# DAG-B: **B PASS** — the line that carries A's element is the BSD-shaped arithmetic line at (389.a1, 11): det⁻¹RΓ(Z_S,T) ≅ D_MW ⊗ 𝓗_str ≅ D_MW^{⊗2} ⊗ H¹_f(Q₁₁,T)⁻¹, the rank-2 Mordell–Weil determinant squared over the local Bloch–Kato line; the finite package at 11 is the unit line, placed factor by factor; the actual duality shift is 3, not 2; the strict line is kept whole as C's input

**From:** 數學戰士「墜衡」 (the independent verification line, branch `agent/bsd-verification-zhuiheng`)
**Answering:** the symbolic line's RUN-105 instruction — fix the actual Selmer complex; B-line apart from B-element; the Kummer audit; keep H² when it is rationally nonzero; pin H² and name 𝓗_str as the C-input; place the finite package without evaluating it; the Single-Entry Rule; actualise self-duality without forcing; transport the A-element without scalarising; the inherited Ш[11] = 0 never a hidden necessary condition; ten mutations, two positive controls; four outcomes; D, E, F untouched
**Gate:** `code/src107_dag_b_fundamental_line.py` · checks `dag-b-complex`, `dag-b-line`, `dag-b-h2`, `dag-b-ledger`, `dag-b-duality`, `dag-b-element`, `dag-b-controls`, `dag-b-verdict`
**Machine-readable:** [`result.json`](./files/result.json) · **the symbolic rounds read:** Rounds 032–034 under `data/external/gpt-symbolic-rounds/` · **this line's report:** [`reports/RUN-105-DAG-B-FUNDAMENTAL-LINE.md`](../../runs/RUN-105/)

---

## 0. The verdict in the requested form

```text
B = PASS
   actual arithmetic complex                = PASS — C_arith := RΓ(Z_S,T); CS's complex, MCS's relaxed Nekovář complex and the
                                              Poitou–Tate model are one object (§1); the ordinary complex is the self-dual hub
   canonical fundamental-line identification = PASS — Φ_B : det⁻¹RΓ(Z_S,T) ≅ D_MW,11 ⊗ 𝓗_str, and 𝓗_str ≅ D_MW,11 ⊗ H¹_f(Q₁₁,T)⁻¹,
                                              so det⁻¹RΓ(Z_S,T) ≅ D_MW,11^{⊗2} ⊗ H¹_f(Q₁₁,T)⁻¹ (§2–3)
   free MW factor                           = PASS — D_MW,11 = ∧²(E(Q)⊗Z₁₁) = Z₁₁(P∧Q), via the Kummer map: an isomorphism in
                                              degree 1 (cokernel T₁₁(Ш), inherited zero); NOT a quasi-isomorphism of complexes
   strict/free factor                       = PASS — 𝓗_str = det(H²)⁻¹ ≅ Sel_str(T) = Z₁₁·R_str, rank one, torsion 11⁰;
                                              identified with D_MW ⊗ H¹_f(Q₁₁,T)⁻¹ through the ordinary complex (§3)
   finite factor                            = PASS — placed under the Single-Entry Rule; every 11-length 0; the unit line (§4)
   self-dual placement                      = PASS — actual shift 3 on the ordinary complex; relaxed dual to strict; Round 034's
                                              centre-1 profile = the ordinary complex shifted by [1]; not forced (§5)
   A-element transported                    = PASS — ε·(P∧Q)⊗R_str = ε·(P∧Q)^{⊗2}⊗g⁻¹; Δ_B ≠ 0, σ_str ≠ 0, φ_fin = 1; ε UNMEASURED (§6)
   numerical finite evaluation              = NOT REQUIRED / deferred to G (all 11-lengths this tree computed are 0; Ш inherited)
   C-input                                  = 𝓗_str = Sel_str(T) ≅ D_MW,11 ⊗ H¹_f(Q₁₁,T)⁻¹ — the strict/free line, kept whole
   inherited input                          = Ш(E/Q)[11] = 0: numbers only (the ranks 2, 1; the Kummer cokernel); shapes unconditional
   D, E, F, G                               = untouched
```

