01 · The thesis
Pieta evolved. Now it solves problems.
Pieta started as an invention engine for patents — generating structurally novel ideas grounded in real prior art, scoring them, packaging them. Over 18 months and 67+ attorney-ready packages across Metsigen, Cooling, the EUV lithography stack, and others, its substrate hardened. The same machinery now points at a 7-Millennium-Problem, BSD. The platform did not change. Its job did.
“BSD over ℚ is a Millennium Problem. We work the function-field analogue — where the theory is already theorem — which is exactly why it's the right place to build and prove a certification pipeline before attacking the harder case.”
02 · What BSD actually says
The conjecture, correctly stated.
For an elliptic curve E over ℚ, BSD links the rank of its Mordell–Weil group E(ℚ) to the order of vanishing at the centre of its Hasse–Weil L-function:
| Statement | Form | Status |
|---|---|---|
| Weak BSD. E(ℚ) is infinite ≡ ords=1 L(E, s) ≥ 1 | Existence-of-rational-points ↔ order of vanishing | SOURCED — known |
| Strong BSD. ords=1 L(E, s) = rank E(ℚ); leading coefficient = |Sha| · Reg · Ω · ∏cv | Equality; all invariants specified | OPEN over ℚ |
| Function-field analogue. Same two statements, over 𝔽q(t) | Kato–Trihan: theorem, conditional on ℓ-primary Sha finiteness | PROVEN (Kato–Trihan, Inv. Math. 153, 2003) |
We state the conjecture and its function-field analogue exactly. No theorem over ℚ is implied here, stated or unstated.
03 · Four foundation stones — laid, verified
The work that exists on disk today.
Every item below has a hash-committed artifact, a pre-registered bar, and a published tripwire record. This is the foundation, not a projection.
A function-field certification engine — tested and passing
An exact L-polynomial generator for elliptic curves over 𝔽q(t) passed its certification pilot 20/20: independent brute-force point counting matched the generator, rank agreement was perfect across the batch, and Wachs's published Example 3.1 reproduced exactly (S1 = −22, L = Φ102). It is the core of the warranted-claim service we currently run.
A ground-truth pilot that told the truth twice
The in-domain held-out AUC reached 0.984 — near-perfect rank prediction from L-function features within the certified family. The naive transfer to ℚ failed with exactly the same rigor: AUC 0.5451 ≤ 0.55, H0 by pre-registered rule. The diagnosis: rank encodes differently in the two worlds — cyclotomic multiplicity here, parity coherence there. We published both verdicts with equal rigour.
A self-auditing method — the scar map
The system's real invention may be its honesty machinery: numeric kill-bars written before every experiment, RUN RECORDs that close only on executable evidence, three independent adversarial readers auditing every idea batch (one reader was caught rewarding costume vocabulary over mathematics — r = +0.33 vs +0.12 — and the best ideas ended up ranked last, by its own score), planted-fake shadow sets, an import-provenance doctrine that forced us to declare every borrowed theorem's transfer condition (one idea collapsed on this exact failure; the lesson is banked), and a public inventory counter that went down when our own audit overturned a win.
A diagnosed monoculture — with the fix designed
The generator, optimising toward its own judge, converged to a single template (93% of output). We diagnosed the cause: the reward, not the fuel, was the bottleneck — the loop found its own optimum. The redesign — judge v3 with leverage + family-novelty + movement, plus a known-theorem detector and a research-before-anchor knowledge decomposition — is specced and waiting on a build slot.
04 · What we found in the data
Three claims, carefully narrowed.
Each result below is qualified, pre-registered, and reproducible from the underlying artifact. Each is also explicitly bounded — the bounds belong to the claim.
4.1 — The λ-strata law (in-family)
The certified curves organize into exactly 113 cyclotomic strata, and within this family a sharp law holds: rank equals the cyclotomic type of Lunit(z) — 2,937 / 2,937. Rank stops being a hidden number; it is readable from the L-function's factorisation. IN-FAMILY ONLY — verified within the certified 𝔽q(t) family
4.2 — Murmuration detection at certified scale
Murmurations — the oscillating patterns in elliptic-curve data discovered over ℚ (He–Lee–Oliver–Pozdnyakov 2022, proved for modular forms by Zubrilina 2025) — had never been measured on certified labels. We ran the detection on our corpus:
| Place degree d | z (rank-0 − rank-1) | Permutation p | Bonferroni ×4 | Note |
|---|---|---|---|---|
| 1 | −1.78 | 0.074 | 0.296 | Flats — 37.1% atom-dominated; can't oscillate |
| 2 | +2.69 | 0.007 | 0.028 | Survives permutation calibration |
| 3 | +0.55 | 0.585 | 1.0 | Flat |
| 4 | +0.00 | 0.999 | 1.0 | Scaling model over-predicts (10.22 expected) ⇒ naive model rejected, not the data |
DETECTION, NOT REPLICATION — we explicitly do not claim replication at Wachs's published scale (~5 × 105). What we claim is the first two-witness certified murmuration detection (bounded priority claim). The pre-registered z ≥ 15 replication bar was not met.
4.3 — Theorem-grade rank, unbounded
Over ℚ, the highest elliptic-curve rank unconditionally certified is 20 — the field's leaderboard records exact rank only up to 20. Using Legendre/Kummer towers and the Conceição–Hall–Ulmer closed-form rank theorem, we built a truth base where rank is proven by formula — reaching rank 59,048 at the p=3, f=10 tower level, with towers extending unboundedly at higher levels (9,765,624 and ≈ 4 × 107 at p=5 and p=7 by the same formula). This is the rank-≥-2 training-and-verification ground that did not exist for ℚ. CONTROLLED — 4/4 published-known-answer controls pass
05 · The portfolio of certified assemblies
One certified spine, seven configurations.
The corpus and the pipeline do not ship as a single product. They ship as a set of instruments, each tuned for a different use case.
The Certifier — flagship
Reality gate + tested generator + calibration loop + adversarial verification harness. Degrades gracefully: if the model fails its gate the honest no-model configuration ships.
The Atlas
Family-level maps of where certification works and what it costs. Honest about gaps, no false coverage.
Triage pipeline (policy)
Never spend expensive exact computation where cheap verified signals suffice. Written down, not a habit.
Rank-Proxy Validator
Audit other teams' rank heuristics against exact strata. Independent second witness on heuristic-only claims.
Descent Auditor
Independent Selmer cross-checks on published descent computations. Bounds the failure modes in any team's rank-via-2-descent claim.
Graph-Spectral Screen
Isogeny-graph spectral health. Cheap first screen; expensive follow-up only where the screen is silent.
Conditionality-Ledger Audit
A formal review service for any team's published claim-of-conditionality. Reads each cited claim against the T-66 led
06 · The honest ledger
What we ran, what we wrote down, what we killed.
A certification body that hides its errors is a contradiction. The failure record is the product.
