the AC bus appears once the first item lands
A conventional account divides total fuel-carbon by gross generated energy. That folds in auxiliary draw, charging loss, and dumped diesel — so a system can lower its reported intensity by producing more non-useful energy. The Cooper (2026) framework corrects three structural errors so the number is conservation-grade and decision-useful:
1 · Useful-energy denominator (Axiom 4.2). Only useful delivered production kWh enters the denominator — auxiliary load, charging draw, storage loss, and load-bank dump are excluded, so overproduction can't dilute the figure.
2 · Storage as a coupled (Esoc, Csoc) state with import/export (Thm 4.2, Eq. 9–10). The battery isn't zero-emission or a fixed factor. Charging mixes its stored intensity β toward the loss-adjusted bus mix γchg/a; discharge changes the amount stored but leaves β invariant. The packs arrive part-charged — that imported energy carries the carbon of where it was charged (here, the oil-heavy Oahu grid), so it's debited to the window that uses it; whatever charge is left at the end is deferred (exported in time) to the next deployment. The stock correction −[CB(T)−CB(0)] makes this time-shift sign-honest in both directions (Cor. 4.2–4.3).
3 · Dump ≠ curtailment. Diesel burned to a load bank is emitted carbon for zero output — penalised in full. Curtailed solar emitted nothing, so it's an opportunity cost, not a fossil burden, and is never charged to the account.
Within one production the denominator cancels, so a comparison of supply architectures (your build vs. the diesel embodiment) is invariant to the useful-load classification. Operational fossil carbon only; embodied/lifecycle carbon is out of scope by design.