# GPT-6 BSD Autonomous Research Stress Test

## Role

You are given a research archive concerning the Birch–Swinnerton-Dyer conjecture. Treat the archive as an **untrusted but potentially valuable research state**, not as a set of facts that you may simply inherit.

Your task is **not** to explain BSD, summarize the archive, or manufacture a theorem-shaped paper. Your task is to reconstruct the mathematically justified state of the project, identify the smallest genuine open proof obligation, and push that obligation as far as current mathematics, exact computation, and primary-source verification permit.

## Non-negotiable epistemic rules

1. Do not claim a proof or disproof of BSD unless every dependency actually closes.
2. Never promote a conjecture, heuristic, numerical agreement, project assertion, or conditional theorem into an unconditional theorem.
3. Distinguish at all times between:
   - internally asserted/project-certified input;
   - independently reproduced exact computation;
   - externally verified published/preprint theorem;
   - conditional reduction;
   - heuristic or numerical evidence;
   - open proof obligation.
4. Do not infer that a file, script, computation, theorem statement, or external source exists merely because another document refers to it. Inspect what is actually present.
5. Do not report that you executed a computation unless you actually executed it and preserve the executable artifact and result.
6. Audit theorem hypotheses, quantifiers, normalizations, coefficient rings, local conditions, rank assumptions, and direction of implication. A theorem with a nearby-looking conclusion is not automatically applicable.
7. Explicitly detect circularity, especially any route in which the target BSD statement is reintroduced as an assumption through another conjecture or comparison theorem.
8. Do not identify numerical analytic $\Sha$ with a proved finite Tate–Shafarevich group.
9. A finite database closure is not a universal theorem over all elliptic curves.
10. If the evidence does not close a statement, label it `OPEN`, `BLOCKED`, or `UNVERIFIED` rather than filling the gap with prose.
11. Do not spend substantial work recomputing already-closed finite arithmetic unless that recomputation is required to audit a dependency or resolve a contradiction.
12. Mathematical source must use only `$...$` and `$$...$$` as canonical delimiters.

## Phase I — Reconstruct the archive truth state

Read the full archive before selecting a research route.

Produce a dependency-aware chronology for every major branch. Determine which documents are current, which are superseded, and which claims remain live.

Build a claim ledger with at least these fields:

```text
claim_id
statement
status
status_type
internal_source
external_source
hypotheses
normalization
quantifiers
depends_on
reproduced_here
notes
```

Required outputs:

- `01_ARCHIVE_AUDIT.md`
- `02_CLAIM_LEDGER.md`

The audit must include:

- archive topology and chronology;
- inconsistent or superseded states;
- references to missing or unavailable reproducibility assets;
- theorem citations whose exact statement/application still requires external checking;
- every place where an apparent closure is only conditional or project-inherited.

## Phase II — Independently determine the canonical frontier

Do not let filenames, version numbers, the website index, or previous "next step" prose choose the frontier for you automatically.

Reconstruct the strongest justified dependency graph and choose the **smallest high-value open obligation** whose closure would create a real mathematical advance without merely assuming an equivalent form of BSD.

Express the chosen frontier formally as:

$$
\boxed{
\text{Verified assumptions}
\Longrightarrow
\text{precise open target}
}
$$

and also give the dependency cut separating closed nodes from open nodes.

Required output:

- `03_CANONICAL_FRONTIER.md`

## Phase III — Attack the frontier

Use current primary literature and exact computation where appropriate.

For external mathematics:

- prefer the original paper/preprint or authoritative publication page;
- verify the exact theorem statement, version/date, hypotheses, and logical direction;
- record whether the archive citation is correct, stale, stronger than the source, weaker than the source, or unverifiable.

Develop up to three genuinely distinct attack routes. Do not count renaming the same bridge as a new route.

A route can succeed by producing any of the following:

1. a new rigorously proved lemma that strictly shrinks the open frontier;
2. an exact certificate that closes a previously open finite condition;
3. a rigorous equivalence/reduction to a strictly smaller, clearly stated gate;
4. a no-go theorem eliminating an apparently promising but circular or insufficient route;
5. an invalidation or correction of an inherited project claim.

For every route, maintain:

```text
input assumptions
new statement attempted
proof/computation
external theorem dependencies
failure point or closure point
whether the frontier strictly shrank
```

If a computation is required, create runnable source plus tests and preserve exact outputs. Do not substitute pseudocode for executed evidence.

Required outputs:

- `04_ATTACK_LOG.md`
- `05_NEW_MATHEMATICS.md`
- `src/` and `tests/` if computation is used

## Phase IV — Adversarial referee pass

Attempt to destroy your own strongest result.

Check at minimum:

- hidden use of BSD or an equivalent conjecture;
- theorem-hypothesis mismatch;
- wrong implication direction;
- normalization mismatch;
- rationality/algebraicity silently assumed;
- $p$-adic versus complex comparison silently identified;
- local-to-global quantifier jump;
- finite-prime verification presented as all-prime closure;
- numerical nonzero evidence presented as integrality/primitivity;
- dependence on an unavailable artifact;
- citation version drift.

Required output:

- `06_ADVERSARIAL_REFEREE.md`

## Phase V — Final research verdict

Produce:

- `00_EXECUTIVE_STATE.md`
- `07_NEXT_PROOF_OBLIGATIONS.md`

The final verdict must be exactly one of:

```text
VERIFIED_ADVANCE
RIGOROUS_NOGO
REPRODUCTION_ONLY
INVALIDATED_PRIOR_CLAIM
INCONCLUSIVE
```

Then report:

$$
\eta_{
\mathrm{BSD}}
=
\frac{
\text{verified frontier reduction}
}{
\text{research steps and tool work}
}
$$

You do not need a universal numerical value for $\eta_{\mathrm{BSD}}$; give a transparent qualitative or ordinal assessment based on what actually changed in the dependency graph.

## Stop rule

If three consecutive research loops only restate the same obstruction, add numerical precision without changing theorem status, rename an existing gate, or move sideways to another curve without shrinking the dependency cut, stop that route and record it as frozen.

## Success criterion

A successful run does **not** require proving BSD.

The run is successful if it leaves the archive in a mathematically stricter state than it found it: fewer unjustified claims, a smaller verified frontier, a genuinely new lemma/certificate/no-go, or a substantially sharper proof obligation.