Take a top row of independent standard-exponential numbers and form the Gilbreath array of absolute differences, a(i+1,j) = |a(i,j) − a(i,j+1)|. The decay constant cn = E a(n,j) is the expected entry at depth n. This applet estimates cn by Monte Carlo — averaging the left diagonal a(n,0) over many random pyramids — and animates it converging, against the exactly-known values and Ross’s digit-sum law. A companion to the Gilbreath array explorer.
Constants and model from “Gilbreath’s conjecture: a Cramér random model and a deterministic analysis” (Chase, Hunter, Tao) — arXiv:2607.08712; exact c4–c6 and the digit-sum law cn ≈ C λs2(n)/n from M. M. Ross, “Empirical Structure of the Gilbreath Decay Constants” (Zenodo). Runs entirely in your browser.
| n | s2 | Monte Carlo cn | ± s.e. | exact | Δ |
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One random pyramid (depth 20). The outlined left diagonal a(n,0) is exactly what the Monte-Carlo average estimates.
The model (Chase–Hunter–Tao). With iid standard-exponential top row, the array is translation-invariant in j, so E a(n,j) depends only on the depth — call it cn. CHT prove Σi≤n ci ≥ log(n+e) (so the constants cannot decay exponentially), compute c0…c3 exactly, and note they cannot even prove (cn) is bounded.
The digit-sum law (Michael M. Ross). A later Monte-Carlo study to depth 8192, anchored by new exact rationals for c4, c5, c6 (computed by a sign-cone decomposition), found that the binary digit sum s2(n) modulates the leading order: cn ≈ C λs2(n)/n, with λ drifting through ≈1.14–1.20. This explains the non-monotone “sawtooth” (already c2 < c3; the exact weave continues c3 > c4 < c5 > c6). See M. M. Ross, “Empirical Structure of the Gilbreath Decay Constants” (Zenodo 10.5281/zenodo.21326026; code & exact certificates).
Method & scope. Each “simulation” draws a fresh width-(n+1) exponential top row and records the diagonal a(i,0); cn is the running mean, with the standard error shown. This is exactly the estimator in the paper’s cn.py. The browser reaches modest depths (tens), where a linear cn-vs-n plot shows the convergence and sawtooth clearly; the log–log strands of Ross’s Figure 1 only separate over the far larger range (to 8192) reachable with dedicated compute. All computation is deterministic given the seed, runs in your browser, and makes no network calls.