# Hydrogen Matrix Lab 0.8.0 — forward observable compiler

## Bounded question

Can one declared harmonic state generate several observer channels without
adding a new hidden variable whenever a requested display is missing?

The frozen state is the ideal fixed-proton, non-relativistic coherent packet

\[
\Psi(t)=\frac{|1s\rangle+e^{-i\omega t}|2p_z\rangle}{\sqrt2}.
\]

The compiler applies explicit operators to this one state. It does not infer
that the state is complete reality.

## Generated channels

The executable produces synchronized position density, probability current,
electron charge density/current, electric dipole, far-zone coherent electric
and magnetic fields, Poynting flux, and an electric-dipole-filtered harmonic
catalog. Its registry also exposes channels that are approximate, blocked, or
non-identified.

At one metre in the dipole radiation zone, the declared packet gives a peak
equatorial coherent field of `1.5179677846e-4 V/m`. The instantaneous peak
dipole power is `5.1240475064e-10 W`; the cycle average is
`2.5620237532e-10 W`.

Eight frozen checks pass:

- analytic charge continuity residual: `0`;
- finite-grid neutrality error: `0.00106955` relative;
- radiation transversality error: `1.33921e-16`;
- `|B|=|E|/c` error: `4.44089e-16`;
- angular power-integral error: `4.53077e-12`;
- cycle-average power error: `2.01789e-16`;
- the parity-silent `1s + 2s` control has zero electric-dipole field;
- the exact semiclassical/quantum power boundary is recovered.

## Exact incompleteness result

Let \(d=e|\langle1s|z|2p_z\rangle|\). The expectation dipole is
\(p(t)=-d\cos\omega t\), so its cycle-average classical radiation power is

\[
\overline P_{\rm coh}=\frac{d^2\omega^4}{12\pi\epsilon_0c^3}.
\]

The electric-dipole spontaneous-emission rate is

\[
A=\frac{\omega^3d^2}{3\pi\epsilon_0\hbar c^3}.
\]

The same equal packet has excited population \(1/2\), hence its initial
quantum E1 energy-loss rate is

\[
P_{\rm quantum}=\frac12\hbar\omega A
=\frac{d^2\omega^4}{6\pi\epsilon_0c^3}.
\]

Therefore

\[
\boxed{\overline P_{\rm coh}/P_{\rm quantum}=1/2}.
\]

This is a written conditional result, not a fitted discrepancy. A classical
projection of the expectation dipole is not the complete spontaneous-photon
channel. The missing structure is a quantized electromagnetic field with
vacuum fluctuations and photon statistics. It is not evidence for one unique
fourth- or fifth-dimensional cause.

## Independent verification and limits

A separate implementation imports none of the production compiler modules and
passes 12 checks against independently reconstructed constants and saved
arrays. The ten new unit tests pass. The full package regression is recorded
in `outputs/release_0.8_test_log.txt`.

The output is model-generated, not a measured movie. It omits spin magnetic
current, near fields, self-consistent Maxwell--Schrodinger backreaction,
radiation damping, recoil, full QED, finite proton size, and external fields.
The higher-dimensional generator remains non-identified because no coupling or
observer operator has been supplied.

Primary methodological boundary: Chen et al., “Ehrenfest+R Dynamics I,”
*J. Chem. Phys.* 150, 044102 (2019),
https://doi.org/10.1063/1.5057365. Their mixed quantum--classical analysis also
shows why ordinary Ehrenfest fields do not contain all spontaneous-emission
physics.

## Next test

Add a two-level quantized-radiation channel with density-matrix decay and
photon-number statistics. Freeze checks for total probability, emitted energy,
the coherent/incoherent split, and the pure-excited-state counterexample before
rendering another hidden-dimensional layer.