# RUN-019 — does the witness-network generalisation admit anything, and the corpus's own stop rule applied to this arm **Arm:** 數學戰士「墜衡」 / AMRAL Research Lab **Subject:** `31_Witness_Network_Criterion_v0.2` and `00_GCD_Witness_Lemmas` — the gcd lemmas, the leave-one-out condition, and whether "strictly stronger" is non-empty **Tools:** `src22_witness_network.py`, `src11_gate_drill.py` **Logs:** `src22-witness-network.json`, `src11-gate-drill.json` **Result: the generalisation is not vacuous, and the gap is larger than the document's own sentence claims. Over 4,063 small models meeting the certificate's arithmetic conditions, `30` reaches 3,925 and `31` reaches all 4,063 — 138 models, 3.4%, that only `31` can phrase. Every one of the 138 fails the same hypothesis, (T5), and 100 of them do so while `31`'s exceptional set stays empty. Of the 38 that carry an odd exceptional prime, 31 were testable on the analytic side, 14 have analytic rank 0, and 13 of those also satisfy (T1)'s `ord₂ L^alg(E,1) = 0` — so the set of curves meeting the *whole* certificate that `30` cannot state is non-empty, not merely the arithmetic half. Lemmas 2.1/2.2 hold on 203,745 instances with no failure; the leave-one-out condition itself fails at 2,457 fixed primes over the base, which is a real cost of the generalisation and is reported as one.** --- ## What `31` changes `30`'s certificate pins `v_ℓ(Δ_E) = 1` at two named primes. `31` replaces that with gcd conditions on the whole multiplicative witness set: $$g_{\rm mult}=\gcd_{\ell\in\mathcal M}n_\ell,\qquad g_-=\gcd_{\ell\in\mathcal M^-}n_\ell,\qquad n_\ell=v_\ell(\Delta_E),$$ and takes `R_mult ∪ R_-`, the odd primes dividing them, as a **finite exceptional set** rather than a condition to be avoided. §3 states the point: > This is strictly stronger than requiring both gcds to be powers of 2. "Strictly" is a claim about non-emptiness, and non-emptiness is measurable. ## The lemmas Lemma 2.1 and 2.2 are elementary: "some `ℓ` has `p ∤ n_ℓ`" fails exactly when `p` divides every `n_ℓ`, which is exactly `p | gcd`. There is nothing to approximate. They are still checked rather than asserted, because the statement quietly needs `M ≠ ∅` for the gcd to exist at all, and the leave-one-out form needs `M∖{p} ≠ ∅` — and an implementation that silently returns "holds" on an empty set would pass every population test. **203,745 instances over the 40,749 base curves that have a multiplicative prime, no failure.** The gcd form and the "some `ℓ` misses `p`" form agree on 110,146 leave-one-out instances and disagree on none. ## What the leave-one-out condition costs `LOO(p)` is the condition that lets the network route a prime `p` through the witness set with `p` itself removed. It is not free: > **it fails at 2,457 fixed primes over the base population.** Where it fails, the network has to route `p` by hand. That is a fact about the population rather than about the theorem, and it is the honest counterweight to the non-emptiness result below: `31` reaches more curves than `30`, and it also has a failure mode `30` does not have. ## The gcd statistic over the base The 40,749 base curves are entirely semistable (RUN-016), so **neither criterion applies to any of them**. The gcd statistic is still computable, and this is the largest population this arm has: | quantity | distribution | | --- | --- | | odd part of `g_mult` | `1`: 40,733 · `3`: 16 | | odd part of `g_-` | `1`: 29,301 · `3`: 2,659 · `5`: 1,222 · `7`: 693 · `9`: 322 (plus 6,053 with no nonsplit prime at all) | | size of `R_mult ∪ R_-` | `0`: 35,350 · `1`: 5,313 · `2`: 86 | **13.2% have a non-empty exceptional set**, and it is essentially always `g_-` that produces it — `g_mult` has odd part 1 on 40,733 of 40,749. The exceptional set is a real cost, and it is concentrated on the nonsplit side. ## Is the generalisation non-empty — and by how much A direct search over small models (`a₁ ∈ {0,1}`, `a₂ ∈ {−1,0,1}`, `a₃ ∈ {0,1}`, `|a₄| ≤ 30`, `|a₆| ≤ 30`) for curves meeting the certificate's *arithmetic* conditions — `Δ < 0`, no rational 2-torsion, `S₃` two-division field, additive reduction only at 2, `M` and `M⁻` both non-empty — returns **4,063 candidates**. Classifying each by which criterion can phrase it at all: | | count | | --- | --- | | `30` and `31` both apply | **3,925** | | `31` only, exceptional set **empty** | **100** | | `31` only, with an odd exceptional prime | **38** | | `30` applies but the exceptional set is non-empty | **0** | The fourth row must be empty — (T4) and (T5) each supply a valuation 1, which forces both gcds to 1 — and it is measured rather than assumed. **Every one of the 138 fails on (T5), and none fails only on (T4).