auto: 2026-07-24T18:34:02Z [skip ci]
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<title id="mark-title-growing-radius">The same radius-2 ball with a soft circumscribing arc, a reverse-shooting orbit of growing dots leading from a terminal wall at right back toward the root, and a printer's signature stub beneath</title>
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<desc>A frontispiece mark for "From Path Tubes to a Near-Critical Domination Bound" — the narrative companion to the preprint. The central figure is the same radius-2 ball in the 3-regular tree used by the preprint mark, so the two marks read as a matched pair. To the right of the ball, a small vertical bar marks the terminal parameter; from there, a sequence of dots grows leftward-and-downward toward the root, each successively larger than the last, encoding the reverse-shooting orbit that reconstructs the stationary point from a terminal parameter. Beneath the ball, a small printer's signature stub with an open circle: the piece's whole point is auditable code as part of the mathematics, and the stub is the visual echo of the SHA-256 signing hash the paper insists on. The mark says: here is the ball, here is how it was found, here is the artifact of the process.</desc>
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After Width: | Height: | Size: 3.6 KiB |
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@ -26,7 +26,7 @@ peer-status: unreviewed
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result-shape: mixed
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---
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This page is a companion to the preprint [**Near-Critical First-Moment Lower Bounds for Growing-Radius Domination in Random Regular Graphs**](/papers/growing-radius-domination-paper.pdf) — [the full theorem-and-proof form is here](/essays/near-critical-growing-radius-domination.html). A complete, runnable version — [`growing-radius-domination-demo.py`](/papers/growing-radius-domination-demo.py) — ships alongside this page, together with [the CSV of diagnostic output](/papers/growing-radius-domination-demo-output.csv) it produces.
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This page is a companion to the preprint [**The Annealed Critical Window for Growing-Radius Domination in Random Regular Graphs**](/papers/growing-radius-domination-paper.pdf) — [the full theorem-and-proof form is here](/essays/near-critical-growing-radius-domination.html). A complete, runnable version — [`growing-radius-domination-demo.py`](/papers/growing-radius-domination-demo.py) — ships alongside this page, together with [the CSV of diagnostic output](/papers/growing-radius-domination-demo-output.csv) it produces.
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The main theorem is the following. Fix a degree $d\ge 3$ and let
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<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 280 280" role="img" aria-labelledby="mark-title-near-critical">
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<title id="mark-title-near-critical">A radius-2 ball in the 3-regular tree with a soft circumscribing arc, above a horizontal threshold ruler marking the near-critical whisker between log B_h and log B_h minus 2 log log B_h minus W_h</title>
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<desc>A frontispiece mark for "Near-Critical First-Moment Lower Bounds for Growing-Radius Domination in Random Regular Graphs." The central figure is a radius-2 ball in the 3-regular tree — root at the bottom, three children at the first level, six grandchildren at the second level (B_2 = 10 for d=3), enclosed by a soft circumscribing arc. Above the ball, a horizontal ruler runs from left to right with two calibrated tick marks: the leftmost tick at the theorem's reachable coordinate log B_h - 2 log log B_h - W_h, and the rightmost tick at the predicted critical coordinate log B_h. Between them, a small horizontal bracket marks the diverging whisker W_h — the bounded critical window the theorem cannot close. The mark says: here is the ball, here is where the theorem stops, and here is what remains open.</desc>
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