# Track B — the Loeffler–Rivero hypotheses instantiated; C4's direction closed; C1's partner-independence identified as a theorem

**From:** 數學戰士「墜衡」 (the independent verification line, branch `agent/bsd-verification-zhuiheng`)
**Answering:** the Track B instruction — minimal theorem burden, hypotheses instantiated on both partners, existence kept apart from computability, and no assumption of full mod-11 surjectivity.
**Source read:** D. Loeffler and Ó. Rivero, *Eisenstein degeneration of Euler systems*, arXiv:2201.02078v2; J. reine angew. Math. **814** (2024), 241–282. Read 2026-09-15; the statements used are recorded — as paraphrases with the paper's own numbering — in `data/external/literature/loeffler-rivero-2201.02078v2/EXTRACTS.md`.
**Gate:** `code/src96_lr_hypotheses_and_c4_direction.py` · checks `lr-identification`, `lr-decency`, `lr-ext-group`, `lr-partner-dictionary`, `lr-bf-existence`, `lr-verdict`
**Machine-readable:** [`result.json`](./files/result.json) · **this line's report:** [`reports/RUN-094-LR-HYPOTHESES-AND-C4-DIRECTION.md`](../../runs/RUN-094/)

---

## 0. A correction to RUN-093, first

RUN-093 reported C4's nonvanishing as `UNMEASURED`, on the grounds that every witness listed needs the overconvergent θ-critical eigenspace. **That was wrong.** One of the listed witnesses is the *reducibility ideal*, and it is a theorem — no overconvergent computation is required. Its hypotheses are arithmetic conditions on our own auxiliary form, and they are computed here.

Everything else in RUN-093 stands: the four vacuous local conditions, the extension dictionary, and `v₁₁(L(−1,χ₈)) = 0`. That last one keeps its meaning — it closes the *classical cuspidal-congruence* route. The witness that works comes from the family's pseudocharacter instead.

## 1. TRACK B RESULT

```text
theorem used:
  Loeffler-Rivero, "Eisenstein degeneration of Euler systems", arXiv:2201.02078v2
    Theorem C1.4    existence of the Beilinson-Flach family class
    C1.1 Setup      the hypotheses on the partner
    Thm C1.9 / Remark C1.11 / Thm C1.13   the auxiliary constant and its partner-independence
    Def A1.1, Def A1.2, Thm A2.1, Thm A4.5, Prop A4.6(ii), Thm A6.1, section A6.2, Cor A6.3(i)

389.a1:
  hypotheses PASS = [ ordinary at 11 (a_11 = -4, unit root 7 mod 11) ]
  UNMEASURED      = [ ]
  FAIL            = [ ]

19.a1:
  hypotheses PASS = [ ordinary at 11 (a_11 = 3, unit root 3 mod 11) ]
  UNMEASURED      = [ ]
  FAIL            = [ ]

auxiliary side (shared):
  hypotheses PASS = [ 11-decent (Def A1.2, computed);
                      non-critical (Prop A4.6(ii), from r = 0 + decent);
                      smoothing d = 5 with (d, 6S) = 1 (Thm C1.4's hypothesis, computed) ]

NOT REQUIRED by the theorem:
  full mod-11 surjectivity of the residual representation. C1.1 asks only "g is ordinary".

BF existence:
  BF_E = PASS-EXISTENCE          BF_0 = PASS-EXISTENCE
  (one theorem — C1.4 — applied to the same auxiliary f with the two different g)

finite-level precursor:
  E = UNMEASURED                 0 = UNMEASURED

common auxiliary data:
  the Eisenstein family f = E_k(1, chi_8) and its weight disc V_1
  the critical stabilisation beta = -11 (slope 1)
  the smoothing integer d = 5
  the cyclotomic coordinate gamma = 12
  the canonical one-dimensional Eisenstein quotient
  c_f(k), omega_f, and C = c*_f(r)

partner-specific data:
  a_11 and the unit root
  lambda_i = L_p(g (x) chi_8)
  D_i = L_p(Ad g)
  the Kato class kappa(g x psi)

C1 smallest remaining blocker:
  NOT partner-independence (now a theorem) and NOT existence (now a theorem).
  It is the ABSENCE OF COORDINATES: no computed representative of kappa_d(f,g),
  hence none of B_{E,2} or r_{0,1}.

next executable object:
  a finite-level Rankin-Eisenstein / Beilinson-Flach class _dBF[f_k, g_l] at an integer
  weight k != r -- exactly what Theorem C1.4 interpolates.
  This tree cannot build it, and the missing engine is nameable: etale cohomology of
  modular curves with coefficient sheaves (a motivic / etale Rankin-Eisenstein class
  engine). It is NOT a modular-symbol gap -- the modular-symbol engines here are
  sufficient for everything else in this round.
```

Against the instruction's Case A–D: this is **Case A, with one qualification**. The BF classes exist canonically for both partners by theorem; every actual hypothesis passes; the auxiliary construction is theorem-level common. But the remaining C1 blocker is **not** a Perrin-Riou/Coleman normalisation comparison — it is that no representative has ever been computed.

