Closed-Form Figure Placement — Training-Free Document Layout

Abstract

Where should a figure go? Most layout engines either drop figures at the end of a document, spread them mechanically, or anchor them to the first place the text says "Figure 1". We ask whether the *document's own statistics* can decide, in closed form, without any training. In the spirit of Reverse Synthetic Networks — where the data synthesises the model directly — we synthesise each figure's position from three statistics of the manuscript itself: the TF-IDF affinity between the figure's caption and each block of running text, the proximity to any explicit lexical reference, and a small set of layout priors. There is no loss function, no gradient, and no learned parameters. On a controlled 600-document corpus (1,903 figures) drawn from the Aitele Commons' research domains, the closed-form placer reaches 92.5% top-1 placement accuracy and a mean position error of 0.18 sections, against 60.9% for a reference-only baseline and ~22% for content-blind baselines. The decisive result is on unreferenced figures — those the prose never names: here the closed-form affinity recovers 85.1% of placements, where the reference-only method collapses to 21.7% (essentially chance). Closed-form caption–text affinity, in other words, supplies the reference the author forgot to write. The same machinery groups related figures into multi-panel composites (pair-F1 0.99, 97.7% of groups recovered exactly), assigns column-spanning width for a balanced two-column page, and — extended to in-text citations — places each reference at the sentence it supports with 90.6% accuracy (vs ~15% for position-based baselines) and splices the grouped markers into the prose. The library ships with a 34-test suite (unit, edge-case, randomised-invariant, determinism, and a 1,200-block stress test). We report the method, the experiment, and its limitations honestly: the corpus is topically separated, the metrics are layout proxies rather than human aesthetic judgement, and page-overflow is out of scope.