# RUN-048 — The schema, the sieve, and the control curve that was supposed to fail **Arm:** 數學戰士「墜衡」 / AMRAL Research Lab **Subject:** `13_Candidate_NonSemistable_Strong_BSD_Family` and `14_Candidate_Sieve` — a schema with six obligations, and the sieve that produced the anchor **with a control that fails** **Tools:** `src50_candidate_schema_and_sieve.py`, `src11_gate_drill.py` **Logs:** `src50-candidate-schema.json`, `src11-gate-drill.json` **Result: `14`'s odd local table for the anchor recomputes exactly — `2` additive `II*` with `v_Δ = 11`, `3` split multiplicative `I₁` with `v_Δ = 1`, `29` **nonsplit** multiplicative `I₁` with `v_Δ = 1`, giving `W_mult^odd = {3, 29}` and `W₋ = {29}` — and its boxed criterion holds. **The control is the point of this round, and the control was run.** `14` eliminates a conductor-116 curve as `FAIL_FIXED_MULTIPLICATIVE_WITNESS`; the gate **found a curve of conductor 116 by search rather than quoting a label** — one model in 7,479 scanned, `[0, 1, 0, −4, 4]` — and confirmed the failure for the stated reason: `2` additive `IV*`, `29` nonsplit multiplicative, so `W_mult^odd = {29}` is a **single** reservoir and the leave-one-out at `p = 29` has **no distinct candidate at all**. It has a nonsplit prime and still fails, which is exactly `14`'s point: the criterion is not "has a nonsplit prime" but **至少兩個 odd multiplicative reservoirs,其中至少一個 nonsplit**. And `13`'s **B1 and B2 are RUN-033's lemma 1 and RUN-037's premise under other names** — both computed here before this document was read. Of `13`'s six obligations: **1 cited, 3 partly, 2 open, none closed**, and `13`'s own rule — 六項完成前,不升級為 theorem — keeps the label where `20` and `27` put it.** --- ## `14`'s table, recomputed | `p` | reduction | Kodaira | `v_p(Δ)` | | | ---: | --- | --- | ---: | --- | | 2 | additive | `II*` | 11 | | | 3 | multiplicative | `I₁` | 1 | **split** | | 29 | multiplicative | `I₁` | 1 | **nonsplit** | `W_mult^odd = {3, 29}`, `W₋ = {29}`, `g_mult^odd = 1`, `g₋ = 1` — both powers of two, which is `13`'s B1 and B2. RUN-033 computed the same table from the discriminant without reading either document; RUN-042 compared it against `15`'s certificate. This is the third independent arrival at the same three rows. ## The control, found rather than quoted `14` names its control by LMFDB label. **A label is a name, not arithmetic**, and RUN-030 caught this arm quoting three numbers from memory. So the gate searched an a-invariant box for curves of conductor 116 and found exactly one: ```text [0, 1, 0, -4, 4] 2 additive IV* vΔ = 8 29 multiplicative I1 vΔ = 1 nonsplit ``` which is the local structure `14` states for its control. Whether this is the curve LMFDB labels `116.b1` is a question about labels and is not settled here — what is settled is that a curve of that conductor with that structure exists and behaves as `14` says. ## It has a nonsplit prime and still fails | | 696.e1 | the control | | --- | --- | --- | | `W_mult^odd` | `{3, 29}` | `{29}` | | `W₋` | `{29}` | `{29}` | | at least one nonsplit | ✔ | **✔** | | at least two reservoirs | ✔ | **✘** | | leave-one-out at each fixed multiplicative `p` | `3 → 29`, `29 → 3` | `29 → ` **nothing** | | boxed criterion | **met** | **not met** | The control passes the test people would think of first — it *has* a nonsplit multiplicative prime — and fails the one that matters. That is `14`'s boxed conclusion: > 至少兩個 odd multiplicative reservoirs，其中至少一個 nonsplit。 and it is why RUN-033's *one witness, no spare* mattered. The anchor has two reservoirs and one of them is nonsplit; the control has one reservoir that happens to be nonsplit, and one reservoir cannot supply a **distinct** witness to itself. RUN-033's lemma 2 is the same statement, and the control is the first curve in this tree where it returns nothing. **A criterion whose failing side is never exhibited has not been tested.