# Capacity, recovery and finite biochemical operation

The full retained-state model answers three different questions: does a stationary joint operating point exist, when does a preparation recover, and how long does its finite supply sustain service? A stationary reduction is a useful capacity benchmark, not a substitute for transient private states or a donor account.

## Run and edit

Use Python 3.11+ in a virtual environment:

```sh
python -m pip install -r requirements.txt
python -m unittest -v
python example.py --output outputs
```

The top of `example.py` exposes the supply law, initial donor, conversion/affinity scales, source strength, matched repair speed, quotas, deadline, horizon and all eight preparation offsets. Concentrations are µM and time is seconds. The default preparation `p` is loaded as an exact rational center from `modal_box.json`; the configured offset is applied to it. The default linear supply starts with 120 µM and uses conversion 1000. That conversion is a concentration-to-source scale, **not a geometric volume**.

`configured_mission.json` and its trajectory belong to your edited inputs. Reference certificates and reference trajectories are fixed casebooks. The configured result gates applicability of the specific maintained/120-µM success witnesses; other inputs remain numerical explorations unless independently certified. Saturating supply has separate fixed-band conditions, which are exposed by `fixed_band_stock`. A favorable sampled trajectory is not promoted to a uniform preparation guarantee.

## Components and model

`branches.py` retains the GPx, Trx, regeneration, stationary-response and eight-state kinetic components from companion example 49, with local provenance. `dynamics.py` adds a matched repair-rate family, supply-law interface, maintained/sinusoidal supply, linear and saturating finite donors, and full ODE integration with service, regeneration, storage and shortfall accounts. No external example needs to be installed.

State order is `x,z,e1,e2,zT,h,w,v`: reduced NADPH and seven private branch states. Conserved pools reconstruct reduced glutathione, free peroxidase, reduced thioredoxin and active peroxiredoxin. Hyperoxidized Prx `w` returns through slow repair. The repair multiplier scales both damage and repair currents, preserving their stationary ratio while changing the transient clock. The model keeps these private states; it does not substitute stationary branch currents into a transient carrier equation.

Peroxide is maintained. Only the regeneration donor becomes finite; the repair-energy donor is still maintained. Coefficients are the paper's declared reduced kinetics, not individually fitted measurements or clinical thresholds. The model is not a thermochemically closed cell.

`ModalCertificate` performs exact preparation membership, recomputes the transformed field/Jacobian/remainder, and checks every nominal and source-perturbed face inequality. `TubeCertificate` validates the structure and replays each supplied moving-tube witness. `tube_kernel.py` is adapted from the manuscript verifier: the numerical builder and write-on-run CLI are removed, the input path is local, and an explicit certificate argument allows the wrapper to validate and verify the same object. The data files are unmodified rational witnesses; their proposed reference trajectories have no evidential authority until the inequalities pass.

## Capacity and hidden recovery time

The stationary NADPH floor is exactly `460180/489387`; fresh rational root enclosures put the minimum source scale between 0.11371266 and 0.11371268. The demand includes GPx overdelivery, not merely the quota sum 14.

Scaling both hyperoxidation and repair by any positive sigma leaves the stationary equations unchanged. It is not a rescaling of time for the whole field. A 95%-hyperoxidized preparation has the necessary delay about `70.877/sigma` seconds before Trx service can reach 4, at any source strength. The example integrates this deliberately damaged preparation separately. Numerical crossings are later and are not certified upper bounds; this preparation is outside the positive recovery theorem.

## Four-millisecond recovery

The modal regions R and S are correlated parallelepipeds. Their coordinate projections do not define an admissible Cartesian product. Exact membership uses `|M^-1(center-c)| + |M^-1|*halfwidths <= radius`. The data's center is not assumed stationary and the modal matrix is not assumed to diagonalize the Jacobian exactly; residuals and off-diagonal terms are recomputed.

