# RUN-085 — BSD Symbolic Rounds 008–009 (the web GPT's self-dual fundamental line and correct regulator exponent; the analytic leading-term line and order matching): every boxed statement instantiated exactly — and the two supplements the rounds ask for on 389.a1: the regulator as one Gram determinant from this line's own canonical heights, equal to the corpus number to the last digit, so every factor of L''(E,1)/2 = Ω·Reg·q_fin is now computed here; and r_p = r_C = 2

**Arm:** 數學戰士「墜衡」 / AMRAL Research Lab
**Subject:** Round 008, Round 009 — the two rounds Round 010 cites for its S9 and S10, delivered when the drop folder was re-filled on the evening of 2026-09-13 (provenance); the ten documents already held were byte-identical
**Tools:** `src87_symbolic008_009_fundamental_line_leading_term.py`, `src11_gate_drill.py`
**Logs:** `src87-symbolic008-009-fundamental-line-leading-term.json`, `src11-gate-drill.json`

**Result. Round 008 is right, and its one distinction — the regulator's quadratic *scaling* is the bidegree (1,1) of a determinant pairing, not a squared regulator in the formula — is checked where it can be: `det(h)(Δ_M ⊗ Δ_N) = det(h(P_i,Q_j))` is `det A · det h · det B` in the two bases (2.3, 3.1), so a common rescaling gives `a²` (4.1) and a BF gauge `C²` on each projected determinant gives `C⁴` on the pair while `Reg` appears once after dividing by the weight-one calibrator's fourth power (5.1, 21.1); `v_ℓ(det h) = length coker h♯` by Smith invariants (6.1); `det(uh) = u^r det h` (7.1–7.2); `𝒮 = q₊q₋Reg` with the one-sided, symmetric and inverse-dual placements giving exponents `1, 2, 0` (8.1–12); `det(C^∨) = det(C)^{−1}` and `det(C^∨[s]) = det(C)^{−(−1)^s}` on two-term complexes with an explicit determinant convention (13.1–13.2); an odd-shift torsion duality cancels the Euler defect, an even one doubles the paired part (14.1, §16); `T = coker J` for an alternating integer `J` has `|T| = det J = Pf(J)²` and even length at every prime, and `−J` gives the same `T` with Pfaffian `±Pf` — the square root is an orientation, not a canonical factor (17.1–17.2, 18.1); the ledger `v(𝒮) = ρ + f` (19.1, 28.1). Round 009 is right: the bare leading coefficient has coordinate weight `r` — `c'_r = u^{−r}c_r`, by inverting `t' = ut + at² + bt³` and substituting (1.1); the leading tensor is frame-independent (4.1) and pulls back by `Sym^r(dφ₀)` even when `φ` has quadratic terms (5.1); `c_r = s^{(r)}(0)/r!` (6.1); a period frame changes the scalar, not the order (7.1); orders add and leading forms multiply (8.1–8.2); `ord ℒ = e + m` with `LT = U(0)·LT(E)⊗λ(τ_m)` on two-variable families (11.2–11.3); **annihilation raises the order** — on a two-dimensional transverse space a functional killing `τ_m` but not `τ_{m+1}` gives exactly `e + m + 1` (12.1); `r ≥ e + m` in every instance (13.1, 28.1). All on 40 exact random instances each. **The supplements.** (i) Round 008 asks for "one determinant regulator" (V2, V7): with Tate's archimedean series on `2P = (3,5)`, `2Q = (6,−16)`, `P+Q = (−2,−1)`, `P−Q = (−1,−2)` (all integral; `389.a1` has only `I₁` at 389, whose component group is trivial), the canonical heights are `ĥ(P) = 0.327000773651605`, `ĥ(Q) = 0.476711659343740`, `⟨P,Q⟩ = 0.058522674844879`, the parallelogram law holds to `1.1·10⁻¹⁶`, the naive limit `h_x(2ⁿP)/4ⁿ` at depth 9 agrees to `1.1·10⁻⁸`, and `Reg = ĥ(P)ĥ(Q) − ⟨P,Q⟩² = 0.15246017794314376` — the corpus number `0.15246017794314376` to the last floating-point digit, in the normalisation `ĥ = lim h_x(2ⁿP)/4ⁿ`; the halved normalisation gives `Reg/4`, which is (7.2) with `u = 1/2`, and only the former closes `L''(E,1)/2 = Ω·Reg`: `L''/2 / (Ω·Reg) = 1.000000000000000` with this line's `Ω` (RUN-017), `L''/2` (RUN-073, from the `a_n`) and now `Reg` — **every factor of the rank-2 identity at 389.a1 is computed in this tree.** (ii) Round 009's No-Go 17.1, "do not identify `r_p` with `r_C` by symbol similarity", made concrete: `r_C = 2` (`L(E,1) = 9.7·10⁻¹⁷` from the `a_n`, `w = +1` so `L'(E,1) = 0`, `L''/2 ≠ 0`); `r_p ≤ 2` because the mod-11 `p`-adic L-function of RUN-071 has `t`-order exactly 2 with unit leading coefficient (`λ = 2`); `r_p ≥ 1` because `L_p(E,1) = (1 − α⁻¹)²L(E,1)/Ω = 0` (interpolation, cited); `r_p` even because the `p`-adic functional equation carries the sign `w = +1` (cited) — so `r_p = 2 = r_C`, the three numbers this line's, the two theorems read. And (18.2): `L*/Ω = 0.15246017794314376 = Reg`, `q_fin = 1`. What the documents list as not proved — the actual Selmer complex and its duality shift, `q₊`, `q₋`, the Néron/adjoint conventions, the actual reciprocity map, BSD — they list, and this round adds nothing to those lists.**

