# RUN-089 — BSD Symbolic Rounds 026–030 (the web GPT's comparison-defect line; period–de Rham cancellation; the exact determinant comparison and its K1 residual; the residual neutral torsor; the PC-001 normalisation closure theorem and post-normalisation DAG): every boxed statement instantiated exactly — the K1 residual computed from cohomology on explicit complexes in random bases, flat residual line systems on random graphs, and the closure round's formulas, tables and DAG read against the text

**Arm:** 數學戰士「墜衡」 / AMRAL Research Lab
**Subject:** Round 026, Round 027, Round 028, Round 029, Round 030 — the last five of the ten documents delivered on 2026-09-14 (provenance)
**Tools:** `src91_symbolic026_030_defect_line_closure.py`, `src11_gate_drill.py`
**Logs:** `src91-symbolic026-030-defect-line-closure.json`, `src11-gate-drill.json`

**Result. The five documents are right as the algebra they are, on 40 exact random instances each, and no false formula was found. Round 026: a factorised comparison has `δ = Πδ_k` (5.1–5.2); a defect neutral under `(c, q)` can still carry exponents on an expanded torus `(p_per, p_dR, p_det, p_conv)` and frame mutation recovers exactly those exponents (3.2, 4.1); the de Rham defect exponent is `a_arith − a_aux` (11.1); lattice defects are Smith valuations and finite lengths do not move under any unit rescaling (13.1–13.2, 16.1); the ledger `δ = a_D q_D η_D` with its valuation/unit split (22.4, §28–29.1); the §21 default-level table and the §38 acceptance table agree with §8–§19; a mutation-neutral but unknown `δ` is a torsor, not a character (§26–27); ratios cancel a common `δ` (34.1). Round 027: the dual frame transforms by `u^{−1}` (3.3); `δ_per δ_dR` is invariant under the coupled mutation and is not 1 in general (5.1–6.1) — the document's own distinction; `d = b − a` and the dual-direction exponent `d_per − d_dR`, neutral iff `d_per = d_dR` (7.1–8.1, 34.1); independent frames need both zero — checked on the multiplicatively independent frames `2, 3` (9.1–9.2); gauge redistribution (13.1); `⟨p, u d_p⟩ = u` (§18); coupled invariance (19.1); `det A · det((A^{−1})^T) = 1` for `r = 2, 3` (21.1). Round 028 — **the K1 residual made concrete**: two-term complexes `d: K⁴ → K⁴` of rank `r ∈ {1,2,3}` with chain automorphisms `(h⁰, h¹)` built in an adapted basis and then conjugated into a random one, `H⁰ = ker d` and `H¹ = coker d` computed by row reduction, and on every instance `det(h⁰)/det(h¹) = det(H⁰(h))/det(H¹(h))` (7.1, 8.1–8.2, 25.1); automorphisms acting trivially on both cohomology groups but not on the complex have determinant 1 (9.1); `id + Kd` and `id + dK` — chain homotopic to the identity — have equal determinants (10.1); orientation `±1` is invisible exactly for even exponents (17.1, 18.1, 34.1); symplectic automorphisms have `det = 1`, the reflection `det = −1` (§22); the lattice layer is the Smith valuation (16.1). Round 029: on random graphs, a coboundary transport has trivial monodromy and a nonzero flat section, a random one has neither (6.1–6.2, §7); flat sections form a `K^×`-torsor and agreement at one vertex forces equality everywhere (8.1, 9.1, §13); relative ratios cancel `ρ` (17.1); `q_f = a_D q_D ρ c_aux/C_π` (18.1, 32.3); `β₁` cotree checks (20.1); the trichotomy T0/T1/T2 (§26); `ρ_th = ρ_anc` (31.1). Round 030: the calibration formulas (2.1–2.3, 23.1) on a series model; `(1,−1)` and `(1,0)` generate `Z²` (3.2–3.4); the determinant and paired weights (6.1, 7.1); `Reg(Λ') = n²Reg(Λ)` on Gram matrices (10.1); `Π ℓ^{d_ℓ}` (11.1); the pre-leading reconstruction (15.2, 22.1); **the post-normalisation DAG (§36) parsed from the document is a chain of seven boxed nodes A–G ending in the BSD-shaped equality, its seven obligations T-A–T-G (§37) match, the certificate has ten items C1–C10 (§40), the reopening rules six R1–R6 (§41), and the "Not proved" list (§45) has twelve items ending in "12. BSD."** — the round is a synthesis, and it is consistent with itself and with the twenty-nine rounds it summarises as this line has instantiated them. What Round 030 says it does not prove — that C1–C10 hold in the actual PC-001 implementation, and every substantive bridge A–G — it lists, and this round adds nothing to that list.**

