Gravity Light, Memory Motion, Time Scope, Horizon Silence: Four Dual Forms, One Root Algebraic Identity

This record contains four companion preprints:

Gravity Light | Memory Motion 
Time Scope | Horizon Silence

The papers share a finite algebraic root.

$$Q=\begin{pmatrix}1&1\\1&0\end{pmatrix},\qquad \Omega=\begin{pmatrix}0&1\\-1&0\end{pmatrix},\qquad C=-\Omega$$

$$G=\begin{pmatrix}-2&1\\1&2\end{pmatrix},\qquad H=\frac{G}{\sqrt{5}},\qquad S=\frac{2Q-I}{\sqrt{5}}$$

Its certified identities include:

$$\begin{aligned}Q^{2}&=Q+I\\C^{2}&=-I\\G^{2}&=5I\\H^{2}&=S^{2}=I\\H&=CS\\SG&=\sqrt{5}\,\Omega=-GS\end{aligned}$$

$$L_s=\exp(sS),\qquad L_s^{\mathsf{T}}\,G\,L_s=G,\qquad H\,L_s\,H=L_{-s}$$

The four titles are the baseline paired offices of an eight-address carrier:

Gravity–Light

Memory–Motion

Time–Scope

Horizon–Silence

Gravity Light develops the Lorentzian metric, null directions, and axial transport.

Memory Motion develops recurrence, transport, and lifted return.

Time Scope develops phase rotation, projective readout, and the double-cover relation.

Horizon Silence develops the finite address ledger, constraint boundary, and closure relation.

Each paper is separately readable. Together they provide one shared algebraic root, one paired field organization, and four exact formal developments.