# ATTACK C1 — the actual-object construction

**From:** 數學戰士「墜衡」 (the independent verification line, branch `agent/bsd-verification-zhuiheng`)
**Answering:** the instruction after RUN-090 — stop summarising, stop extending the symbolic framework, turn the first blocker into a measurable arithmetic object; ideally along `𝒯_X(g) mod X²` with a stated pseudocharacter identification; if it cannot be done, say so and name the smallest missing construction; do not idle — drop to Attack B.
**Project:** `389.a1`, `p = 11`; calibrator `19.a1` (LMFDB) = `19a2` (Cremona)
**Gate:** `code/src93_attack_c1_actual_object.py` · drill `code/src11_gate_drill.py` (checks `c1-auxiliary-family`, `c1-verdict`, `attack-b-congruence`)
**Machine-readable:** [`result.json`](./files/result.json) · **evidence:** [`evidence/`](./files/evidence/) · **this line's report:** [`reports/RUN-091-ATTACK-C1-ACTUAL-OBJECT.md`](../../runs/RUN-091/)

---

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

```text
C1 still UNMEASURED

  exact missing object =
      R_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 themselves.
      This is now the ONLY missing input: the auxiliary half is constructed below.

  exact missing theorem =
      that the leading constant of the meromorphic Eichler–Shimura map is partner-independent
      once the auxiliary frame is fixed (Loeffler–Rivero C1.9–C1.13, uniqueness C1.11).
      CITED_THEOREM. Its auxiliary-side hypotheses are now instantiated on actual objects
      (decency, the slope-1 refinement, the canonical Eisenstein quotient); its partner-side
      hypothesis — that the two projections exist — is not, because the classes do not exist.

  smallest constructible precursor =
      R_0(X,t) for the rank-0 calibrator ALONE. PC-001 §4: U_0(0,0) ≠ 0, so R_0(X,0) and c_f(X)
      have the same X-vanishing order — the calibrator by itself probes n = ord_X c_f, which is
      one of the two factors of C_π = c_lead/q_f.

  next executable attack =
      Attack B's valuation (done here), then C4's Θ_aux. For C1 itself the next step is
      R_0(X,t), which needs slope-1 overconvergent modular symbols at level lcm(19, 8, 11) —
      a Sage/Magma-scale computation, not a stdlib one.
```

No item was promoted. `C_{π,E}` and `C_{π,0}` still do not exist as separate objects, so `C_{π,E} = C_{π,0}` has no truth value on actual objects and is not asserted.

---

## 1. What IS constructed here — the auxiliary side, in full

`COMPUTED_HERE.` The auxiliary family is the Eisenstein family through `g = E₂(1,χ₈)` in the weight direction. Eisenstein series are explicit at every weight, so unlike a general Coleman family this one is writable down:

```text
a_n(X) = Σ_{d|n} χ₈(d) · d · ⟨d⟩^X
```

**The pseudocharacter identification.** The Galois pseudocharacter of the family is the sum of the two Jordan–Hölder characters of the reducible representation attached to `E₂(1,χ₈)`, deformed in the weight direction:

```text
𝒯_X(Frob_ℓ) = 1 + χ₈(ℓ)·ℓ^{1+X}        (ℓ ≠ 11)
             = char₁ + char₂ ,   char₁ = 1 ,  char₂ = χ₈·χ_cyc
```

**The trace jet.** Expanding `ℓ^{1+X} = ℓ·⟨ℓ⟩^X = ℓ(1 + X·log₁₁ℓ + O(X²))`:

```text
𝒯_X(g)(Frob_ℓ) ≡ 1 + χ₈(ℓ)ℓ  +  χ₈(ℓ)·ℓ·log₁₁(ℓ) · X       (mod X²)
```

computed at eight Frobenius elements mod `11³` (the Iwasawa log lands in `11Z₁₁`, so `11¹` would show nothing):

| `ℓ` | constant `1 + χ₈(ℓ)ℓ` | `log₁₁(ℓ)` mod `11³` | `X`-coefficient mod `11³` |
| --- | --- | --- | --- |
| 3 | 1329 | 847 | 121 |
| 5 | 1327 | 44 | 1111 |
| 7 | 8 | 220 | 209 |
| 13 | 1319 | 374 | 462 |
| 17 | 18 | 935 | 1254 |
| 19 | 1313 | 517 | 825 |
| 23 | 24 | 1111 | 264 |
| 29 | 1303 | 517 | 979 |

Checks carried by the gate: the constant term equals the **actual** q-expansion coefficient of `E₂(1,χ₈)` at every one of those primes (RUN-077's numbers, recomputed); the logarithm is additive (`log(ab) = log(a)+log(b)` on six pairs) and lands in `11Z₁₁`. (`log₁₁(19) = log₁₁(29)` mod `11³` is a genuine coincidence — `⟨19⟩ ≡ ⟨29⟩ mod 11³` but not mod `11⁵`.)

**The canonical Eisenstein quotient — PC-001 §4's hazard, discharged.** PC-001 warns that a generic auxiliary Galois representation does **not** provide a one-dimensional Eisenstein quotient, so the order of operations matters. Here the two Jordan–Hölder characters are **distinct**:

```text
Frob_3 :  char₁ = 1 ,  char₂ = χ₈(3)·3 = 8   (mod 11)      distinct
```

so the residual representation is multiplicity-free, the one-dimensional Eisenstein quotient exists and is **canonical**. A canonical quotient cannot be chosen per partner — which closes the one route by which the quotient transport could have been partner-dependent. (Distinctness means the characters differ at *some* Frobenius; at `ℓ = 23` they happen to agree mod 11, which is not a counterexample.)

