# 04 | External Research Pathway Matrix

| Pathway | Strongest Natural Output | Cumulativity | Common Bottleneck | Phase 1 Verdict |
|---|---|---:|---|---|
| Gross–Zagier–Kolyvagin | rank $0/1$ weak BSD | High | Does not extend to higher ranks | Baseline |
| p-adic zeta / Iwasawa | $p$-part, rank $0/1$ strong form | High | local hypotheses, all primes | Green light |
| Strong-BSD twist families | Infinite family theorem | Very High | applicability criteria | **Top Choice** |
| p-converse | Selmer rank $\Rightarrow$ analytic rank | High | residual/local conditions | Green light |
| generalized Kato / higher GZ | Higher rank bridge | Med-High | non-vanishing, uniformity | Yellow light |
| exact computational BSD | Finite set full proof | Very High | saturation, $\Sha$ exactness | Green light |
| Selmer arithmetic statistics | rank/family distribution | High | Does not rule out universal exceptions | Auxiliary |
| numerical BSD atlas | evidence / anomaly detection | High | Not equivalent to proof | Data layer only |
| Lattice rank convergence | Theoretically attempts to link rank and zero order | Low | Hidden equality, discontinuity of discrete quantities | Red light |
| Faithful certificate frontier | Global research control | High | Does not provide BSD theorem | Control layer |

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# 1. Top Choice: Twist-family reproduction

Reasons:

1. 2024 external theorems already exist;
2. 2026 algorithmic papers already exist;
3. LMFDB conductor $\le500{,}000$ range is complete;
4. Results can be replayed item by item;
5. Agents can compile hypotheses into predicates;
6. Both successes and failures can build an applicability atlas.

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# 2. Higher Rank Pathway

Do not immediately demand "proving rank $2$ BSD".

First ask:

- Which higher Kato class must be non-zero?
- Which Selmer rank equality is known?
- Which $p$-converse is applicable?
- Which of the regulator, points, and descent for rank $2$ are already exact?
- Which prime part of $\Sha$ remains unknown?

The output is a dependency DAG, rather than a proof pretending to be closed.

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# 3. Representation Escape

For each candidate BSD bottleneck, test at least:

- complex $L$-function;
- $p$-adic $L$-function;
- Selmer group;
- Euler system;
- Heegner / Kato classes;
- descent;
- Iwasawa main conjecture;
- twist family;
- explicit computation.

If a certain "difficulty" exists only in a single representation, it cannot be called a global bottleneck.

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# 4. Initial MCDM Evaluation

$$
\mathfrak D_{\mathrm{BSD}/\mathbb Q}
\approx
(G5,U3,X2\text{--}X3,P4\text{--}P5).
$$

Where:

- $G5$: Multiple main methods lead back to higher ranks, $\Sha$, and all primes;
- $U3$: No complete globally closed pathway;
- $X2$–$X3$: Curve exceptions can be faithfully enumerated, but theorem coverage is still often family/statistical;
- $P4$–$P5$: Certificates, data, and theorem applicability are highly cumulative.

This is a research routing evaluation, not a mathematical theorem.