATTACK-C1-ACTUAL-OBJECT/) — [REPORT.md](../../packages/ATTACK-C1-ACTUAL-OBJECT/), result.json, INPUTS.json, MANIFEST.json, `evidence/`, `logs/`, `code/`

**Result. C1 is still UNMEASURED, but it is no longer a blocker with nothing behind it: the auxiliary half of the construction is built here, explicitly, and the missing half is one named object. The auxiliary family is the Eisenstein family through `g = E₂(1,χ₈)` in the weight direction, and because Eisenstein series are explicit at every weight the family is writable down — `a_n(X) = Σ_{d|n} χ₈(d)·d·⟨d⟩^X`. Its Galois pseudocharacter is the sum of the two Jordan–Hölder characters of the reducible representation, deformed: `𝒯_X(Frob_ℓ) = 1 + χ₈(ℓ)ℓ^{1+X}` for `ℓ ≠ 11`, with `char₁ = 1` and `char₂ = χ₈·χ_cyc`. Expanding `ℓ^{1+X} = ℓ(1 + X log₁₁ℓ + O(X²))` gives the trace jet `𝒯_X(g)(Frob_ℓ) ≡ 1 + χ₈(ℓ)ℓ + χ₈(ℓ)·ℓ·log₁₁(ℓ)·X (mod X²)`, computed at eight Frobenius elements mod `11³` — `ℓ = 3` gives constant `1329` and `X`-coefficient `121`, `ℓ = 5` gives `1327` and `1111`, and so on — whose constant term is the **actual** q-expansion coefficient of `E₂(1,χ₈)` at every one of them (RUN-077's numbers, recomputed), with the Iwasawa logarithm verified additive on six pairs and landing in `11Z₁₁`. Three things then cohere on actual objects. **The two characters are distinct** (`Frob_3`: `1` vs `χ₈(3)·3 = 8` mod 11), so the residual representation is multiplicity-free and the one-dimensional Eisenstein quotient exists and is **canonical** — which is exactly the hazard PC-001 §4 raises ("generic 輔助 Galois representation 不提供 Eisenstein 的一維商"), now discharged, and a canonical quotient cannot be chosen per partner, closing the one route by which the quotient transport could have been partner-dependent. **At `ℓ = 11` the two characters take the values `1` and `χ₈(11)·11 = −11`, exactly the two roots of the Hecke polynomial `U² + 10U − 11 = (U − 1)(U + 11)`** — so the critical slope-1 refinement `f_β` that PC-001 and Attack 09 actually use is the `χ₈χ_cyc` branch: the pseudocharacter and the refinement are one object seen twice. **Both partners are ordinary at 11** (`a₁₁ = −4`, unit root 7; `a₁₁ = 3`, unit root 3). What is still absent is `B̂_i` itself — the partner's `X`-leading Beilinson–Flach/Rankin class pushed through that quotient — i.e. `R_i(X,t) = c_f(X)U_i(X,t)`, which needs slope-1 overconvergent modular symbols and is not a stdlib computation; neither peer line has it and both say so. So `C_{π,E}` and `C_{π,0}` still do not exist as separate objects and nothing about their equality is asserted. **Attack B moved for real.** With `D_i = Π_i·E_i` and Hida's theorem that the adjoint `p`-adic L-value generates the congruence ideal, `v₁₁(Π_i) = v₁₁(η_{f_i})`. At level 389 this gate computes the **generalised** mod-11 eigenspace of `f`'s Hecke system in the dual Manin space (dim 65 = 2g+1, g = 32) — `ker((T_q − a_q)^8)` for `q = 2,3,5`, intersected — and finds it **2-dimensional, equal to the ordinary eigenspace**: the `±` pair of `f` alone, so the localised Hecke algebra is `Z₁₁`, no cusp form is congruent to `f` mod 11, `v₁₁(η) = 0`. At level 19, `dim S₂(Γ₀(19)) = 1` — one newform, nothing to be congruent to — so `v₁₁(η) = 0` too. Hence **`Π_E/Π_0` is an 11-adic unit**, and `A_0(0)/A_E(0) ≡ 6·(Π_E/Π_0) (mod 11)` is a **nonzero unit** — the non-vanishing PC-001's calibration needs, now resting on a computation. The residue itself stays UNMEASURED and the gate refuses to claim one: Hida pins the ideal, not the value, and the period normalisation PC-001 §3 declines to fix, the Manin constants, and the non-optimality of `19.a1 = 19a2` all sit in the way. The round's required mutation — `C_{π,E} ↦ u·C_{π,E}` with `C_{π,0}` fixed — is carried and goes red.**

