# RUN-087 — BSD Symbolic Rounds 017–020 (the web GPT's flatness from naturality with H¹ and H² obstructions; the normalisation fibration for cross-rank calibration; the universal normalisation line versus partner factors; the character-lattice one-calibrator criterion): every boxed statement instantiated exactly — natural comparisons on random graphs, a genuine H² obstruction over Q, calibration by tensor contraction with the three mutation drills the documents prescribe, the maximal calibratable lattice by valuations, and the exact and valuation-level one-calibrator criteria with explicit witnesses and Smith normal forms

**Arm:** 數學戰士「墜衡」 / AMRAL Research Lab
**Subject:** Round 017, Round 018, Round 019, Round 020 — the four documents that reached the drop folder late on 2026-09-13 ("現在已經到了20了吧") (provenance); the sixteen already held were byte-identical
**Tools:** `src89_symbolic017_020_naturality_fibration_lattice.py`, `src11_gate_drill.py`
**Logs:** `src89-symbolic017-020-naturality-fibration-lattice.json`, `src11-gate-drill.json`

**Result. The four documents are right as the groupoid cohomology, tensor bookkeeping and integer linear algebra they are, and are checked as such on 40 exact random instances each. Round 017: with honest line functors `𝒜, ℬ` on a random graph and a natural `Ξ`, the relative transport `ρ_f = β_f/α_f` is the coboundary `x_Y/x_X` (4.1), every loop has trivial relative holonomy (5.1) while `𝒜` and `ℬ` carry the same nontrivial absolute monodromy (6.1); the anchor-extended `Ξ_X = ℬ(f)Ξ₀𝒜(f)⁻¹` is path-independent exactly when the relative monodromy is trivial — equal along the tree and through each cotree edge for the natural `ℬ`, unequal for a random one (7.1–7.3, 8.2); projective defects multiply along composites (10.1), move by coboundaries under rescaling (11.1), are removable iff their holonomy class is trivial (12.1, 13.1) and always on a tree (14.1); naturality on generators extends (15.1); **the H² layer made concrete** — projective functors of `Z/n` with carry multipliers `t^{⌊(a+b)/n⌋}` satisfy the cocycle condition and `δc = α_ℬ/α_𝒜` (24.1–24.2), and for `n = 2`, `t_𝒜 = 2`, `t_ℬ = 3` no rational comparison scalars exist because `c₁² = 3/2` has no solution in `Q` (25.1), while equal multipliers give an ordinary 1-cocycle (§26). Round 018: `∧²g` acts on `det H₂` by `det g`, so no `GL(H₂)`-natural nonzero map `det H₀ → det H₂` exists (2.1); `Θ̂ = Θ^{(n)} ⊗ n^{⊗w}` is intrinsic with `Θ^{(cn)} = c^{−w}Θ^{(n)}` (5.1); the calibrator anchor and the contraction `Θ^{cal} = (id ⊗ τ_{n₀}^{⊗w})Θ̂` are frame-independent (6.1, 7.1), the calibrator square is `𝒩^{⊗2}` (8.1, 9.1); normalisation transports compose (10.2) and one anchor calibrates every weight path-independently along a flat component (23.1); a partner factor absorbed into `𝒩` leaves `A_E/A₀` in the calibrated ratio (15.1); the declared weight is audited by a reframe (§32); Round 001's ratio and Round 002's square are contractions (§33–34). Round 019: the ten-row factor table (§14) agrees with the boxed classifications of §5–§13; E- versus L-universality — three factors with equal valuations at every prime up to 13 but differing by `−17` and `1/17` cancel at valuation level and not exactly (15.1–16.1, 17.1); **the maximal calibratable lattice (18.1, 19.1) computed as the integer kernel of the partner-ratio valuation matrix** on designed families `f₁ = a_i`, `f₂ = a_i²b`, `f₃ = b_i`, `f₄ = C` comes out as `Z(2,−1,0,0) ⊕ Z(0,0,0,1)`, and `f₁²/f₂ = 1/b` is indeed partner-independent; calibration removes exactly the universal character and leaves `J^{a_E−ka₀}S^{b_E−kb₀}` and the partner coefficients (22.1, 23.1); the vector-level formula `κ_E = (A₀ℓ₀/A_E)·B_E/b₀` (24.1) and the determinant level where `b₀²` removes `C_π²` but not `P_{E,1}P_{E,2}` (25.1); the three mutation drills of §32–34 run as written — a common reframe scales the raw objects by `c^{w_i}` and leaves the calibrated ones fixed, a partner rescale survives calibration, a coordinate change moves `j_lead` by `u^{−1}` and the anchor not at all. Round 020: `w = kw₀ ⇒ χ_w = χ_{w₀}^k` (4.1–4.3) and, for `w ∉ Zw₀`, explicit witnesses in the kernel of `χ_{w₀}` that move `χ_w` — a `2`-power witness when `w` is rationally independent, the sign witness (one odd coordinate of `p` negated) when `w₀ = 2p` (7.1, 8.2, §11); the valuation criterion `w ∈ Sat(Zw₀)` with `c = (k/d₀)c₀` (10.1–10.2); two calibrators of coprime weights recover the primitive character by Bezout, `u = y₁^a y₂^b`, and `(2, 4)` do not (13.1–13.2); **the multi-calibrator criteria decided by a Smith normal form with transforms** — `UWV = D` verified, `d₁ | d₂ | …`, a target in the row lattice reconstructed as `Π χ_{w_j}^{a_j}` with `a` read from the transforms (14.2–14.3), a target in `Sat(L_W) ∖ L_W` whose valuation is still `Σ(U_{i₀j}/d_{i₀})c_j` (15.2), a target outside the saturation with an integer kernel witness (20.2–20.3), the kernel structure `(G_m)^{s−r} × Πμ_{d_i}` (19.1); `|d₀| = 1` for the full rank-one lattice and the restricted family `{2p, 4p}` (22.1, §23); `w_E, 2w₀, 4w₀ ∈ Zw₀` (§24–26); `(1,0) ∉ Z(1,−1)` with witness `(2,2)` (§27); F1 versus F2 (§34–35). The gate also records Round 020's own correction of Round 019's (35.1): `w₀ = 2p, w = p` has observable rank one and is not exactly determined. No false formula found. What the documents list as not proved — that any actual BSD transport is a coherent functor or natural comparison, the actual universal factors of PC-001 and the weight vectors of `19.a1` and `389.a1`, that `q_f` is universal, BSD — they list, and this round adds nothing to those lists.**