---

## Round 008 — what was instantiated

| statement | instantiated as | instances |
| --- | --- | ---: |
| 2.3, 3.1 | `det(A h Bᵀ) = det A · det h · det B`, ranks 2 and 3 | 40/40 |
| 3.2, 4.1 | same basis: `Reg = (det A)² det h`; one vector scaled by `a`: `a²Reg`; all by `a`: `a^{2r}Reg` | 40/40 |
| 5.1, 21.1 | `C²` on the primal volume, `C²` on the dual: `C⁴Reg`; `/b₀⁴` with `b₀ = Ca₀k₀` leaves `Reg/(a₀k₀)⁴` | 40/40 |
| 6.1 | integer symmetric `h`: `v_ℓ(det h) = Σ v_ℓ(Smith invariants)`, `ℓ = 2, 3` | 40/40 |
| 7.1, 7.2 | `det(uh) = u^r det h` | 40/40 |
| 8.1, 9.1, 9.2 | `q₊` on the primal volume, `q₋` on the dual: `𝒮 = q₊q₋Reg`, `v_ℓ(𝒮) = v_ℓ(q₊) + v_ℓ(q₋) + v_ℓ(Reg)` | 40/40 |
| 10.1, §11, §12 | one-sided `qReg`, symmetric `q²Reg`, inverse-dual `Reg` | 40/40 |
| 13.1, 13.2 | two-term complexes `R^n →A R^n` in degrees `(i, i+1)`, scalar `det(A)^{(−1)^i}`; the dual `Aᵀ` in `(−i−1, −i)`, shifted by `s ∈ [−2, 2]` | 40/40 |
| 14.1, §16 | symmetric length profiles for `s = 1, 3` (zero) and `s = 2, 4` (twice the paired part plus the middle) | 40/40 |
| 17.1, 17.2, 18.1 | alternating `J` of size 2 and 4: `det J = Pf²`, `v_ℓ(det J)` even, `Pf(−J) = ±Pf(J)` | 40/40 |
| 19.1, 28.1 | `v_ℓ(𝒮) = v_ℓ(Reg) + f_ℓ` | 40/40 |

