# RUN-014 — the Phase 2 anchor: `r_an = 0` computed, and `L(E,1) = Ω` to the last bit **Arm:** 數學戰士「墜衡」 / AMRAL Research Lab **Subject:** `21_Base_BSD_Anchor_Repair` — the two arithmetic facts the repair moves the whole Phase 2 family theorem onto **Tools:** `src15_phase2_anchor.py`, `src11_gate_drill.py` **Logs:** `src15-phase2-anchor.json`, `src11-gate-drill.json` **Result: both facts hold. `N = 696` is selected by the functional equation over its nearest competitor by a factor of 339, without Tate's algorithm. The root number is `+1` and `L(696.e1, 1) = 1.631727740072 ≠ 0`, so `r_an = 0`. And `L(E,1)/Ω = 1.0000000000000002` — one, to within a single unit in the last place — which with trivial torsion and `c₃ = c₂₉ = 1` forces `c₂ · #Ш = 1`.** --- ## What the repair moved the weight onto `21` records a careful correction. v0.3 argued from Banwait–Huang's historical remark that Miller verified full BSD for **most** rank 0/1 curves below conductor 5000 — from which `696 < 5000` does not follow. The repair cites Creutz–Miller Theorem 1.1 instead: > `N < 5000` and `r_an ≤ 1` ⟹ full BSD. That is the right kind of repair — it replaces an inference from "most" with a theorem that quantifies over all. And it moves the load onto two arithmetic facts about `696.e1`: **`N = 696`** and **`r_an = 0`**. RUN-009 checked neither. It verified the discriminant, the reduction types and the non-semistability, and took the conductor from the note. ## The root number is measured, not assumed — and 37a1 is why The convenient formula $$L(E,1)=2\sum_{n\ge1}\frac{a_n}{n}e^{-2\pi n/\sqrt N}$$ is valid **only when `w = +1`**. `37a1` shows what happens if that is forgotten: it has rank 1, so `L(E,1) = 0` — and the same sum evaluates to `0.19`. A nonzero sum is no evidence of a nonzero `L`-value. So the sign is read off instead of assumed. At `s = 2` the Dirichlet series `Σ a_n n^{−2}` converges absolutely and needs no functional equation at all, while the smoothed formula for `Λ(2)` depends on `w`. Evaluating both signs and keeping whichever matches determines `w` numerically: | curve | `Σ a_n n^{−2}` | smoothed, `w = +1` | smoothed, `w = −1` | picked | known | | --- | ---: | ---: | ---: | :-: | :-: | | `11a1` | 0.546045816 | 0.546048036 | 0.212403460 | **+1** | +1 ✓ | | `37a1` | 0.381578708 | 0.573749146 | 0.381575408 | **−1** | −1 ✓ | | `389a1` | 0.360097631 | 0.360092864 | 0.430502866 | **+1** | +1 ✓ | | **`696.e1`** | **1.068637376** | **1.068630094** | 0.942865445 | **+1** | — | For the anchor the residual is `7.28 × 10⁻⁶` and the gap to the other sign is `0.1258` — **seventeen thousand times larger**. ## The conductor, selected rather than assumed That comparison closes only with the right `N`, so running it over candidate conductors picks one out: | `N` | residual | | ---: | ---: | | 174 | 2.085 × 10⁻² | | 348 | 1.287 × 10⁻² | | **696** | **7.283 × 10⁻⁶** | | 1044 | 4.816 × 10⁻³ | | 1392 | 6.446 × 10⁻³ | | 2088 | 6.457 × 10⁻³ | | 4176 | 2.469 × 10⁻³ | `N = 696` wins by a factor of **339** over the runner-up. That is a verification of the conductor by a route that never runs Tate's algorithm — the functional equation simply does not close on a wrong one. ## The anchor | | | | --- | --- | | `p = 2` | additive, `a₂ = 0` | | `p = 3` | split multiplicative, `a₃ = +1` | | `p = 29` | non-split multiplicative, `a₂₉ = −1` | | root number `w` | **+1** | | real period `Ω` | 1.631727740071569 | | `L(E,1)` | **1.631727740071569** | | **`r_an = 0`** | **yes** | | `#E(Q)_tors` | **1** (gcd of `#E(F_p)` over good `p ≤ 120`) | | `c₃ · c₂₉` | 1 | | **`L(E,1)/Ω`** | **1.0000000000000002** | `L(E,1)` and `Ω` agree to every digit float64 has; their ratio differs from 1 by `2.2 × 10⁻¹⁶`, which is one unit in the last place. ## What that ratio forces At rank 0 the strong BSD formula reads $$\frac{L(E,1)}{\Omega}=\frac{\#Ш\cdot\prod_p c_p}{\#E(\mathbf Q)_{\rm tors}^2}.