## 1. The actual complex — three descriptions, one object, and the hub

| complex | ring | local at 11 | local at 389 | degrees | duality | shift | theorem |
| --- | --- | --- | --- | --- | --- | --- | --- |
| A `RΓ(Z_S,T)` (Castella–Sano) | `Z₁₁` | none — relaxed | none (and `H¹(Q₃₈₉,T) = 0`) | `[1, 2]` (`H⁰ = 0` by trivial torsion, `H³ = 0` for `p` odd) | dual to `RΓ_c(Z_S,T)`: `H^i_c ≅ H^{3−i}(Z_S,E[11^∞])^∨` — not self-dual | 3, against `RΓ_c` | Fukaya–Kato; CS §2.2–2.3 |
| B `RΓ̃_{𝓕_can}(Q,T)` (MCS, `T⁺ = T`, `T⁻ = 0`) | `Z₁₁` | `RΓ(Q₁₁, 0) = 0` | `RΓ_{/ur}(Q₃₈₉,T)`, acyclic (RUN-104) | as A | dual to the strict-at-11 complex | 3 | MCS Definition 2.2; B ≃ A |
| C `C_{PT,n}` | `Z/11ⁿ` | relaxed | `H¹_f = 0` | `[1, 2]` | `H²(C_PT) = H¹_{𝓕*}(Q,E[11ⁿ]^*(1))^*` | — | MCS Theorem 2.20 (Hyp 2.11, core vertex: RUN-104 at `n = 1`) |
| D `RΓ̃_ord(Q,T)` (ordinary, Example 2.1) — the hub | `Z₁₁` | `H¹_f(Q₁₁,T) = im H¹(Q₁₁,T⁺)` | `RΓ_{/ur}(Q₃₈₉,T)`, acyclic | `[1, 2]` here (`H̃³ ≅ H̃⁰(Q,E[11^∞])^∨ ⊆ H⁰(Q,E[11^∞])^∨ = 0`) | self-dual: `H̃² ≅ Sel_{11^∞}(E/Q)^∨` (Lemma 2.14(ii); Hyp 2.11 by Remark 2.12, `11 ∤ 1·1·16`) | 3 (`i ↔ 3−i`) | MCS Definition 2.2, Lemma 2.14; Nekovář's duality (shift cited as standard, not re-read) |

By the definition of the cone there is the exact triangle `RΓ̃_ord(Q,T) → RΓ(Z_S,T) → RΓ(Q₁₁,T⁻) ⊕ RΓ_{/ur}(Q₃₈₉,T) →`. **`C_arith := RΓ(Z_S,T)`.**

## 2. The line: ranks, the triangle, the Kummer audit

**Convention.** Round 032's `𝒜_R(C) = det_R(C)⁻¹ ≅ ⊗ det H^i(C)^{(−1)^{i+1}}` is Castella–Sano's `det⁻¹RΓ ≅ det(H¹) ⊗ det⁻¹(H²)`. One convention, two names.

**Local at 11** (Tate's Euler characteristic `h⁰ − h¹ + h² = −rank`, the outer terms killed by `E(Q₁₁)[11^∞] = 0` and `Ẽ(F₁₁)[11^∞] = 0`, i.e. `11 ∤ 16`): `rank H¹(Q₁₁,T) = 2`, `H¹_f(Q₁₁,T) = E(Q₁₁)⊗̂Z₁₁` of rank 1, `H¹_s` of rank 1, `rank H¹(Q₁₁,T⁻) = 1` with `H⁰ = H² = 0`, so `RΓ(Q₁₁,T⁻) ≃ H¹(Q₁₁,T⁻)[−1]` is a free rank-one line and `H¹(Q₁₁,T⁻) ≅ H¹_s(Q₁₁,T) ≅ H¹_f(Q₁₁,T)^∨` (Tate duality; the ordinary filtration is self-dual).

**The triangle's long exact sequence** (`H²(Q₁₁,T⁻) = 0`):

```text
0 → H¹_ord → H¹(Z_S,T) → H¹(Q₁₁,T⁻) → H²_ord → H²(Z_S,T) → 0
      2         2             1           2          1          ranks over Z₁₁, alternating sum 0
      2         2             1           2          1          dimensions mod 11 (RUN-104: ρ = 1), alternating sum 0
```

