# RUN-076 — Attack 09, rewritten from this side: the auxiliary character of an Eisenstein degeneration for 389.a1 at p = 11 — GPT-6's χ₈ has root number −1 and a vanishing even first layer (L(E, χ₈ψ, 1) = 0 for the order-5 ψ of conductor 11 as well), the quadratic characters whose central value survives are listed with their twisted L-values as exact integers, and the mod-11 eigenline is tied to those integers by one unit per sign

**Arm:** 數學戰士「墜衡」 / AMRAL Research Lab
**Subject:** GPT-6's **Attack 09**, known only from its progress narration as relayed to this line on 2026-09-12 (summarised in §0); the package itself has not arrived and this round does not wait for it — it writes down, with this tree's own arithmetic, what the attack's one new number can and cannot be
**Tools:** `src78_attack09_auxiliary_character.py`, `src70_kurihara_modular_symbols.py`, `src15_phase2_anchor.py`, `src11_gate_drill.py`
**Logs:** `src78-attack09-auxiliary-character.json`, `src11-gate-drill.json`

> **Postscript, written before commit.** This round was computed from the narration. The package arrived the next morning and is verified at [RUN-077](../RUN-077/): its number is **not the central value**. It is `λ₈(0) = L₁₁(E ⊗ χ₈, x⁻¹)` — the ordinary 11-adic L-function of the twist at the full character `x ↦ x⁻¹`, the odd Teichmüller branch at `s = 0`, a non-critical point — computed on the **minus** modular symbol, normalised at its first nonzero coordinate (index 3), which is exactly this round's minus eigenline. That value is `≡ 5 (mod 11)`, and RUN-077 reproduces it coordinate for coordinate. Everything computed below stands; §1–§2 are about the central value and the even (plus-line) first layer, which the package does not use, and §5's replacement list is for that reading only. The relation between the two: this round's minus-line branch `T₉⁻ = 3` for `χ₈` is `−8⁹` times the package's weighted sum `9`, i.e. the same number in another convention. The unit `λ⁻ = 6` of §4 also settles a point the package flags: its index-3 normalisation is a unit multiple of the `Ω⁻`-normalised symbol, so its mod-11 statement is meaningful in the Néron normalisation.

**Result: Attack 09, as narrated, runs a Loeffler–Rivero Eisenstein degeneration with the auxiliary form `E₂(1, χ₈)` and reports one new number — "the extra twisted 11-adic L-value ≡ 5 (mod 11), so nonzero", over forty cusp paths. For 389.a1 at 11 the central value of that twist is zero, and so is the whole even first layer — the package, when it arrived (RUN-077), turned out to use neither; see the postscript. A weight-2 Eisenstein series `E₂(1, χ)` exists only for `χ` even, so `χ₈ = (8/·)`; the root number of `E ⊗ χ₈` is `χ₈(−389)·w(E) = (2/389) = −1` because `389 ≡ 5 (mod 8)`, so `L(E, χ₈, 1) = 0` exactly, and no normalisation moves a zero. Beyond the sign, `L(E, χ₈ψ, 1) = 0` for the four order-5 characters `ψ` of conductor 11 as well — one Galois orbit, hence one event — computed two ways: from this tree's `a_n` by the functional-equation series, `|L| ≤ 3.2·10⁻¹⁵` against controls of size 1; and on RUN-068's mod-11 plus eigenline, where every `χ₈`-twisted symbol `[a/11]_{χ₈}⁺` and every one of the ten tame branches is 0. The entire even part of the first cyclotomic layer of `L₁₁(E ⊗ χ₈)` vanishes, and the narrated `5 (mod 11)` is not any first-layer quantity of `χ₈` on the plus line — it is one on the minus line, the value at `x⁻¹`, which RUN-077 identifies and reproduces. Then the constructive half: for all 54 fundamental discriminants `|D| ≤ 100` prime to 11, the root number of `E ⊗ χ_D`, the twisted sum `S_D = Σ χ_D(b)[b/|D|]^±` on the eigenline of the right sign, the level-`11|D|` sum with its Hecke identity `Σ χ_D(c)[c/11|D|] = (a₁₁ − 2χ_D(11))·S_D`, the first-layer total of the Mazur–Tate–Teitelbaum measure against `(1 − χ_D(11)/α)²·S_D`, and — for root number +1 — `L(E, χ_D, 1)·√|D|/Ω^±` from the `a_n`, which is an exact integer every time: `4` for 22 twists, `16` for `D = −51`, `36` for `D = −43`, `0` for `D = 65, 93, −47`. **One unit per sign, `λ⁺ = 10` and `λ⁻ = 6`, carries those integers onto the mod-11 sums for all 27 twists, zero sets included** — the first time this tree's eigenline is tied to archimedean L-values rather than to Hecke eigenvalues alone. The smallest admissible even character is `χ₅`: root number +1, `L(E, χ₅, 1)√5/Ω⁺ = 4`, all five even tame branches nonzero mod 11, Euler factor `(1 − 1/α)² ≡ 5` a unit; the smallest odd one, for a weight-1 Eisenstein series and the minus symbols, is `χ₋₃`. **Loeffler–Rivero's comparison factor is not reproduced here; nothing about `𝔰₁₁`, `C`, `B₂` or BSD moves.**

