Theorem: Sequential physicality and energy accounting

Equations (eq:firstemission)–(eq:completedmap) define a consistent sequence of isometries on the declared residual spaces. They conserve energy and the specified charges, retain every vacuum and unsuccessful branch, and perform no reset.

For $t\ge 1$, the accumulated isometry is

Every branch has total energy $M$. The mean pre-endpoint residual energy is

Proof. For pending inputs, the two terms in (eq:attemptmap) are orthogonal, and their squared norms sum to the input norm. For distinct key labels the residual summands are orthogonal. A newly completed branch has a different residual label from every previously completed branch. Thus the maps preserve all input inner products.

A pending successful term changes residual energy by $e$ and emits energy $e$. An unsuccessful or previously completed term emits vacuum and retains its residual energy. The first emission changes residual energy by $e_{\mathrm{K}}$ and emits that amount. The same reasoning applies to the declared charges.

Induction on $t$ proves (eq:accumulated): a pending amplitude is multiplied by $\sqrt{1-{\lambda }_t}$, and its newly completed amplitude is multiplied by $\sqrt{{\lambda }_t}$. The earlier amplitudes are unchanged. Their probabilities are (eq:weights) and sum to one. The energy average follows.