** (T5) asks for a nonsplit multiplicative prime of valuation exactly 1; among the 38 with an odd exceptional prime, (T5) is satisfiable for **0** of them and (T4) for 34. That identifies precisely which hypothesis the generalisation is buying out. It also shows the document's §3 sentence understates its own result. "Strictly stronger than requiring both gcds to be powers of 2" is a claim about the 38. But the 100 in the middle row have `g_-` a power of 2 — a single nonsplit prime of valuation 2 in 94 cases and 4 in 6 — so they are reached by `31` **with no exceptional prime at all**, and still cannot be stated by `30`. All 100 satisfy (T4); it is only (T5) that stops them. The gcd reformulation buys more than the sentence claims, and most of what it buys is not in the exceptional set. ## The analytic side, which is the half that could still have been empty Arithmetic non-emptiness is the weaker claim. The certificate also demands an analytic input — analytic rank 0 and `ord₂ L^alg(E,1) = 0` — and a family that met the arithmetic conditions and never the analytic ones would leave `31` with an empty domain regardless. So the 38 were run through the L-series machinery: | outcome | count | | --- | --- | | conductor above the gate's cap, not tested | 7 | | root number undecided at this truncation | 9 | | `w = −1`, so the rank is odd | 7 | | `w = +1` but `L(E,1)` vanishes — rank ≥ 2 | 1 | | **analytic rank 0, numerics consistent** | **14** | | **…and of those, `ord₂ L^alg(E,1) = 0`** | **13** | The three negative outcomes are kept apart on purpose. An undecided root number is not evidence of positive rank; it says only that this gate did not resolve the sign at 4,000 terms, and collapsing those nine into "rank ≥ 1" would be a free result. The one curve that reaches rank 0 and still fails the certificate is worth naming. `[0,−1,0,0,−27]`, conductor 17,592, has `c₂ = 2` and `L/Ω = 2`. Its BSD quotient is 1 — odd — while the quantity (T1) actually names is even. Testing the quotient instead of `L^alg` would have counted it, and the drill now plants exactly that defect. **This round does not compute Ш.** The quotient `L·|E_tors|²/(Ω·∏c_p)` is used only as a consistency test on the numerics: a value that is not a positive square integer means the period, the L-value, the torsion bound or a Tamagawa number is wrong, and the row is not counted. Calling it an order of Ш is the substitution the corpus's own stop rule names, and the next section is about that. ## The stop rule, applied to this arm Phase 0's `00_BSD_Global_Enclosure_Consensus` §6 freezes any route that for three consecutive rounds only increases numerical precision, restates BSD, renames the same gap, replays the same formula on a new curve, or **substitutes analytic Ш for actual Ш** — without new theorem applicability, a new exact certificate, a new excluded domain, a new family, or a barrier escape. The rule is written for the research line. It applies to the verification arm too, and nobody else is going to run it against me. **RUN-017's headline was of the forbidden kind.** "285 curves close on `#Ш = 1`" is a formula replayed on new curves, and the `#Ш` in it is the analytic order. Both patterns, in one sentence. It was reported as a result, and under §6 it is not one. The three rounds since: | round | §6 verdict | | --- | --- | | RUN-017 | the `#Ш = 1` headline is **replay + analytic-for-actual**. What survives is the two errors it found in my own work — a real period wrong for three rounds, and a root-number confidence that was not one. Error-finding is not on §6's progress list either. | | RUN-018 | **new exact certificate.** `[K_E:Q] = 16` and `e_E = 2` by exact linear algebra rather than measurement, and the reason RUN-015's redundancy exists. | | RUN-019 | **new excluded domain and a new family.** 138 models `30` cannot phrase and `31` can, with the failing hypothesis identified as (T5); 2,457 primes where the leave-one-out condition fails; 13 curves meeting the whole certificate. | **The line is not frozen**, and the reason it is not is RUN-018 and this round rather than RUN-017. Recording that is the point of running the rule: the alternative is a line that keeps producing analytic-Ш rows and never notices, because the rows all look like results. ## The drill **80 defects, 80 caught by the check named for each — none uncaught, none caught by the wrong check. 