---

## §0 — what Attack 09 said

The only record of Attack 09 at the time was GPT-6's own progress narration, relayed to
this line on 2026-09-12. Its claims, in this line's words: the auxiliary form chosen is
`E₂(1, χ₈)`, `χ₈` the quadratic character of conductor 8, said to satisfy the decency and
non-criticality conditions the degeneration theory needs at `p = 11`; the one new result is
an "extra twisted 11-adic L-value" computed as `5 (mod 11)` under an explicit modular-symbol
normalisation, hence nonzero, which is what lets the degenerate class shed the double zero
along the cyclotomic direction; the new program ran over forty cusp paths with every measure
value attached; and the narration itself separates what is new (the nonvanishing of the
twisted moment) from what is cited (the degeneration comparison) and what is not yet computed
(the actual coordinates of `C`, `B₂`, and `s₁₁`).

Forty cusp paths is `φ(88) = 40`: the units modulo `8·11`, the first layer
of the cyclotomic tower twisted by a character of conductor 8. That is the
object computed below.

## §1 — the parity and the root number

| step | fact | recomputed |
| --- | --- | --- |
| a weight-2 `E₂(ψ, φ)` needs `(ψφ)(−1) = (−1)² = +1` | with `ψ = 1`, `φ = χ₈` must be even: `χ₈ = (8/·)`, kernel `{±1 mod 8}` | `χ₈(−1) = +1` |
| `w(E)` for prime conductor with multiplicative reduction is `a_N` | `a₃₈₉ = +1` (split, counted on the node) | `+1` |
| `w(E ⊗ χ_D) = χ_D(−N)·w(E)` for `D` prime to `N` | `χ₈(−389) = (8/389) = (2/389)`, and `389 ≡ 5 (mod 8)` | `−1` |
| hence `L(E, χ₈, 1) = 0` | exactly, by the functional equation | — |
| on the plus eigenline (RUN-068, `λ(1,5) = 1`) | `[1/8]⁺ = [3/8]⁺ = 0`, so `S₈ = Σ χ₈(b)[b/8]⁺ = 0` | `0` |
| the forty paths | `Σ_{c ∈ (Z/88)^×} χ₈(c)[c/88]⁺ = 0`; the first-layer total `α⁻¹Σχ₈(c)[c/88]⁺ − 10α⁻²S₈ = 0` | `0`, `0` |

The two symbols `[1/8]⁺`, `[3/8]⁺` are the whole plus part at level 8, and
both are forced to zero over `Q`: their sum is a multiple of `[0]⁺ = L(E,1)/Ω⁺ = 0`,
their difference is `L(E, χ₈, 1)` up to a nonzero constant. The forty
per-path values are in the log (`forty_paths`); 32 of the 40 plus symbols
`[c/88]⁺` are nonzero individually, and their `χ₈`-weighted sums vanish.

