Source
J. J. Hopfield (1982), PNAS 79:2554–2558, LIT-918, read in full in NOTE-819.
What was actually shown
- Derived. For Tᵢⱼ = Tⱼᵢ and asynchronous threshold updates, E = −½ Σᵢ≠ⱼ TᵢⱼVᵢVⱼ changes by ΔE = −ΔVᵢ Σⱼ TᵢⱼVⱼ ≤ 0 at each update (Eqs. 7–8), so the state ends at a local minimum. With storage Tᵢⱼ = Σₛ (2Vᵢˢ − 1)(2Vⱼˢ − 1), a stored random pattern’s local field has mean ±N/2 and crosstalk of rms [(n − 1)N/2]^½ (Eqs. 3–4, 10).
- Simulated, N = 30 and 100.
- Capacity. “About 0.15 N states can be simultaneously remembered before error in recall is severe”: at N = 100, five patterns are almost always stable; at fifteen, about half end within 5 bits and the rest far away (Fig. 2).
- Basins. At N = 30, n = 5, a start within 5 bits of a memory reaches it more than 90% of the time, falling to 0.2 at 12 bits.
- Merging. At N = 100, an eighth pattern placed 30 bits from another leaves both stable; at 20 the minima are displaced; at 10 they are “often fused”.
- Overload. Past capacity, new memories make “all memory states irretrievable unless there is a provision for forgetting”; saturating couplings keep the recent and lose the distant past.
Each simulated result could have come out otherwise: the stored patterns need not have been stable, the basins could have been a bit wide, and nearby patterns could have stayed distinct. The capacity figure was also checked against the Gaussian estimate (predicted 0.40 perfect recall at n = 10, N = 100; observed 0.6).
What this does not say
- It does not give a sharp capacity. 0.15N is where about half the memories are well retained at these sizes. The paper expects it “for all large N” from the noise estimate, but shows it only at N ≤ 100.
- Spurious states are not a symptom of overload. At n = 5, N = 30, 10% of random starts end in “stable states of no obvious meaning”. Reading them as what appears past capacity, as the record did before this reading, is wrong.
- Merging and overload are different failures. Fusion comes from two memories being close, at any load. Overload makes every memory unreachable, not merged.
- Nothing about addresses. The model stores by content only. Any contrast between retrieval by content and by code (CLAIM-180) needs another source; Kanerva’s random-access-memory limit (LIT-921) is one.
- Not a model of a brain, a person or a record. Hopfield offers the biology as motivation. Readings of an argument record through it are analogy.
Connections
- CLAIM-180 rests on this model for its basin picture. This account supports that picture and corrects the claim’s attribution of merging to capacity (see NOTE-819).
- CLAIM-179: the overload result is the sharper form of the flooding analogy in LIT-918’s standing: loss of all memories, unless forgetting is provided.
- THEORY-261 (sparse distributed memory) states the related capacity for Kanerva’s memory, which grows with the number of hard locations, not with N.
- THEORY-136 holds what evidence shows alternative stable states in a field system; this account is the designed case, where the attractors are written in.
Note of 2026-10-20: what was measured, and the two checks
Checked for the promotion audit (batch 051, ADR-047 v3).
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The energy argument. Take V in {0, 1}, T symmetric and Tᵢᵢ = 0. Updating unit i changes E = −½ Σᵢⱼ TᵢⱼVᵢVⱼ by ΔE = −ΔVᵢ Σⱼ TᵢⱼVⱼ. The threshold rule raises Vᵢ only when that sum is positive and lowers it only when the sum is negative, so every change lowers E. The state space is finite, so asynchronous updating ends at a fixed point.
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The crosstalk estimate. For a stored random pattern, the local field has a signal of ±N/2 and n − 1 noise terms. Each noise term is a sum of about N/2 terms of ±1, so the noise has rms [(n − 1)N/2]^½ (Eq. 10). At n = 10 and N = 100: σ ≈ 21.2, z ≈ 2.36, a bit errs with probability about 0.009, and a perfect 100-bit recall has probability about e^−0.92 ≈ 0.40. That agrees with the paper; its simulation gave 0.6.
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What the title keeps.
- At N = 100, five patterns are almost always stable; at fifteen, about half are recalled within 5 bits.
- At N = 30 with five patterns, a start within 5 bits of a memory reaches it more than 90% of the time.
- At N = 100, an extra pattern 10 bits from another was “often fused” with it.
- At N = 100 with n = 500, no stored pattern was stable.
Whether 0.15N holds at large N, and how the fusion distance scales, is QUESTION-201.
Note of 2026-10-20: the wider claim, filed Proposed
Under ADR-047 v4 the narrowing-loss sweep (batch 069) filed THEORY-800 as Proposed, extending this entry. It carries Hopfield’s large-N capacity of about 0.15N and his forgetting clause, which this entry’s title carried before batch 051 narrowed it to his simulations at N = 30 and 100. This entry is its evidence and is not changed by the filing.