Memory weighed at its breaking point

If you store memories in the Hebbian weights of a recurrent network of N binary neurons, how many can it hold before interference destroys recall — and is the collapse at the Amit–Gutfreund–Sompolinsky critical load α_c ≈ 0.138 a sharp first-order cliff rather than graceful decay?

Units: dimensionless critical load α_c = P_max/N (AGS 1985 replica value; secondary known value m_c = 0.967, the retrieval overlap at the first-order jump)

▶ Run this simulationRead how it works

The finding

Memory weighed at its breaking point: a from-scratch zero-temperature Hopfield network — Hebbian sums and sign updates ONLY, no 0.138, no replica theory, no erfc in the generator — returns the storage-capacity transition α_c(∞) = 0.13886 ± 0.00254 vs the Amit–Gutfreund–Sompolinsky replica value 0.138 (rel 0.62%, 0.34 SE) by N^(−1/2) finite-size scaling of blind recall crossings at N = 500–4000 (width slope −0.58 confirms the exponent independently), the cliff edge holds the AGS first-order jump m = 0.9704 vs 0.967 with an EMPTY forbidden band (0/665 trials at intermediate overlap), and the naive signal-to-noise rival is falsified by 0.49 absolute in m at α = 0.2 while its own one-step premise erfc(1/√(2α))/2 is separately CONFIRMED to 0.1–3% — error feedback, not one-step noise, sets how much a network can remember

Method

Generator: a from-scratch zero-temperature Hopfield network — P = αN random ±1 patterns, Hebbian couplings W_ij = (1/N)Σ_μ ξ_i^μ ξ_j^μ with zero diagonal, asynchronous random-order sign updates descending the energy to a fixed point; local fields via the exact integer overlap identity N·h_i = Σ_μ ξ_i^μ M_μ − P·s_i (no N×N matrix, no float-tie ambiguity; exact ties keep state). No 0.138, no replica theory, no erfc in the generator. Recovery channels: (1) recall probability R(N, α) over a blind α grid 0.115–0.205 at N = 500/1000/2000/4000 (40/24/14/7 instances × 5 patterns each); pseudo-critical crossing α_c(N) at R = ½, extrapolated against N^(−1/2) — the exponent fixed a priori by the CLT rounding argument and checked independently by the transition-width scaling — with bootstrap SE (200 replicas); (2) the AGS first-order jump: recalled-trial overlap in the two grid bins bracketing the crossing, plus a forbidden-band (m ∈ [0.55, 0.85]) bimodality count; (3) the one-step Gaussian-crosstalk law at four loads (N = 4000, up to 4×10⁴ events/point); (4) rival falsification at α = 0.2; (5) variance-matched random symmetric Gaussian matrix control; (6) 30%-corrupted-cue basin perturbation; (7) verbatim HopfieldModule._measure mirror (N = 400, Float32 W, h ≥ 0 → +1 tie rule, module seed formula, 4 seeds). Gates A–I, all seeded, ~53 s.

The law it recovers

α_c(N) = 0.1769 → 0.1650 → 0.1576 → 0.1525 over N = 500 → 4000, strictly monotone, extrapolating on N^(−1/2) to 0.13886 ± 0.00254 (0.34 SE from 0.138); transition width 0.0358 → 0.0110 with log-log slope −0.581 (CLT rounding ~N^(−1/2), the independent check of the extrapolation exponent); recalled-trial overlap at the cliff edge 0.9704 vs AGS m_c = 0.967 with ZERO of 665 trials at intermediate overlap m ∈ [0.55, 0.85] — the collapse is first-order, never graceful; one-step unstable-bit fraction matches ½erfc(1/√(2α)) to 3.0/0.3/0.3/0.1% at α = 0.1/0.15/0.2/0.3; 30%-corrupted cues complete to m ≥ 0.9 with R = 1.00 at α = 0.05, shrinking to 0.46 at α = 0.11 and 0.00 at α = 0.14 (the basin contracts as the cliff approaches).

