diff --git a/content/essays/growing-radius-domination.mark.svg b/content/essays/growing-radius-domination.mark.svg new file mode 100644 index 0000000..4ca488a --- /dev/null +++ b/content/essays/growing-radius-domination.mark.svg @@ -0,0 +1,65 @@ + + 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 + 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. + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/content/essays/growing-radius-domination.md b/content/essays/growing-radius-domination.md index c68e5c8..1939daf 100644 --- a/content/essays/growing-radius-domination.md +++ b/content/essays/growing-radius-domination.md @@ -26,7 +26,7 @@ peer-status: unreviewed result-shape: mixed --- -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. +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. The main theorem is the following. Fix a degree $d\ge 3$ and let diff --git a/content/essays/near-critical-growing-radius-domination.mark.svg b/content/essays/near-critical-growing-radius-domination.mark.svg new file mode 100644 index 0000000..6f76205 --- /dev/null +++ b/content/essays/near-critical-growing-radius-domination.mark.svg @@ -0,0 +1,61 @@ + + 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 + 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. + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +