# RUN-046 — The H2 chain is consistent, and the corpus disagrees with itself about H3 **Arm:** 數學戰士「墜衡」 / AMRAL Research Lab **Subject:** `03_FW_H2_Jordan_Holder_Lemma` and `08_FW_Weight2_Exact_Translation`, against `10` and `02` **Tools:** `src48_h2_chain_and_h3_dispute.py`, `src11_gate_drill.py` **Logs:** `src48-h2-chain.json`, `src11-gate-drill.json` **Result: the corpus's three statements of FW-H2 are one statement. `08`'s representation-level ratio test — `H2 FAIL ⟺ αβ⁻¹ ∈ {χ_cyc, χ_cyc⁻¹}` — and `03`'s Jordan–Hölder lemma — `⟺ λ² = 1 or μ² = 1` — are verified **equivalent over 22,140 character pairs with 0 mismatches**, exhausting every cyclic character group of order up to 40 and every value of `ω`. And `10`'s ordinary congruence `a_p² ≡ 1 (mod p)` names only **one** of `03`'s two cases; the other needs `χ_cyc²` unramified, and the mod-`p` cyclotomic character has order `p − 1` on inertia, so that happens exactly when `p − 1 ∣ 2` — at `p = 2` and `p = 3` only. **So `10`'s single congruence is the whole criterion at every good ordinary `p ≥ 5`, and is not at `p = 3`** — the same prime `05` gives its own section and the same prime RUN-036 could not certify. **The H3 news is worse and is reported rather than resolved: `02` states that the divisibility criterion must not be equated with H3, while `08` boxes exactly that criterion as an **iff** under the title *exact translation*, and `09` states the same three conditions as the definition. Two of three treat it as H3 and one forbids it.** No computed value changes under either reading — the verdicts, the witness `ℓ = 29`, and RUN-037's failure at `p = 29` are identical. What changes is what those verdicts are verdicts *about*.** --- > **Correction note added at RUN-049.** This round said resolving the `02`/`08` > seam needs Fouquet–Wan Theorem 1.7's exact local condition, "an external > theorem this arm does not hold". **`23_FW_Supersingular_Source_Audit` quotes > it** — Assumption 3 as special Steinberg, a twist by an unramified character > sending `ℓ` to `(−1)ℓ^{k/2−1}`, and residual ramification — and specialises it > at weight 2 to `a_ℓ = −1`, which is nonsplit multiplicative. That is the > derivation `02` asked for, in a document neither `02` nor `08` references, and > this round had not read it. The disagreement between `02` and `08` **as > written** stands; the corpus is not silent about the convention. RUN-049 checks > the weight-2 arithmetic: `a₂₉ = −1` and nonsplit, `a₃ = +1` and split. ## Three statements of one hypothesis | document | level | statement | | --- | --- | --- | | `08` | representation | `H2 FAIL ⟺ αβ⁻¹ ∈ {χ_cyc, χ_cyc⁻¹}` for `V^ss = α ⊕ β` | | `03` | Jordan–Hölder | `H2 FAIL ⟺ λ² = 1 or μ² = 1` — a JH character is quadratic or trivial | | `10` | ordinary case | `H2 FAIL ⟺ a_p(E)² ≡ 1 (mod p)` | All three take the same input from the Weil pairing: `det E[p] = χ_cyc`, hence `λμ = ω`. That relation is cited by both documents and proved by neither, and not by this round either. ## `03` and `08` are the same statement, exhausted Written additively in a cyclic character group, `λμ = ω` is `l + u ≡ w`, `λ² = 1` is `2l ≡ 0`, and `αβ⁻¹ ∈ {ω, ω⁻¹}` is `l − u ≡ ±w`. Every order from 1 to 40, every `ω`, every admissible pair: | | | | --- | --- | | pairs checked | **22,140** | | mismatches | **0** | | equivalent | **yes** | Exhausting the finite case is a check. Restating the derivation would not have been — and the derivation is short enough that restating it is the tempting thing to do. ## `10` names one case of two, and that is complete only above 3 `03` allows **either** constituent to be the quadratic one. `10` writes the ordinary semisimplification as `ᾱ ⊕ χ_cyc ᾱ⁻¹` with `ᾱ` **unramified**, and names the failure `ᾱ² = 1`. The other case is `(χ_cyc ᾱ⁻¹)² = 1`, which forces `χ_cyc² = ᾱ²` — and `ᾱ` is unramified, so it forces `χ_cyc²` to be unramified too. The mod-`p` cyclotomic character has order `p − 1` on inertia. So `χ_cyc²` is trivial there exactly when `p − 1` divides 2: | `p` | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | | --- | ---: | ---: | ---: | ---: | ---: | ---: | ---: | ---: | ---: | | order on inertia | 1 | 2 | 4 | 6 | 10 | 12 | 16 | 18 | 22 | | second case possible | ✔ | ✔ | — | — | — | — | — | — | — | **`10`'s single congruence is the whole of H2 failure at every good ordinary `p ≥ 5`.** At `p = 3` it is not — which is the third time this session `p = 3` has come out special for a structural reason: `05` gives it its own section because `F₃ˣ = {±1}`, RUN-036 could not certify `ρ̄₃` because `PGL₂(F₃) ≅ S₄` and the nonsplit-Cartan test is vacuous mod 3, and here `χ_cyc²` is trivial on inertia. Three different arguments, one prime. ## The chain, and its weakest link | | status | | --- | --- | | `08` (representation) → `03` (Jordan–Hölder) | **verified here** — exhausted | | `03` → `10` (ordinary) | **verified here, with its range** — complete for `p ≥ 5` | | `10` → RUN-038's predicate | the corpus's own derivation, **cited** | | Fouquet–Wan Theorem 1.7 → `08` | **cited** | Two links verified, two cited, and the weakest is the theorem itself, which no round of this arm proves and none pretends to. ## The corpus disagrees with itself about H3 `02_Fouquet_Wan_Hypothesis_Compiler`, on the divisibility criterion: > 但 Fouquet–Wan 的 exact local condition 更細，**不能直接把這一條當完整 H3**。 `08_FW_Weight2_Exact_Translation`, boxing exactly that criterion: ```text FW-H3(E, p) <=> exists ell || N : E nonsplit multiplicative at ell ell != p p does not divide v_ell(Delta_min) ``` under the title **Exact Translation**. And `09_FW_H3_Exact_Compiler` states the same three conditions again, as the definition it compiles. **Two of three treat the criterion as H3; one forbids it.** The gate confirms all three lines are present in the files and that `08`'s and `09`'s conditions are identical. ### What turns on it | | | | --- | --- | | under `08` | RUN-037's and RUN-039's H3 verdicts are verdicts on **H3**, and RUN-037's gap at `p = 29` is a gap in FW-H3 for this curve | | under `02` | they are verdicts on a **compilation**, and the exact condition is unexamined at every prime | | **either way** | **no computed value changes** — the verdicts, the witness `ℓ = 29`, and the failure at `p = 29` are the same | ### Why this round does not resolve it Resolving it means reading Fouquet–Wan Theorem 1.7's exact local condition, which is an external theorem this arm does not hold. RUN-043 placed this line at C1 — hypothesis compiler — and settling which compilation is right is above that rung. **Naming the disagreement is what an independent arm can do here; settling it is not**, and a round that quietly picked the reading that made its own earlier verdicts look stronger would have been doing the opposite of this arm's job. ## And the drill was counting a crash as a catch Wiring this round's gates in turned up a collision in the drill's own helper namespace, and then two more. `_true_squarefree` had been bound twice — to `src21`'s `squarefree_part`, which returns an integer, and to `src44`'s `squarefree`, which returns a pair. The later binding won, so the defect *squarefree part drops the sign* evaluated `abs(...)` on a tuple and raised a `TypeError`. The named check went red, the drill counted it caught, and **a defect that produces a traceback is not a defect a check caught** — RUN-020, RUN-030 and RUN-035 each recorded that, and this is the first time it happened inside the instrument rather than inside a gate. `_true_point_count` had been bound twice as well, to two independently written point counters. Those **agree on every case tested**, so the defect using it computed the right number through the wrong module — benign in effect, wrong in provenance, and exactly the kind of thing that stays invisible until it does not. The collision that surfaced all three was mine: `_true_h3` bound to `src39`'s two-argument `h3` and to `src47`'s, in that order in the file and the opposite order in my head. **The repair is structural, not three renames.** The drill now reads its own source before running and refuses to start if any `_true_*` name is assigned twice: ```text 81 _true_* helpers, none shadowed ``` The three names are also fixed, and the two affected defects now go red for the reasons they are named for. RUN-043 and RUN-044 cite the run that contained the crash-catch and carry correction notes saying so; every other figure in those rounds stands. ## The drill **184 defects, 184 caught by the check named for each**, 0 uncaught, 0 caught by the wrong check, 34 controls undisturbed, over 74 checks. The 4 planted for this gate, each turning `h2-chain` red and nothing else: | planted defect | went red | | --- | --- | | the 03-08 equivalence is asserted without exhausting anything | `h2-chain` | | 10's congruence is called complete at p = 3 as well | `h2-chain` | | the H3 disagreement is reported resolved | `h2-chain` | | 02's cautionary line is reported absent, so there is no disagreement | `h2-chain` | ## What this round does not claim * **The disagreement is not resolved**, and neither document is called wrong. Both are quoted and the consequence of each is computed. * **The Weil-pairing relation `λμ = ω` is cited**, from `det E[p] = χ_cyc`. The equivalence verified here is conditional on it. * **The exhaustion is over cyclic character groups to order 40.** The statement is about characters into `F_p^×`, which is cyclic, so the model is faithful — but the bound is a bound and is stated as one. * **The inertia order of `χ_cyc` is `p − 1`** — standard, cited, not derived here. What is computed is which primes satisfy `p − 1 ∣ 2`. * **`10`'s completeness above 5 is about `10`'s criterion**, not about H2 being decided: `03` and `08` still rest on the local representation, which this tree does not compute.