---

## Round 017 — naturality and its obstructions

Graphs: a random spanning tree on 6 vertices plus 4 cotree edges; `𝒜` random scalars on the generating edges (an honest functor on the free groupoid); `Ξ_X(a_X) = x_X b_X` with random `x`; the natural `ℬ` defined by `β_f = x_Y α_f / x_X` on generators; a second random `ℬ'` for the projective case.

| statement | instantiated as | instances |
| --- | --- | ---: |
| 4.1, 5.1 | `ρ_f = x_Y/x_X` on every edge; every fundamental cycle has `ρ`-holonomy 1 | 40/40 |
| 6.1 | `𝒜` and `ℬ` have equal monodromy on every cycle, nontrivial on some | 40/40 |
| 7.1–7.3, 8.2 | `Ξ_X` extended from `Ξ₀` along the tree vs through each cotree edge: equal (`= x_X`) for `ℬ`; for `ℬ'` equal iff the relative cycle holonomy is 1 | 40/40 |
| 10.1 | `c_f = β'_f x_X/(x_Y α_f)`: the defect of a composite path equals the product of edge defects | 40/40 |
| 11.1, 12.1, 13.1 | `Ξ' = uΞ` moves `c` by `u_X/u_Y`, cycle products unchanged; a coboundary defect is removed by tree potentials; a random one is not | 40/40 |
| 14.1 | on the tree edges every defect is removed | 40/40 |
| 15.1 | naturality holds along a random composite path | 40/40 |
| 24.1–24.2 | `Z/n` carry multipliers satisfy the 2-cocycle condition; `𝒜(a) = s_𝒜^a`, `ℬ(a) = s_ℬ^a` give `c_a c_b/c_{a+b} = α_ℬ/α_𝒜` for three rational pairs and `n ∈ {2,3,4}` | all |
| 25.1, §26 | `(n, t_𝒜, t_ℬ) = (2,2,3)`, `(4,3,2)`: no rational `c` (`t_ℬ/t_𝒜` not an `n`-th power in `Q`); `(3,2,2)`, `(2,5,5)`: `c` exists, the defect is an ordinary cocycle | all |