** `14` supplied the failing side; this round ran it. ## `13`'s schema | | | | | --- | --- | --- | | **B0** | 2-part anchor, and 建議 `c_E = 1` | **partly** — `v₂(L^alg) = 0`, trivial torsion, `Δ < 0` computed (RUN-014/030); `BSD(E,2)` cited; `c_E = 1` is RUN-039's open item | | **B1** | `W_mult^odd ≠ ∅`, `g_mult^odd` a power of two | **computed** — RUN-033's lemma 1 | | **B2** | `W₋ ≠ ∅`, `g₋` a power of two | **computed** — RUN-037's premise | | **B3** | fixed additive odd primes: exact H1, H2, period/Manin | **open** — RUN-045 marks H2 `UNKNOWN` there | | **B4** | fixed multiplicative: per-prime check + distinct ramified witness | **computed** — RUN-033's leave-one-out | | **B5** | twist support restrictions | **computed** — RUN-032's five conditions, RUN-036's images | B1 and B2 are worth naming twice: this tree computed both **before** reading `13`, from `00_GCD_Witness_Lemmas` and `09_FW_H3_Exact_Compiler`. The schema and the lemmas were written by different hands and ask for the same two gcds. ## The six obligations, scored | | status | where | | --- | --- | --- | | fixed additive primes' exact H1 backend | **partly** | RUN-036 certifies surjectivity at 38 primes; `p = 2` is not among them | | fixed additive primes' exact H2 local residual backend | **open** | RUN-034 found `05`'s no-go silent there; silence is not a PASS | | fixed multiplicative branch formal theorem table | **cited** | `27`'s router names Skinner Theorem C | | Manin-period compatibility | **open** | RUN-039's `c_E = 1`; RUN-040 checked the weaker `p ∤ c` | | all support restrictions' Chebotarev / CRT simultaneous compatibility | **partly** | RUN-044 computed the Chebotarev half | | final all-prime cover proof | **partly** | RUN-047 ran `27`'s router as a partition; the branch theorems are cited | **1 cited, 3 partly, 2 open — none closed.** `13`'s own rule is 六項完成前, 不升級為 theorem, and this arm's scoring keeps the label exactly where `20` and `27` put it. That agreement is the finding: the schema's stopping rule and the referee's label were set independently and point at the same rung. ## The drill **192 defects, 192 caught by the check named for each**, 0 uncaught, 0 caught by the wrong check, 37 controls undisturbed, over 76 checks. The 4 planted for this gate, each turning `candidate-schema` red and nothing else: | planted defect | went red | | --- | --- | | the control curve cannot be found, so the failing side is never shown | `candidate-schema` | | the sieve criterion drops the two-reservoir clause | `candidate-schema` | | additive primes are counted as multiplicative reservoirs | `candidate-schema` | | one of 13's six obligations is reported closed | `candidate-schema` | ## What this round does not claim * **The control's identity is not established.** A curve of conductor 116 with the stated local structure was found; whether it carries the label `116.b1` is a question about a database, not about arithmetic, and this arm does not answer it. * **The search box is a box.** Exactly one conductor-116 model was found in it; that is not a claim that the isogeny class has one curve. * **"Partly" is not "nearly".** Three obligations are scored partly because a piece of each was computed; none is close to closed, and the two open ones are open at the same place they were at RUN-039. * **`13` is a schema, not a theorem**, and says so in its first line. Scoring its obligations measures distance, not progress toward a proof. * **B0's `BSD(E,2)` and the Manin constant stay cited**, as they have since RUN-030's certificate.