# RUN-088 — BSD Symbolic Rounds 021–025 (the web GPT's PC-001 character lattice and the late versus pre-leading routes; the second anchor versus a direct transport theorem; non-circularity of a rank-zero pre-leading anchor; the source-line search; the auxiliary leading tensor as native anchor): every boxed statement instantiated exactly, the PC-001 degeneration formula built as actual series so that every normalisation weight in the documents' table is measured rather than read — and the one number this tree can put into Round 021's partner ratio

**Arm:** 數學戰士「墜衡」 / AMRAL Research Lab
**Subject:** Round 021, Round 022, Round 023, Round 024, Round 025 — five of the ten documents (021–030) that reached the drop folder by the afternoon of 2026-09-14 ("現在到30了。這邊確實需要先驗證了") (provenance); the twenty already held were byte-identical; a non-BSD file in the same folder was not taken
**Tools:** `src90_symbolic021_025_pc001_lattice_second_anchor.py`, `src11_gate_drill.py`
**Logs:** `src90-symbolic021-025-pc001-lattice-second-anchor.json`, `src11-gate-drill.json`

**Result. These five rounds are the first to touch PC-001's actual formula, and they are right as far as they go — which is: conditional character bookkeeping on `B̂_i(t) = C_π A_i(t) j(t) z_i(t)`, not a computation of any of its factors. The gate builds that formula as genuine truncated series in `Q[[t]]` — `A_i = 𝒮₅(t)λ_i(t)/D_i`, `C_π = c_lead/q_f`, `j(t) = log(1+t)/L`, `z_E = tκ_E + O(t²)`, `Col₀` a linear functional on coefficients — with the normalisation coordinates `(c_lead, q_f)` live, and then *measures* every weight the documents assert by reframing `c ↦ αc`, `q ↦ βq`: `B_{E,2} = [t²]B̂_E = C_πA_E(0)j₁κ_E` and `r_{0,1} = [t]Col₀(B̂₀) = C_πA₀(0)j₁ℓ₀` (3.1, 4.1) both have weight `(1,−1)`, the late ratio `(0,0)`, the determinant of two late classes `(2,−2)`, the paired scalar `(4,−4)`, and `R_i^lead = q_f Col_i(B̂_i)/j = c_lead A_i Col_i(z_i)` — checked as a series identity — has weight `(1,0)` (2.1, 5.1, 10.1, 14.1–14.2): the seven rows of Round 021's §20 table, measured, are the table. `W_late` has Smith form `diag(1,0)`, `W_ext = [[1,−1],[1,0]]` has determinant 1 and Smith form `diag(1,1)` (5.1, 11.1–11.4); `(1,0) ∉ Z(1,−1)` (12.1); the calibration formula `κ_E = (A₀(0)ℓ₀/A_E(0))·B_{E,2}/r_{0,1}` recovers `κ_E` on every instance (7.1) and is invariant under a calibrator-side Coleman rescaling (§9); `A₀(0)/A_E(0) = λ₀(0)D_E/(λ_E(0)D₀)` (§8); a hidden third direction `p` raises the rank exactly when some observable sees a different `p`-exponent (§19). Round 022: `ker χ_π` is the diagonal (2.1), `χ_{(a,b)}(u,u) = u^{a+b}` (3.1), `det W(a,b) = a + b` (5.1), Smith form `diag(1,|a+b|)` (7.1), the inversions `c = y_v y_π^b`, `q = y_v y_π^{b−1}` for `a + b = 1` (12.1–12.2), only `c^d` for `|a+b| = d` (§13), the triangle `y_c = y_π q` (19.1), primitive kernel restriction iff `|a+b| = 1` (22.1–23.1). Round 023: weight certification by reframing (5.1–5.3); **the point of the round instantiated** — the circular construction `n_π ⊗ n_Q` has weight `(1,0)` and passes the `q`-mutation test (it is just `c`), and only the dependency-graph audit rejects it, because a path from `q` reaches it, while a construction from a `q`-blind source is accepted (No-Go 4.1, Definition 12.1, 13.1); the new-source theorem — monomials in inputs of weight `∈ Z(1,−1)` and neutral data never reach `(1,0)` (9.1); a partner prefactor carrying `q`-weight spoils the anchor (15.1); `q_rec = y_C/y_π` (23.1–23.2). Round 024: `c_m ↦ u^{−m}c_m` (§7); the empty determinant is neutral (10.1); the §23 candidate table's residual lines agree with §5–§22 (seven rows); two `q`-blind anchors differ by one scalar (§26). Round 025: `c_m(dX)^{⊗m}` is coordinate-invariant — `c_m ↦ u^{−m}c_m` by series inversion, `(dX')^{⊗m} = u^m(dX)^{⊗m}` (3.1); native weight `(1,0)` (4.1–4.2); `c_arith = δc_aux` with `w(𝒟) = (0,0)` (6.1–9.1); `ev_{v^{⊗m}}` and the `u^{−m}` law (12.1, §13); relative ratios cancel a common `δ`, the absolute `q = δc_aux/C_π` does not, and `δ = qC_π/c_aux` (19.1–21.1); a unit `g` scales the leading tensor by `g(0)` (§33). No false formula found. **Supplied:** Round 021 §26 lists "actual values of `λ_i(0)/D_i`" as unproved. This tree has the `λ` half: `λ_E(0) ≡ 5` and `λ₀(0) ≡ 4 (mod 11)` (RUN-077, 080, 081 — the constant terms of the twisted odd-branch series), so `λ₀(0)/λ_E(0) ≡ 3 (mod 11)`; the adjoint period ratio `D_E/D₀` is not computed here — PC-001 says `D₀` was not computed, and RUN-081 repeated PC-002's warning that its cup ratio `J₀/J_E = 3` is *not* `D₀/D_E`. That the two threes coincide is a coincidence until a theorem says otherwise. What the documents list as not proved — exact partner-independence of `C_π`, `c_lead`, `q_f`, the actual BF classes, the auxiliary tensor's identification with the arithmetic pre-leading line, BSD — they list, and this round adds nothing to those lists.**

