# RUN-040 — The odd-additive period barrier, met at one prime per member and cleared structurally **Arm:** 數學戰士「墜衡」 / AMRAL Research Lab **Subject:** `01_Odd_Additive_Period_Barrier` — an obstruction that is not Iwasawa-theoretic but arithmetic-of-periods, and the published sufficient condition that clears it **Tools:** `src42_odd_additive_period_barrier.py`, `src11_gate_drill.py` **Logs:** `src42-odd-additive-barrier.json`, `src11-gate-drill.json` **Result: the barrier is **empty for the base curve** — 696.e1's only additive prime is 2, and `01`'s obstruction is about a fixed *odd* additive prime — and each of the 19 members meets it at exactly one prime, `q` itself. There the Kodaira type is `I₀*` for **19 of 19**, because the twist forces the valuations: `d` squarefree, coprime to `N` and good for `E` scales `c₄` by `d²`, `c₆` by `d³` and `Δ` by `d⁶`, so at `p = d` they are exactly `(2, 3, 6)` — measured on every member, and `(2,3,6)` is `I₀*`, never II, III or IV. With `q ≥ 241 ≥ 11` and optimality cited, `01`'s sufficient condition is met throughout, and `p ∤ c` follows from the cited Manin-constant result. **The excluded types are shown to be reachable rather than assumed non-empty**: `y² = x³ + 11` is type II, `y² = x³ + 11x` is type III, `y² = x³ + 121` is type IV, and the condition correctly fails at each. This is the piece RUN-039 left open, approached from the weaker side `01` actually needs.** --- ## The obstruction is about periods, not Iwasawa theory `01` makes a distinction worth keeping: Fouquet–Wan permits arbitrary reduction type at `p`, so a fixed odd additive bad prime is **not** excluded at the theorem level. What fails is period normalisation — FW Corollary 1.10 uses the modular period, which differs from the Néron period by the Manin constant. At a good supersingular `p` that is harmless, since the Manin constant's prime divisors sit at additive primes. At a fixed **additive** `p` that escape is unavailable, because `p` is exactly such a prime. So every odd-additive extension needs an explicit period certificate, and `01` names a published sufficient condition: * `p ≥ 11`; * the local reduction is **not** additive potentially ordinary of Kodaira type II, III or IV; * the twist is optimal. Then `p ∤ c`. ## The base curve never meets it 696.e1 is additive only at 2. `01`'s obstruction is about an **odd** additive prime, and `12_Hybrid_Odd_Prime_Router` sends `p = 2` to Banwait–Huang at P0, which never reaches this period argument. So the barrier's domain is empty for the base — reported as empty, with the reason, rather than as "passed". ## Each member meets it exactly once, and clears it by construction | | measured | | --- | --- | | the odd additive prime is exactly `q` | **19 of 19** | | Kodaira type there is `I₀*` | **19 of 19** | | valuations `(v_q(c₄), v_q(c₆), v_q(Δ))` are `(2, 3, 6)` | **19 of 19** | | `p ≥ 11` (smallest member is 241) | **19 of 19** | | `01`'s condition met | **19 of 19** | The third row is why the second is not a coincidence. A quadratic twist by `d` scales `c₄ → d²c₄`, `c₆ → d³c₆`, `Δ → d⁶Δ`; when `d` is squarefree, coprime to the conductor and good for `E`, the valuations at `p = d` are forced to `(2, 3, 6)` exactly. That is Kodaira `I₀*` — and `I₀*` is not among II, III, IV for any member, at any bound. ## The excluded types are reachable, and the condition fails there A condition whose excluded set is never exhibited has not been tested. Rather than search, the gate constructs candidates and lets Tate's algorithm say what they are: | type | curve | `p` | condition met? | | --- | --- | ---: | --- | | **II** | `y² = x³ + 11` | 11 | **no** | | **III** | `y² = x³ + 11x` | 11 | **no** | | **IV** | `y² = x³ + 121` | 11 | **no** | All three of the excluded types occur, at a prime satisfying `p ≥ 11`, and the condition correctly refuses each. So "no member is II, III or IV" is a finding about the family and not about the emptiness of the category. ## What is not computed is said "Potentially ordinary" is part of the excluded description, and this arm does not compute the ordinarity of a potential good reduction. **It is not needed here**: the Kodaira type alone already places every member outside the excluded set, whatever the ordinarity turns out to be. The gate records `potentially_ordinary_computed: false` on every row rather than leaving the reader to assume it was checked. And `p ∤ c` itself is a **cited** theorem. What this round checks is that its hypotheses hold. ## What this closes of RUN-039 RUN-039 left `c_E = 1` open — `11_Derived_Supersingular_FW_Bridge`'s own safe period condition, proposed to avoid the modular-versus-Néron splice. `01` asks for less and gets it at the prime that needs it: **`p ∤ c`**, not `c = 1`. The `p`-part of BSD needs `p` not to divide the Manin constant, not the constant to be 1, so the weaker statement is the one the argument actually consumes. The Manin constant is still not computed anywhere in this tree, and that has not changed. What has changed is that for the odd-additive branch the corpus's own cited route is now checked rather than assumed: its three hypotheses hold on every member. ## The drill **160 defects, 160 caught by the check named for each**, 0 uncaught, 0 caught by the wrong check, 28 controls undisturbed, over 68 checks. The 4 planted for this gate, each turning `odd-additive-barrier` red and nothing else: | planted defect | went red | | --- | --- | | the barrier's p >= 11 floor is raised above the family | `odd-additive-barrier` | | the excluded Kodaira types come back unreachable | `odd-additive-barrier` | | the even additive prime is counted as an odd one | `odd-additive-barrier` | | the twist valuations are read off the untwisted model | `odd-additive-barrier` | ## What this round does not claim * **The Manin-constant theorem is cited, not verified**, and `c_E` is still not computed anywhere in this tree. * **Optimality is cited**, as it has been since RUN-031 — RUN-036 closed the premise the audit's optimality argument rests on, not optimality itself. * **"Potentially ordinary" is not computed.** The type clears the condition without it here; on a curve where the type were II, III or IV, this arm could not decide the condition at all, and the gate says so per row. * **The `(2,3,6)` forcing is measured on 19 members** and argued in general from the twist scaling. It holds for `d` squarefree, coprime to `N` and good for `E` — all three being membership conditions of `𝒫`, which is the fourth time this session those conditions have carried weight. * **The base's `p = 2` is out of scope**, not cleared: `01` is about odd additive primes, and 2 is routed elsewhere.