# Hydrogen Matrix Lab 0.2.0 — waves and 4D projection

## Correction to the first visual

The release 0.1.0 `1s` cloud was correctly static: for one energy eigenstate, the time factor is a global complex phase and cancels from `|psi|^2`. It therefore could not show interacting waves producing a changing 3D image.

Release 0.2.0 adds two distinct mechanisms rather than confusing them:

- a standard quantum reference in which the fourth coordinate is time;
- a synthetic extra-spatial-dimension rotor hypothesis.

## Quantum reference result

The equal coherent superposition of ideal fixed-proton hydrogen `1s` and `2p_z` states has density

\[
\rho(\mathbf r,t)=\frac12\left(\psi_{1s}^2+\psi_{2p_z}^2+2\psi_{1s}\psi_{2p_z}\cos\frac{(E_2-E_1)t}{\hbar}\right).
\]

The cross term moves probability between the positive-`z` and negative-`z` sides. At relative phases `pi/2` and `3pi/2`, the cross term vanishes. Phases `0` and `pi` are exact mirror images in the analytic model.

Using the ideal Coulomb energies and the NIST 2022 CODATA Rydberg energy:

| Quantity | Result |
|---|---:|
| Energy gap | 0.375 Hartree |
| Angular beat frequency | 1.5503015001e16 rad/s |
| Density cycle | 405.287959 attoseconds |
| Ideal-model transition wavelength | 121.502273 nm |
| Saved time matrix | `24 x 49 x 49 x 49` |

Finite `129^3` basis checks:

| Check | Result |
|---|---:|
| `1s` norm | 1.000658729139 |
| `2p_z` norm | 0.999924863898 |
| `1s`–`2p_z` overlap | -1.0842e-18 |
| `2p_z` nodal-plane maximum | 0 |
| `2p_z` odd-parity error | 0 |
| Phase-`pi` mirror error | 0 |
| Phase-`2pi` periodicity error | 0 |

The independent implementation reduced both exact three-dimensional normalization integrals to separate radial quadratures and directly re-audited the saved matrices. All eight independent wave checks passed. Across the static, quantum-wave, and synthetic-projection engines, all 18 unit-test groups passed.

## Touching 4D rotor result

The synthetic engine places two equal 3-spheres in `R4` at opposite centers `+R u(t)` and `-R u(t)`, with `u(t)` rotating in the `x-w` plane. Written geometry proves:

- center distance is always `2R`;
- the 4D origin is on both surfaces, so they remain tangent there;
- the contact projects to the 3D origin.

The finite engine generated a `24 x 35 x 35 x 35` sequence of projected wave-interference matrices. Minimum intensity was zero within floating arithmetic, and the maximum one-cycle periodicity error was `3.11e-15`.

This is a successful implementation of the proposed geometry, **not evidence that the geometry generates hydrogen**. The displayed local projected source envelopes are an approximation to the full projection of each 3-sphere.

## Evidence classification

- **Written mathematics:** hydrogen basis formulas, superposition density, beat period, 4D tangency, and contact projection.
- **Finite computation:** basis matrices, time frames, projection frames, renderings, and regression tests.
- **Synthetic test:** the touching-rotor wave generator.
- **Measured physical evidence:** none collected.
- **Hypothesis:** a deeper 4D spatial generator could underlie the observed hydrogen density.
- **Theological interpretation:** unconstrained by these results.

## Next discriminating test

Fit only the synthetic rotor parameters on part of the quantum reference cycle, freeze them, and score the remaining phases voxel-by-voxel. Compare against a radial baseline. If the 4D generator cannot reproduce the `2p_z` nodal plane, mirror relation, and full held-out cycle simultaneously, reject this version of the mechanism rather than tuning it after seeing the test frames.