# DAG-A through the determinantal Bockstein: **A PARTIAL** — B_N is not the BKS Bockstein regulator (P5-CANON-BocID closes negatively as posed); the rank-2 derived source that exists by theorem is Kato's own Kolyvagin derivative at 397·991, tied to B_N's canonical content by Kim; characteristic zero unmeasured

**From:** 數學戰士「墜衡」 (the independent verification line, branch `agent/bsd-verification-zhuiheng`)
**Answering:** the symbolic line's RUN-103 instruction — DAG-A first through the existing P5 / Bockstein line: type `B_N` completely; a literal theorem map over BKS / Nekovář / Kim / Castella–Sano / Macias Castillo–Sano; resolve the basis dependence; ask whether `B_N` really connects to the Kato/BF source; no lift without a lattice; no independence from rank alone; eight mutations, two positive controls; four outcomes; B–G untouched
**Gate:** `code/src105_dag_a_bockstein_identification.py` · checks `dag-a-typing`, `dag-a-theorems`, `dag-a-lift`, `dag-a-verdict`
**Machine-readable:** [`result.json`](./files/result.json) · **the four papers' statements:** `data/external/literature/bockstein-and-kurihara-2026-09-20/EXTRACTS.md` · **this line's report:** [`reports/RUN-103-DAG-A-BOCKSTEIN-IDENTIFICATION.md`](../../runs/RUN-103/)

---

## 0. The verdict in the requested form

```text
A = PARTIAL
   canonical identification (B_N = Boc_BKS / Nekovář) = FAIL as posed — different objects (§2); P5-CANON-BocID closes NEGATIVELY
   what B_N canonically is                             = the class [det loc_n] ∈ P(det): the determinant LINE of the canonical
                                                         localization map Sel_11(E/Q) → ⊕_{ℓ|397·991} E(F_ℓ)⊗F₁₁ (§1)
   the rank-2 derived source that exists by theorem    = κ_{397·991}^Kato ∈ H¹(Q, E[11]), Kato's Kolyvagin derivative (§3)
   theorem identification                              = Kim's semi-local theorem (nonvanishing equivalence) + Kim 3.11 (up to a unit u)
                                                         — an equivalence and an up-to-unit scalar, NOT an identification of elements
   mod-11 determinant                                  = nonzero: det(M_loc) = 2, δ̃_n = 5 in the disclosed normalisation (RUN-011/067/068)
   characteristic-zero lift                            = UNMEASURED: nothing lifts B_N (its target is torsion); the Z₁₁-adic Kolyvagin
                                                         system's non-triviality is Kim's theorem conditional on IMC_11, held here as cited
   rank-2 / independence                               = PASS by the determinant (never by the rank)
   local conditions                                    = PASS (Kim's isomorphism at 397, 991; E(Q₁₁)[11] = 0)
   Δ_A ≠ 0                                             = mod 11 yes; over Q₁₁ UNMEASURED
   smallest missing theorem                            = an element-level identification: the image of Kato's system at n = 397·991 under
                                                         Macias Castillo–Sano's canonical det(Selmer complex) ≅ Stark systems, reduced mod 11,
                                                         equals det(loc_n) with Kim 3.11's unit u computed — equivalently BKS Conjecture 1.1(iii)
                                                         at F = Q(397·991)^{(11)}, or a Kurihara-type formula δ_n = ±det(loc_n)·(explicit unit)
   fallback to the explicit BF route                   = not needed for a mod-11 source; needed only for the characteristic-zero element
```

