# 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

```bash
/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.