# RUN-086 — BSD Symbolic Rounds 013–016 (the web GPT's line-by-line descent audit, observable normalisation character, direct fundamental-line comparison, and flat transport with holonomy): every boxed statement instantiated exactly — the bookkeeping character (8.1) verified on a scalar pipeline that performs each transformation, the audit table read against the text, and the holonomy calculus run on random finite graphs

**Arm:** 數學戰士「墜衡」 / AMRAL Research Lab
**Subject:** Round 013, Round 014, Round 015, Round 016 — the four documents that arrived with the re-filled drop folder on the evening of 2026-09-13 (provenance)
**Tools:** `src88_symbolic013_016_character_direct_map_holonomy.py`, `src11_gate_drill.py`
**Logs:** `src88-symbolic013-016-character-direct-map-holonomy.json`, `src11-gate-drill.json`

**Result. The four documents are correct as the linear algebra, group theory and graph theory they are, and they are checked as such — on 40 exact random instances each. Round 013: a frame change `(c_i)` acts on `⊗L_i^{⊗m_i}` by `χ(c) = Πc_i^{m_i}` and kernel elements built explicitly are invisible (3.1–3.2); two non-principal unit ideles with exponents `(1, −1)` give a principal total, so line-by-line principalisation is sufficient and not necessary (4.1, 5.1); the regulator sees `a₊a₋` (8.1); `h ↦ AᵀhB` moves `det h` by `det A · det B` (9.1, 10.1); a unimodular change of the comparison matrix leaves every lattice defect (11.1, 12.1); the BF scalar cancels against `b₀⁴` (13.1); `λ ↦ cλ` keeps the order and scales the coefficient (15.1, 16.1); the period–reciprocity and primal–dual cancellations (23.1, §24); the twelve rows of the §25 table agree with the boxed statements of §6–§21; (31.2) is Round 014's `χ_E` because `det(c_h h) = c_h² det h`. Round 014: primitive exponent vectors have Bezout sections with `χ(s(z)) = z` and `g = k·s(χ(g))` with `k ∈ ker χ` (4.1–4.3); a non-primitive vector `d·m₀` has image inside `d`-th powers (§5, by valuations); **the eight-coordinate bookkeeping character (8.1), `(−r, 0, 1, 1, 2, 1, 1, −1)`, is not taken from the text but recovered from a scalar pipeline that actually performs each transformation** — series reparametrisation, the calibrated BF gauge, primal and dual frames on a determinant pairing, the height scale on a Gram matrix, reciprocity, cross-completion, the period frame — for `r = 1, 2, 3`, and it is primitive; G1–G5 lie in the kernel and with the BF direction are six independent vectors (§10); the BF direction does not move the pipeline (11.1); `m_arith` and `(2, 1, 1, −1)` are primitive with the Bezout choice `(0, 1, 0, 0)` (13.1–15.1); any tuple gauge-fixes to `(1, η, 1, 1)` (17.1); the valuation dot product (18.1); `u_rec = u_∞` non-principal with the others trivial gives `η = 1` (19.1–21); Smith normal form of a primitive row is `(1)`, of `d·row` is `(d)`, a two-observable matrix has rank 2 (§27–28). Round 015: torsors of line maps, internal gauge conjugation, endpoint-invariant map ratios, transported maps and the direct calibration theorem, cocycles and closed loops, the commutative square, two-path ratios as `χ_{m₂−m₁}`, and **naturality forcing one ratio per connected component** — checked on random connected graphs with random transports (18.1–19.1); holonomy gauge-invariant (27.1–28.1); one-vector certificates (34.1). Round 016, on random graphs (6 vertices, a spanning tree plus 4 cotree edges, random edge ratios in `Q^×`): path transport (2.1–2.3); holonomy gauge invariance (4.1); **Theorem 6.1 in both directions** — a coboundary transport is flat, path-independent and gauge-trivialisable, a random one fails all three with the failing fundamental cycle exhibited; one anchor propagates path-independently exactly when flat (7.1–7.2, 8.1, 9.1); the spanning-tree gauge sends tree edges to 1 and cotree edges to their fundamental holonomies (10.1–11.1); `β₁ = |E| − |V| + 1` fundamental cycles suffice — random closed walks have trivial holonomy once they do (12.1–13.1); two components need two anchors and `β₁ = |E| − |V| + 2` (14.1–14.2); two transports with equal fundamental holonomies are gauge-equivalent (16.1–17.1); valuation holonomy is additive and `h = −1` is valuation-flat at every prime yet not flat (18.1, 19.1); a loop tuple in `ker χ_E` is invisible to observable holonomy (21.1–22.1); the cotree defect `ε_e = ρ^{meas}/ρ^{pred}` is the fundamental holonomy, in the document's orientation (28.1–29.3, 30.1); complete graphs have `β₁ = (n−1)(n−2)/2` (§32); a single bad edge is located from the syndrome whenever its cycle-membership pattern is unique — 22 of 40 trials on these small graphs (§35). No false formula found. What the documents list as not proved — the actual fundamental-line character, the actual height, reciprocity, comparison and period scalars, that any actual transport is flat or that calibrator and target share a component, BSD — they list, and this round adds nothing to those lists.**

