# Independent analytic check: the cyclic lossless storage model

Internal Codex derivation; not external peer review and not a novelty claim.

Let r[t] be demand minus supply in MW, with a declared interval weight dt hours. Define D[0]=0 and D[k]=dt*sum(r[1:k]), in MWh. Assume a closed balancing system with no spillage/shortfall, no storage losses, unlimited charge/discharge power, freely chosen initial state of charge S[0], and a cyclic boundary condition. Then S[k]=S[0]-D[k]. Exact sum(r)=0 is necessary for S[N]=S[0].

For every feasible capacity E and initial charge, 0<=S[0]-D[k]<=E for all k. Thus S[0]>=max(D), S[0]-E<=min(D), and E>=max(D)-min(D). Conversely, choosing S[0]=max(D) gives S[k]=max(D)-D[k] in [0,max(D)-min(D)]. Exact cycle balance closes the trajectory. Hence the minimum capacity is exactly max(D)-min(D), including the initial zero prefix.

With dt=0.5 hours, [1,1,-1,-1] MW gives D=[0,0.5,1,0.5,0] MWh and capacity1MWh; [1,-1,1,-1] gives D=[0,0.5,0,0.5,0] and capacity0.5MWh. Both have mean0, population variance1MW^2, and equal positive/negative peaks. Temporal order therefore matters even when these marginal summaries match.

In experiment v2, dt is explicitly a nominal slot weight; MWh-equivalent is an accounting unit of the constructed source-label model, not proof of physical elapsed-time energy coverage. Permuting complete day blocks retains the entire residual sample multiset and therefore marginal variance, mean and peaks. It changes only between-day ordering within each month, while retaining month order and within-day order. A nonzero change in cumulative range is a conditional chronology sensitivity, not a causal claim or a refutation of the source mechanism that already includes duration.