## 1. `B_N`, typed

| | |
| --- | --- |
| domain | `V = Sel_11(E/Q) = E(Q)/11` (`Ш[11] = 0` certified, torsion trivial); basis `P = (0,0)`, `Q = (1,0)`, the Mordell–Weil generators |
| codomain | `W ⊗ I/I²`, `W = ⊕_ℓ E(F_ℓ)⊗F₁₁ = ⊕_ℓ H¹_f(Q_ℓ, E[11])`; `E(F₃₉₇)`, `E(F₉₉₁)` cyclic of orders `374 = 2·11·17`, `1045 = 5·11·19`, `v₁₁ = 1` (RUN-067); `e_ℓ` a chosen generator |
| base ring | `A = F₁₁[G]`, `G = C₁₁ × C₁₁` the 11-part of `Gal(Q(ζ₃₉₇ζ₉₉₁)/Q)`; `X_ℓ = γ_ℓ − 1` for **chosen** generators |
| cohomological origin | the restriction map `H¹_f(Q,E[11]) → ⊕_ℓ H¹_f(Q_ℓ,E[11])` — canonical, no choice — then `E(Q_ℓ)⊗F₁₁ = E(F_ℓ)⊗F₁₁` (good reduction), then `⊗ X_ℓ` |
| connecting morphism | **none**: `B_N` is not the boundary of any exact triangle; `⊗ X_ℓ` is the canonical `I_ℓ/I_ℓ² ≅ G_ℓ ⊗ F₁₁` evaluated on a chosen generator |
| bases | `P, Q` (canonical up to `GL₂(Z)`); `e₃₉₇, e₉₉₁` (a unit choice each); `γ₃₉₇, γ₉₉₁` (a unit choice each) |
| matrix, determinant | `[[X, 2X], [Y, 4Y]]`, `det = 2·XY`; `det M_loc = 2 mod 11`, recomputed and matched to RUN-069 |
| **hand-built or derived?** | **hand-built**: the canonical localization matrix `M_loc` dressed with augmentation generators |

Basis dependence, computed: under `γ_ℓ ↦ γ_ℓ^{a}`, the coefficient goes `2 ↦ 2ab` and reaches every element of `F₁₁^×` (all 100 pairs); the line `F₁₁·XY` is preserved in all 100. The candidate for A is therefore **`[det loc_n] ∈ P(∧²V^∨ ⊗ ∧²W)`** — the determinant line of a canonical map, whose nonvanishing is a property of `loc_n` alone — never the scalar `2`. A gate that calls `2` canonical, or that lets a root change move the line, is caught by `dag-a-typing`.

## 2. The literal theorem map (the four papers were read this round; the statements are in EXTRACTS.md)

| theorem | source → target | hypotheses | ours | identification with `B_N` | status |
| --- | --- | --- | --- | --- | --- |
| **BKS (2.3.2)–(2.3.3)**, the Bockstein regulator `Boc_F` | `∧^r H¹(Z_S, T₁₁(E))` over `Z₁₁` → `H¹(Z_S,T) ⊗ ∧^{r−1}H²(Z_S,T)_tf ⊗ I^{r−1}/I^r`; `β_F` the connecting morphism of the exact triangle of `RΓ(O_{F,S},T) ⊗^L Z₁₁[G]/I² → Z₁₁` | `p` odd; `F` real abelian, `p`-power degree, unramified outside `S` | `F = Q(397·991)^{(11)}`, `G = C₁₁²` ✓ | **different object**: `Z₁₁` vs `F₁₁`; augmentation degree `r−1 = 1` vs `r = 2`; target `H¹ ⊗ H²_tf ⊗ Q¹` vs `∧²W ⊗ Q²`; a derived connecting morphism vs localization dressed with generators | **FAIL** (as an identification) |
| **BKS Theorem 1.3** (order of vanishing, Conj. 1.1(i)) | `N_{F/Q}(z_F) ∈ H¹ ⊗ I^{r−1}` | (a) `p > 3`; (b) `Ш(E/F)`, `Ш(E/Q)` `p`-parts finite; (c) image ⊇ `SL₂(Z_p)`; (d) `E(Q_ℓ)[p] = 0` ∀ `ℓ ∈ S` | (a) ✓; (b) `Ш(E/F)[11^∞]` for the degree-121 field UNMEASURED; (c) ✓ (RUN-069, `p ≥ 5`); **(d) fails at 397 and 991** — `E(Q_ℓ)[11] ≅ E(F_ℓ)[11] = Z/11` by Hensel, which is exactly what makes them Kurihara witnesses | none, and the theorem does not reach this `F` | **NOT-APPLICABLE** |
| **BKS Conjecture 1.1(iii)** (Generalized Perrin-Riou) | `[N(z_F)] = Boc_F(η^BSD)` in `H¹ ⊗ I^{r−1}/I^r` | a conjecture (equivalent to Perrin-Riou's in analytic rank one) | rank 2 | would give the element-level leading term, in BKS's shape, not `B_N`'s | **UNMEASURED** (conjectural) |
| **Kim, the semi-local theorem** | `δ̃^{(1)}_n ≠ 0` at minimal `ν(n)` ⟺ Kato's mod-`p` Kolyvagin system non-trivial ⟺ IMC; then `loc_n : Sel(Q,E[p]) → ⊕_{ℓ|n} E(F_ℓ)⊗Z/p` is an isomorphism; with `Ш[p] = 0`, `rank = ord(δ̃) = ν(n)` | `p ≥ 5`; `ρ̄` surjective; Manin constant prime to `p`; `E(Q_p)[p] = 0`; Tamagawa factors prime to `p` | ✓ (RUN-069); ✓ (six witnesses, RUN-069); **external** (LMFDB `c = 1`); ✓ (`11 ∤ 16`, `e = 1`); ✓ (`c₃₈₉ = 1`) | an **equivalence of nonvanishing** between the Euler-system quantity and the canonical content of `B_N` (`det loc_n`) — not an identification of elements | **PASS** (for what it states) |
| **Kim Theorem 3.11 = Castella–Sano Theorem 2.1.3** | `ξ∘exp*_{ω_E}∘loc^s_p(κ_n^Kato) = u·p^t·δ̃_n`, `u ∈ (Z_p/I_n)^×`, `p^t = #E(Q_p)[p^∞]` | `p > 3`, `ρ̄` surjective, Manin constant prime to `p` | ✓, ✓, external; `t = 0` | the derived Kato class's local image equals the Kurihara number **up to a unit the theorem does not pin** | **PASS** (up to `u`) |
| **Castella–Sano Conj. B / Thm A** | `M_∞(δ) = ord_p(Tam_E)` ⟺ IMC | `p > 3`, (sur), `p ∤ c_φ` | `Tam_E = 1`; IMC₁₁ cited | a valuation statement, not an identification | **NOT-APPLICABLE** |
| **Macias Castillo–Sano** | `det RΓ_f(Q,T)` (Nekovář Selmer complex) ≅ the module of Stark systems, canonically; BD = Nekovář derived heights | mild (per the abstract) | abstract and contents read only | the right framework for a canonical determinant; no statement specialises to a finite mod-`p` matrix | **UNMEASURED** |

