# RUN-105 — DAG-B: the line that carries RUN-104's element is the BSD-shaped arithmetic fundamental line at `(389.a1, 11)` — `det⁻¹RΓ(Z_S,T) = D_MW ⊗ 𝓗_str`, and through the self-dual ordinary Selmer complex (Nekovář shift `[3]`, determinant `D_MW^{⊗2}`) the strict line is `𝓗_str ≅ D_MW ⊗ H¹_f(Q₁₁,T)⁻¹`, so the whole line is the rank-2 Mordell–Weil determinant squared over the local Bloch–Kato line at 11; the Kummer map is an isomorphism in degree 1 but not a quasi-isomorphism of complexes, so Round 032's concentration criterion fails and `H²` is kept, not killed; the finite package at 11 is the unit line with every factor placed once; the actual duality shift is 3, not 2, and Round 034's centre-1 profile is the ordinary complex shifted by `[1]`: **B PASS; C-input = the strict line** **Arm:** 數學戰士「墜衡」 / AMRAL Research Lab **Subject:** the symbolic line's instruction after [RUN-104](../RUN-104/) — fix the actual Selmer complex (three descriptions itemised); B-line apart from B-element; the Kummer audit; H² kept when rationally nonzero; H²'s identity at theorem level and `𝓗_str := det(H²_free)⁻¹` as the C-input; finite package placed, not evaluated; B versus G; the Single-Entry Rule; self-duality actualised, not forced; the A-element transported without scalarising; the inherited Ш[11] = 0 never a hidden necessary condition; ten mutations, two positive controls; four outcomes; D, E, F untouched **Tools:** `src107_dag_b_fundamental_line.py`, `src11_gate_drill.py` **Logs:** `src107-dag-b-fundamental-line.json`, `src11-gate-drill.json` **Deliverable:** [`handoff/DAG-B-FUNDAMENTAL-LINE/`](../../packages/DAG-B-FUNDAMENTAL-LINE/) · **the symbolic rounds read this round:** Round 032, 033, 034 · **the papers' statements:** `EXTRACTS.md` **Result. The actual arithmetic perfect complex is fixed: Castella–Sano's `RΓ(Z_S,T)`, Macias Castillo–Sano's relaxed Nekovář complex and the Poitou–Tate model are one object (RUN-104's acyclicity at 389 and Theorem 2.20), itemised with ring, local conditions at 11 and 389, degrees `[1, 2]`, duality and shift; the ordinary Greenberg complex `RΓ̃_ord(Q,T)` is the self-dual hub, related to it by the triangle of Definition 2.2. Round 032's arithmetic determinant `𝒜 = det⁻¹` is Castella–Sano's convention, so `𝒜(C_arith) = det(H¹) ⊗ det⁻¹(H²)`. The Kummer map `E(Q)⊗Z₁₁ → H¹(Z_S,T)` is audited: integral, injective, cokernel `T₁₁(Ш)` (inherited zero, numeric only), an isomorphism in degree 1 — and NOT a quasi-isomorphism of complexes, because `H²(Z_S,T) ⊗ Q ≠ 0` (rank 1): Round 032's concentration hypothesis (6.1) fails and the cone of the Kummer map is a free line, so `D_MW,11 = ∧²(E(Q)⊗Z₁₁) = Z₁₁(P∧Q)` is a factor of the line, not the line; `H²` is kept as `𝓗_str := det(H²)⁻¹`. `H²` is pinned by Lemma 2.14(ii): `Sel_str(E[11^∞])^∨`, free of rank one, torsion `11⁰` (RUN-104's index), dual `Sel_str(T) = Z₁₁·R_str`, `R_str ≡ P + 4Q`. The triangle's long exact sequence is a rank ledger — `0 → H¹_ord → H¹(Z_S,T) → H¹(Q₁₁,T⁻) → H²_ord → H²(Z_S,T) → 0` with ranks `2, 2, 1, 2, 1` over `Z₁₁` and dimensions `2, 2, 1, 2, 1` mod 11, alternating sums 0 — whose reading is that the ordinary `H²` (`≅ Sel^∨`, rank 2) is an extension of the relaxed `H²` (`≅ Sel_str^∨`, rank 1) by the local singular line at 11. Since `H²_ord ≅ Sel(T)^*` canonically, `𝒜(RΓ̃_ord) = D_MW^{⊗2}` — the square is the duality square between degrees 1 and 2, the one a height pairing trivialises — and hence `𝓗_str ≅ D_MW ⊗ H¹_f(Q₁₁,T)⁻¹`, checked mod 11: `(P∧Q) ⊗ g⁻¹ ↦ u_Q P − u_P Q = 6P + 2Q = 6(P + 4Q)`, on the strict line. So `det⁻¹RΓ(Z_S,T) ≅ D_MW^{⊗2} ⊗ H¹_f(Q₁₁,T)⁻¹`: the rank-2 determinant squared over the local Bloch–Kato line at 11 — the BSD-shaped line. The finite package at 11 is placed factor by factor under the Single-Entry Rule (torsion, Ш, `c₃₈₉`, the local torsion at 11, the index of `loc₁₁`, the unramified terms at 389 — each with its exact sequence, degree, sign, source) and is the unit line: every 11-length is 0, Ш's zero labelled inherited and numeric only; nothing is named Ш × Tamagawa. Duality: the relaxed complex is dual to the strict one, not to itself; the ordinary complex is self-dual with the actual shift `[3]` (`i ↔ 3−i`), not `[2]`; Round 034's centre-1 profile `H⁰ = T, H¹ = M, H² = T^∨` is the ordinary complex shifted by `[1]`, and the exponent flips sign with the shift (`2ℓ(T) − ℓ(M)` against `ℓ(M) − 2ℓ(T)`, checked on a toy profile) — nothing forced. The A-element transports to `𝔷_{Q∞}(1) = ε·(P∧Q) ⊗ R_str = ε·(P∧Q)^{⊗2} ⊗ g⁻¹`, `Δ_B` and `σ_str` nonzero, the finite element 1, `ε ∈ Z₁₁^×` unmeasured, nothing scalarised, the orientation a sign absorbed into `ε`. Verdict B PASS; C-input `𝓗_str = Sel_str(T) ≅ D_MW ⊗ H¹_f(Q₁₁,T)⁻¹`. Ten mutations and both controls planted; D, E, F, G not opened.** --- ## 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 ```