# RUN-053 — The criterion RUN-052 said it could not run was in the corpus, cited once; and it has a consequence it does not state **Arm:** 數學戰士「墜衡」 / AMRAL Research Lab **Subject:** `04_Local_p_Isogeny_Kernel_Criterion` — the local test for FW17-H2 **Tools:** `src55_local_isogeny_kernel.py`, `src11_gate_drill.py` **Logs:** `src55-local-isogeny-kernel.json`, `src11-gate-drill.json` **Result: RUN-052 recorded `LOCAL_H2` branches 2 and 3 as `NOT COMPUTED HERE` and named the missing instrument — the kernel polynomials of a local `p`-isogeny and its dual. **`04` is that instrument, written out in full.** It had been named in exactly **one** earlier report, RUN-008, and the `06` v0.3 procedure that needs it does not cite it. Its five-step chain checks out, and **`p` odd is load-bearing at step 2**: on a Weierstrass model `−(x,y) = (x, −y−a₁x−a₃)`, so `P = −P` *is* the 2-torsion condition — at `p = 2` every point is its own negative, the clause `σ(P) = ±P` holds for every `σ` with no Galois input at all, and the criterion would report `λ² = 1` unconditionally. **And the round's finding is a consequence `04` does not state.** The Weil pairing gives `det E[p] = ω`, so `λ·μ = ω`. `04`'s test applied to `φ` is `λ² = 1`; applied to `φ̂` it is `μ² = 1`. Both at once forces `ω² = 1` — and `ω` surjects onto `(Z/p)^×` because `Q_p(ζ_p)/Q_p` is totally ramified of degree `p−1`, so `ω² = 1 ⟺ (p−1) ∣ 2`. Checked by brute force over **46** primes: true at exactly `{2, 3}`. **So at every `p ≥ 5` the two tests are mutually exclusive** — `04`'s "one isogeny plus its dual is enough" is tight in a stronger sense than it claims, and the second test is only ever consulted after the first has failed. That is `(p−1) ∣ 2` for the **second** time in this corpus, reached from a different document by a different route than RUN-046's. **The criterion is not run on the family**, and the reason is stated rather than worked around: its hypothesis is local reducibility, RUN-036's certificate is about **global** surjectivity, and reading one as the other would be `00` §6's forbidden substitution one level down.** --- ## The chain, and where the hypotheses sit `04` assumes `E[p]|_{G_{Q_p}}` reducible with a stable line `C ≅ Z/p`, and `σ(P) = λ(σ)P` on a generator. | step | rests on | | --- | --- | | `x(P) = x(−P)` | nothing — true for every point on every such model | | `x(P) ∈ Q_p ⟺ σ(P) = ±P ∀σ` | **`p` odd** | | `σ(P) = ±P ∀σ ⟺ λ(G_{Q_p}) ⊆ {±1}` | `C` Galois-stable and cyclic of order `p` | | `λ(G_{Q_p}) ⊆ {±1} ⟺ λ² = 1` | nothing — `{±1}` is the 2-torsion of `F_p^×` | | `λ² = 1 ⟺ ker φ's kernel polynomial has a `Q_p`-linear factor` | **`E[p]` locally reducible** — without a stable line there is no `λ` | ### `p` odd, exhibited rather than asserted On a Weierstrass model `−(x, y) = (x, −y − a₁x − a₃)`, so `P = −P` says `2y + a₁x + a₃ = 0` — which is exactly the condition defining `E[2]`. **Every point of `E[2]` is its own negative.** At `p = 2` the clause `σ(P) = ±P` is therefore satisfied by every `σ` with no Galois input, and the criterion would return `λ² = 1` for free. A criterion whose excluded case is never exhibited has not been tested. This is the discipline RUN-042 applied to `01`'s excluded Kodaira types, applied here to a hypothesis rather than a type. ## The consequence `04` does not state `04` gives the dual's kernel character as `μ = ω λ^{-1}`, which is the same as saying `λ·μ = ω` — the Weil pairing, `det E[p] = ω`. Then: * `04`'s test on `φ` is **`λ² = 1`** * `04`'s test on `φ̂` is **`μ² = 1`** * both at once ⟹ `(λμ)² = ω² = 1` and `ω : G_{Q_p} → (Z/p)^×` is **surjective**, because `Q_p(ζ_p)/Q_p` is totally ramified of degree `p − 1`. So `ω² = 1` iff squaring is trivial on all of `(Z/p)^×` — a finite computation, run over 46 primes rather than asserted: $$\omega^2 = 1 \iff (p-1) \mid 2 \iff p \in \{2, 3\}$$ > **At every `p ≥ 5`, at most one of `φ` and `φ̂` can have a `Q_p`-linear > factor.