---

## What was asked, and what "construct" was allowed to mean

The instruction was explicit: not another audit of Round 030's algebra, but an attempt to make the first blocker into an arithmetic object; the `𝒯_X(g) mod X²` route named as the ideal one; a different construction allowed if more direct; and — if it cannot be done — `C1 UNMEASURED` with the smallest missing construction named, then straight to Attack B rather than idling.

So this round did not build a new model. It asked what `B̂_i` *is* and built every part of it that can be built. `B̂_i` is the image of the partner's `X`-leading Beilinson–Flach class under the auxiliary family's Eisenstein quotient, in a fixed frame. That decomposes into an **auxiliary** half (the family, its deformation, the quotient it is projected through) and a **partner** half (the class being projected). The auxiliary half turns out to be entirely explicit, because the auxiliary family is Eisenstein.

## The auxiliary side, constructed

| object | what it is here | label |
| --- | --- | --- |
| the family | `E_k(1,χ₈)` through `k = 2`: `a_n(X) = Σ_{d|n} χ₈(d)·d·⟨d⟩^X` | COMPUTED_HERE |
| the pseudocharacter | `𝒯_X(Frob_ℓ) = 1 + χ₈(ℓ)·ℓ^{1+X}` (`ℓ ≠ 11`) — the sum of `char₁ = 1` and `char₂ = χ₈·χ_cyc`, deformed | COMPUTED_HERE |
| the trace jet | `≡ 1 + χ₈(ℓ)ℓ + χ₈(ℓ)ℓ·log₁₁(ℓ)·X (mod X²)`, at 8 Frobenius elements mod `11³` | COMPUTED_HERE |
| the quotient | one-dimensional, and **canonical** because `char₁ ≠ char₂` | COMPUTED_HERE |
| the refinement | slope-1 `U₁₁ = −11 = char₂(Frob₁₁)`, the `χ₈χ_cyc` branch | COMPUTED_HERE |

The trace jet, mod `11³`:

| `ℓ` | 3 | 5 | 7 | 13 | 17 | 19 | 23 | 29 |
| --- | --- | --- | --- | --- | --- | --- | --- | --- |
| constant `1 + χ₈(ℓ)ℓ` | 1329 | 1327 | 8 | 1319 | 18 | 1313 | 24 | 1303 |
| `log₁₁(ℓ)` | 847 | 44 | 220 | 374 | 935 | 517 | 1111 | 517 |
| `X`-coefficient | 121 | 1111 | 209 | 462 | 1254 | 825 | 264 | 979 |

The gate checks that the constant row **is** the q-expansion of `E₂(1,χ₈)` at those primes, that `log₁₁` is additive and lands in `11Z₁₁` (so `11¹` would show nothing — the jet is invisible mod 11 and only appears from `11²`), and that `χ₈` agrees with RUN-077's. `log₁₁(19) = log₁₁(29)` mod `11³` is a real coincidence — `⟨19⟩ ≡ ⟨29⟩` mod `11³` but not mod `11⁵` — not a bug; the logarithm's additivity is checked separately.

## The partner side, named

```text
missing object   :  R_i(X,t) = c_f(X)·U_i(X,t), the two-variable Rankin p-adic L-function of
                    (E_i, g) over the slope-1 Coleman family through g — equivalently the
                    Beilinson–Flach classes. THE ONLY missing input; the auxiliary half is above.
why not here     :  slope-1 overconvergent modular symbols at level lcm(N_i, 8, 11); no stdlib
                    path. PC-001 §2 「仍要求真正構造 B_{E,2} 與 r_{0,1}」; Attack 09 「未宣稱計算出
                    Coleman 族」.
smallest precursor:  R_0(X,t) for the rank-0 calibrator alone — PC-001 §4 notes U_0(0,0) ≠ 0, so
                    R_0(X,0) and c_f(X) share their X-vanishing order, and the calibrator by
                    itself probes n = ord_X c_f, one of the two factors of C_π = c_lead/q_f.
```

Consequently `C_{π,E}` and `C_{π,0}` are still not two objects, `C_{π,E} = C_{π,0}` has no truth value on actual objects, and **C1 is UNMEASURED** — not PASS, not FAIL.

