# Hydrogen Matrix Lab 0.4.0 — bounded rescue tests

## Result 1: extra same-family rotors do not rescue the model

The release-0.3 six-parameter geometry was frozen. A prespecified extension allowed up to four nonnegative phase-shifted copies chosen from twelve offsets. Global simplex weights and component choices were learned on the same eight training phases; the four-component rescue candidate was then evaluated on the sixteen previously untouched phases.

| Components | Training H² | Holdout H² |
|---:|---:|---:|
| 1 | 0.204041584 | 0.172620216 |
| 2 | 0.203640581 | 0.171868367 |
| 3 | 0.203586339 | 0.172032715 |
| 4 | 0.203586339 | 0.172032715 |

The fourth component receives zero weight. Relative holdout improvement versus one component is 0.3403%, below the predeclared 20% minimum. The four-component model remains far worse than the frozen time-mean baseline (`H² = 0.022196552`). Held-out axial-centroid correlation is `-0.058760`, dynamic nodal leakage is `0.058831`, and opposite-phase mirror TV is `0.043823`; all fail. Only full-cycle periodicity passes (`1.08e-18`). Five of six gates fail, so this bounded rescue is rejected.

This is a synthetic model-comparison result. It does not prove that no missing variable exists. It shows that “add more copies of the same frozen rotor at other phases” is insufficient.

## Result 2: temperature changes populations, not phase direction

The NIST-clocked two-level gap is `10.198836391 eV`. In the declared bounded two-level truncation:

| Temperature | Excited/ground ratio | Density-matrix change under unitary evolution |
|---:|---:|---:|
| 300 K | `4.64611e-172` | `6.33e-188` |
| 10,000 K | `7.24457e-6` | `9.55e-22` |
| -10,000 K | `1.38034e5` | `1.47e-16` |

The tiny nonzero changes are floating-point roundoff. Analytically, `rho_beta = exp(-beta H)/Z` commutes with `H`, so it is stationary. Negative temperature reverses the level populations only because the test truncation has an upper energy bound. The coherent control has phase slope `+1.5494760135e16 rad/s`; reversing time changes it to the negative value. Temperature does not enter that unitary.

Positive temperature also has no finite “maximum Kelvin” endpoint. In inverse-temperature coordinates, `T -> +infinity` means `beta -> 0+`. Negative-temperature equilibrium is a special population-inverted regime for upper-bounded spectra, not a colder continuation below absolute zero.

Melting delay does not identify rotation direction. It can arise from sensible heat, latent heat, the applied heat-transfer rate, geometry, pressure, and material properties.

## Verification and evidence boundary

All 29 unit-test groups pass (23 inherited plus 6 new). A separate implementation reconstructs the spectroscopic gap, optical period, all three thermal population ratios, time-reversal sign, saved latent-component metrics, simplex weights, and the six-gate decision: 11/11 checks pass.

These are written mathematical statements plus finite computation and synthetic tests. The NIST wavelengths are measured/compiled evidence for the clock only. No measured time-resolved 3D hydrogen density, fourth spatial dimension, unrestricted hidden-variable family, or theological cause is established.

## Next test

Do not add unconstrained components to this holdout. Specify a genuinely new interaction law—preferably a Hamiltonian or continuity equation—that predicts both spectral and spatial observables with a fixed parameter budget. Freeze it before using a new experimental preparation with a matching measured spatial observable.