# RUN-008 — the half with no evidence: 36,687 kept curves, and what the base actually is **Arm:** 數學戰士「墜衡」 / AMRAL Research Lab **Subject:** the curves Algorithm 1 **kept** — the direction the census's own checks are structurally blind to — and the base's own selection criterion **Tools:** `src09_kept_curves_removal_gate.py`, `src08_modular_curve_confirmation.py` **Logs:** `src09-kept-curves.json` **Result: all 110,061 isogeny determinations on the kept curves decided, zero hits, zero undecided. With RUN-007 that closes Algorithm 1's removal criterion in both directions over the entire 40,749-curve base — 122,247 determinations, no disagreement anywhere. Two things came out of doing it that the pass itself does not say: the base is not one population but two, split exactly by 2-torsion; and RUN-007's agreement tested nothing about the twist-invariance it relies on, because the base contains no twist pairs at all.** --- ## The direction with no evidence in it Every round so far audited the 4,062 **removed** curves. That is where the evidence is — each removed curve carries a row saying why. RUN-007 named the other half and left it: > an error in that direction is the one the census's own > `algorithm1_all_removed_explained` check can never see: a missed isogeny > wrongly *keeps* a curve, and a keep leaves no evidence to audit. Algorithm 1 removes a base curve for a rational 3-, 5- or 7-isogeny, or for `|a₃| = 3`, and reports `UNEXPLAINED = 0`. The contrapositive is a claim about every survivor: > **kept ⟹ no rational 3-, 5- or 7-isogeny, and `|a₃| ≠ 3`.** 36,687 curves, 110,061 isogeny determinations, and not one row in any results file. `algorithm1_all_removed_explained` cannot reach them: it audits removals, and a wrongly-kept curve produces no removal to audit. The check is sound and it is blind in exactly one direction. ## The result | | kept but **has** a rational isogeny | confirmed none | undecided | | --- | ---: | ---: | ---: | | **n = 3** | **0** | 36,687 | 0 | | **n = 5** | **0** | 36,687 | 0 | | **n = 7** | **0** | 36,687 | 0 | `|a₃| = 3` among the kept: **0**. `j = 0` or `1728`: **0**, and the carve-out branch is reachable — `27a1` and `32a1` trigger it. Together with RUN-007's 12,186 determinations on the removed curves, Algorithm 1's criterion is now verified in **both directions across the whole base**: **122,247 isogeny determinations, zero disagreements, zero undecided.** ## Three things measured rather than read **The partition.** `kept ⊔ removed = the old base` was recomputed here from the label file and the census CSV — 36,687 ⊔ 4,062 = 40,749, no overlap, arithmetic present for every kept curve — rather than taken from the package's own `checks` block. A package's self-audit is a claim about the package. **The factorisations.** `j`'s denominator reaches **48 digits** among the kept curves, past what Pollard rho reaches in any reasonable time. It never had to: `den(j)` divides `Δ`, and for a minimal model the primes of `Δ` are the primes of the conductor — which the package supplies as `discriminant_valuations`. So the factorisation is handed over. This gate reconstructs `∏ p^{v_p}` and compares it against the `Δ` it computes from the a-invariants itself, and refuses the shortcut on any curve where the two differ. > **40,749 agree, 0 disagree.** Not one curve needed a general factorisation. A supplied factorisation that has been checked is data. One that has not is an assumption wearing a number's clothes — and it would have silently shrunk the candidate set, turning an unfinished search into a false "no rational point." **a₃**, straight off the minimal model. ## The base is two populations, not one The base is not "the elliptic curves of conductor below 500,000." It is: | source | curves | rational 2-torsion | undecided | | --- | ---: | ---: | ---: | | `Zha16_no_2_tors` | 37,002 | **0** | 0 | | `CLZ20` | 3,747 | **3,747** | 0 | A rational 2-isogeny is exactly a rational 2-torsion point, and `X₀(2)` has the same shape as the others — `j = (t+256)³/t²`, with `N(0) = 2²⁴` again a pure power of `n`, and `E[2]` unchanged by a quadratic twist because `χ_d` lands in `{±1}` and `−1 = +1` in `F₂`. So the same complete method decides it. The first label is exactly true: **zero** of its 37,002 curves has 2-torsion. The second is the perfect complement: **every one** of its 3,747 does. Nothing is wrong here — but the base is a union of two disjoint families defined by their 2-part, and only one of them says so in its name. Any argument that treats the 40,749 as a uniform population is wrong on 9.2% of it in a way the labels do not announce. Two further properties of the population, both exact and neither obvious: * **No curve in the base has an integral `j`.** All 40,749 have potentially multiplicative reduction at some prime — a real constraint, and one worth knowing for the Phase 2 line, which is about non-semistable families. * **All 40,749 `j`-invariants are distinct.** ## What RUN-007's agreement did not test That last line is not a curiosity. RUN-007 and this round both decide isogenies from `j` alone, and the step that licenses it is Phase 2's `03_Quadratic_Twist_Invariance_Bridge` Candidate Lemma A — irreducibility of `ρ̄_{E,p}` is invariant under twisting by a character. If that failed, two curves sharing a `j` could differ in isogeny status, and a `j`-only method could not match a per-curve source on both. The base contains **zero** pairs of curves sharing a `j`. So: > **RUN-007's 12,186 agreements tested nothing about twist invariance.** The > method used it as a proven input, and the population gave it no opportunity to > fail. This does not weaken RUN-007 — Lemma A is standard representation theory, cited rather than claimed — but "the j-only method matched a per-curve source everywhere" would be worthless as *evidence for* twist invariance, and it is worth saying so before anyone reads it that way. This arm's own work gets the vacuity check too. ## An agreement running the other way RUN-007 derived, independently, why RUN-006's method stops at `n = 3`: for `n ≥ 5` a Galois-stable subgroup has `(n−1)/2` distinct x-coordinates and Galois need only preserve the set, so an isogeny is a rational *factor* of `ψ_n`, not a rational root. Phase 2's `04_Local_p_Isogeny_Kernel_Criterion` states the same thing at the local level: `x(P) ∈ Q_p ⟺ λ² = 1`, so the kernel polynomial has a linear factor exactly when the kernel character is quadratic — which for `n = 3` is automatic, the kernel polynomial being linear already. The corpus already contained the obstruction RUN-007 spent a round rediscovering. Recording it as corroboration in both directions: the derivation was reached without the document, and the document says it first. ## What this round does not claim * **Nothing about BSD.** * **It verifies that Algorithm 1 applied its stated criterion correctly, in both directions. It does not verify that the criterion is the right one** — that removing isogeny-carrying and `|a₃| = 3` curves is the correct thing to do is the census's design, not its arithmetic. * **It says nothing about the twist maps**, `247,391` admissible twist pairs, or Algorithm 2. Those are separate outputs with separate evidence. * **The 2-torsion table is a fact about the base's composition, not a defect.** No claim is made that the census intended, or should have intended, a uniform population.