$$ With `#tors = 1` and `c₃ = c₂₉ = 1`, the measured ratio gives $$\boxed{c_2\cdot\#Ш = 1,}$$ and both are positive integers, so `c₂ = 1` and `Ш` is trivial. **This is a consistency reading, not an independent proof of either factor.** It uses the BSD formula that Creutz–Miller supplies for this curve. Read the other way — if `c₂` and `#Ш` were known independently to be 1 — the same measurement would be a verification of strong BSD for `696.e1` to sixteen digits. The round does not compute `c₂`: the reduction at 2 is additive and its Tamagawa number needs Tate's algorithm, which this gate does not implement. ## Every piece validated before the anchor was touched The gate refuses to run on `696.e1` until it reproduces three curves whose answers are fixed outside it: * **`11a1`** — torsion `Z/5`, `c₁₁ = 5`, `Ш` trivial, so `L/Ω` must be exactly `5/25 = 1/5`. Measured: `0.20000000000000004`, off by `2.8 × 10⁻¹⁷`. Real period `1.2692093042795534`. Torsion bound `5`. * **`37a1`** — rank 1: `w = −1` and `L(E,1) = 0`. * **`389a1`** — rank 2 with `w = +1`: `L(E,1) = 0` *despite* the sign being positive. This is the case that separates "the sum vanishes" from "the sign kills it", and it is the one that makes the `37a1` lesson precise rather than anecdotal. ## The drill, and three checks that could not see a defect **59 defects, 59 caught by the check named for each. 14 controls, none disturbed.** Gate 15's eight defects include the Hecke recursion losing its `−p·a_{p^{k−1}}` term, `L(E,1)` losing its factor of two, an inverted split/non-split test, and `E₁` implemented without the Euler–Mascheroni constant. `E₁` is written in the gate with no dependencies at all; **mpmath appears only in the drill**, as the independent implementation to check it against — agreement to `2.1 × 10⁻¹⁵` relative across nine arguments spanning `0.05` to `120`. Three of the eight escaped on the first run, and all three for the same reason: **the check's tolerance was looser than the quantity's precision.** Undisturbed, `|L/Ω − 1/5|` is `2.8 × 10⁻¹⁷`; a one-step AGM shifts the real period by `1.4 × 10⁻¹²` and an inverted split test shifts the ratio by `2.6 × 10⁻¹⁰` — both invisible inside the `1e-9` window the first version used. The third had no assertion on the real period for a curve with `Δ > 0` at all, so dropping the second component of `E(R)` changed nothing it looked at. A tolerance chosen for looking round rather than for matching the computation is a check that cannot see a real defect. The windows are now set to the measured precision, and the reason is written where the numbers are. ## What this round does not claim * **Nothing about BSD in general**, and nothing about the family theorem. It checks the two arithmetic facts the anchor repair rests on. * **It does not verify Creutz–Miller Theorem 1.1**, which is a published theorem this arm has not read. It verifies that `696.e1` satisfies its hypotheses. * **`c₂` is not computed** — additive reduction needs Tate's algorithm. Only the product `c₂ · #Ш` is measured. * **`Ш` is not computed.** That it must be trivial follows from BSD holding, not from anything measured here. * **The `L`-values are floating-point.** They are reported at the precision they have — sixteen digits for the anchor ratio, `7.3 × 10⁻⁶` for the root-number residual — and no claim is made beyond it.