First-Hit Interacting Optimizer MVP

Implementation

first_hit_toy.py implements the paper's solvable one-dimensional first-passage mechanism. Particles begin at distance A=1 from an absorbing target and use exact inverse-Gaussian first-hit samples. It compares independent search, bounded normalized drift, unnormalized coherent drift alpha*(N-1), and antisymmetric pair-kick forcing represented by the paper's exact marginal diffusivity D_eff = D + kappa*(N-1).

The experiment uses fixed seed 1729, 30,000 Monte Carlo trials per population size, and N={4,8,16,32,64,128,256}. The script also sweeps coherent interaction strength and pair-kick strength.

Quantitative mechanism checks

Predictions and observations from results.json:

1. Bounded/normalized interaction remains logarithmic. Regression of log E[T_N] against log(log N) gives slope −1.095 for normalized coupling (independent baseline: −1.445, with finite-size corrections). This is consistent with the predicted 1/log N class and does not show an algebraic population gain.
2. Coherent unnormalized force gives algebraic acceleration. With alpha=1, the high-population log-log slope is −0.876, versus predicted −1. At N=128, increasing alpha through {0.8,1.6,3.2,6.4} gives alpha*E[T] = {0.00545,0.00607,0.00655,0.00692}, approaching the deterministic prediction A/(N-1)=0.00787. In the zero-noise control, all four products equal exactly 0.00787.
3. Pair kicks give the 1/(N log N) law. The high-N log-log slope is −1.232, versus predicted −1; N log(N) E[T] at N={64,128,256} is {1.562,1.552,1.535}, approaching the predicted asymptotic constant A^2/(4*kappa)=1.25 for kappa=0.2. The strength sweep gives kappa*N*log(N)*E[T] values {0.279,0.299,0.311,0.317} for kappa={0.05,0.1,0.2,0.4}, close to the predicted A^2/4=0.25 with finite-size/image corrections.

Baseline comparison

At N=128, the independent baseline has mean first-hit time 0.06591. Pair kicks have 0.00250, about 26.4x lower in this toy, while coherent coupling has an algebraic trend but is not uniformly better at every finite population because it competes with Brownian extreme hits.

Interpretation

The mechanism manifests: normalized bounded coupling stays in the logarithmic extreme-search class, while unnormalized coherent accumulation and enhanced pairwise fluctuations produce distinct algebraic accelerations. The coherent exponent is not exactly asymptotic at the tested population sizes, but its −0.876 slope and alpha scaling are a clear signal.

Reproduction

/home/maxwelhelp/main/bin/python3 first_hit_toy.py

The command writes results.json and prints the full measurements.

Limitations

This is a solvable first-passage toy, not a neural-network optimizer or MNIST experiment. Pair kicks are not discretely simulated as shared antisymmetric Brownian increments; their exact effective marginal diffusivity from the paper is sampled directly, so cross-label correlations are not tested. The independent and normalized logarithmic fits are finite-N regressions, and the coherent result includes a Brownian-to-drift crossover.