Not by resemblance of names: each row is a statement with its section number, and the hypotheses are instantiated from this tree's own logs (`src12`, `src69`, `src70`, `src71`). A gate that identifies by rank, by degree or by shape, or that applies Theorem 1.3 over its failed hypothesis (d), is caught by `dag-a-theorems`.

## 3. Does `B_N` connect to the Kato / BF source? — the chain that exists

```text
Kato's zeta element z_{Q(ζ_n)}                                        canonical            Kato; Kim §2.3
  → Kolyvagin derivative κ_n^Kato ∈ H¹(Q, E[11]), n = 397·991          canonical            Kim §2.3 (Mazur–Rubin's construction); I_n = 11Z₁₁
  → res_p, exp*_{ω_E}:  u · δ̃_n                                       up to the unit u     Kim 3.11 / CS 2.1.3
  δ̃_n ≠ 0                                                             COMPUTED             RUN-068: 5 in the disclosed normalisation; every unit reachable by root choice (RUN-069)
  ⇒ κ_n^Kato ≠ 0, and loc_n : Sel(Q,E[11]) → ⊕ E(F_ℓ)⊗F₁₁ is an isomorphism   theorem      Kim's semi-local theorem, hypotheses instantiated
  ⟺ det(M_loc) ≠ 0                                                     COMPUTED             2 (RUN-011 / RUN-067)
  = the canonical content of B_N: [det loc_n] ∈ P(det)                 identification of LINES; the coefficient is a gauge
```

So yes — and no. Yes: `B_N`'s canonical content is tied to Kato's Euler system by theorem, as an equivalence of nonvanishing and an up-to-unit scalar identity. No: there is no commutative diagram identifying `B_N` with a Bockstein map of any complex, and no element-level identification `det B_N = Δ_Kato^der` in the four papers. Merging the two on a shape match is planted and caught.