---

## Round 021 — the weights, measured

The model: `L, 𝒮₅(t), λ_E(t), λ₀(t), D_E, D₀, κ_E, z₀(t)` random with nonzero constant terms, `Col₀` a random unit acting coefficientwise; `(c, q)` random units; the same inner randomness is replayed with `c ↦ 2c` and with `q ↦ 3q` and the exponents read off exactly.

| observable | formula in the model | measured weight | table §20 |
| --- | --- | ---: | ---: |
| `C_π` | `c/q` | `(1,−1)` | `(1,−1)` |
| `B_{E,2}` | `[t²](C_π A_E j z_E)` | `(1,−1)` | `(1,−1)` |
| `r_{0,1}` | `[t](col·C_π A₀ j z₀)` | `(1,−1)` | `(1,−1)` |
| late ratio | `B_{E,2}/r_{0,1}` | `(0,0)` | `(0,0)` |
| target determinant | two late classes wedged | `(2,−2)` | `(2,−2)` |
| paired late scalar | primal ⊗ dual | `(4,−4)` | `(4,−4)` |
| `R_i^lead` | `q·Col_i(B̂_i)/j` | `(1,0)` | `(1,0)` |

| statement | instantiated as | instances |
| --- | --- | ---: |
| 3.1, 4.1 | the two leading coefficients as series coefficients | 40/40 |
| 5.1, 11.1–11.4, 12.1, 14.1–14.2, §17, §19 | Smith forms `diag(1,0)`, `diag(1,1)`, `det = 1`; `(1,0) ∉ Z(1,−1)`; rank one for repeated `(1,−1)`; rank two only with a different `p`-exponent | 40/40 |
| 7.1, §8, §9 | `κ_E` recovered; the partner ratio; Coleman rescaling cancels | 40/40 |
| 10.1 | `R₀^lead = c_lead A₀ Col₀(z₀)` as series (to `t⁴`) | 40/40 |
| 15.1–16.2 | Routes A/B: `c = y_R`, `q = y_R/y_π` | 40/40 |

