# Reality Revision 1.53 — a taxonomy of projected harmonies

The five-coordinate proposal now has an exact forward law and an explicit
observation boundary. For a fixed real skew generator,

\[
\dot z=\Omega z,\qquad y=Pz,qquad P=[I_3\;0],
\]

every three-dimensional output is a constant plus at most two elliptical
sine/cosine harmonies. This is the complete linear norm-preserving family in
five coordinates; it is not a claim that nature uses five dimensions.

Matter Harmony 1.0.0 classifies eight reference cases and 512 randomized
generators by spectral relation, coupling rank, known-generator observability,
single-history Hankel rank, projected span, exposure loss, and tick aliasing.
All 512 bounded spectral labels are recovered when the generator is provided,
with maximum frequency error \(1.11\times10^{-15}\).

Three corrections are decisive:

1. Coupling rank is not observability. Two rank-one couplings produce
   observability ranks four and five depending on hidden-hidden mixing.
2. Known-system observability is not generator identification. A 101-angle
   hidden-basis gauge family changes the generator by as much as 4.0736 while
   preserving every 3D sample within \(5.17\times10^{-14}\).
3. The observer changes the available information. A one-mode-period exposure
   lowers the \((1,\sqrt2)\) history rank from five to three; a common-period
   exposure lowers the 3:2 case to one. The passive observer does not change
   the latent state, so no quantum collapse is claimed.

An independent implementation checks the catalog, all randomized cases, 6,464
gauge-time states, two exposure erasures, and 81 alias samples. These are
written mathematics and finite noiseless synthetic computations—not measured
physical evidence, proof of a fifth dimension, or theological identification.

The next test will infer bounded mode counts from noisy 3D trajectories alone
and must reject three-tone, nonlinear, and drifting controls as outside the
five-coordinate skew-linear class.