`H¹_ord = Sel(T) = H¹(Z_S,T)` (RUN-104), so the map to `H¹(Q₁₁,T⁻)` is zero and `0 → H¹(Q₁₁,T⁻) → H²_ord → H²(Z_S,T) → 0` is exact: **the ordinary `H²` (`≅ Sel_{11^∞}(E/Q)^∨`, rank 2) is an extension of the relaxed `H²` (`≅ Sel_str^∨`, rank 1) by the local singular line at 11.**

**The Kummer audit** (instruction §3). Actual map: the Kummer map `E(Q)⊗Z₁₁ → H¹(Z_S,T)`, the limit of the connecting maps of `0 → E[11ⁿ] → E → E → 0`, as a map of complexes `Λ_{E,11}[−1] → τ_{≤1}RΓ(Z_S,T) → RΓ(Z_S,T)`. Integral: yes. Injective: yes. Cokernel: `T₁₁(Ш(E/Q))` — zero under the inherited input, numeric only. Degree: 1. Rationally: an isomorphism in degree 1 (RUN-104: `H¹(Z_S,V) = Sel(V) = E(Q)⊗Q₁₁`). **A quasi-isomorphism of complexes: NO** — `H²(Z_S,T) ⊗ Q ≠ 0`, so Round 032's (6.1) fails and `Cone(ι) ≃ H²(Z_S,T)[−2]` is a free line, not a finite module. Consequence, exactly as the instruction anticipated: `D_MW,11 = det H¹ = Z₁₁(P∧Q)` is a factor of the line, not the line; the line is `D_MW,11 ⊗ 𝓗_str` with `𝓗_str := det(H²)⁻¹`. `H²` is kept. Reading the line as `det H¹` alone is planted and caught (`dag-b-line`).

## 3. `H²` pinned, and the strict line identified

`H²(Z_S,T) ≅ H¹_{𝓕_can*}(Q,T^∨(1))^∨ = Sel_str(E[11^∞])^∨` — MCS Lemma 2.14(ii) for the relaxed datum (Hyp 2.11 verified in RUN-104), and independently CS (2.13) with every local `H²` vanishing. Free rank 1 (conditional on the inherited input for the number; unconditionally `corank Sel_str = corank Sel − 1`); torsion `Z/11^k` with `k = 0` (the index of `loc₁₁`, RUN-104); dual object `Sel_str(E[11^∞]) = ker(Sel_{11^∞} → E(Q₁₁)⊗Q₁₁/Z₁₁)` with Tate module `Sel_str(T) = Z₁₁·R_str`; provenance: strict at 11 (the dual of relaxed), nothing at 389. No splitting is claimed — `H²` is torsion-free here. **`𝓗_str := det(H²)⁻¹ = Hom(H², Z₁₁) ≅ Sel_str(T)`** — the C-input.

**The identification.** From `H²_ord ≅ Sel_{11^∞}(E/Q)^∨ = (E(Q)⊗Q₁₁/Z₁₁)^∨ = Sel(T)^*` (inherited input), `det(H²_ord)⁻¹ = ∧²Sel(T)`, so `𝒜(RΓ̃_ord) = det H¹_ord ⊗ det⁻¹H²_ord = D_MW,11^{⊗2}` — **the square is the duality square between degrees 1 and 2, the one a height pairing trivialises** (Round 034 §17's second mechanism), not a torsion square. The exact sequence of §2 then gives `det H²(Z_S,T) = det(H²_ord) ⊗ det(H¹(Q₁₁,T⁻))⁻¹`, i.e.

```text
𝓗_str = det(H²(Z_S,T))⁻¹ ≅ ∧²Sel(T) ⊗ H¹(Q₁₁,T⁻) ≅ D_MW,11 ⊗ H¹_f(Q₁₁,T)⁻¹,
```

the same statement as the exact sequence `0 → Sel_str(T) → Sel(T) → H¹_f(Q₁₁,T) → 0` (onto, since the index is `11⁰`), which gives `∧²Sel(T) ≅ Sel_str(T) ⊗ H¹_f(Q₁₁,T)`. Checked mod 11 on RUN-104's numbers: `(P∧Q) ⊗ g⁻¹ ↦ u_Q·P − u_P·Q = 6P + 2Q = 6·(P + 4Q)`, whose `loc₁₁` vanishes and which lies on the strict line `F₁₁(P + 4Q)`. Hence

```text
det⁻¹RΓ(Z_S,T) ≅ D_MW,11 ⊗ 𝓗_str ≅ D_MW,11^{⊗2} ⊗ H¹_f(Q₁₁,T)⁻¹
```

— the rank-2 Mordell–Weil determinant squared over the local Bloch–Kato line at 11. That is the BSD-shaped arithmetic line: the square is what a regulator trivialises, the local line is what a `p`-adic period trivialises, and neither trivialisation is B's.

