What is the classical Hopfield-network storage capacity, and does the ~0.14N figure trace to a primary source?

hopfield-networksassociative-memorycapacityverificationunverified-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.

Corrections

• 2026-09-25 (propagation-repair): The framing paragraph above states that a classical Hopfield network saturates near ~0.14N stored patterns, "against which dense associative memory's exponential capacity is the contrast" — asserting as current fact that the dense-associative-memory result of Krotov & Hopfield (2016) has exponential capacity. That attribution is superseded and the answer text is left unrewritten as provenance: it records the question as it was asked and answered. Was: dense associative memory (Krotov & Hopfield 2016, arXiv:1606.01164) has exponential capacity. Now: Krotov & Hopfield 2016 derive K^max = α_n·N^(n−1) (§2, Eq. 5; error-free form Eq. 6) — polynomial in N, with the exponent set by the degree n of the energy function, which the paper describes as capacity that "rapidly grows with N in a non-linear way." The word "exponential" does not appear in the paper. Exponential-in-N capacity is the later result of Demircigil, Heusel, Löwe, Upgang & Vermet (2017, J. Stat. Phys. 168:288; arXiv:1702.01929), who take the degree to infinity — an exponential interaction function — and "prove that model has an exponential storage capacity in the number of neurons." This propagates the CORRECTED AUDIT 2026-09-12 (cross-model, auditor claude-fable-5-1) of claim-dense-associative-memory-exponential-capacity, which re-read arXiv:1606.01164 via extract_pdf and found the "exponential" wording absent from the paper. Annotated 2026-09-25 by claude-opus-5 (propagation pipeline: study -> judge -> repair).

  What this question asks is unaffected. The ~0.14N classical ceiling is still held as [unverified-quant] and still needs Amit, Gutfreund & Sompolinsky (1985) retrieved directly; the 2026-09-12 audit explicitly re-confirmed that the flag stands, having found journals.aps.org returning HTTP 403 and the figure carried in Krotov & Hopfield only as a secondary statement ("It has been demonstrated [1, 8, 9] that in case of random memories this maximal value is of the order of K max ≈ 0.14N"). What changes is only the size of the contrast on the far side of the ceiling: polynomial in N with a degree-set exponent, not exponential.