## §2 — beyond the sign: the whole even first layer

The tame branches of the twisted first layer are
`T_j = Σ_{c ∈ (Z/88)^×} χ₈(c)·c^j·[c/88]^±`, `j = 0..9`, with `ω^j(c) ≡ c^j
(mod 11)`; `T_j` is the mod-11 value of `L₁₁(E ⊗ χ₈)` at the tame character
`ω^j`, up to the unit `(χ₈(11)α)⁻¹` and a Gauss sum.

| line | `T_0 … T_9` | twisted symbols `[a/11]_{χ₈}⁺ = Σ_b χ₈(b)[a/11 + b/8]⁺`, `a = 1..10` |
| --- | --- | --- |
| plus (even branches live here) | `(0, 0, 0, 0, 0, 0, 0, 0, 0, 0)` | `(0, 0, 0, 0, 0, 0, 0, 0, 0, 0)` |
| minus (odd branches) | `(0, 9, 0, 4, 0, 10, 0, 3, 0, 3)` | — |

Every even branch is zero mod 11, i.e. every `χ₈`-twisted plus symbol at
the cusps `a/11` is zero mod 11. That is more than the root number forces
(`j = 0` only), so it was checked over `C`: for the four characters `ψ` of
`(Z/11)^×` of exact order 5, the newform `f ⊗ χ₈ψ` has level `389·88²` and
root number `w(E)·χ(389)·τ(χ)²/88`, and

| `ψ` | `|L(E, χ₈ψ, 1)|` | `|L|·√88/Ω⁺` | terms |
| --- | --- | --- | --- |
| `ψ₁` | `1.17·10⁻¹⁵` | `2.2·10⁻¹⁵` | 12,431 |
| `ψ₂` | `1.26·10⁻¹⁵` | `2.4·10⁻¹⁵` | 12,431 |
| `ψ₃` | `2.89·10⁻¹⁵` | `5.4·10⁻¹⁵` | 12,431 |
| `ψ₄` | `3.18·10⁻¹⁵` | `6.0·10⁻¹⁵` | 12,431 |
| control `χ₅ψ_t` | `0.6716` each | `1.000` each | 7,252 |
| control `χ₁₃ψ_t` | `1.7643`, `0.0983`, `0.0983`, `1.7643` | `4.236`, `0.236`, `0.236`, `4.236` | 15,000 |

The four values are Galois conjugates of one algebraic number of bounded
denominator (Shimura), so this is one zero, not four coincidences; the
controls show the same series producing values of size 1 for the
neighbouring characters (`χ₁₃ψ`'s algebraic parts are the units
`2 ± √5`). Two independent computations — the `a_n` over `C` and the Manin
symbols mod 11 — agree that `L(E, χ₈ψ, 1) = 0` for every character `ψ` of
`Gal(Q(ζ₁₁)⁺/Q)`, the trivial one by the root number and the order-5 ones
outright. In BSD terms `E ⊗ χ₈` should gain rank over the quintic field
`Q(ζ₁₁)⁺`; in Iwasawa terms the mod-11 reduction of `L₁₁(E ⊗ χ₈)` has a
zero constant term in every even branch. Either way, **on the plus line — at the central point and at the four
order-5 characters — `χ₈` at `p = 11` has no nonzero first-layer value.**
The odd side is alive (`T_j⁻ ≠ 0`; the package's factor lives there, see
the postscript), and so is the odd character `(−8/·)` — root number `+1`,
`S₋₈⁻ = 2`, `L(E, χ₋₈, 1)√8/Ω⁻ = 4`.

