# Fixed Grid String Lab 0.3.0 — 1S→2P benchmark results

The frozen `radial-coulomb-e1-dirichlet-v1` benchmark passed every declared
gate on its first run. It fits zero parameters and uses no measured validation.
The result verifies an implementation of imported nonrelativistic quantum
mechanics; it is not evidence for strings, extra dimensions or a physical grid.

Protocol SHA-256:

    0eafcae663b9a53c93734acb6b336a3ef2a57b67957386a1337e24da92f10277

## Refinement results at R=160 a₀

| Grid a/a₀ | Gap (Hartree) | Radial integral (a₀) | Oscillator strength | Relative f error |
|---:|---:|---:|---:|---:|
| 0.40 | 0.356029069945 | 1.359023447440 | 0.438377343138 | 5.3294% |
| 0.20 | 0.369992762914 | 1.307695101009 | 0.421808147112 | 1.3483% |
| 0.10 | 0.373730129725 | 1.294639278893 | 0.417603770303 | 0.3381% |
| 0.05 | 0.374681377112 | 1.291360476703 | 0.416548743989 | 0.08458% |
| Exact nonrelativistic | 0.375 | 1.290266201960 | 0.416196717980 | — |

The absolute errors decrease strictly at every refinement. Empirical orders for
the final refinement are 1.995 for the gap, 1.999 for the radial integral and
1.999 for the oscillator strength, consistent with second-order spatial
convergence. This is a finite convergence observation, not a general proof for
all observables or spacings.

At the primary spacing \(a=0.05a_0\), changing the outer wall from 80 to
160 \(a_0\) changes the gap, radial integral and oscillator strength by at most
\(7.60\times10^{-14}\) in their respective atomic-unit quantities. The largest
eigenvector residual across all 16 calculated eigenstates is
\(8.26\times10^{-14}\).

## Selection rules and independent replay

The exact angular implementation gives

- 1S→2P, \(m'=0\), z polarization: \(1/\sqrt3\);
- 1S→2S: zero;
- 1S→3D: zero;
- 1S→2P, \(m'=1\), z polarization: zero.

A separate audit imports no production functions. It uses sparse shift-invert
eigenpairs instead of the production tridiagonal eigensolver, direct
Gauss–Legendre/Fourier angular quadrature, and adaptive continuum radial
quadrature. It passed **58/58 checks**. All **32 package unit tests** passed.
Both implementations share NumPy/SciPy and the declared quantum model, so this
is implementation verification, not independent physical replication.

## Claim ceiling and next test

This release calculates the standard E1 coupling after that coupling is
supplied. It does not derive electromagnetism from four- or five-dimensional
motion, and it does not predict the measured Lyman-α wavelength at precision.
NIST lists the resolved strong H I components near 1215.67 Å; those external
data were not used for tuning or validation.

The next physical-development gate is not another refinement of this same
baseline. It is one explicit project-specific interaction term that produces a
numerical contrast from the standard model. Before evaluating unused measured
data, the model must also include conventional reduced-mass, relativistic,
radiative, recoil, nuclear and measurement terms and freeze its uncertainty and
decision rule. If the new variables leave all observation probabilities
unchanged, they remain observationally unidentifiable.