---

## Round 013

| statement | instantiated as | instances |
| --- | --- | ---: |
| 3.1, 3.2 | `n ∈ [2,5]` lines, exponents in `±{1,2,3}`, frames in `Q^×`: the induced frame scalar is `Πc_i^{m_i}` | 40/40 |
| ker χ | `c_i = u^{m_j}, c_j = u^{−m_i}`: `χ = 1`, the frame unchanged | 40/40 |
| 4.1, 5.1 | unit ideles at `{2,3,5,7,11,13,∞}`: `u₁ = u₂` non-principal, `u₁u₂^{−1}` principal | 40/40 |
| 8.1 | `det(h)(a₊Δ₊ ⊗ a₋Δ₋) = a₊a₋Reg₀` | 40/40 |
| 9.1, 10.1 | `det(AᵀhB) = det A · det h · det B` | 40/40 |
| 11.1, 12.1 | `v_ℓ(det M) = Σ v_ℓ(Smith)`; `M·U` for unimodular `U`: the same | 40/40 |
| 13.1, 15.1, 16.1, 23.1, §24, 31.2 | scalar identities on random data | 40/40 |
| §25 | the table's twelve `(structure, scalar)` pairs against (6.1), (7.1), (9.1)/(10.1), (11.1), (12.1), (13.1), (14.1), (15.1)/(16.1), (17.1), (18.1), (19.1), §21 | identical |

## Round 014 — the character, from the pipeline

The pipeline's scalar is `t'^r)) · det(h)(Δ̂₊ ⊗ Δ̂₋)/b₀⁴ · c_rec · c_pc / c_∞` with `t' = c_t t + at²`,
`Δ̂₊ = c_BF² c₊ Δ₊`, `Δ̂₋ = c_BF² c₋ Δ₋`, the pairing matrix `c_h·h`, and `b₀ = c_BF a₀k₀`. Its ratio to the
untransformed scalar, on random `(c_t, c_BF, c₊, c₋, c_h, c_rec, c_pc, c_∞)`, equals
`c_t^{−r} c_BF^0 c₊ c₋ c_h² c_rec c_pc c_∞^{−1}` on every instance for `r = 1, 2, 3`. That is (8.1); its gcd is 1.

| statement | instantiated as | instances |
| --- | --- | ---: |
| 2.1, 3.1, 4.1–4.3 | random primitive `m ∈ Z^n`: Bezout `n` with `Σm_in_i = 1`, `χ(s(z)) = z`, `g = k·s(χ(g))`, `χ(k) = 1` | 40/40 |
| §5 | `d·m`: `v_ℓ(χ) ≡ 0 (mod d)` at `2, 3, 5, 7`, so `2 ∉ image` | 40/40 |
| 7.1, 8.1, primitive | the pipeline ratio equals the stated character; gcd 1 | 40/40 |
| §10 | G1–G5 in the kernel; rank of `{BF, G1, …, G5}` is 6 | 40/40 |
| 11.1 | `c_BF` alone leaves the pipeline fixed | 40/40 |
| 13.1–13.2, 14.1–15.1 | `(0,1,1,2,1,1,−1)` and `(2,1,1,−1)` primitive; `2·0 + 1 + 0 − 0 = 1`; `s(z) = (1, z, 1, 1)` | 40/40 |
| 17.1, 18.1 | `k = (c_h⁻¹, η/c_rec, c_pc⁻¹, c_∞⁻¹) ∈ ker`, `kg = (1, η, 1, 1)`; `v_ℓ(η) = 2v_ℓ(c_h) + v_ℓ(c_rec) + v_ℓ(c_pc) − v_ℓ(c_∞)` | 40/40 |
| 19.1–21 | `(1, u, 1, u)` with `u` non-principal: `η ≡ 1` | 40/40 |
| §27–28 | Smith invariants `(1)`, `(d)`, rank 2 | 40/40 |

## Round 015

| statement | instantiated as | instances |
| --- | --- | ---: |
| 2.1, 3.1, 34.1 | maps between lines as scalars; equality decided on one vector | 40/40 |
| 4.1–4.2, 5.1 | chains of 2–5 maps under internal conjugation `φ_i ↦ g_iφ_i, φ_{i+1} ↦ φ_{i+1}g_i⁻¹` | 40/40 |
| 7.1, 8.1 | `(u_out Ξ_B u_in⁻¹)/(u_out Ξ_A u_in⁻¹) = ρ_{B/A}` | 40/40 |
| 9.1–11.1 | `Ξ_E = T_an⁻¹(ρ_E Ξ₀)T_alg` | 40/40 |
| 12.1–13.1, 14.2 | potentials on four vertices: cocycle, inverse, closed loop; the square | 40/40 |
| 15.1–17.1 | `χ_{m₂}(g)/χ_{m₁}(g) = χ_{m₂−m₁}(g)` | 40/40 |
| 18.1, 19.1 | random connected graph, random `T_alg, T_an` per edge, two natural families: `Ξ'_X/Ξ_X` constant | 40/40 |
| 22.1, 27.1, 28.1 | the kernel of `χ_E`; fundamental-cycle holonomy under a random vertex gauge | 40/40 |