## Round 022–025 — the kernel, the anchor, the source, the tensor

| round | statement | instantiated as | instances |
| --- | --- | --- | ---: |
| 022 | 2.1, 3.1, 4.1, 5.1, 6.1–7.1 | `χ_π(u,u) = 1`, `χ_π(u,1/u) ≠ 1`; `u^{a+b}`; `det = a + b`; Smith `diag(1,|a+b|)` or `diag(1,0)` | 40/40 |
| 022 | 12.1–12.2, §13, 19.1, 22.1–23.1 | the inversions for `a + b = 1`; `c^d` for `d = 2, 3`; `c = y_π q`; primitive iff unimodular, informative iff rank 2 | 40/40 |
| 023 | 5.1–5.3, 13.1 | `Y_C ↦ αY_C`, `Y_C ↦ Y_C`; `Y_π ↦ β⁻¹Y_π`; the circular `Y_π q` also invariant | 40/40 |
| 023 | 4.1, 12.1 | dependency graphs: `{candidate ← n_π, n_Q; n_Q ← q}` rejected, `{candidate ← R̂₀, β₀; R̂₀ ← aux, P₀}` accepted | 40/40 |
| 023 | 9.1, 15.1, 23.1–23.2 | random monomials in `Z(1,−1)`-weights stay there; a `q`-weighted prefactor fails; `q_rec = q` | 40/40 |
| 024 | §7, 10.1, §23, §26 | `u^{−m}` by series inversion with a quadratic term; empty determinant; the table; one scalar | 40/40 |
| 025 | 3.1, 4.1–4.2, 6.1–9.1, 12.1–13, 19.1–21.1, §24, §33 | coordinate invariance of `c_m(dX)^{⊗m}`; `(1,0)`; `δ`, its neutral weight; `ev_{v^{⊗m}}`; relative vs absolute; projective ambiguity; `g(0)` | 40/40 |

## The one number

| factor of Round 021's partner ratio `λ₀(0)D_E/(λ_E(0)D₀)` | status in this tree |
| --- | --- |
| `λ_E(0)` | `≡ 5 (mod 11)` — RUN-077 (`λ₈(0)`), RUN-081 (constant term of `λ_E`) |
| `λ₀(0)` | `≡ 4 (mod 11)` — RUN-080 (`L₁₁(E₀ ⊗ χ₈, x⁻¹)`), RUN-081 (constant term of `λ₀`) |
| `λ₀(0)/λ_E(0)` | `≡ 4·5⁻¹ ≡ 3 (mod 11)` |
| `D_E/D₀` | not computed — PC-001 §3: "其完整 period 比值 `D₀` 沒有在本輪算出"; RUN-081: the cup ratio 3 is not `D₀/D_E` |
| `𝒮₅(0)` | `26 ≡ 4`, common to both, cancels in the ratio (§8) |

## The labels

| document | header | what it says of itself |
| --- | --- | --- |
| 021 | "no claim that BSD is proved" | §26 "Not proved", ending "10. BSD." |
| 022 | "no claim that BSD is proved" | §37 "Not proved", ending "10. BSD." |
| 023 | "no claim that BSD is proved" | §28 "Not proved", ending "10. BSD." |
| 024 | "no claim that BSD is proved" | §33 "Not proved", ending "10. BSD." |
| 025 | "no claim that BSD is proved" | Theorem 34.1 is an "if and only if" on hypotheses S1–S8 |

## 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 8 planted for this gate, each turning `symbolic021-025-pc001-lattice` red and nothing else:

| planted defect | went red |
| --- | --- |
| Symbolic 021's late character is taken as (1, 1) instead of (1, -1) | `symbolic021-025-pc001-lattice` |
| the pre-leading character is taken as the late one, (1, -1) | `symbolic021-025-pc001-lattice` |
| the two-anchor determinant is written a - b | `symbolic021-025-pc001-lattice` |
| the restriction of (a, b) to the diagonal kernel is written u^(a-b) | `symbolic021-025-pc001-lattice` |
| the provenance audit no longer forbids q upstream, so the circular anchor is accepted | `symbolic021-025-pc001-lattice` |
| the auxiliary leading coefficient is said to transform by u^(+m) | `symbolic021-025-pc001-lattice` |
| the comparison defect is dropped from q_f = delta c_aux / C_pi | `symbolic021-025-pc001-lattice` |
| a calibrator-side Coleman rescaling is said to change kappa_E | `symbolic021-025-pc001-lattice` |


## What this round does not claim

* **That `C_π`, `c_lead` or `q_f` is partner-independent** — the character bookkeeping assumes it (Round 021 L1); nothing here tests it, and nothing in this tree can until an actual second partner's `C_π` is computed.
* **`D_E/D₀`**, the actual BF classes `B_{E,2}`, `r_{0,1}`, the auxiliary leading tensor or its comparison with the arithmetic pre-leading line.
* **Nothing about `𝔰₁₁` or BSD.**