## Attack B, computed

`D_i = Π_i·E_i` (Attack 09 (14)–(15)), `E_E ≡ 3`, `E_0 ≡ 7` (RUN-090). Hida: `L_p(Ad f)(0)` generates the congruence ideal, so `v₁₁(Π_i) = v₁₁(η_{f_i})` — CITED_THEOREM.

| level | computed here | result |
| --- | --- | --- |
| 389 | generalised mod-11 eigenspace in the dual Manin space (65 = 2g+1, g = 32): `⋂_q ker((T_q − a_q)^8)`, `q = 2,3,5`, `a_q ≡ 9, 9, 8` — **dim 2**, equal to the ordinary eigenspace; `f` mod 11 is not Eisenstein (`a₂ = 9 ≠ 3`) | `v₁₁(η) = 0` |
| 19 | `dim S₂(Γ₀(19)) = 1` (genus 1; dual Manin dim 3, cusp dim 2) — a single newform | `v₁₁(η) = 0` |

so **`Π_E/Π_0` is an 11-adic unit**, and the partner ratio `A_0(0)/A_E(0) ≡ 6·(Π_E/Π_0)` is a nonzero unit mod 11.

The residue is **not** claimed. Hida's theorem pins the ideal; the value needs Petersson norms against a stated period — which PC-001 §3 declines to fix — plus the Manin constants and the fact that the calibrator `19.a1` (LMFDB) `= 19a2` (Cremona) is **not** the `X₀`-optimal curve of its isogeny class, so its period differs from the modular one by an isogeny factor nobody has stated. A planted defect that claims a residue turns the check red.

## The drill

**448 defects, 448 caught by the check named for each**, 0 uncaught, 0 caught by the wrong check, 114 controls undisturbed, over 131 checks.

The 7 planted for this gate, over its three checks `c1-auxiliary-family`, `c1-verdict`, `attack-b-congruence` — each caught by the check named for it (a defect in the auxiliary side also turns `c1-verdict` red, since the verdict rests on it):

| planted defect | named check | went red |
| --- | --- | --- |
| the late factor is split on the C1 model, C_pi,E = 2 C_pi,0, while (2) is still asserted exact | `c1-verdict` | `c1-verdict` |
| C1 is promoted from UNMEASURED to PASS with the partner side unbuilt | `c1-verdict` | `c1-verdict`, `attack-b-congruence` |
| the auxiliary trace jet is built without log_11, so <d>^X is read as 1 | `c1-auxiliary-family` | `c1-auxiliary-family`, `c1-verdict` |
| the Eisenstein coefficients are taken as sum chi_8(d) d^0 instead of d^1 | `c1-auxiliary-family` | `c1-auxiliary-family`, `c1-verdict` |
| the canonical Eisenstein quotient no longer requires the two characters to be distinct | `c1-auxiliary-family` | `c1-auxiliary-family`, `c1-verdict` |
| a generalised eigenspace of dimension 2 is read as evidence of a mod-11 congruence | `attack-b-congruence` | `attack-b-congruence` |
| a residue mod 11 is claimed for Pi_E/Pi_0, which Hida's theorem does not give | `attack-b-congruence` | `attack-b-congruence` |


## What this round does not claim

* **Not that `C_{π,E} = C_{π,0}`.** The auxiliary side being canonical removes one mechanism by which they could differ; it does not make them exist.
* **Not a residue for `Π_E/Π_0`** — only `v₁₁ = 0` on both sides.
* **Not `D_E/D_0` as a value**, not the BF/Kato classes, not `B_{E,2}`, `r_{0,1}`, `C`, `𝔰₁₁`, not any bridge A–G.
* **BSD remains open.**