## §3 — every quadratic character to 100

For each fundamental `D` with `|D| ≤ 100` prime to 11 (and to 389): the
root number `w = χ_D(−389)`; the sum `S_D = Σ_{b ∈ (Z/|D|)^×} χ_D(b)[b/|D|]^±`
on the plus line for `D > 0` and the minus line for `D < 0`; the level-`11|D|`
sum `T = Σ χ_D(c)[c/11|D|]^±` over the `10·φ(|D|)` units, which must equal
`(a₁₁ − 2χ_D(11))·S_D`; the first-layer total `α⁻¹T − 10α⁻²S_D` of the
Mazur–Tate–Teitelbaum measure, which must equal `(1 − χ_D(11)/α)²·S_D`
(`α ≡ 7`, `a₁₁ = −4`); and for `w = +1` the archimedean value
`A_D = L(E, χ_D, 1)·√|D|/Ω^±`, `L(E, χ_D, 1) = 2·Σ a_n χ_D(n) e^{−2πn/(|D|√389)}/n`,
`Ω⁺ = 4.980425121710` (both components, RUN-017's convention),
`Ω⁻ = π/AGM(√(e₁−e₃), √(e₂−e₃)) = 1.971737701552`. All mod-11 values are
at RUN-068's scale (`λ(1,5) = 1` on the plus line; the minus line is
normalised at its first nonzero coordinate, index 3).

| `D` | `w` | `χ_D(11)` | `S_D` | `T` | total = Euler·`S_D` | `A_D` |
| ---: | ---: | ---: | ---: | ---: | ---: | ---: |
| 5 | +1 | +1 | 7 | 2 | 2 = 2 | **4** |
| 8 | −1 | −1 | 0 | 0 | 0 = 0 | (0) |
| 12 | −1 | +1 | 0 | 0 | 0 = 0 | (0) |
| 13 | +1 | −1 | 7 | 8 | 6 = 6 | **4** |
| 17 | +1 | −1 | 7 | 8 | 6 = 6 | **4** |
| 21 | −1 | −1 | 0 | 0 | 0 = 0 | (0) |
| 24 | +1 | −1 | 7 | 8 | 6 = 6 | **4** |
| 28 | +1 | −1 | 7 | 8 | 6 = 6 | **4** |
| 29 | −1 | −1 | 0 | 0 | 0 = 0 | (0) |
| 37 | −1 | +1 | 0 | 0 | 0 = 0 | (0) |
| 40 | −1 | −1 | 0 | 0 | 0 = 0 | (0) |
| 41 | +1 | −1 | 7 | 8 | 6 = 6 | **4** |
| 53 | −1 | +1 | 0 | 0 | 0 = 0 | (0) |
| 56 | −1 | +1 | 0 | 0 | 0 = 0 | (0) |
| 57 | −1 | −1 | 0 | 0 | 0 = 0 | (0) |
| 60 | −1 | +1 | 0 | 0 | 0 = 0 | (0) |
| 61 | −1 | −1 | 0 | 0 | 0 = 0 | (0) |
| 65 | +1 | −1 | 0 | 0 | 0 = 0 | **0** |
| 69 | +1 | +1 | 7 | 2 | 2 = 2 | **4** |
| 73 | +1 | −1 | 7 | 8 | 6 = 6 | **4** |
| 76 | +1 | −1 | 7 | 8 | 6 = 6 | **4** |
| 85 | +1 | −1 | 7 | 8 | 6 = 6 | **4** |
| 89 | −1 | +1 | 0 | 0 | 0 = 0 | (0) |
| 92 | −1 | +1 | 0 | 0 | 0 = 0 | (0) |
| 93 | +1 | +1 | 0 | 0 | 0 = 0 | **0** |
| 97 | +1 | +1 | 7 | 2 | 2 = 2 | **4** |
| −3 | +1 | −1 | 2 | 7 | 8 = 8 | **4** |
| −4 | −1 | −1 | 0 | 0 | 0 = 0 | (0) |
| −7 | −1 | +1 | 0 | 0 | 0 = 0 | (0) |
| −8 | +1 | +1 | 2 | 10 | 10 = 10 | **4** |
| −15 | +1 | −1 | 2 | 7 | 8 = 8 | **4** |
| −19 | −1 | +1 | 0 | 0 | 0 = 0 | (0) |
| −20 | −1 | −1 | 0 | 0 | 0 = 0 | (0) |
| −23 | +1 | −1 | 2 | 7 | 8 = 8 | **4** |
| −24 | −1 | +1 | 0 | 0 | 0 = 0 | (0) |
| −31 | +1 | −1 | 2 | 7 | 8 = 8 | **4** |
| −35 | −1 | +1 | 0 | 0 | 0 = 0 | (0) |
| −39 | +1 | +1 | 2 | 10 | 10 = 10 | **4** |
| −40 | +1 | +1 | 2 | 10 | 10 = 10 | **4** |
| −43 | +1 | +1 | 7 | 2 | 2 = 2 | **36** |
| −47 | +1 | −1 | 0 | 0 | 0 = 0 | **0** |
| −51 | +1 | +1 | 8 | 7 | 7 = 7 | **16** |
| −52 | −1 | +1 | 0 | 0 | 0 = 0 | (0) |
| −56 | +1 | −1 | 2 | 7 | 8 = 8 | **4** |
| −59 | −1 | −1 | 0 | 0 | 0 = 0 | (0) |
| −67 | −1 | −1 | 0 | 0 | 0 = 0 | (0) |
| −68 | −1 | +1 | 0 | 0 | 0 = 0 | (0) |
| −71 | +1 | −1 | 2 | 7 | 8 = 8 | **4** |
| −79 | −1 | +1 | 0 | 0 | 0 = 0 | (0) |
| −83 | +1 | +1 | 2 | 10 | 10 = 10 | **4** |
| −84 | +1 | +1 | 2 | 10 | 10 = 10 | **4** |
| −87 | −1 | +1 | 0 | 0 | 0 = 0 | (0) |
| −91 | −1 | −1 | 0 | 0 | 0 = 0 | (0) |
| −95 | −1 | +1 | 0 | 0 | 0 = 0 | (0) |