**Coherence with the refinement PC-001 and Attack 09 actually use.** At `ℓ = 11` the two characters take the values `1` and `χ₈(11)·11 = −11`, which are exactly the two roots of the Hecke polynomial `U² + 10U − 11 = (U − 1)(U + 11)`. The critical (slope-1) refinement `f_β = f(q) − f(q¹¹)` with `U₁₁f_β = −11f_β` is therefore the `χ₈·χ_cyc` branch. The pseudocharacter and the refinement are the same object seen twice.

## 2. What is NOT constructed — and it is one thing

`UNMEASURED.` `B̂_i` is the image of the partner's `X`-leading Beilinson–Flach / Rankin class for the pair `(E_i, g)` under that canonical quotient, in the fixed frame. Both partners are ordinary at 11 (`a₁₁ = −4`, unit root 7; `a₁₁ = 3`, unit root 3 — recomputed), and the frame and coordinate are fixed and shared (RUN-090's C3 PASS). What is absent is the class itself: `R_i(X,t) = c_f(X)U_i(X,t)`.

It is not constructible here: a slope-1 Coleman family needs overconvergent modular symbols of positive slope at level `lcm(N_i, 8, 11)`; there is no stdlib path, and neither peer line has the classes — PC-001 §2 (「仍要求真正構造 `B_{E,2}` 與 `r_{0,1}`」) and Attack 09 (「未宣稱計算出 Coleman 族」) say so themselves.

So the gap has gone from *"nothing on either side"* to *"the auxiliary side is built; the partner side is one named object."*

## 3. Attack B — the valuation, computed

`COMPUTED_HERE` for the valuation, `CITED_THEOREM` for the identification, `UNMEASURED` for the residue.

With `D_i = Π_i·E_i` (Attack 09 (14)–(15); `E_i` the adjoint Euler multiplier, `3` and `7` mod 11 from RUN-090), Hida's theorem says the adjoint `p`-adic L-value `L_p(Ad f)(0)` generates the congruence ideal of `f`, so `v₁₁(Π_i) = v₁₁(η_{f_i})`.

| level | what was computed | `v₁₁(η)` |
| --- | --- | --- |
| **389** | 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 — **dimension 2**, equal to the ordinary eigenspace, i.e. the `±` pair of `f` alone. So the localised Hecke algebra is `Z₁₁` and no cusp form is congruent to `f` mod 11. (`f` mod 11 is also not Eisenstein: `a₂ = 9 ≠ 3`.) | **0** |
| **19** | `dim S₂(Γ₀(19)) = 1` (genus 1, cusp dimension 2 in the dual Manin space): a single newform, so there is no cusp form for it to be congruent to. | **0** |

Therefore:

```text
Π_E/Π_0  is an 11-adic UNIT
⇒  A_0(0)/A_E(0) ≡ 6·(Π_E/Π_0) (mod 11)  is a nonzero unit
```

which is exactly the non-vanishing PC-001's calibration needs, and it now rests on a computation rather than on hope.

**The residue stays UNMEASURED, and the gate refuses to claim one.** Hida's theorem pins the *ideal*, not the value. Recovering the residue needs the actual Petersson norms against a stated period — the normalisation PC-001 §3 explicitly declines to fix (「其完整 period 比值 `D_0` 沒有在本輪算出」) — plus the Manin constants and the fact that `19.a1` = Cremona `19a2` is **not** the `X₀`-optimal curve of its isogeny class, so its period is not the modular one without a stated isogeny factor. A planted defect that claims a residue turns `attack-b-congruence` red.

## 4. The mutation this round was required to carry

```text
C_{π,E} ↦ u·C_{π,E},   C_{π,0} fixed     ⇒  check `c1-verdict` goes RED
```

Carried, and red. On the model of PC-001 (1)–(2) with the two late factors kept **separate**, the defect of identity (2) is exactly their ratio — `RHS(2)/κ_E = C_{π,E}/C_{π,0}` on all 40 instances — so a split common factor is caught directly rather than inferred. Six further defects: `UNMEASURED` promoted to `PASS`; the trace jet built without `log₁₁`; the Eisenstein coefficients taken as `Σχ₈(d)d⁰`; the canonical quotient no longer requiring distinct characters; a generalised eigenspace of dimension 2 read as a congruence; a residue claimed for `Π_E/Π_0`. Four controls (another seed, more trials, `11⁴` precision, ten Frobenius elements) stay green.

## 5. Claim labels

`COMPUTED_HERE`: the auxiliary family and its weight deformation; the pseudocharacter's two characters and their distinctness; the trace jet mod `X²` at eight primes; the coincidence of the characters at 11 with the Hecke roots and the slope-1 branch; the Iwasawa logarithm and its additivity; both partners' ordinarity; the generalised eigenspace at 389; the cuspidal dimension at 19.
`CITED_THEOREM`: Loeffler–Rivero C1.9–C1.13 and C1.11 (the ES uniqueness); Hida (adjoint L-value generates the congruence ideal); Bellaïche–Dasgupta / Loeffler–Rivero A4.5 (non-criticality).
`DERIVED_FROM_VERIFIED_INPUTS`: that a canonical quotient cannot be partner-dependent; that `A_0(0)/A_E(0)` is a nonzero unit.
`UNMEASURED`: `B̂_E`, `B̂_0`, `C_{π,E}`, `C_{π,0}`, `R_i(X,t)`, the residue of `Π_E/Π_0`.
`FAILED`: nothing.

## 6. Not claimed

That `C_{π,E} = C_{π,0}`; that C1 passes; `D_E/D_0` as a value; the BF/Kato classes; `B_{E,2}`, `r_{0,1}`; `C`; `𝔰₁₁`; any bridge A–G. **BSD remains open.**