Numerical verification protocol, fixed before execution, 7 October 2026 (Asia/Shanghai).
Purpose: verify the unified continuous-time state equations, interval integration, solver convergence and finite-horizon uncertainty propagation under completely specified illustrative conditions. This is a numerical verification experiment, not an additional physical measurement or a phone-parameter calibration.

State: x=[z,v1,v2], dz/dt=-I/(3600Q), dvj/dt=-vj/(RjCj)+I/Cj. I>0 means discharge. Q=4Ah; z(0)=0.95; v1(0)=v2(0)=0V. R0=.040ohm, R1=.012ohm, C1=2500F (tau1=30s); R2=.025ohm, C2=9600F (tau2=240s). The algebraic observation is Vt=Uoc(z)-IR0-v1-v2, Uoc(z)=3.0+1.2z volts. All parameter values are illustrative design choices; none are attributed to a measured device.

Input: duration7200s; I(t)=1.5A for0<=t mod600<300s, and0.3A for300<=t mod600<600s. At event times use the right-continuous current for terminal voltage; integration of the preceding interval uses its own constant current. Steps h=[20,10,5,2,1,0.5]s divide every300s switch interval exactly. Do not let RK stages use the next segment's current at an endpoint. Each scheme receives the same exact segment current and switch schedule.

Solvers: forward Euler, classical RK4, and exact piecewise-constant recurrence. Reference is the event-level closed-form solution at each requested timestamp: vj(t+dt)=exp(-dt/tauj)vj(t)+RjI[1-exp(-dt/tauj)]; z(t+dt)=z(t)-I dt/(3600Q). Reference integrates at input events, independent of the tested timestep recurrence. Report maximum absolute nodewise v1,v2,Vt,z errors against this reference and errors at the final timestamp for every solver/step. Estimate apparent orders from the three finest prescribed steps[2,1,.5]using log-log regression of maximum voltage error. Euler SOC should be exact up to rounding for this input; do not invent SOC truncation error.

Uncertainty checks use analytical propagation on a1s reporting grid, no numerical solver error: (a) current offset+0.020A with exactQ; (b) Qhat=3.8Ah with exact current; (c) both. InitialSOC is exact. For each case compute e(t)=zhat-z and bound B(t)=a*integral|Ihat-I|dt+|ahat-a|*integral|Ihat|dt, a=1/(3600Q),ahat=1/(3600Qhat). Report supremum(|e|-B), ending error, ending bound and all physical state ranges. The inputs are unidirectional and offsets chosen positive, so this triangle-inequality bound is expected to be tight; do not generalize equality to arbitrary signed perturbations.

Output: numerical_results.json, solver_convergence.csv, uncertainty_propagation.csv, numerical_payload.json with manuscript-ready prose, one6.2x3.0inch figure with convergence and SOC error-bound panels. Independent verification recomputes the closed-form endpoint from a geometric-series formula, checks1.8Ah true throughput, units and state ranges, recomputes exact bias/bound endpoints, and checks RK4/Euler scalar amplification formulas against saved errors. All runs/cases retained; no data-driven step or parameter selection. No end-to-endTTE or physical-device accuracy is inferred.