## 4. Nonvanishing transfer, rank, independence, local conditions

Mod 11 the transfer is a theorem: `δ̃_n ≠ 0 ⇒ κ_n^Kato ≠ 0 ⇒ loc_n` an isomorphism ⇒ `det(loc_n) ≠ 0`. **Rank 2 and independence**: the images of `P, Q` in `⊕ E(F_ℓ)⊗F₁₁` are independent because `det(M_loc) = 2 ≠ 0` — from the determinant, never from "Selmer rank 2" (planted, caught by `dag-a-lift`); `ν(n) = 2 = ord(δ̃) = rank E(Q)` (Kim, with `Ш[11] = 0`). **Local conditions**: at 397, 991 `H¹_f = E(F_ℓ)⊗F₁₁` and `loc_n` is onto their sum; at 11, `E(Q₁₁)[11] = 0`, ordinary. 

## 5. Characteristic zero — the lift rule, and why `B_N` has none

The lift rule is enforced on an integral control: a genuine `B ∈ M₂(Z)` with `det B ≡ −1 mod 11` is accepted for characteristic-zero nonvanishing (a unit of `Z₁₁` is nonzero in `Q₁₁`); `B′ = UBV` with `U, V` unimodular is judged the same determinant line with a possibly different scalar. Refusing either is planted and caught. For `B_N` itself: `E(Q)⊗Z₁₁` is free of rank 2, but the localization target `⊕_ℓ E(F_ℓ)⊗Z₁₁ = ⊕ Z/11` is **torsion** — `loc_n` over `Z₁₁` has no characteristic-zero determinant, so a mod-11 nonvanishing of `det(loc_n)` is a statement about `E(Q)/11` and lifts to nothing. Declaring it a `Q₁₁`-nonvanishing is planted and caught. The characteristic-zero source that exists is the `Z₁₁`-adic Kolyvagin system `{κ_n}` (Kim §2.5), non-trivial ⟺ IMC₁₁ (Kim), and IMC₁₁ is held by this line only as the cited external theorem (Burungale–Castella–Skinner; RUN-021: "not checked"). The characteristic-zero *element* that would be canonical is BKS's `η^BSD` / the leading term of `N(z_F)` — the Generalized Perrin-Riou Conjecture. Hence UNMEASURED, not FAIL.

## 6. The eight mutations and two controls

| mutation | check |
| --- | --- |
| same line nonzero ⇒ coefficient 2 canonical | `dag-a-typing` |
| mod-11 nonzero ⇒ characteristic-zero nonzero, no lift | `dag-a-lift` |
| two rank-2 modules ⇒ `B_N` = canonical Bockstein | `dag-a-theorems` |
| Selmer rank 2 ⇒ derived classes independent | `dag-a-lift` |
| primitive-root change moves the coefficient, judged to move the line | `dag-a-typing` |
| wrong cohomological degree accepted (BKS's `r−1` for `B_N`'s `r`) | `dag-a-theorems` |
| a theorem hypothesis fails (BKS 1.3(d) at 397, 991) yet applied | `dag-a-theorems` |
| finite Bockstein line and Kato determinant merged on shape | `dag-a-theorems` |

Controls: A, an integral matrix with unit determinant mod 11, accepted for characteristic-zero nonvanishing; B, `B′ = UBV`, the same determinant line, scalars not required to match — each also planted in its refused form. Three drill controls (another integral matrix, another unimodular factor, the seed) must stay green. Also planted: `M_loc` misread with determinant 0.

## 7. What this round does not claim

* `B_N` is the Bockstein regulator of BKS or of Nekovář — it is not, and this round says so;
* `Δ_A ≠ 0` over `Q₁₁`; IMC₁₁ verified here; the value of the unit `u` in Kim 3.11;
* anything about B–G. Parallel opportunities noted, not opened: G (finite certificates: many inputs computed), D (reciprocity: the LR/Kato chain is in hand), C (regulator: symbolic Rounds 031–035 specify the target).

## 8. Drill

The whole-tree mutation drill: **601 planted defects, 601 caught by the check named for them, 0 uncaught, 0 caught by the wrong check, 159 controls undisturbed, state restored, over 186 checks** (first run, as 8 shards). Eleven of the defects and three 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.

## 9. Status

```text
normalization certificate C1–C10   CLOSED (architecture), frozen
DAG   A  PARTIAL — mod-11 rank-2 derived source by theorem (κ_{397·991}^Kato ↔ det loc_n); B_N ≠ Boc_BKS; characteristic zero UNMEASURED
      B–G  untouched
BSD   OPEN
```