Only the second modal radius needs to shrink to meet the quota. Exact face checks certify capture from all of R into S in 0.004 seconds, GPx service above 10.35 throughout, Trx above 3.91 initially and above 4.01 afterwards. The shortfall allowance is 0.00036 µM. The stored-account range on S is less than 0.1 µM, giving later regeneration at least `14.36*T - 0.1`. These arguments certify physicality, existence and capture; they do not certify contraction, convergence or global attraction.

An independent 0.1% box around `p` is **not contained in R**. Its positive result comes from a separate moving tube, which enters S near 0.2477 seconds and then connects to the maintained invariant theorem. The nominal tube covers all intervening times and certifies the smaller shortfall allowance 0.00011 µM. A coordinate-wise halfwidth around `p` that fits R is only about 0.633 pM; the example makes the geometric distinction explicit.

## Finite duration and the same-trajectory account

The branch buffers satisfy `BG' = HG - psiG` and `BT' = HT - psiT`. With a nonnegative normalized store W, `W' = regeneration - HG - HT`. A finite donor obeys `Q' = -regeneration`, hence `(Q+W)' = -HG-HT`. This rules out positive service forever without replenishment. It is a stoichiometric storage identity, not a thermodynamic theorem or a tight lower bound on short-mission stock.

Three complete 1,206-step witnesses are freshly verified in rational arithmetic:

* Maintained source: the independent 0.1% preparation box recovers and joins the invariant region.
* Linear donor, 120 µM: all preparations in that box, with ±0.001 µM donor tolerance, meet both quotas from 0.004 to 1.004 seconds. The Trx lower floor is above 4.034; donor spent on that service window is between 14.034 and 14.045 µM.
* Linear donor, 60 µM: the Trx upper bound at the mission end is below 3.9132, proving failure for every admitted preparation.

The checker includes moving-center drift, signed modal faces, donor faces, physical margins, preparation inclusion and endpoint accounts. It does not just validate values at grid points. Saved slice CSVs are for plotting; connecting their endpoints is illustrative, while the face inequalities cover the intervals between them. The shortfall field in the Q60 verifier is only **before the recovery deadline**; later failure adds further shortfall and must not be omitted from a full-mission account.

During the successful 120-µM mission, source strength drops below the stationary threshold after about 0.434 seconds. This persists because donor decreases. Exact storage drawdown is between 0.700 and 0.751 µM: stored carrier pays for service above current regeneration. The depleted trajectory leaves S, so the old 0.1-µM storage range cannot be reused.

The 1,927,680-µM sufficient stock from the linear fixed-band argument keeps the source within a very narrow uncertainty band. It is not the mission's consumption or a necessary stock. Under the **same linear law**, the moving tube certifies 120 µM. A saturating donor is instead a different model; exact sufficient examples `(stock,affinity)=(200,0.01)` and `(1500,1)` keep it inside the band for all preparations in R. These preparation sets and supply laws must not be merged into one optimal-stock curve. The 60–120 bracket is not a minimal-stock computation.

## Verification and reuse limits

Seven scientific tests check the literal field/Jacobian and exact storage identity, matched stationary responses and nonuniform transient scaling, nominal barriers/preparation geometry, fresh moving-tube replay and invalid-witness rejection, supply-law inequalities, numerical depleted ledgers and invalid models. Main execution replays all three tubes. Float ODE diagnostics use the complete retained state and integrated accounts; source/stock are not calculated afterwards from an undepleted trajectory.

The exact field and rational inequalities are rerun. Lean is not rerun, and the conventional moving-box, ODE continuation and comparison arguments are used from the paper. Source hashes, input hashes, the manuscript hash, all rational witnesses, complete trajectories and figure outputs are included. The model offers reusable components; a new enzyme inventory, preparation region, supply law or arbitrary rate perturbation requires its own relevant certificate.

MIT is proposed for newly authored code, pending owner confirmation; no license grant is asserted here. Preserve provenance for adapted source and witness data.