## 2. C4 — the direction closes

| step | statement | source | label |
| --- | --- | --- | --- |
| (a) | our form is `E₂(1,χ₈)`: `ψ = 1`, `τ = χ₈`, `r = 0`, `N = 8`; `α = 1`, `β = −11` | Def. A1.1 | **COMPUTED_HERE** |
| (b) | 11-decency: `ℓ = 2` → `2 \| 8`; `ℓ = 11` → `χ₈(11) = −1 ≠ 1` | Def. A1.2 | **COMPUTED_HERE** |
| (c) | `f_β` non-critical | Prop. A4.6(ii) | CITED + verified hypotheses |
| (d) | reducibility ideal `= 𝔪` | Thm. A6.1 (= [BC06, Thm 1]) | CITED_THEOREM |
| (e) | all four representations non-split | §A6.2 | CITED_THEOREM |
| (f) | `V(f_β)` = case (3): non-split, **crystalline at p** | Cor. A6.3(i), Thm. A2.1 | CITED_THEOREM |
| (g) | case (3)'s class lies in `H¹(Q, Q₁₁(χ₈)(1))` | Ext bookkeeping | **COMPUTED_HERE** |
| (h) | `dim H¹_f = 1` spanned by `κ(ε)`; local conditions vacuous | RUN-092/093 | **COMPUTED_HERE** |

```text
crystalline  =>  the class is in H^1_f
non-split    =>  the class is nonzero
dim H^1_f = 1  =>  [Theta_aux] = c_ES/Kum * kappa(epsilon),  c_ES/Kum in Q_11^x

C4-direction     CLOSED
C4-exact-scalar  OPEN
```

The Ext bookkeeping is confirmed independently by the paper itself: Theorem A4.5 phrases non-criticality as the non-vanishing of the localisation out of `H¹_f(Q, ψτ⁻¹ε^{1+r})` — the same group.

## 3. C1 — partner-independence is the source's own result

* **Thm C1.9**: `c_f(k)` is "a meromorphic function on `V₁` **alone**" — `V₁` being the *auxiliary* family's disc.
* **Thm C1.10**: there exist `n ⩾ 0` and a **unique** isomorphism `ω_f` over `O(V₁)`.
* **Remark C1.11**: the partner family `g` is used in constructing `ω_f`, but it "is uniquely determined by `f`".
* **Thm C1.13**: the degeneration formula, whose proof opens by taking `C = c*_f(r)`, with `C` nonzero.

Theorem C1.13 **is** PC-001's formula (1). The dictionary, checked complete in both directions:

| LR Theorem C1.13 | PC-001 (1) |
| --- | --- |
| `C` (= `c*_f(r)`, nonzero) | `C_π` |
| `C_d(f_β, g, j)` | `𝒮₅(t)` |
| `L_p(g ⊗ τ, j−1−r)` | `λ_i(t)` |
| `L_p(Ad g)` | `D_i` |
| `log^{[r+1]}` | `j(t)` |
| `κ(g × ψ)` | the Kato class |

So `C_π = C = c*_f(r)` is a functional of the auxiliary family alone: **partner-independent, and nonzero.** RUN-090 carried this as certificate item C1's unproved core; it is not an assumption.

## 4. The notation trap

LR's **`f`** is the **Eisenstein** family and its **`g`** is the **cuspidal partner** — the reverse of PC-001's letters. And LR normalises by `tr ρ(Frob_ℓ⁻¹) = a_ℓ`, so its second Jordan–Hölder character is `χ₈ε^{−1−r}` where RUN-091's is `χ₈ε^{1+r}`. The conventions are dual; the gate checks that they give the **same** Ext group, so RUN-093's dictionary stands unchanged.

## 5. Claim labels

`COMPUTED_HERE`: the identification in LR's notation against the actual q-expansion at ten primes, with parity and both stabilisation roots; 11-decency (with `cond(ψ/τ)` *derived*, not asserted); the Ext-group bookkeeping and the duality of conventions; `(d, 6S) = 1` for `d = 5`; both partners' ordinarity; and RUN-092/093's Selmer dimension and local conditions.
`CITED_THEOREM`: every LR statement listed above, and [BC06, Thm 1] through it.
`DERIVED_FROM_VERIFIED_INPUTS`: non-criticality from `r = 0` + decency; the C1.13 ↔ PC-001 dictionary.
`UNMEASURED`: the BF coordinates and any finite-level representative; `c_ES/Kum`; `Π_E/Π_0`.
`FAILED`: nothing.

## 6. Status

```text
C1  partner-independence CITED_THEOREM (hypotheses instantiated); BF existence PASS-EXISTENCE;
    coordinates UNMEASURED -- that is now C1's whole remaining gap
C2  UNMEASURED -- Pi_E/Pi_0 still open
C3  PASS
C4  direction CLOSED, exact scalar OPEN
C5-C7  unchanged     C8  PARTIAL     C9-C10  not yet ready     BSD  OPEN
```