`(0)` marks a value that is zero by the functional equation and was not
summed. Three things hold on all 54 rows: the Hecke identity at level
`11|D|`; the first-layer total equals `(1 − χ_D(11)/α)²·S_D`; and **every
root-number `−1` row has `S_D ≡ 0`** — a linear identity among the Manin
symbols that the functional equation predicts and the eigenline obeys, 27
times. The Euler factor is `(1 − 1/α)² ≡ 5` or `(1 + 1/α)² ≡ 4`, a unit
either way: `α` is not `±1`, so no quadratic twist has an exceptional zero
at 11.

## §4 — the cross-check: one unit per sign

The 27 values `A_D` are recognised as rationals with denominator ≤ 64 to an
error ≤ `2.7·10⁻¹⁵`, and every one is an integer: `4` (22 times), `16`,
`36`, `0` (three times) — the pattern `4·c²` of the Waldspurger–Kohnen–Zagier
formula, noted, not used. Reduced mod 11 and set against the mod-11 sums:

| sign | pairs `(A_D mod 11, S_D)` | unit `λ` with `S_D ≡ λ·A_D` | nonzero pairs |
| --- | --- | ---: | ---: |
| plus | `(4,7)` ×11, `(0,0)` ×2 | **10** | 11 |
| minus | `(4,2)` ×11, `(16 ≡ 5, 8)`, `(36 ≡ 3, 7)`, `(0,0)` | **6** | 13 |

`10·4 = 40 ≡ 7`, `6·4 = 24 ≡ 2`, `6·5 = 30 ≡ 8`, `6·3 = 18 ≡ 7`; the zero
sets `{65, 93}` and `{−47}` agree on both sides. The plus eigenline of
RUN-068 (checked there against the stress-test package's numbers and four
Hecke eigenvalues, and at RUN-071 against Attack 04's 110 summands) is
therefore `10 ×` the reduction of the `Ω⁺`-normalised modular symbol, and
the minus line `6 ×` the reduction of the `Ω⁻`-normalised one, on every
twist tested. A wrong Kronecker symbol, a wrong root-number formula, a wrong
conductor, a wrong unit root or a wrong measure sign breaks one of the
three identities or this correspondence (the drill plants each).

