# RUN-044 — The Chebotarev audit, every field-theoretic step computed, and the compatibility that could have failed **Arm:** 數學戰士「墜衡」 / AMRAL Research Lab **Subject:** `25_Chebotarev_Referee_Audit` — eight steps from a cubic to a density, none of them computed anywhere in the corpus **Tools:** `src46_chebotarev_audit.py`, `src11_gate_drill.py` **Logs:** `src46-chebotarev.json`, `src11-gate-drill.json` **Result: every step runs. `f₂ = x³ + x² + 8x − 16` has `disc = −11136 = −2⁷·3·29`, no rational root and a non-square discriminant, so `Gal(L/Q) = S₃`; the squarefree part is `−174`, giving `F₀ = Q(√−174)`. `−174 = −6 × 29`, and `Q(√−6)` sits inside `Q(ζ₂₄)` because its discriminant `−24` divides 24 while `Q(√29)`'s does not and `F₀`'s `−696` does not either — so `F₀ ⊂ K = Q(ζ₂₄, √29)` and `[K:Q] = φ(24)·2 = 16`. `S₃`'s normal subgroups have orders `{1, 3, 6}`, exactly one nontrivial proper one, so `L` has exactly one nontrivial proper Galois subfield and `L ∩ K = F₀`, giving `[LK:Q] = 6·16/2 = 48`. The 3-cycles form a class of size 2, and **`δ = 2/48 = 1/24`**. **The compatibility step is shown against a case where it fails**: a 3-cycle has sign `+1` and acts trivially on `F₀`, agreeing with the identity on `K`; a transposition has sign `−1`, contradicts it, and that condition would give density **0**. Three routes now give the same number — this derivation, RUN-018's formula `δ = e_E/(3[K_E:Q]) = 2/48`, and RUN-009's measurement of `0.99900 × 1/24` below `2×10⁷`.** --- > **Correction note added at RUN-046.** One of the 176 defects in the drill this > round cites — *squarefree part drops the sign* — went red by raising a > `TypeError`, not by dropping a sign. Two drill helpers named `_true_squarefree` > had been bound to functions with different return types, and the later binding > shadowed the one that defect needed. **A defect that produces a traceback is > not a defect a check caught**, so one of the 176 was a crash-catch. The names > are repaired, the drill now refuses to run with any shadowed `_true_*` helper, > and RUN-046's run re-exercises that defect as a real catch. Every other figure > in this round stands. ## The chain, computed | step | | | | --- | --- | --- | | `disc(f₂)` | `−11136` | `= −2⁷·3·29` ✔ | | rational roots | none | irreducible | | discriminant a square | no | with irreducibility, `Gal(L/Q) = S₃` | | squarefree part | `−174` | `F₀ = Q(√−174)` | | `−174 = −6 × 29` | ✔ | | | `Q(√−6) ⊂ Q(ζ₂₄)` | disc `−24`, and `24 ∣ 24` | ✔ | | `Q(√29) ⊂ Q(ζ₂₄)` | disc `29`, and `29 ∤ 24` | **no** — which is why it is adjoined | | `F₀ ⊂ Q(ζ₂₄)` | disc `−696`, and `696 ∤ 24` | **no** — but `F₀ ⊂ K` | | `[K:Q]` | `φ(24) × 2 = 8 × 2` | **16** | | `S₃` normal subgroup orders | `{1, 3, 6}` | exactly one nontrivial proper, of index 2 | | `L ∩ K` | `F₀` | `K` abelian, so the intersection is abelian and Galois, hence the unique degree-2 subfield — which is inside `K` | | `[LK:Q]` | `6 × 16 / 2` | **48** | | class size | 2 | the 3-cycles | | **`δ`** | **`2/48`** | **`1/24`** | The containment test is the only field theory imported rather than computed: `Q(√m) ⊂ Q(ζ_n)` exactly when `|disc(Q(√m))|` divides `n`. Everything else — the discriminants, the subgroup lattice, the degrees — is arithmetic done here. ## The compatibility could have failed, so both sides are shown The support condition is *identity on `K`* and *a 3-cycle on `L`*. Those are conditions on two different fields, and they are only jointly satisfiable if they agree on the intersection `F₀`. `25` says a 3-cycle fixes `F₀`, and that is exactly right — but a check that has only ever succeeded has not been tested: | element | sign | trivial on `F₀`? | consequence | | --- | ---: | --- | --- | | 3-cycle | `+1` | **yes** | compatible, class size 2, `δ = 1/24` | | transposition | `−1` | **no** | contradicts the identity on `K`; class empty, **`δ = 0`** | `F₀` is the fixed field of `A₃`, so the action on it is the sign character. The 3-cycles are in `A₃` and act trivially; the transpositions are not and do not. If the corpus's support condition had asked for a transposition, the density would be zero and the whole family would be empty — so this is the step the derivation most needed and least showed. ## Three routes, one number * **this derivation**: `δ = 2/48 = 1/24`; * **RUN-018's formula**: `δ = e_E / (3[K_E:Q]) = 2 / (3 × 16) = 1/24`, from the multiquadratic degree `[K_E:Q] = 16` and `e_E = 2`; * **RUN-009's measurement**: `0.99900 × 1/24` over primes below `2×10⁷`. RUN-018 also explained the redundancy the three-cycle condition creates — `e_E = 2` is exactly why the density is `1/24` and not `1/48` — and this round recovers the same 2 as a conjugacy-class size. The two derivations meet at the same integer for the same reason, from opposite ends. ## An observation, recorded and not explained `|disc(F₀)| = 696`, which is the conductor of `E`. The odd part agreeing is structural — the 2-division field ramifies only at bad primes, and the odd bad primes are 3 and 29 — but the 2-part agreeing exactly is not explained by anything in this tree. It is recorded as an observation in the gate's log rather than as a finding. ## The drill **176 defects, 176 caught by the check named for each**, 0 uncaught, 0 caught by the wrong check, 32 controls undisturbed, over 72 checks. The 4 planted for this gate, each turning `chebotarev-audit` red and nothing else: | planted defect | went red | | --- | --- | | every quadratic field is called a subfield of Q(zeta_24) | `chebotarev-audit` | | the support condition asks for a transposition | `chebotarev-audit` | | the squarefree part is not extracted | `chebotarev-audit` | | every subgroup is counted as normal | `chebotarev-audit` | ## What this round does not claim * **Chebotarev's theorem is cited.** What is computed is the field data the theorem consumes — degrees, the intersection, the class size — and the resulting density; the theorem itself is not proved here. * **`Q(√m) ⊂ Q(ζ_n) ⟺ |disc| ∣ n` is imported**, being the quadratic case of Kronecker–Weber. Everything else in the chain is computed. * **`L ∩ K = F₀` uses that `K` is abelian**, which is true because `K` is a compositum of abelian extensions; the gate records the reason but does not verify abelianness by computation. * **RUN-009's `0.99900` is a measurement over a finite range**, not a proof of the density; the agreement is evidence that the derivation and the count are about the same set. * **The `|disc(F₀)| = 696` coincidence is an observation**, not a claim that it generalises.