# Hydrogen Matrix Lab 0.3.0 — spectroscopy calibration and frozen rotor holdout

## Bounded question

Can the release-0.2 tangent-3-sphere projection family be calibrated on selected phases and then reproduce unseen phases of the declared hydrogen `1s + 2p_z` reference better than simpler frozen baselines?

The protocol, phase split, parameter bounds, optimizer seed, metrics, and five acceptance gates were fixed in `REAL_DATA_PROTOCOL_0.3.json` before the fit. The sixteen holdout phases were not used by the objective.

## Real-world calibration

The NIST Hydrogen strong-lines table lists two vacuum Lyman-alpha fine-structure wavelengths:

| NIST wavelength | Listed relative intensity |
|---:|---:|
| 121.566824 nm | 1000 |
| 121.567364 nm | 500 |

Their listed-intensity-weighted centroid is `121.567004 nm`, corresponding to an optical period of `405.503877 as`. The doublet frequency separation is `10.954243544 GHz`, corresponding to a `91.288823 ps` two-frequency beat period.

The release-0.2 nonrelativistic fixed-proton Coulomb prediction was `121.502273411 nm`, or `0.064730589 nm` shorter than the NIST centroid, a relative error of `-0.0532468%`. Release 0.3 therefore supplies `hydrogen_1s_2pz_spectroscopy_clocked.npz`, whose phase timestamps use the NIST centroid. Its spatial voxels remain the ideal model; they are not relabeled as measured electron-density data.

The NIST table is spectral evidence, not a 3D movie of one atom. Stodolna et al. directly observed nodal structure in photoionization microscopy of hydrogen Stark states (Phys. Rev. Lett. 110, 213001, 2013), but that externally driven excited-state measurement is not interchangeable with this field-free `1s + 2p_z` voxel reference.

## Frozen train/holdout result

Eight of 24 phases were used for fitting and sixteen were held out. The optimizer made 1,862 objective evaluations and then froze six parameters. On the independent `33^3` evaluation grid:

| Predictor on 16 unseen phases | Mean squared Hellinger | Mean total variation | Mean relative L2 |
|---|---:|---:|---:|
| Fitted tangent-3-sphere rotor | 0.172620216 | 0.432929726 | 0.821171870 |
| Frozen training-phase mean | 0.022196552 | 0.131144467 | 0.188746349 |
| Spherical radial baseline | 0.072178684 | 0.259993679 | 0.225927917 |

The rotor was worse than both baselines on all three aggregate metrics. Its held-out axial-centroid correlation was `-0.086074`, rather than the required minimum `0.95`.

## Structural gates

| Predeclared gate | Threshold | Result | Pass? |
|---|---:|---:|:---:|
| Holdout Hellinger below time-mean baseline | strict improvement | 0.172620 vs 0.022197 | No |
| Held-out `z`-centroid correlation | at least 0.95 | -0.086074 | No |
| Opposite-phase dynamic nodal leakage | at most 0.02 | 0.057650 | No |
| Opposite-phase mirror TV error | at most 0.02 | 0.053451 | No |
| Full-cycle probability repeat error | at most `1e-10` | `4.34e-19` | Yes |

Only periodicity passed. The exact release-0.2 projected-envelope rotor family is therefore rejected as an adequate generator of the declared quantum reference. Tangency and periodicity remain mathematically valid properties of the construction; they are insufficient to reproduce hydrogen.

## Verification

- All 23 unit-test groups pass: 18 inherited and 5 new.
- The independent 0.3 implementation reproduces the NIST-derived quantities, both saved matrices, the train/holdout split, aggregate scores, centroid correlation, all five decisions, and the final rejection.
- Existing release-0.1 and release-0.2 artifacts are retained unchanged inside the package.

## Evidence classification

- **Measured or measurement-derived evidence:** the critically compiled hydrogen spectral values underlying the NIST reference.
- **Written mathematics:** ideal hydrogen basis functions, tangent 4D geometry, normalization, and scoring definitions.
- **Finite computation:** wavelength conversions, optimization, matrices, scores, and audits.
- **Synthetic test:** voxel comparison against the ideal quantum reference.
- **Not obtained:** a measured time-resolved 3D density for the exact prepared state.
- **Rejected hypothesis:** this exact tangent-sphere projected-envelope family adequately generates the declared hydrogen reference.
- **Still unconstrained:** other higher-dimensional mechanisms and theological interpretation.

## Next test

Do not add more rotor parameters to this holdout. Derive the next candidate from a Hamiltonian or continuity equation first. Freeze its spatial basis before any new comparison, then test it against both the two NIST Lyman-alpha components and a separately sourced spatial observable from a matching experimental preparation. A candidate that merely imports the hydrogen basis by construction is not an explanatory win.

## Sources

- NIST, *Strong Lines of Hydrogen*: https://physics.nist.gov/PhysRefData/Handbook/Tables/hydrogentable2.htm
- A. Kramida, *A critical compilation of experimental data on spectral lines and energy levels of hydrogen, deuterium, and tritium* (2010): https://www.nist.gov/publications/critical-compilation-experimental-data-spectral-lines-and-energy-levels-hydrogen
- A. S. Stodolna et al., *Hydrogen Atoms under Magnification: Direct Observation of the Nodal Structure of Stark States* (2013): https://doi.org/10.1103/PhysRevLett.110.213001