## Round 018 — the normalisation fibration

| statement | instantiated as | instances |
| --- | --- | ---: |
| 2.1 | random `g ∈ GL₂(Q)` with `det g ≠ 1`: `ge₁ ∧ ge₂ = det(g)·e₁ ∧ e₂ ≠ e₁ ∧ e₂` | 40/40 |
| 5.1, §32 | `Θ^{(cn)} = c^{−w}Θ^{(n)}` for `w ∈ [1,4]`; the reframe audit of the declared weight | 40/40 |
| 6.1, 7.1 | the contraction by `τ_{n₀}^{⊗w}` computed in the frame `n` and in `cn`: equal | 40/40 |
| 8.1, 9.1 | `cal(k₁k₂, w = 2) = cal(k₁, 1)·cal(k₂, 1)`; `cal(k, 1) = k·n/n₀` | 40/40 |
| 10.2, 23.1 | potentials on a random graph: `ν` composes; one anchor transported along the tree or through a cotree edge gives the same calibrated value at every weight | 40/40 |
| 15.1 | `raw_E = C A_E θ_E`, `raw₀ = C A₀ ℓ₀`: `(A₀ℓ₀/A_E)·raw_E/raw₀ = θ_E`, while treating `A` as common leaves `θ_E·A_E/A₀` | 40/40 |
| §33–34 | `(A₀ℓ₀/A_E)·B_E/b₀ = θ_E`; `C²θ_Ek / (b₀/(A₀j_eℓ₀))² = θ_Ek` | 40/40 |

## Round 019 — the universal line

| statement | instantiated as | instances |
| --- | --- | ---: |
| §14 | the table's ten `(class, calibratable?)` pairs against (5.1), (6.1), (7.1), (8.1), (9.1), (10.1), (11.1), (12.1), (13.1) | identical |
| 15.1–15.2, 16.1, 17.1 | `u_L = (u, −17u, u/17)`: equal valuations at `2 … 13`, ratios `≠ 1`; `u_E` constant: ratios `1` | 40/40 |
| 18.1, 19.1 | `f₁ = 2^{i+1}3^{r_i}`, `f₂ = f₁²b`, `f₃ = 5^{i+1}7^{s_i}`, `f₄ = C` on three partners: the integer kernel of the ratio-valuation matrix has rank 2, contains `(2,−1,0,0)` and `(0,0,0,1)`, and `f₁²/f₂ = 1/b` for every partner | 40/40 |
| 22.1, 23.1 | `X_E/X₀^k = J^{a_E−ka₀}S^{b_E−kb₀}P_EA_E/(P₀A₀)^k` | 40/40 |
| 24.1, 25.1 | the vector-level and determinant-level formulas | 40/40 |
| §32, §33, §34 | common reframe `n ↦ cn`: raw `×c^{w_i}`, calibrated fixed; partner rescale `P_E ↦ uP_E`: anchor fixed, calibrated `×u`; `t = t'/u`: `j` coefficients `/u`, `/u²`, anchor fixed | 40/40 |

## Round 020 — the character lattice

`s = 3` universal unknowns, `g ∈ (Q^×)³` random; `w₀` random nonzero, `p = w₀/gcd`.

