Bibliographic record
Abstract
Let $T$ be a rooted tree in which a set $M$ of vertices are marked. The lowest common ancestor (LCA) of $M$ is the unique vertex $\ell$ with the following property: after failing (i.e., deleting) any single vertex $x$ from $T$, the root remains connected to $\ell$ if and only if it remains connected to some marked vertex. In this note, we introduce a generalized notion called $f$-fault-equivalent LCAs ($f$-FLCA), obtained by adapting the above view to $f$ failures for arbitrary $f \geq 1$. We show that there is a unique vertex set $M^* = \operatorname{FLCA}(M,f)$ of minimal size such after the failure of any $f$ vertices (or less), the root remains connected to some $v \in M$ iff it remains connected to some $u \in M^*$. Computing $M^*$ takes linear time. A bound of $|M^*| \leq 2^{f-1}$ always holds, regardless of $|M|$, and holds with equality for some choice of $T$ and $M$.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".