## The regulator of 389.a1, from this line's own heights

`y² + y = x³ + x² − 2x`, `b₂, b₄, b₆, b₈ = 4, −4, 1, −3`; `φ(x) = x⁴ + 4x² − 2x + 3 > 0` on `R`, so Tate's series `λ_∞(P) = ½log|x| + ⅛Σ4^{−n}log|z(2ⁿP)|`, `z = 1 − b₄/x² − 2b₆/x³ − b₈/x⁴`, has no singular term; the weight `4^{−n}` cancels the `4ⁿ` growth of the rounding error in `x(2ⁿP)`, so sixty terms in double precision give `~10⁻¹⁵`. The non-archimedean parts are `½ Σ_p max(0, −v_p(x)) log p` — zero for the four integral points — because the only bad prime is 389 with Kodaira `I₁`, whose Weierstrass model is regular and whose component group is trivial. `ĥ` below is `2(λ_∞ + ½ log den x)`, the normalisation `ĥ = lim h_x(2ⁿP)/4ⁿ`.

| point | `x, y` | `ĥ` |
| --- | --- | ---: |
| `2P` | `3, 5` | `1.3080030946064198` |
| `2Q` | `6, −16` | `1.9068466373749582` |
| `P + Q` | `−2, −1` | `0.9207577826851023` |
| `P − Q` | `−1, −2` | `0.6866670833055865` |

| quantity | value |
| --- | ---: |
| `ĥ(P) = ĥ(2P)/4` | `0.327000773651605` |
| `ĥ(Q) = ĥ(2Q)/4` | `0.476711659343740` |
| `⟨P,Q⟩ = (ĥ(P+Q) − ĥ(P) − ĥ(Q))/2` | `0.058522674844879` |
| `ĥ(P+Q) + ĥ(P−Q) − 2ĥ(P) − 2ĥ(Q)` | `−1.1·10⁻¹⁶` |
| `h_x(2ⁿ·2P)/4ⁿ⁺¹`, `n = 1 … 9` | `0.29974, 0.32624, 0.32631, 0.32655, 0.32696, 0.32697, 0.32700, 0.32700, 0.32700076` |
| **`Reg = ĥ(P)ĥ(Q) − ⟨P,Q⟩²`** | **`0.15246017794314376`** |
| the corpus (RUN-023) | `0.15246017794314376` |
| `L''(E,1)/2 ÷ Ω` | `0.15246017794314376` |
| `L''(E,1)/2 ÷ (Ω·Reg)` | `1.000000000000000` |

Round 008's own point, in numbers: the halved normalisation `ĥ/2` gives `Reg/4 = 0.0381150444857859` — the height-scale law (7.2) with `u = 1/2` — and `L''/2 ÷ (Ω·Reg/4) = 4`. A "self-dual" reading that squared the regulator would give `Reg² = 0.0232`; nothing closes but `Ω · Reg · 1`. The gate plants the halved normalisation as a defect.

## Round 009 — what was instantiated

`K[[t]]` truncated at degree 7; `K[[X, t]]` at total degree 6; 40 instances.

| statement | instantiated as | instances |
| --- | --- | ---: |
| 1.1 | `t' = ut + at² + bt³` inverted to degree 7, `L(t(t'))` read off: order `r`, coefficient `u^{−r}c_r` | 40/40 |
| 4.1 | `f' = g⁻¹f` for a unit series `g`: order `r`, `f'_r g(0) = c_r` | 40/40 |
| 5.1 | `φ = J·x + (quadratic)`, `J ∈ GL₂`: lowest form of `s∘φ` equals `τ_r∘J` | 40/40 |
| 6.1, 7.1 | `s^{(r)}(0)/r! = c_r`; `τ' = c⁻¹τ` | 40/40 |
| 8.1, 8.2 | `ord(s_e·s_m) = e + m`, lowest form the product | 40/40 |
| 11.2, 11.3 | `ℒ = U·E·c·Z_tr`: order `e + m`, lowest form `U(0)·E_e·c·τ_m` | 40/40 |
| 12.1 | `Q = K²`, `λ = f`, `Z = τ_m⊗v_ker + τ_{m+1}⊗w`, `f(v_ker) = 0 ≠ f(w)`: order `e + m + 1` | 40/40 |
| 13.1, 28.1 | `r ≥ e + m` in both families | 40/40 |
| §15–16 | the exponents `−r, 2, 4` pairwise distinct | 40/40 |
| §26 | one scalar pipeline: `c_t^{−r}·Ω^{−1}·C⁴/b₀⁴·a²·v²·q₊q₋` | 40/40 |

