Source algebra and frozen couplings

Let $N=2n\ge 4$, and let ${\chi }_1,\dots ,{\chi }_N$ be Majorana operators satisfying

They act on

This normalization is fixed throughout; conversion to another SYK normalization must rescale the couplings explicitly.

Freeze a real array $\mathbf{J}=(J_{ijkl}{)}_{i<j<k<l}$. Set

Products of four distinct Majoranas are self-adjoint and have norm one. Consequently,

The added constant fixes an energy reference. It is not a gravitational mass normalization.

No disorder average is taken. Theorems below hold for each frozen array. They do not imply that every array has a useful recovery regime or a gravitational interpretation.

Define commuting source spins

For a fixed vector $s^{\prime }\in \{-1,1{\}}^n$, set

Then $O_{s^{\prime }}$ is even, self-adjoint, and has norm at most one.

Fix real coefficients $\alpha ,\zeta$, and define

Thus

This polynomial coupling is independently specified before channel certification. It is not selected after seeing a favorable decoder.

For notational flexibility, the analytical estimates also permit a fixed finite family of even self-adjoint operators $A_a$. The model used for the scalar comparison may take all $A_a=A$.