19 controls, none disturbed, over 36 checks. Eleven minutes and sixteen seconds.** Gate 22 contributes three checks and seven planted defects. Two of the checks are new this round and exist because this round's **headline numbers** come out of them: `criterion-reach` covers the classifier that produces 3,925 / 100 / 38, and `analytic-classification` covers the four-way analytic verdict. Both are drilled the same way as everything else — reading (T5) against `M` instead of `M⁻`, or letting it accept any odd valuation rather than valuation one, each turns `criterion-reach` red, and so does a candidate landing in the impossible cell. One planted defect did not survive contact and moved to the controls with its reason: The runtime is the one number that went the wrong way. RUN-018 cut the drill from fifteen minutes to eight; the three new checks put it back to nineteen, because the drill runs **every** check once per defect and a 1.5-second scan inside a check is a two-minute tax on the run. That was fixed before the run that produced these numbers rather than after — the candidate list is now taken once at baseline (a scope statement, written down: neither new check can see a defect planted inside the search, and neither claims to), the analytic fixture runs at **1,200 terms because 1,200 is measured** — the smallest truncation at which all four verdicts agree with the 4,000-term run — and the inherited-contract assertion moved to the cheapest curves that produce both of its branches. 5.56 seconds per sweep became 1.13, and the drill lands at 11:16 instead of 19 minutes. Every one of the four defects still turns its named check red. > Dropping the root-number requirement from `is_analytic_rank_zero` changes no > verdict, because `src15`'s `analyse` already encodes the sign **into** the > L-value — it returns `L = None` when the sign is undecided and `L = 0.0` > exactly when `w = −1`. So "L is a non-zero number" and "decided, `w = +1`, > `L ≠ 0`" are the same predicate, and no fixture can separate them. That is worth stating rather than omitting, because it says something true about the code: the sign condition in gate 22 is **inherited, not independently enforced**. If `src15` ever returned a raw partial sum for `w = −1`, gate 22 would silently begin counting odd-rank curves as rank 0. So the check now asserts that contract directly — `L` must be `None` when the sign is undecided and exactly `0.0` when `w = −1` — and stubbing `l_value_at_one` to return a non-zero constant turns it red. ## What this round does not claim * **Nothing about BSD**, and nothing about `31`'s routing. It checks the elementary lemmas and measures whether the generalisation has anything in it. * **The search is over small models only.** Absence in that range is not absence, and the non-emptiness it finds is narrow in one specific way: **all 138 have exactly one nonsplit multiplicative prime**, so `g_-` is a single valuation and not a gcd at all. What is exercised is "a nonsplit valuation greater than 1 is allowed", not "a gcd over several nonsplit primes rescues the certificate". The second is the interesting case, and this range contains no example of it. * **The base population is semistable**, so neither criterion applies to any of its curves; the gcd statistic is computable there and nothing more. * **Ш is not computed, here or anywhere in this arm.** The BSD quotient is a numerics check, and 13 curves satisfying (T1) is a statement about `L^alg`, not about Ш. * **Seven candidates were not tested at all** — conductor above 60,000 — so the analytic tally is a lower bound on the 38, not a census of them. * **The search itself is not drilled.** `criterion-reach` and `analytic-classification` cover the classifiers that read a candidate, not `certificate_candidates`, which produces one; both take the candidate list at baseline and neither could see a defect planted inside the scan. Its conditions are each independently checked elsewhere — discriminant, reduction type and conductor by gates 16 and 18, the two-division field by gate 21 — but the scan that composes them has no drill of its own, and this round's 4,063 rests on that.