Measurements, controls & cross-checks

Recovered se

0.00254

Rival

Name

naive signal-to-noise capacity theory (the textbook pre-AGS argument: the Gaussian crosstalk is static, so bit errors stay at their one-step value and recall survives past α = 0.2 with m ≈ 0.975)

Premise confirmed

the one-step error rate IS the Gaussian law — measured vs ½erfc(1/√(2α)) agrees to 0.1–3% across α = 0.1–0.3 (gate D)

Conclusion falsified

at α = 0.2 the rival predicts m ≥ 0.9747; the full dynamics avalanche to mean m = 0.482 with recall probability 0.000 — off by 0.49 absolute

Verdict

falsified — error FEEDBACK, not one-step noise, sets the capacity; that is why α_c = 0.138 sits far below the naive ~0.15–0.19

Structure control

a variance-matched random symmetric Gaussian W (sd √P/N, zero diagonal) under the identical settle dynamics yields only |m| = 0.160 mean / 0.196 max toward the probe — the known zero-T dynamical remanence of a random-matrix quench, decisively below the 0.9 retrieval threshold — at the equivalent of α = 0.05 where the Hebbian network recalls perfectly (R = 1.00): retrieval lives in the outer-product STRUCTURE of W, not in the settling dynamics or the coupling statistics

Module systematic

Mirror alphaStar by seed

• 0.15
• 0.15
• 0.15
• 0.138

Note

the module's on-screen capacity estimate (N = 400, 5 trials, largest grid-α with mean m ≥ 0.9) reads 0.138–0.15 depending on seed — one grid step ABOVE α_c in 3/4 seeds, exactly the +N^(−1/2) finite-size shift this oracle quantifies (a50(400) ≈ 0.138 + 0.84/√400 ≈ 0.18 for the R=½ criterion; the module's stricter mean-m criterion reads lower). The on-screen 'cliff right at α_c≈0.138' is honest at the module's own grid granularity (m ≥ 0.99 through α = 0.12, collapsing by 0.17); the seed-to-seed wobble of its 5-trial curve and the +1-grid-step bias are disclosed here rather than hidden.

Definitional systematic

re-running the whole extrapolation with recall thresholds m ≥ 0.8 / 0.95 moves the intercept by ±0.0032 (each a50 moves ≤ 0.0013; extrapolation amplifies) — measured and covered by the 2.5% tolerance

What it reduces to

The Amit–Gutfreund–Sompolinsky storage-capacity transition of the Hopfield model (PRL 55, 1530, 1985; Ann. Phys. 173, 30, 1987): zero-temperature retrieval states vanish at α_c ≈ 0.138 with a discontinuous jump from m_c ≈ 0.967, and the one-step error rate is the textbook Gaussian-crosstalk law erfc(1/√(2α))/2 (Hertz–Krogh–Palmer ch. 2; Hopfield 1982 for the model). Non-circular because the generator contains only Hebbian sums and sign updates: the capacity emerges from finite-size scaling of blind recall counts (0.138 and 0.967 live only in the scorer), the N^(−1/2) extrapolation exponent is fixed by the CLT rounding argument and independently confirmed by the width scaling (slope −0.58), and the same machinery both CONFIRMS the rival's one-step premise (to 0.1–3%) and falsifies its conclusion (by 0.49 in m) — the avalanche is measured, not assumed.

Confidence & reproduction

Confidence

high

Validation

derisk-pass

Re-run the check

npm run derisk -- hopfield (scripts/hopfield-derisk.mjs — 9/9 gates (A capacity FSS, B size scaling, C first-order cliff, D one-step law, E rival falsified, F random-matrix control, G basin contraction, H threshold robustness, I module mirror), deterministic, ~53 s)

Oracle

scripts/oracles/hopfield.reference.json

Sources

D. J. Amit, H. Gutfreund & H. Sompolinsky, Phys. Rev. Lett. 55, 1530 (1985); Ann. Phys. 173, 30 (1987); J. J. Hopfield, PNAS 79, 2554 (1982); J. Hertz, A. Krogh & R. G. Palmer, Introduction to the Theory of Neural Computation (1991), ch. 2