# RUN-078 — GPT-6's Attack 10: the circular unit in Z[ζ₈], its 11-adic logarithm to 11¹⁰, the Leopoldt value L₁₁(1, χ₈) ≡ 5, the filtered-Frobenius entry q_bot ≡ 9, the Bernoulli cross-check, and the first-order trace algebra on 676 pairs — every finite statement recomputed and identical; the 1-motive, the Selmer identification and the family inputs read

**Arm:** 數學戰士「墜衡」 / AMRAL Research Lab
**Subject:** `BSD_Proof_Attack_10_Unit_and_Trace_Bridge`, with its `attack10_result.json` — the eighth and last package from the web GPT-6 session, received 2026-09-13 (provenance)
**Tools:** `src80_attack10_unit_and_trace_bridge.py`, `src11_gate_drill.py`
**Logs:** `src80-attack10-unit-and-trace-bridge.json`, `src11-gate-drill.json`

**Result: Attack 10 builds two things around Attack 09's unknown leading period `C`: a norm-one 1-motive `[Z → Res¹_{Q(√2)/Q} G_m]`, `1 ↦ 17 − 12√2`, whose 11-adic filtered Frobenius matrix it calibrates exactly, and a trace-reconstruction formula in a weight family that names the null-homotopy of the cup product `b_ε ∪ β`. Its finite content recomputes exactly, all of it: in `Z[ζ₈]`, `(1 − ζ₈)(1 − ζ₈⁷) = 2 − r` and `(1 − ζ₈³)(1 − ζ₈⁵) = 2 + r` with `r = ζ₈ − ζ₈³`, `r² = 2`, so `u₈ = 3 − 2r = ε⁻²`, `u_bot = u₈² = 17 − 12r = ε⁻⁴`, `N(ε) = −1`, `N(u₈) = N(u_bot) = 1`, `G(χ₈) = 2r`; `L(1, χ₈) = log(1 + √2)/√2 = 0.623225240140…` by the Gauss-sum logarithms, the Dirichlet series and the class-number formula; `u_bot⁶ = 768398401 − 543339720√2`, its logarithm from the series to `m = 19` exactly in `Q(√2)` with the omitted terms of valuation `≥ 19`, giving `L_bot ≡ 8662686351`, `L_ε ≡ 17287396863 (mod 11¹⁰)`, `v₁₁(L_ε) = 1`, `L_ε/11 ≡ 1571581533`, `L₁₁^std(1, χ₈) = (12/11)L_ε ≡ 2353344559`, `q_bot = −(12/11)L_bot ≡ 2339535163 (mod 11⁹)` — all nineteen term residues and all five values identical to the package's — with `L₁₁^std(1, χ₈) ≡ 5`, `q_bot ≡ 9 (mod 11)`, `q_bot = 4·L₁₁^std(1, χ₈)`, and `L_ε ≡ 55 (mod 121)`, Attack 09's `log₁₁ ε ≡ 55√2` read from the other end of the same series; `B_{10,χ₈} = 28730410`, `−B₁₀/10 ≡ 5 (mod 11)`, the Kubota–Leopoldt value at `−9` agreeing mod 11 with the value at 1 as it must; the framed Frobenius matrix `[[−1/11, q],[0, 1]]`, `q = −(12/11)L_bot`; and the model `ρ(a), ρ(b), ρ(J)` over `Q[X]/(X²)` on the 26 reduced words of length ≤ 2 in `Free(a,b) ∗ C₂`, where on all 676 pairs `a_g = (T(g) − T(Jg))/2`, `Q(g,h) = b_ε(g)v(h)`, `b_ε ∪ β = −δF`, both cocycle relations, `rank Q ≤ 1`, `Q(a,a) = 15`, `Q(a,b) = 21`, `v(a) = 5`, `v(b) = 7` from either pivot, and the variant with lower entries `8X, 9X` giving `24, 27` — the package's 676-row table and 26 reconstructions matched row by row. **The package computes no Coleman trace jet, no `n`, no `C`, no `B₂`, no `𝔰₁₁`; it rejects its own shortcut ("把式 (14) 的 unit 稱為 C … 不成立"); its status is "BSD 尚未證明；canonical frontier 不更新". Nothing moves.**