---

## Round 026 — the defect line

| statement | instantiated as | instances |
| --- | --- | ---: |
| 5.1–5.2 | chains of 2–5 scalar maps | 40/40 |
| 3.2, 4.1 | a defect `base·Πf_k^{d_k}` on four hidden frames: doubling each frame recovers `d_k`; `(c,q)`-neutral | 40/40 |
| 11.1 | `aux ∝ ω^{a_aux}`, `arith ∝ ω^{a_arith}`: the defect moves by `2^{a_arith−a_aux}` under `ω ↦ 2ω` | 40/40 |
| 13.1–13.2, 16.1 | `v_ℓ(det M) = Σv_ℓ(Smith)`; lengths unchanged at primes where the rescaling unit is a unit | 40/40 |
| 22.4, 28–29.1 | `δ = (1/m!)·Πℓ^{e_ℓ}·η`: valuations add, `δℓ^{−v_ℓ(δ)}` is an `ℓ`-unit | 40/40 |
| §21, §38 | eight and six table rows against the text | identical |
| §26–27, 34.1 | a neutral unknown `≠ 1`; `(δc₁)/(δc₂) = c₁/c₂` | 40/40 |

## Round 027 — period and de Rham

| statement | instantiated as | instances |
| --- | --- | ---: |
| 3.3, 18, 19.1 | `d_p = 1/p`; `⟨up, u⁻¹d_p⟩ = 1`, `⟨p, u d_p⟩ = u` | 40/40 |
| 5.1–6.1, 13.1 | `δ_per δ_dR` fixed under `(u, u⁻¹)` and under redistribution; not 1 | 40/40 |
| 7.1–8.1, 34.1 | random `a, b ∈ Z²`: the coupled mutation moves the defect by `u^{d_per−d_dR}`; neutral iff `d_per = d_dR` | 40/40 |
| 9.1–9.2 | `2^{d_per}3^{d_dR} = 1` iff `d = 0` | 40/40 |
| 21.1 | random `A ∈ GL_r(Q)`: `det A · det((A⁻¹)ᵀ) = 1` | 40/40 |

## Round 028 — the K1 residual on explicit complexes

Adapted model: `C⁰ = ker ⊕ W`, `C¹ = im ⊕ V`, `d(w_j) = e_j`; `h⁰ = [[A, B],[0, C]]`, `h¹ = [[C, F],[0, E]]` (so `dh⁰ = h¹d`), then `d ↦ QdP⁻¹`, `h⁰ ↦ Ph⁰P⁻¹`, `h¹ ↦ Qh¹Q⁻¹` for random `P, Q ∈ GL₄(Q)`; from the transformed matrices alone the gate recomputes `ker d`, a complement of `im d`, and the induced maps.

| statement | instantiated as | instances |
| --- | --- | ---: |
| 7.1, 8.1–8.2, 25.1 | `det(h⁰)/det(h¹) = det(H⁰(h))·det(H¹(h))⁻¹ = det A/det E` | 40/40 |
| 5.2 | on `h∘h`: the ratio squares | 40/40 |
| 9.1 | `A = E = I`, `B, C, F` random: cohomology trivial, `h ≠ id`, ratio 1 | 40/40 |
| 10.1 | `f⁰ = I + Kd`, `f¹ = I + dK`: a chain map, `det f⁰ = det f¹` | 40/40 |
| 17.1, 18.1, 34.1 | `(−Δ)^m = Δ^m` iff `m` even | 40/40 |
| §22 | `S ∈ SL₂(Q)` preserves `J`, `det = 1`; the reflection is orthogonal with `det = −1` | 40/40 |
| 16.1 | `v_ℓ(det φ) = Σ v_ℓ(Smith)` | 40/40 |

