Manuscript VII-G@1 · VII-G.tex · proposition · 54f561e8

Proposition: Resource balance

Balance laws and admissibility without postselection › A separate resource ledger · lines 1371–1394 · proof included or linked

VII-G@1#prop:resource, VII-G@1#eq:expected-resource

On every supported history,

r (g_{n}) + t = 1 \sum^{n} c_{g_{t} ∣ g_{t - 1}} = r (g_{0}) .

Its expectation satisfies

(F^{*} r)_{g} = r (g) I - h \sum c_{h ∣ g} E_{h ∣ g} .

Proof. The history identity telescopes (eq:path-resource). For the expected identity,

(F^{*} r)_{g} = h \sum r (h) E_{h ∣ g} = h \sum (r (g) - c_{h ∣ g}) E_{h ∣ g},

and the effects sum to $I$ .

Displayed formulas in this passage

VII-G@1#eq:expected-resource · line 1378
$(F^{*} r)_{g} = r (g) I - h \sum c_{h ∣ g} E_{h ∣ g} .$
unlabeled · line 1374
$r (g_{n}) + t = 1 \sum^{n} c_{g_{t} ∣ g_{t - 1}} = r (g_{0}) .$
unlabeled · line 1389
$(F^{*} r)_{g} = h \sum r (h) E_{h ∣ g} = h \sum (r (g) - c_{h ∣ g}) E_{h ∣ g},$