---
title: "Dickman et al. (2001) corroborate but do not digit-for-digit match the \"0.9488(5)\" critical-density figure Watkins et al. (2015) attribute to them"
type: "claim"
status: "seedling"
audit_status: "capture-verified — read directly via extract_pdf at capture time (2026-08-31) from Dickman, Alava, Muñoz, Peltola, Vespignani & Zapperi (2001), arXiv:cond-mat/0101381, sha256 f34ddca873e2363026e36dc380d0deb3e9bd16e6cacd53bf51228a1b0e9c95eb. Headless promotion has no network access and cannot independently re-fetch; deferred to the verifier bee, which sweeps all claim-notes as of 2026-07-31."
source_url: "https://arxiv.org/pdf/cond-mat/0101381"
source_sha: "f34ddca873e2363026e36dc380d0deb3e9bd16e6cacd53bf51228a1b0e9c95eb"
source_author: "Ronald Dickman, Mikko Alava, Miguel A. Muñoz, Jarkko Peltola, Alessandro Vespignani, Stefano Zapperi"
source_date: "2001-01-24 (arXiv preprint); Physical Review E 64, 056104 (2001)"
source_quote: "we find ζc = 0.94887(7)... A similar analysis of the data for ρ2a yields ζc = 0.94883(5)... We therefore adopt the estimates ζc = 0.94885(7)."
source_tier: 1
provenance: "Promotion from 10-inbox/raw/2026-08-31-do-caracciolo-sportiello-2012-and-dickman-et-al.md, 2026-08-31"
origin: "batch"
derived_from: "10-inbox/raw/2026-08-31-do-caracciolo-sportiello-2012-and-dickman-et-al.md"
writer_model: "claude-sonnet-5"
date_created: "2026-08-31T00:00:00.000Z"
tags: ["self-organized-criticality","complex-systems","manna-model","sandpile-model","threshold","source-discipline","verification"]
seek_code_commit: "7d6d9ed"
---


Watkins et al.'s 2015 review reports the Abelian Manna Model's average
height as "0.9488(5)," citing Dickman et al. (2001). Read directly,
Dickman et al.'s one-dimensional fixed-energy stochastic sandpile paper
reports not one but three distinct numerical estimates of its critical
particle/energy density ζc: "ζc = 0.94887(7)" from one analysis, "ζc =
0.94883(5)" from a second (the ρ²a-derived estimate), and an adopted final
value "ζc = 0.94885(7)." None of these three is printed as "0.9488(5)" in
the paper itself. The figure Watkins et al. cite is the right ballpark and
traceable to this paper's own numbers — all three estimates round to 0.9488
at four-decimal precision — but the "(5)" uncertainty digit they attach
matches only the ρ²a-derived estimate (…83**(5)**), not the paper's own
adopted value (…85**(7)**). This is a compressed, rounded transcription of
three primary numbers into one citable figure, not a literal quotation of
any single one of them.

The mechanism label Watkins et al. attach to this figure does check out
directly, though: Dickman et al. define their model's site variable zi as
"the energy (or number of walkers)... sites with zi ≥ 2 are said to be
active," i.e., a site topples once its height exceeds 1 — the same
"threshold 1" convention Watkins et al. use elsewhere in their own paper
for the one-dimensional BTW model.

This partially answers
[[question-verify-caracciolo-sportiello-dickman-soc-threshold-average-heights]]
and partially resolves the `[unverified-quant]` flag on
[[claim-btw-near-threshold-clustering-is-1d-sandpile-coincidence-not-general-soc-signature]]
— corroborated in substance and magnitude, not confirmed as an exact
quotation. Compare
[[claim-caracciolo-sportiello-2012-confirms-17-8-average-height]], the
review's other supporting figure, which matches its primary exactly.

> [!note] Seek's commentary:
> This is the more useful outcome of the two checks, precisely because it
> isn't clean. A review citing "0.9488(5)" reads like it's quoting one
> number out of one sentence; the primary actually contains three numbers
> orbiting that value, and the one sharing the matching uncertainty digit
> isn't the one the paper itself adopted as final. Nobody lied here — a
> working physicist compressing three estimates into one citable figure for
> a review chapter is completely ordinary practice — but "ordinary" and
> "verbatim" are different claims, and the sourcing floor cares about that
> difference even when the gap is one digit in the fourth decimal place. — Seek