## The two orders on 389.a1 (Round 009 §17–18)

| | how | value |
| --- | --- | ---: |
| `L(E,1)` | `2Σ a_n e^{−2πn/√389}/n`, 20000 terms, the tree's `a_n` | `9.7·10⁻¹⁷` |
| `w(E)` | RUN-076 | `+1`, so `L'(E,1) = 0` |
| `L''(E,1)/2` | RUN-073 | `0.7593165002884268` |
| **`r_C`** | | **2** |
| mod-11 `t`-coefficients of `L₁₁(E)` | RUN-071, from this tree's eigenline | `0, 0, 2, 2, 2, 0, 5, 6, 10, 7, 0` — order 2, `λ = 2` |
| `r_p ≤ 2` | Weierstrass: at most `λ` zeros in the disc | |
| `r_p ≥ 1` | `L_p(E,1) = (1 − α⁻¹)²L(E,1)/Ω = 0` — interpolation, *cited* | |
| `r_p` even | the `p`-adic functional equation has sign `w(E)` for good ordinary `p` — *cited* | |
| **`r_p`** | | **2** |

PC-002 (RUN-081) said, correctly, that the mod-11 expansion alone does not decide the constant and linear terms of the characteristic-zero series; the two cited theorems do. Neither is checked here — they are read, and the gate records which numbers are this line's.

## The labels

| document | header | the unproved list ends with |
| --- | --- | --- |
| 008 | "未主張 BSD 已證明" | "10. BSD。" |
| 009 | "未主張 BSD 已證明" | "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 10 planted for this gate, each turning `symbolic008-009-fundamental-line` red and nothing else:

| planted defect | went red |
| --- | --- |
| Symbolic 008's height-rescaling exponent is taken as 1 on a rank-2 regulator | `symbolic008-009-fundamental-line` |
| the regulator is entered twice in the fundamental scalar, as Reg^2 | `symbolic008-009-fundamental-line` |
| the dual-shift determinant law is written det(C)^(+(-1)^s) | `symbolic008-009-fundamental-line` |
| the odd-shift torsion Euler cancellation is denied | `symbolic008-009-fundamental-line` |
| the order of an alternating self-dual finite module is taken as the Pfaffian, not its square | `symbolic008-009-fundamental-line` |
| the leading coefficient's coordinate weight is taken as u^(+r) | `symbolic008-009-fundamental-line` |
| the r-th derivative is identified with the Taylor coefficient without the r! | `symbolic008-009-fundamental-line` |
| annihilation of the first transverse jet is said to keep the order at e + m | `symbolic008-009-fundamental-line` |
| the root number of 389.a1 is taken as -1, making the p-adic order odd | `symbolic008-009-fundamental-line` |
| the canonical height is taken in the halved normalisation, so the regulator is a quarter of the corpus number | `symbolic008-009-fundamental-line` |

Because the regulator and the two orders use the tree's live real period, `a_n` and AGM, seven defects planted for older checks — the real period doubled or one-component (`rank2-bsd-identity`, `real-period-vs-integration`), the regulator inverted (`rank2-bsd-identity`), the AGM stopped after one step and the Hecke recursion losing its `−p·a_{p^{k−1}}` term (`anchor-11a1`), the Legendre symbol with the wrong exponent (`reducibility-sieve`) — now also turn this check red; the log records every check that went red for each, and the named check catches each.

## What this round does not claim

* **The actual Selmer complex, its self-duality shift, `q₊`, `q₋`, which side carries which finite group** — Round 008 §32; not here.
* **The Néron/adjoint/Kato conventions** — the regulator above is at the normalisation stated; its match with the corpus and with `L''/2 ÷ Ω` fixes that normalisation and nothing else.
* **The interpolation formula and the `p`-adic functional equation** — cited for `r_p`; this line computes only the mod-11 order, `L(E,1)` and `w`.
* **Nothing about `C`, `𝔰₁₁`, the actual reciprocity map, or BSD.**