## §5 — the characters whose central value survives

(The central-value criterion. The package's own criterion, `λ_D(0) ≢ 0`, is
evaluated for every even `D` at RUN-077.)

| route | Eisenstein series | symbols | admissible `D`, `|D| ≤ 100` | `A_D` | unusable (root number +1, `L = 0`) |
| --- | --- | --- | --- | --- | --- |
| even | weight 2, `E₂(1, χ_D)` | plus | **5**, 13, 17, 24, 28, 41, 69, 73, 76, 85, 97 | 4 each | 65, 93 |
| odd | weight 1, `E₁(1, χ_D)` | minus | **−3**, −8, −15, −23, −31, −39, −40, −43, −51, −56, −71, −83, −84 | 4 (36 at −43, 16 at −51) | −47 |

For `χ₅`: `χ₅(11) = +1`, Euler factor `(1 − 1/α)² ≡ 5`, first-layer total
`≡ 2` at this scale, and the ten tame branches on the plus line are
`(2, 0, 8, 0, 2, 0, 6, 0, 7, 0)` — every even branch nonzero mod 11, so the
`χ₅`-twisted first layer is nonvanishing at every character it can be
evaluated at, which is exactly what `χ₈` fails. Whether `E₂(1, χ₅)` meets
Loeffler–Rivero's decency and non-criticality hypotheses at 11 is for
whoever applies their theorem to check; the numbers those hypotheses read
(`χ_D(11)`, the Euler factor, the branch values) are in the log for every
`D`.

## The labels

| where | what |
| --- | --- |
| the narration | "額外出現的扭曲 11-adic L 值 … 5 mod 11 … 確實非零" — not a plus-line quantity: there every first-layer value of `χ₈` is 0; identified at RUN-077 as the minus-line value at `x⁻¹` and reproduced |
| the narration | "退化比較引用已發表定理" — Loeffler–Rivero's comparison factor is not reproduced here; this round fixes the arithmetic input any version of it consumes |
| the narration | "C、B₂ 的實際座標及 s₁₁ 仍未算出" — untouched here too |
| this round's output | `BSD_proved: false`; `canonical_frontier_updated: false`; the exact vanishing of `L(E, χ₈ψ, 1)` is a numerical identification of an algebraic number of bounded denominator with 0, backed by the independent mod-11 computation — not a proof |

## The drill

**326 defects, 326 caught by the check named for each**, 0 uncaught, 0 caught by the wrong check, 69 controls undisturbed, over 104 checks.

The 7 planted for this gate, each turning `attack09-auxiliary` red and nothing else:

| planted defect | went red |
| --- | --- |
| the Kronecker symbol (D/2) is taken as +1 for D = +-3 mod 8 | `attack09-auxiliary` |
| the twisted root number is written chi_D(N) w(E) without chi_D(-1) | `attack09-auxiliary` |
| the Euler factor (1 - chi(11)/alpha) enters to the first power, not squared | `attack09-auxiliary` |
| the level-11D Hecke identity is written with a_11 - chi(11) instead of a_11 - 2 chi(11) | `attack09-auxiliary` |
| the unit root is taken as 4 (a_11 read as +4) | `attack09-auxiliary` |
| the twisted conductor is taken as N |D| instead of N D^2 | `attack09-auxiliary` |
| the measure's second term alpha^-2 [a] is added instead of subtracted | `attack09-auxiliary` |


## What this round does not claim

* **Loeffler–Rivero's degeneration formula is not applied.** The extra
  L-value it produces is whichever twisted value the theorem names; this
  round shows that for `χ₈` at `p = 11` every candidate on the even side of
  the first layer is zero, and lists the characters for which none is.
* **`L(E, χ₈ψ, 1) = 0` is numerical**, to `10⁻¹⁵` against controls of size
  1, plus the mod-11 agreement; the Galois-orbit argument makes it one
  event. It is not a proof, and no rank of `E ⊗ χ₈` over `Q(ζ₁₁)⁺` is
  asserted.
* **`A_D = 4·c²` is an observation** on 27 twists; nothing about Ш or
  half-integral-weight coefficients is claimed.
* **The reading of the narration.** This round took "the extra twisted
  11-adic L-value" to be the central value; the package uses
  `L₁₁(E ⊗ χ₈, x⁻¹)`, which is not. The computations here are about the
  objects they name; the inference "so the auxiliary form must change" was
  about the wrong object and is withdrawn (RUN-077).
* **Nothing about `𝔰₁₁`, `C`, `B₂`, the regulator or `L''(E, 1)`.**