| statement | instantiated as | instances |
| --- | --- | ---: |
| 4.1–4.3 | `χ_{kw₀} = χ_{w₀}^k` | 40/40 |
| 7.1, 8.2 | `w` rationally independent of `w₀`: `e` with `w₀·e = 0`, `w·e ≠ 0` from the Smith transforms, `g' = g·2^e` fixes `χ_{w₀}`, moves `χ_w`; `w₀ = 2p`: negating one odd coordinate fixes `χ_{2p}`, negates `χ_p`; the integer criterion rejects `p ∈ Z·2p` | 40/40 |
| 10.1–10.2 | `c = (k/d₀)c₀` on random integral `ν` | 40/40 |
| §11 | `(−u)² = u²`; `v_ℓ(u²) = 2v_ℓ(u)` | 40/40 |
| 13.1–13.2 | `(d₁,d₂) ∈ {(2,3),(3,4),(4,5),(2,5),(3,5)}`: `u = y₁^a y₂^b` by Bezout; `(2,4)`: `−u` gives the same `y₁, y₂` | 40/40 |
| 14.2–14.3, 17.1 | random `W ∈ M_{m×3}(Z)`, `m ≤ 3`: `UWV = D` checked, `d_i | d_{i+1}`; a random integer combination of the rows is exact and its coefficients `a = (wV/d)U` reproduce it and `χ_w = Πχ_{w_j}^{a_j}` | 40/40 |
| 15.2 | when some `d_{i₀} > 1`: row `i₀` of `V⁻¹` is in `Sat(L_W) ∖ L_W`, and its valuation is `Σ_j (U_{i₀j}/d_{i₀})·c_j` | when present |
| 20.2–20.3 | when `r < s`: a target outside the saturation, witness `g = 3^e` with `We = 0`, `w·e ≠ 0` | when present |
| 22.1, §23, §24–26, §27, §34–35 | `|d₀| = 1` iff `p ∈ Zw₀`; `{2p, 4p} ⊆ Z·2p`; `w₀, 2w₀, 4w₀ ∈ Zw₀`; `(1,0) ∉ Z(1,−1)` with `χ_{(1,−1)}(2,2) = 1`, `χ_{(1,0)}(2,2) = 2`; F1 vs F2 | 40/40 |
| Round 019 (35.1) | `p` is a rational but not an integer multiple of `2p` — rank one is not enough at the exact level, as Round 020 §1 says | 40/40 |

One remark on `K`-points: Theorem 4.1 is a statement about the torus; over `Q` itself only `μ_2 = {±1}` is available, so the finite ambiguity `μ_{d₀}` is visible on `Q`-points exactly when `d₀` is even — the sign witness above — and invisible for odd `d₀` (a cube root in `Q` is unique). The theorem as stated (over `K̄`) is right; a reader working over `Q` should know which `d₀` bite.

## The labels

| document | header | what it says of itself |
| --- | --- | --- |
| 017 | "no claim that BSD is proved" | §40 "Not proved", ending "10. BSD." |
| 018 | "no claim that BSD is proved" | §38: N1 + N2 needed, N3 "stronger and can be postponed" |
| 019 | "no claim that BSD is proved" | §40 "Not proved", ending "10. BSD." |
| 020 | "no claim that BSD is proved" | §41 "Not proved", ending "10. BSD." |

## The drill

**410 defects, 410 caught by the check named for each**, 0 uncaught, 0 caught by the wrong check, 99 controls undisturbed, over 115 checks.

The 9 planted for this gate, each turning `symbolic017-020-lattice` red and nothing else:

| planted defect | went red |
| --- | --- |
| Symbolic 017's relative transport is written as the coboundary x_X/x_Y instead of x_Y/x_X | `symbolic017-020-lattice` |
| the projective-functor defect equation is written delta c = alpha_A/alpha_B | `symbolic017-020-lattice` |
| a comparison is asserted to exist even when the two H^2 multiplier classes differ | `symbolic017-020-lattice` |
| a reframe n -> cn is said to scale a weight-w coordinate by c^(+w) | `symbolic017-020-lattice` |
| a determinant of two weight-one classes is given normalisation weight 1 | `symbolic017-020-lattice` |
| the partner factor A_X is absorbed into the universal normalisation line | `symbolic017-020-lattice` |
| one calibrator is declared sufficient when the target weight is a rational multiple, not an integer multiple | `symbolic017-020-lattice` |
| the valuation-level criterion is tested against the row lattice instead of its saturation | `symbolic017-020-lattice` |
| two calibrators of weights 2p and 4p are said to determine the primitive character | `symbolic017-020-lattice` |


## What this round does not claim

* **That any actual BSD transport is a coherent functor, that an actual natural comparison exists, or that the actual multipliers have matching `H²` classes** — Round 017 §40.
* **The actual universal factors of PC-001, the actual weight vectors of `19.a1` and `389.a1`, that `q_f` or the period transport is universal, or that the actual observable rank is one** — Rounds 019–020's own lists; Round 020 §44 proposes doing exactly this next, and nothing here pre-empts it.
* **Nothing about `C`, `𝔰₁₁`, or BSD.**