| State | Item | Detail |
|---|---|---|
| VERIFIED | Citation accuracy — 14-row quarantine ledger | Common-error typology logged across every artefact re-fetched: misspelled-author camouflage (e.g. Kato–Trihan as the misspelled namesake), journal-shadow duplication, unverified venue claims. Every caught instance rewritten against the verified SOURCES ledger before publication. |
| VERIFIED | Upper witness = rank on 5 / 5 rank-2 specimens | Prior pass with pre-fix generator mismatched (rank 0) — caught by functional-equation tripwire, halted before contamination. SOURCED sha 99895bd2ff57... |
| VERIFIED | CHU rank theorem controls — 4 / 4 pass | Known-answer tower ranks reproduced before any data is run. |
| VERIFIED | 2,937 / 63 quarantine | Quarantine is named, not silent. The 63 is the pre-declared degree-4/5 Iₙ-at-degree-d stratum. |
| FAILED | Naive ℚ-transfer (H0) | AUC 0.5451 ≤ 0.55 by pre-registered rule. Diagnosed, not refit. |
| FAILED | Matrix-completion recovery claim | Full-rank matrix ⇒ theorem inapplicable. Precondition gate now standard. |
| FAILED | Local epsilon decomposition | Scored 0.68 against its own pre-registered 0.96 bar. Honest re-scoring is now doctrine. |
| FAILED | The assembler (4 pre-registered failures) | Auto-parked per its own kill clause; revival key written; revisit policy documented. |
| VERIFIED | Calibration loop — 950be9 | RUN COMPLETE 2026-09-29. T1 (L-derived trace map) cleared all four pre-registered bars (rank-≥-1 / rank-≥-2 / margin / null); pure-geometry H0 banked (T3 row). Service gate opened; result committed as RUN_RECORD_950be9.md (sha 905bbebd380e…). |
| FLIGHT | Anchor v1.4 (with the landscape-knowledge gate in front) | Prevents named-equation camouflage recurrence by construction. |
| FLIGHT | Batch-3 with diversity-guarded judge | The reward, not the fuel, was the bottleneck — judge v3 resolves the monoculture. |
07 · What comes next
The queue, plainly.
Nothing here is a result. Every item is a stated direction.
Cross-family λ-strata
Test whether rank = cyclotomic type survives on Ulmer towers, where ranks mix and the law has never been tested. Instrument under construction.
Spectral characterisation
Promote the mode-2 oscillation from a detection in a single family to a candidate family invariant — if it survives cross-family.
Deg-7 family extension
Extend the certified corpus past degree-5. Requires extending the trace precomputation out to higher maxima — non-trivial, gated.
Certified Murmurations II
The v2 paper with the cross-family result, written only if the cross-family law survives. v1 is the foundation; v2 is the test.
ℚ-lane certification
The schema already renders the ℚ boundary honestly — ANALYTIC-INFERRED rather than UNCONDITIONAL_IN_FAMILY. The machinery is the long-term target, not an overclaim.
08 · Live, beta
A two-witness certificate — what it looks like in production.
For a curve in the certified 𝔽q(t) family, you ship with two independent proofs and a full provenance chain.
Constructive
Explicit independent points on the curve proving rank ≥ r. Independently verifiable; the points survive recomputation from the committed generator.
Independent from the lower
Order-of-vanishing bound computed from point counts via Honda–Tate. Proves rank ≤ R. Validated 5 / 5 on rank-2 specimens and caught a generator bug pre-contamination.
SHA-256 binding
Every certificate binds data, code, and engine version into a single hash. If a bug is found tomorrow, affected certificates can be precisely recalled — a revocation semantics that survives independently of the issuer.
Honest about the boundary
Each certificate carries a provenance flag — UNCONDITIONAL_IN_FAMILY (the 𝔽q(t) regime) or ANALYTIC-INFERRED (the ℚ regime, rank ≤ 1 only). The schema ships the limit with the claim.
Pricing (founding-beta, first ten buyers): $75 · $120 · $160 for GENERIC · ISOTRIVIAL-CM · ULMER-TOWER respectively. Both demo certificates are free, permanently — including one deliberately honest failure demo over ℚ, so the boundary is visible before any commitment. BETA
09 · The one sentence
Why this matters — straight.
In the neighborhood of BSD, statements whose verification travels with them are rare. What we ship is exactly that: a warranted-claim service that says “this rank claim is certified, here's the engine version, the evidence chain, and the exact boundary where the guarantee holds” — over function fields today, with the ℚ-side gap mapped honestly rather than hand-waved. The same infrastructure — Ground Engine, verification ritual, honest-failure doctrine — is engineered to generalise to any domain where claims need warrants. The methodology is the product. The certified corpus is the ammunition. The paper is the proof the machine works.