---

## §2 — the unit

| the package says | recomputed in `Z[ζ₈]/(ζ₈⁴ + 1)` and `Z[√2]` |
| --- | --- |
| `r = ζ₈ − ζ₈³`, `r² = 2` | `(2, 0, 0, 0)` |
| `u₈ = (1−ζ₈)(1−ζ₈⁷)/((1−ζ₈³)(1−ζ₈⁵)) = 3 − 2r = ε⁻²` | numerator `2 − r`, denominator `2 + r`; `(3 − 2r)(2 + r) = 2 − r`; `(3 − 2√2)(1 + √2)² = 1` |
| `u_bot = u₈² = 17 − 12r = ε⁻⁴`, `Kum(u_bot) = −4Kum(ε)` | `(17, −12)`; `(17 − 12√2)(1 + √2)⁴ = 1` |
| `N(ε) = −1`, field norms of `u₈`, `u_bot` are 1 | `−1, 1, 1` |
| `G(χ₈) = ζ₈ − ζ₈³ − ζ₈⁵ + ζ₈⁷ = 2r` | `(0, 2, 0, −2) = 2r` |
| (8) `L(1, χ₈) = −(1/2r)Σ χ₈(a) log|1 − ζ₈^a| = log(1 + r)/r` | `0.623225240140230` both ways; the series in blocks of 8 to `10⁻¹¹` |
| (7) `log|u_bot|/r = −4 log(1 + r)/r = −4L(1, χ₈)` | identity |

## §3 — the 11-adic calibration

| quantity | the package (mod `11¹⁰` or `11⁹`) | here |
| --- | ---: | ---: |
| `u_bot⁶` | `768398401 − 543339720√2` | same |
| series terms `m = 1..19`, residues | listed | all 19 identical |
| `L_bot = log₁₁(u_bot)/√2` | `8662686351` | `8662686351` |
| `L_ε = −L_bot/4` | `17287396863`, `v₁₁ = 1` | same, `v₁₁ = 1` |
| `L_ε/11` (mod `11⁹`) | `1571581533` | same |
| `L₁₁^std(1, χ₈) = (12/11)L_ε` | `2353344559`, `≡ 5 (mod 11)` | same |
| `q_bot = −(12/11)L_bot` | `2339535163`, `≡ 9 (mod 11)` | same; `= 4·L₁₁^std` |
| `L_ε (mod 121)` | Attack 09: `log₁₁ ε ≡ 55√2` | `55` |

The tail: for `m ≥ 20`, `v₁₁(w^m/m) ≥ m − v₁₁(m) ≥ 19`, checked for `m` to
219. The first coordinate of `log(u_bot⁶)` is `0 (mod 11¹⁰)`, as the norm
forces. The framed matrix: with `φ = diag(−1/11, 1)` on `(d_χ, d₀)` and
`Fil⁰ = ⟨h⟩`, `h = d₀ + L_bot d_χ`, one has `φ(h) = h − (12/11)L_bot d_χ`,
so `q_bot = −(12/11)L_bot` — (12); and `q_bot = 4L₁₁^std(1, χ₈)` — (14) —
follows from `L_bot = −4L_ε` and (13). Bernoulli: `B_{10,χ₈} = 8⁹·Σ_a
χ₈(a)B₁₀(a/8) = 28730410`, `−B₁₀/10 = −2873041 ≡ 5`, and `B_{2,χ₈} = 2`
again.

Not verified: the identification of the Kummer cocycle's Bloch–Kato
logarithm with the unit logarithm (Darmon–Rotger Example 1.6, cited), the
Leopoldt formula itself (Bertolini et al. Theorem 1.1, cited), and
Corollary 10.3's descent argument — read.

## §5–§6 — the trace algebra

`ρ(a) = [[2+7X, 3+13X],[5X, 1+11X]]`, `ρ(b) = [[3+17X, 2+19X],[7X, 1+23X]]`,
`ρ(J) = diag(−1, 1)` over `Q[X]/(X²)`; the 26 reduced words of length ≤ 2
in `a, a⁻¹, b, b⁻¹, J`; `ξ(g) = a_g(0)`, `A(g) = [X]a_g`, `b_ε(g) = b_g(0)`,
`v(g) = [X]` of the lower-left entry, `β = v/ξ`, `F = A/ξ`.

| identity | pairs / words | result |
| --- | ---: | --- |
| (21) `a_g(X) = (T(g) − T(Jg))/2` | 26 | all |
| (22) `Q(g,h) = A(gh) − ξ(g)A(h) − ξ(h)A(g) = b_ε(g)v(h)` | 676 | all |
| (24) `(b_ε ∪ β)(g,h) = −(δF)(g,h)` | 676 | all |
| `b_ε(gh) = b_ε(g) + ξ(g)b_ε(h)`, `v(gh) = v(g)ξ(h) + v(h)` | 676 | all |
| `rank (Q(g_i, h_j)) ≤ 1` (every 2×2 minor) | 26 × 26 | all |
| (23) `β(h)` from pivot `a` (`b_ε = 3`) and from pivot `b` (`b_ε = 2`) | 26 | equal, and equal to `v/ξ` |
| `Q(a,a) = 15`, `Q(a,b) = 21`, `v(a) = 5`, `v(b) = 7` | | same |
| lower entries `8X, 9X`: `Q(a,a) = 24`, `Q(a,b) = 27`, same `X = 0` matrices | | same |

Every one of the package's 676 rows (`Q`, cup, `−δF`) and 26 reconstruction
rows (`β`, `A/ξ`, `v` from either pivot) is reproduced. This is what the
package says it is — "EXACT ALGEBRA MODEL; not arithmetic Galois or Hecke
trace data" — and this round treats it as such.

## The labels

| where | what |
| --- | --- |
| the status line | "尚未計算真實 Coleman 族的 trace jet 或 meromorphic Eichler–Shimura 常數 C" |
| §1 | the unit motive's period "尚未被識別為 Attack 09 的族插值常數 C" |
| §6 | "模型不是 Galois 數據" |
| §7 | "把式 (14) 的 unit 稱為 C，會跳過上表第二個比較箭頭，因此不成立" |
| §8 | "BSD 尚未證明；canonical frontier 不更新" |
| the output | `actual_Coleman_trace_jets`, `meromorphic_ES_order_n`, `period_C`, `Beilinson_Flach_B2_coordinates`, `s11` all null; `BSD_proved: false`; `canonical_frontier_updated: false`; `period_C_computed: false` |

## The drill

**341 defects, 341 caught by the check named for each**, 0 uncaught, 0 caught by the wrong check, 72 controls undisturbed, over 106 checks.

The 8 planted for this gate, each turning `attack10-unit-bridge` red and nothing else:

| planted defect | went red |
| --- | --- |
| sqrt 2 is written as zeta_8 + zeta_8^3, whose square is -2 | `attack10-unit-bridge` |
| the circular unit's numerator and denominator are swapped | `attack10-unit-bridge` |
| the bottom norm factor 1 - chi_8(11) is taken as 1, so u_bot = u_8 | `attack10-unit-bridge` |
| the logarithm series is started from u_bot^5, which is not 1 mod 11 | `attack10-unit-bridge` |
| phi on the subline is taken as +1/11 | `attack10-unit-bridge` |
| Leopoldt's Euler factor is written 1 - 1/11 instead of 1 + 1/11 | `attack10-unit-bridge` |
| the model's lower-left entry of rho(a) is 6X instead of 5X | `attack10-unit-bridge` |
| the Bernoulli cross-check is run at n = 8 instead of n = 10 | `attack10-unit-bridge` |


## What this round does not claim

* **The 1-motive and its Tate realisation (5)–(6), `D_cris(T)` and (9), the
  Selmer space being spanned by `Kum(ε)`, the reducibility ideal `(X)` and
  the reverse extension of §4** are theory, read; the matrix (12) is
  checked as linear algebra given (9) and (11).
* **Theorem 10.4 is checked on the model, as the package checks it**; the
  written proof is elementary algebra on (19) and is read. No arithmetic
  trace `𝒯_X(g)` exists in the package or here.
* **Nothing about `C`, `n`, `B₂`, `𝔰₁₁`** — the package's two open arrows
  in §7 stay open.
* **This is the last web package.** The eight packages
  (RUN-070–075, 077–078) share one shape: every finite computation right,
  every theorem cited, no OPEN gate closed, and each one says so.