## Round 016

Graphs: a random spanning tree on 6 vertices plus 4 random cotree edges (`β₁ = 4`); edge ratios random in `Q^×`;
the fundamental cycle of a cotree edge `(i, j)` is `i → j → (tree) → i`, the orientation of §28–29.

| statement | instantiated as | instances |
| --- | --- | ---: |
| 2.1–2.3, 4.1 | reversed and composed paths; holonomy under a random gauge | 40/40 |
| 6.1 (⇒) | coboundary `ρ_{j/i} = a_j/a_i`: all four conditions hold | 40/40 |
| 6.1 (⇐) | random `ρ`: some fundamental cycle `≠ 1`, the BFS potentials fail on a cotree edge, no gauge trivialises | 40/40 |
| 7.1–7.2, 8.1, 9.1 | anchor propagated along the tree vs through each cotree edge: equal iff flat; a loop multiplies by `h ≠ 1` | 40/40 |
| 10.1, 11.1 | gauge `g_v = a_v⁻¹`: tree edges `1`, cotree edges `= h(C_e)` | 40/40 |
| 12.1, 13.1 | `#cotree = |E| − |V| + 1`; three random closed walks per graph have trivial holonomy under a flat `ρ` | 40/40 |
| 14.1, 14.2 | two disjoint copies: `c = 2`, `β₁ = |E| − |V| + 2` | 40/40 |
| 16.1, 17.1 | `ρ' = gauge(ρ)`: equal fundamental holonomies, and `ρ'/ρ` is a coboundary | 40/40 |
| 18.1, 19.1 | `v_ℓ(h) = Σ ±v_ℓ(ρ_e)`; one cotree edge negated on a flat `ρ`: `v_ℓ(h) = 0` at all six primes, `h = −1` | 40/40 |
| 21.1, 22.1 | a cotree edge carrying `(c_h, c_rec, c_pc, c_∞)` with `c_pc = (c_h²c_rec)⁻¹`: loop tuple `≠ 1`, `χ_E = 1` | 40/40 |
| 28.1–29.3, 30.1 | `ρ^{pred} = a_j/a_i`, `ε = ρ^{meas}/ρ^{pred} = h(C_e)`, valuations too | 40/40 |
| §32 | `K₃ … K₆`: `β₁ = 1, 3, 6, 10` | 40/40 |
| §35 | one edge of a flat `ρ` multiplied by a unit; the failing cycles are those through it; unique when its membership pattern is | 22 of 40 unique |

## The labels

| document | header | what it says of itself |
| --- | --- | --- |
| 013 | "no claim that BSD is proved" | §34 "Not proved", ending "9. BSD." |
| 014 | "no claim that BSD is proved" | §37 "What remains theorem-level" — arithmetic identification, not bookkeeping |
| 015 | "no claim that BSD is proved" | §38 "What remains genuinely substantive" — existence, naturality, one anchor, the classical scalar |
| 016 | "no claim that BSD is proved" | §44 "Not proved", ending "10. BSD." |

## The drill

**401 defects, 401 caught by the check named for each**, 0 uncaught, 0 caught by the wrong check, 95 controls undisturbed, over 114 checks.

The 9 planted for this gate, each turning `symbolic013-016-holonomy` red and nothing else:

| planted defect | went red |
| --- | --- |
| Symbolic 014's bookkeeping vector gives the height scale exponent 1 instead of 2 | `symbolic013-016-holonomy` |
| Symbolic 014's bookkeeping vector gives the period frame exponent +1 instead of -1 | `symbolic013-016-holonomy` |
| the reduced exponent vector is doubled to (2, 2, 2, -2), which is not primitive | `symbolic013-016-holonomy` |
| the height-reciprocity compensation G4 is written (u, u^-1), which is not in the kernel | `symbolic013-016-holonomy` |
| a frame change acts on the tensor line with |m_i| instead of m_i | `symbolic013-016-holonomy` |
| naturality is written with the algebraic and analytic transports swapped | `symbolic013-016-holonomy` |
| the vertex gauge acts as g_j rho g_i, without the inverse | `symbolic013-016-holonomy` |
| beta_1 is taken as |E| - |V|, dropping the component count | `symbolic013-016-holonomy` |
| the predicted cotree ratio is taken as a_i/a_j instead of a_j/a_i | `symbolic013-016-holonomy` |


## What this round does not claim

* **Any actual scalar** — `det h` on the true height pairing, `c_rec`, `c_pc`, `Ω_∞`, `η_BSD` — nor that the actual
  observable idele is principal; the characters are checked as characters.
* **That any actual BSD transport is flat, or that the rank-0 calibrator and the rank-2 target lie in one admissible
  component** — Round 016 §24 calls this an arithmetic identification condition, and it is.
* **The direct fundamental-line theorem** of Round 015 §36–38: existence, naturality, nonvanishing, the anchor.
* **Nothing about `C`, `𝔰₁₁`, or BSD.**