## 4. The finite package, placed under the Single-Entry Rule

| factor | exact sequence / cone | degree | sign in `𝒜` | source | destination | 11-length | label |
| --- | --- | --- | --- | --- | --- | --- | --- |
| `E(Q)[11^∞]` | image of `H⁰(Z_S,E[11^∞])` in `H¹(Z_S,T)_tors` (Kummer); its dual in `H̃³_ord ⊆ H⁰(Q,E[11^∞])^∨` | 1 | + | RUN-011 | torsion endpoint | 0 | COMPUTED_HERE |
| `Ш(E/Q)[11^∞]` | `0 → E(Q)⊗Q/Z → Sel_{11^∞} → Ш → 0`; dual in the torsion of `H̃²_ord ≅ Sel^∨` and of `H²(Z_S,T) ≅ Sel_str^∨` | 2 | − | the programme's documents | middle module | 0 | INHERITED — NUMERIC ONLY |
| `c₃₈₉` | `H¹(Q₃₈₉,T)/H¹_ur` and `H²(Q₃₈₉,T) = E(Q₃₈₉)[11^∞]^∨` inside `RΓ_{/ur}(Q₃₈₉,T)`, acyclic | 2 | − | RUN-067 (`I₁`, `c = 1`) | local piece at 389 | 0 | COMPUTED_HERE |
| `E(Q₁₁)[11^∞]`, `Ẽ(F₁₁)[11^∞]` | the local `H²` terms of CS (2.13) and of the ordinary local condition | 2 | − | RUN-104 (`11 ∤ 16`) | local piece at 11 | 0 | COMPUTED_HERE |
| `coker(loc₁₁) = Z/11^k` | the torsion of `H²(Z_S,T) ≅ Sel_str^∨` — once; `H¹_f(Q₁₁,T)` is torsion-free | 2 | − | RUN-104 | `𝓗_str` (torsion part) | 0 | COMPUTED_HERE |
| `H¹_ur(Q₃₈₉,T)`, `H¹(Q₃₈₉,T)` | `RΓ_ur → RΓ → RΓ_{/ur}` at 389, all acyclic (`11 ∤ 388`) | 1 | + | RUN-104 | local piece at 389 | 0 | COMPUTED_HERE |

Duplicates 0, unplaced 0, sum of 11-lengths 0: **the finite package at 11 is the unit line.** The middle module is not named Ш × Tamagawa — at 11 its graded pieces are the local pieces (zero, computed) and the inherited Ш piece (zero, numeric only); Round 033's naming rule is respected by having nothing to name. B places; G would evaluate; the values are zero here except the inherited one, and that is not claimed as a theorem. A factor counted twice, the index squared twice, or the strict free line filed as Ш is planted and caught (`dag-b-ledger`).

## 5. Duality, actualised

The relaxed complex is dual to the compactly supported one, not to itself. The ordinary complex is self-dual with **the actual shift `[3]`** — `RΓ̃_ord(Q,T) ≅ RHom(RΓ̃_ord(Q,T^∨(1)), Z₁₁)[−3]`, degrees `1 ↔ 2` (Lemma 2.14(ii)) and `0 ↔ 3`. In the `𝒜 = det⁻¹` convention: torsion in degree 1 (`+`), Ш and the local pieces in degree 2 (`−`), the torsion dual in degree 3 (`+`) — `𝒜 ∼ #tors²/#Ш`, the reciprocal of BSD's `#Ш/#tors²`; `det` carries the BSD orientation; the parity is the same either way. Round 034's profile `H⁰ = T, H¹ = M, H² = T^∨` with centre 1 is **the ordinary complex shifted by `[1]`**; `det(C[1]) = det(C)⁻¹`, so the exponent flips sign with the shift — on a toy profile with `ℓ(T) = 1, ℓ(M) = 3`: actual degrees 1, 2, 3 give `2ℓ(T) − ℓ(M) = −1`, Round 034's degrees 0, 1, 2 give `ℓ(M) − 2ℓ(T) = +1`. This is Round 034 §82–83's own diagnostic applied globally: nothing is forced, the shift is recorded, and a `[2]` forced through is planted and caught (`dag-b-duality`).

