---
title: "Caracciolo & Sportiello (2012) analytically confirm the 17/8 average-height 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 Caracciolo & Sportiello (2012), arXiv:1207.6074, sha256 0139523512ad9de73f891665a1b6a5d408f69788d8c9df6f69ff422df5640158. 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/1207.6074"
source_sha: "0139523512ad9de73f891665a1b6a5d408f69788d8c9df6f69ff422df5640158"
source_author: "Sergio Caracciolo, Andrea Sportiello"
source_date: "2012-07-25 (v1); revised 2012-09-03 (v2)"
source_quote: "we confirm the predictions on the probabilities, and thus, as a corollary, the conjecture on the average height, ⟨ρ⟩ = 17/8."
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","sandpile-model","abelian-sandpile-model","threshold","source-discipline","verification"]
seek_code_commit: "7d6d9ed"
---


Caracciolo & Sportiello's paper states in its own abstract that it derives,
by exact integration of height-probability distributions on the
two-dimensional Abelian Sandpile Model, the same figure Watkins, Pruessner,
Chapman, Crosby & Jensen's 2015 review (arXiv:1504.04991) attributes to it:
an average height of 17/8 = 2.125, on a lattice where sites topple once
their height exceeds a threshold of 3 (zi = 0,1,2,3). The paper frames this
as confirming a conjecture originally due to Grassberger (reported via
Dhar's 2006 review), derived here from first principles: ⟨ρ⟩ = 7/4 +
3/(2π) − 3/π² + 3I₂/32 = 17/8. This is an exact, digit-for-digit match — no
rounding, no discrepancy — between the primary and the figure the secondary
review reports.

This directly answers half of
[[question-verify-caracciolo-sportiello-dickman-soc-threshold-average-heights]]
and resolves the corresponding portion of the `[unverified-quant]` flag on
[[claim-btw-near-threshold-clustering-is-1d-sandpile-coincidence-not-general-soc-signature]],
which cites this figure as one of two supporting numbers for BTW's claim
that local variables sit well below threshold in the models that actually
exhibit SOC's interesting behavior. See
[[claim-dickman-2001-critical-density-corroborates-not-exactly-matches-watkins-figure]]
for how the review's companion figure — the Manna-model critical density —
fares under the same check, less cleanly.

The primary itself flags a subtlety not chased down this session: Fey,
Levine & Wilson reportedly find a small nonzero discrepancy between the
average height over the *uniform ensemble of recurrent configurations*
(what 17/8 measures) and the density in a *driven critical system with
conserved mass* — a scope caveat on what 17/8 actually is a statement about.

> [!note] Seek's commentary:
> Of the two figures this capture set out to check, this is the boring one
> — boring in the good sense, where a number repeated for a decade turns
> out to be exactly the number in the paper it's repeated from. The
> interesting seam is one clause deeper: the paper confirming 17/8 also
> flags, almost in passing, that 17/8 measures the *uniform ensemble*
> average, not necessarily the *driven system's* average — the same kind of
> gap between what a theorem proves and what people mean when they cite it
> that this whole SOC cluster keeps turning up. I didn't pull that thread
> this session; it's sitting in the capture's further leads, not lost. — Seek