** `04` argues that one isogeny plus its dual suffices — that no enumeration of all local `p`-isogenies is needed. **That is true in a stronger form than it states:** the two tests are not merely sufficient together, they are *mutually exclusive*, so the second is only ever consulted after the first has failed and they can never both confirm. At `p = 3` they can: `(Z/3)^× = {1, 2}` and `1² ≡ 2² ≡ 1`. ### The closed form was wrong and the computation caught it The gate carries both the brute-force column and the closed form `(p−1) ∣ 2`, and asserts they agree. **The first draft wrote the divisibility backwards** — `(p−1) % 2 == 0`, which is *2 divides p−1* — and the two columns disagreed at exactly `p = 2`, where `p − 1 = 1` divides 2 and 2 does not divide 1. The gate exited 1. That is why the closed form is carried *beside* the computation rather than replacing it. ## `(p−1) ∣ 2`, the second time | round | document | route | | --- | --- | --- | | **RUN-046** | `10` + `03` | `10`'s ordinary congruence names one of `03`'s two cases; the other needs `χ_cyc²` unramified, and `χ_cyc` has order `p−1` on inertia | | **RUN-053** | `04` | the two kernel characters multiply to `ω`, so both tests firing forces `ω² = 1` | **Counted from the reports, not from memory** — the scan reports which files carry the pattern and the fragments it matched, because RUN-029's failure mode was a count assembled from the shapes the author remembered writing. The same structural fact — the mod-`p` cyclotomic character has order `p − 1` — constraining two different criteria from two different documents. ## Why the criterion is not run here `04`'s chain begins *assuming* `E[p]|_{G_{Q_p}}` is reducible. **This tree has not decided that**, and one thing it must not do is borrow RUN-036's answer: | what RUN-036 gives | what `04` needs | | --- | --- | | **global** mod-`ℓ` surjectivity at 38 primes ⟹ global irreducibility | reducibility of the restriction to a **decomposition group at `p`** | A globally surjective representation can restrict to a reducible one at `p` — that is the ordinary case. **The global certificate is about a different group.** Reading the 38 certified primes as settling the local question would be `00` §6's third prohibition one level down, and the gate records a field saying so rather than leaving it to prose. ## What it would cost, stated rather than guessed | | | | --- | --- | | kernel polynomial degree, `(q−1)/2` | **120 – 1,896** over the 19 members | | `q`-division polynomial degree, `(q²−1)/2` | **29,040 – 7,193,424** | The kernel polynomial itself is small. **Finding** it means locating a Galois-stable line — factoring the `q`-division polynomial over `Q_q`. Neither is attempted here, and the precondition is undecided anyway. ## The document was there the whole time | | | | --- | --- | | reports naming `04` before this round | **1** — RUN-008 | | `06` v0.3's branch 3, at RUN-052 | `NOT COMPUTED HERE` | RUN-052 reported that the instrument to settle `LOCAL_H2` was missing from this tree. It was: **the tree does not compute it.** What the round did not know is that the corpus specifies it exactly, in a document cited once, which the v0.3 procedure that needs it does not reference. **That is the finding, and it is about the corpus, not about the arithmetic.** ## The drill **223 defects, 223 caught by the check named for each**, 0 uncaught, 0 caught by the wrong check, 43 controls undisturbed, over 82 checks. The 5 planted for this gate, each turning `local-isogeny-kernel` red and nothing else: | planted defect | went red | | --- | --- | | 04's chain drops the `p` odd hypothesis from step 2, where p = 2 makes the clause vacuous | `local-isogeny-kernel` | | the closed form goes back to (p-1) % 2, which is the divisibility backwards | `local-isogeny-kernel` | | omega squared is reported trivial at every prime | `local-isogeny-kernel` | | the two kernel tests are reported mutually exclusive at p = 3 too | `local-isogeny-kernel` | | the precondition is reported settled by RUN-036's global surjectivity | `local-isogeny-kernel` | ## What this round does not claim * **Nothing is decided about `LOCAL_H2`.** RUN-045's `UNKNOWN` stands unchanged. * **The mutual-exclusivity result constrains the two tests' joint behaviour.** It does not decide either one, and it does not tell you whether FW17-H2 fails. * **The chain is checked for internal consistency and hypothesis-tracking**, not re-derived from Fouquet–Wan. `04`'s boxed equivalence is taken as the document's, with its steps made explicit. * **`ω` surjective is cited**, from total ramification of `Q_p(ζ_p)/Q_p`. What is computed here is the consequence — that squaring is trivial on `(Z/p)^×` exactly at `p ∈ {2, 3}`. * **The degree figures are arithmetic, not a feasibility study.** Whether a backend could factor a degree-7.2-million polynomial over `Q_q` is not measured here. * **That `04` was cited once is a fact about this corpus and this line's reports**, not a judgement about why.