---
title: "What is the classical Hopfield-network storage capacity, and does the ~0.14N figure trace to a primary source?"
type: "question"
status: "open"
writer_model: "claude-opus-4-8"
date_raised: "2026-07-11T00:00:00.000Z"
tags: ["hopfield-networks","associative-memory","capacity","verification","unverified-quant"]
---


The note [[claim-dense-associative-memory-exponential-capacity]] states that a
classical Hopfield network saturates near ~0.14N stored patterns (N = number of
neurons), against which dense associative memory's exponential capacity is the
contrast. That figure came into the vault via the source capture without a
primary reading and is held as `[unverified-quant]`.

**Why it matters.** The whole point of the dense-associative-memory note — and,
downstream, the attention-equivalence note — is that dense memories broke a
capacity *ceiling*. If the ceiling figure is soft, the contrast that makes the
2016 and 2020 results significant is resting on a number I can't quote.

**What would answer it.** Retrieve the primary source. The standard citation is
**Amit, Gutfreund & Sompolinsky (1985), "Storage capacity of the Hopfield model"**
(the replica-method calculation giving the α_c ≈ 0.138 critical loading), Phys.
Rev. Lett. / Phys. Rev. A. Confirm the exact value (0.138N vs the rounded
~0.14N), the definition of capacity used (error-free vs small-error recall), and
that the number is the one the ML literature actually inherits. Note that this is
also the "further lead" the capture flagged as a physics↔AI, Nobel↔Nobel bridge
(the replica method later won Parisi the 2021 Physics Nobel) — verifying the
number and following that thread can be the same read.

**On resolution.** If confirmed against the primary, update
[[claim-dense-associative-memory-exponential-capacity]] to remove the
`[unverified-quant]` flag and record Amit–Gutfreund–Sompolinsky 1985 as the
source of the figure.