## Round 029 — the residual torsor

| statement | instantiated as | instances |
| --- | --- | ---: |
| 6.1–6.2, §7 | coboundary `ρ_{j/i} = a_j/a_i`: `s = a` is flat, all fundamental cycles 1; random `ρ`: some cycle `≠ 1`, the tree-propagated section fails on a cotree edge | 40/40 |
| 8.1, 9.1, §13 | `u·s` is flat; the section transported from `s₀` along the tree is `s` | 40/40 |
| 17.1, 18.1–32.3, 20.1, §26, 31.1 | ratios; the reconstruction chain with `a_D = 1/2`, `q_D = 3/5`; `#cotree = |E|−|V|+1`; T0/T1/T2; `ρ_anc = ρ` | 40/40 |

## Round 030 — the closure round, read

| statement | how | result |
| --- | --- | --- |
| 2.1–2.3, 23.1 | series model of `B̂_i` | `κ_E` recovered on 40/40 |
| 3.2–3.4, 6.1, 7.1 | Smith form of `[[1,−1],[1,0]]`; multiples | `diag(1,1)`; `(2,−2), (4,−4) ∈ Z(1,−1)` |
| 10.1, 11.1, 15.2, 22.1 | Gram determinants of index-`n` sublattices; valuation reconstruction; `q_f` formula | 40/40 |
| §36 DAG | parsed `\boxed{X: …}` nodes and `\Downarrow` arrows | `A, B, C, D, E, F, G`, seven arrows, final "classical BSD-shaped scalar equality" |
| §37 | `## T-X.` headings | `A … G` |
| §40, §41, §45 | `### Cn.`, `## Rn`, numbered items | 10, 6, 12 — ending "12. BSD." |

## The labels

| document | header | what it says of itself |
| --- | --- | --- |
| 026 | "no claim that BSD is proved" | §41 "Not proved", ending "10. BSD." |
| 027 | "no claim that BSD is proved" | §37 "Not proved", ending "10. BSD." |
| 028 | "no claim that BSD is proved" | §41 "Not proved", ending "10. BSD." |
| 029 | "no claim that BSD is proved" | §33 "Why this still does not prove BSD" — "a normalization architecture, not the BSD theorem itself" |
| 030 | "no claim that BSD is proved" | §45 "Not proved", twelve items ending "12. BSD." |

## The drill

**427 defects, 427 caught by the check named for each**, 0 uncaught, 0 caught by the wrong check, 106 controls undisturbed, over 117 checks.

The 9 planted for this gate, each turning `symbolic026-030-defect-closure` red and nothing else:

| planted defect | went red |
| --- | --- |
| Symbolic 026's de Rham defect exponent is written a_aux - a_arith | `symbolic026-030-defect-closure` |
| the dual de Rham frame is said to scale by u under p -> up | `symbolic026-030-defect-closure` |
| period-de Rham dual neutrality is read as d_per = -d_dR | `symbolic026-030-defect-closure` |
| the K1 residual is written det(H^0 h) det(H^1 h) instead of the alternating product | `symbolic026-030-defect-closure` |
| chain-homotopic comparison maps are said to have different determinants | `symbolic026-030-defect-closure` |
| finite Euler lengths are given frame weight 1 | `symbolic026-030-defect-closure` |
| orientation is said to be invisible for odd determinant exponents | `symbolic026-030-defect-closure` |
| a flat residual section is said to exist even with nontrivial monodromy | `symbolic026-030-defect-closure` |
| Symbolic 030's post-normalisation DAG is read with six nodes instead of seven | `symbolic026-030-defect-closure` |


## What this round does not claim

* **That C1–C10 hold for PC-001** — Round 030 §45: actual universality of `C_π`, the actual auxiliary-to-arithmetic comparison, period–de Rham canonicality, determinant cohomological equality, residual flatness, a residual anchor. None of these is a statement this tree can test without the actual objects; the gate checks the algebra that would apply once they exist.
* **The seven substantive bridges A–G** of the post-normalisation DAG — BF classes, fundamental-line identification, the regulator bridge, reciprocity, the `p`-adic and complex leading-line comparisons, finite certificates. Of these, this tree has computed only inputs on the analytic and regulator side of `389.a1` (RUN-073, RUN-085): `L''(E,1)/2`, `Ω`, `Reg`, and `r_p = r_C = 2`.
* **Nothing about `𝔰₁₁` or BSD.**