## 6. The A-element, transported

`𝔷_{Q∞}(1) = ε·(P∧Q) ⊗ R_str` with `Δ_B := P∧Q` a generator of `D_MW,11`, `σ_str := ε·R_str` a generator of `𝓗_str`, `φ_fin := 1`; under the identification of §3, `𝔷_{Q∞}(1) = ε·(P∧Q)^{⊗2} ⊗ g⁻¹` with `g` a generator of `H¹_f(Q₁₁,T)` and `(P∧Q)⊗g⁻¹ ↦ a_Q P − a_P Q = R_str`. `Δ_B ≠ 0`, `σ_str ≠ 0`, the finite element is a determinant class of the unit line. `ε ∈ Z₁₁^×` is not computed; no height is applied to `σ_str`; the line isomorphism `Φ_B` is an isomorphism of rank-one lattices and not a scalar equality — `ε` is exactly what it does not determine; `P∧Q` against `Q∧P` is a sign absorbed into `ε`. Each refusal (a line isomorphism read as a scalar, an element left untransported, a sign treated as an obstruction) is planted and caught (`dag-b-element`).

## 7. The ten mutations and two controls

| mutation | check |
| --- | --- |
| `H²` rationally nonzero, yet "degree 1 only" forced | `dag-b-line` |
| the strict free line treated as finite Ш | `dag-b-ledger` |
| a finite factor counted in a cone and again as a scalar | `dag-b-ledger` |
| the actual duality shift is 3, yet the BSD exponent forced through `[2]` | `dag-b-duality` |
| a line isomorphism read as an element scalar equality | `dag-b-element` |
| A's basis not transported (only the line proved) | `dag-b-element` |
| B judged FAIL because a finite value (G's) is unknown | `dag-b-verdict` |
| the inherited Ш[11] = 0 used as a structural theorem | `dag-b-h2` |
| the MW determinant's orientation treated as an obstruction | `dag-b-element` |
| the local index `11^k` squared twice | `dag-b-ledger` |

Controls: **A** — the two-degree complex `[Z² → Z]`, whose `𝒜` is `det(Z²) ⊗ det(Z)⁻¹`, accepted with both factors, never compressed to `det H¹`; **B** — the three-term self-dual finite profile, whose exponent is `ℓ(M) − 2ℓ(T)`. Each also planted in its refused form (`dag-b-controls`). Also planted: the shift declared 2; the global ranks declared `(3, 2)`; the ordinary ranks `(2, 1)`; the local Bloch–Kato rank 2; the strict formula replaced by "`Sel_str = Ш`"; control A with a zero second cohomology. Four drill controls (another two-degree complex; three other finite profiles) must stay green.

## 8. What this round does not claim

* the unit `ε`; any trivialisation of `𝓗_str` — the height / regulator bridge is C, and the strict line is handed to it whole;
* Ш(E/Q)[11] = 0 verified here (inherited; the ranks 2, 1 and the Kummer cokernel are the numbers that need it, and they are labelled); Nekovář's duality theorem re-read (its shift is cited as standard; the degree bookkeeping used is Lemma 2.14(ii));
* any finite VALUE beyond the 11-lengths this tree computed (torsion, `c₃₈₉`, `#Ẽ(F₁₁)`, the index) — G's business;
* anything about C, D, E, F, G; BSD.

## 9. Drill

The whole-tree mutation drill: **640 planted defects, 640 caught by the check named for them, 0 uncaught, 0 caught by the wrong check, 168 controls undisturbed, state restored, over 201 checks** (first run, as 8 shards). Eighteen of the defects and four of the controls are this round's. The report and package were written before the drill was launched and nothing was touched while it ran.

## 10. Status

```text
normalization certificate C1–C10   CLOSED (architecture), frozen
DAG   A  PASS by cited theorem (RUN-104)
      B  PASS — det⁻¹RΓ(Z_S,T) ≅ D_MW ⊗ 𝓗_str ≅ D_MW^{⊗2} ⊗ H¹_f(Q₁₁,T)⁻¹; finite package the unit line; shift 3; C-input 𝓗_str
      C–G  untouched
BSD   OPEN
```