# Decision-Oriented Optimum Preservation MVP

## Reproduction

```bash
/home/maxwelhelp/main/bin/python3 experiment.py
```

Results are saved to `results.json`. The fixed seed is 2711 and the run uses CUDA when available, with CPU fallback.

## What was implemented

`experiment.py` implements a one-dimensional differentiable surrogate for a nonlinear process, the economic objective, projected unrolled inner optimization, trajectory/data loss, and decision-aware loss. It compares MSE-only training with decision-aware training, then evaluates the predicted economic optimum and 100 multistart gradient runs.

## Numerical math checks

1. **Inner-loop stability.** For a quadratic with Hessian `H=3.7`, the update is `e_(t+1)=(1-rho H)e_t`, so the predicted stability boundary is `rho=2/H=0.54054`. The sweep observed stable contraction through `rho=0.50` (`rho H=1.85`) and divergence at `rho=0.55` (`rho H=2.035`), confirming the boundary to the sweep resolution.

2. **Perturbation and extra minima.** For `J_eps(u)=(u-0.63)^2+eps*cos(6*pi*u)`, the curvature perturbation scale is `2/(6*pi)^2=0.00563`. The sweep has one minimum at `eps<=0.01`, two at `eps=0.02`, and three at `eps>=0.04`. Thus the predicted mechanism (sufficiently large objective perturbations can create extra optima) is observed, although the exact onset is later than the simple curvature scale because the perturbation is not centered at the base optimum and stationarity also matters.

## Controlled surrogate result

The trusted plant optimum is `u*=0.56495`, `J*=0.006122`.

- MSE-only: predicted optimum `u=0.16`, displacement `0.40495`, data MSE `0.07961`, one grid minimum.
- Decision-aware: predicted optimum `u=0.58`, displacement `0.01505`, data MSE `0.08007`, one grid minimum.

The decision-aware objective reduced displacement by about 96.3% with essentially unchanged data MSE. This is a promising direct signal for the intended non-identifiability mechanism. However, the reported multistart counts are not reliable evidence of distinct local minima: the simple fixed-step gradient flow stops at many nearly stationary points in flat regions. A robust root-finding or Hessian-filtered basin-counting procedure would be needed for that claim.

## Verdict

The optimizer stability prediction is quantitatively confirmed, and the extra-minimum mechanism is qualitatively confirmed. The decision-aware loss substantially preserves the trusted optimum in this toy problem, but the experiment does not establish the stronger claim that it removes extra local optima because the basin-counting diagnostic needs refinement.