Gaps in the Chromatic Spectrum of Face-Constrained Plane Graphs
Why this work is in the frame
A frame that forgets how it found something cannot be audited. These are the routes that admitted this work.
Bibliographic record
Abstract
Let $G$ be a plane graph whose vertices are to be colored subject to constraints on some of the faces. There are 3 types of constraints: a $C$ indicates that the face must contain two vertices of a $C$ommon color, a $D$ that it must contain two vertices of a $D$ifferent color and a $B$ that $B$oth conditions must hold simultaneously. A coloring of the vertices of $G$ satisfying the facial constraints is a strict $k$-coloring if it uses exactly $k$ colors. The chromatic spectrum of $G$ is the set of all $k$ for which $G$ has a strict $k$-coloring. We show that a set of integers $S$ is the spectrum of some plane graph with face-constraints if and only if $S$ is an interval $\{s,s+1,\dots,t\}$ with $1\leq s\leq 4$, or $S=\{2,4,5,\dots,t\}$, i.e. there is a gap at 3.
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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 | 0.003 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.002 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.003 | 0.000 |
| 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 it