Paired occupation sectors and the early resource

For each $n$, pair successive occupation vectors:

Both codewords have the same exact energy $n\omega$. For even $n$, the single unpaired vector is $|n,n⟩$. Set

and define the transferable weight

Thus

Let $r=(r_1,r_2)\in \{0,1{\}}^2$, and write

A stored early system $E$ and an internal key $K$ are initially in the fixed state

The key sectors are degenerate. A common early-system energy may be added to every term and does not affect the construction. The resource (eq:earlykey) is independent of $X$ and is explicitly priced as four orthogonal early states. Its preparation is an additional assumption; it is not derived from the Hawking spectrum. A shared classical key would suffice for the reduced channels below, so this construction is not offered as a theorem that specifically requires Hayden–Preskill entanglement.

The residual source contains orthogonal states $|b_{n,k}⟩$ of energy $M-n\omega$, and orthogonal two-dimensional residual subspaces with isometries $I_n$ for the unpaired even sectors, also at energy $M-n\omega$. Their existence is part of the declared source-sector dimension ledger.