The Governing Minimum: Multi-Domain Rate Asymmetry Under Challenge as Functional Reserve

1. Foundation

Functional reserve is the surplus capacity separating independent function from disability. It is demand-dependent. Prevailing operationalisations treat it as a stock measured at rest and combined by weighted summation. Three properties of that architecture are rejected: measurement at rest, substitutability across domains, and the use of levels rather than rates.

A companion paper defines a within-person gain–loss ratio for a single domain over a train–detrain window of weeks (O’Leary, 2026). The present paper defines a related quantity on the acute challenge–recovery cycle (seconds to minutes) and supplies the multi-domain aggregation rule. The two constructs share formal structure but differ in timescale and underlying physiology; this is not a direct generalisation of domain count alone.

Ontology

• Functional reserve — latent physiological construct
• Challenge response — observed perturbation trajectory
• Rate asymmetry — measurable kinetic phenotype
• Governing domain — the eligible domain with the smallest estimated ρ_d

2. Construction

Mandatory core. Force recovery, lactate clearance, and dual-task cost recovery are always required. They were chosen because each admits a matched instrument, none requires laboratory infrastructure beyond ordinary clinical equipment, they span three distinct rate-limiting physiologies, and each literature already contains established challenge protocols. The core is the minimum common panel specified for cross-person comparison; it is not claimed to be exhaustive.

Rate definition. For each eligible domain a standardised challenge is applied. Both rates are fitted constants obtained under the kinetic discipline of the companion paper:

• r_l — degradation rate under load
• r_g — restoration rate after load removal

ρ_d = r_g / r_l

Same-instrument rule. Both rates must be recorded by the identical instrument in identical units. Domains that cannot meet this rule are ineligible. No z-score substitution is permitted.

Primary quantity and uncertainty. ρ_min = min_d of the point estimates (ρ̂_d). Uncertainty is reported as a bootstrap or simultaneous interval for ρ_min itself. Taking the minimum of independent domain-wise lower confidence bounds is rejected: an imprecisely measured domain would systematically win the minimum.

Acute-window boundary. Plateau detection is an independent preprocessing step. After load removal the performance variable is examined for the earliest time t* at which (i) the local first derivative is statistically indistinguishable from zero within the instrument noise model and (ii) the derivative remains non-positive for a consecutive confirmation interval of length τ. All data up to t* are eligible for fitting r_g; data after t* are excluded. Because the plateau is noisy, t* is reported as a band. τ is a free parameter fixed before data collection; sensitivity of ρ_min and of domain ordering to reasonable variation in τ is a required reporting item.

Experimental isolation. Water immersion removes axial load while preserving graded muscular demand through velocity-dependent resistance and producing the same central fluid shift seen in bed rest. It isolates muscular work from joint loading.

3. Interpretation

Why the model predicts ρ < 1 on the acute cycle. Degradation under load is multi-factorial and parallel. Restoration after load is sequential and energy-constrained. The structural asymmetry predicts that the recovery limb will be shallower than the loss limb. The claim is restricted to the acute window.

Governing domain. The domain whose restoration is most constrained relative to its degradation yields the lowest ρ_d. Because domains are non-substitutable, that domain sets the recovery interval the person actually requires under the specified challenge. Asymmetry explains why ρ < 1; non-substitutability explains why the minimum is taken. These remain separate arguments. The governing domain is a rate-asymmetry bottleneck; it is not synonymous with the most clinically abnormal domain.

Timescale distinction. Net adaptation (supercompensation) lies outside the acute window. ρ therefore predicts the required recovery gap rather than the adaptive outcome. The required gap G is a monotonically decreasing function of ρ_min whose exact form is to be determined empirically.

Clinical consequence. Two patients can share the same rehabilitation programme and the same numerical minimum yet diverge. One recovers; the other plateaus. The under-recovered patient was re-challenged before the governing domain returned to baseline and therefore accumulated deficit under a programme that looked identical on paper. Intensity is measured. Volume is measured. Frequency is inherited.

4. Worked Illustration

Illustrative values only. Uncertainty intervals for ρ_min are omitted from this illustration; section 2 requires that they be reported with any measured estimate.

Domain	rl	rg	ρ̂d
Force recovery	12.4 N·s−1	8.1 N·s−1	0.65
Lactate clearance	0.28 mmol·L−1·min−1	0.19 mmol·L−1·min−1	0.68
Dual-task cost recovery	42 ms·min−1	31 ms·min−1	0.74

ρ_min = 0.65 (force domain), over three domains. A programme dosed to the lactate or cognitive figures under-recovers the mechanical domain.

5. Schematic

Figure 1. Acute cycle schematic. Steep loss limb under load, shallower recovery limb that decelerates into a visible plateau, acute-window boundary drawn as a band on that plateau, and supercompensation overshoot (lighter weight) outside the measured window.

Figure 1. Acute cycle schematic. Steep loss limb under load, shallower recovery limb that decelerates into a visible plateau, acute-window boundary drawn as a band on that plateau, and supercompensation overshoot (lighter weight) outside the measured window.

6. Relation to Prior Work

Convertino’s compensatory reserve is the closest precedent. It measures under progressive challenge and reports remaining margin to decompensation. It is the single-domain, single-limb, margin-to-threshold special case of the present framework. Convertino reports the level of remaining capacity; the slope of that trajectory is rl. The later review by Suresh, Convertino and colleagues makes the same object explicit: a single integrated waveform index of remaining compensatory mechanisms in hemorrhage, with multi-system organ dysfunction named only as the far end of that continuum (Suresh et al., 2019). A second near-miss returned by the sparse intersections is Ferrucci’s aerobic resilience in the Baltimore Longitudinal Study of Aging. Mitochondrial oxidative capacity (kPCr) predicts VO2peak and a ratio of peak capacity to the energetic cost of walking, attenuated after age 70 (Trevisan et al., 2026). That is a capacity ratio inside one energetic domain, not a minimum of matched-instrument rate ratios across domains.

Limiting-factor models already embody a bottleneck rule. The novelty claimed here is restricted to the quantity being minimised: a matched-instrument rate asymmetry rather than a peak level.

The stimulus-response programme in physical resilience is the nearest live work and shares this paper’s premise. Varadhan and colleagues (2008) proposed characterising loss of resilience in homeostatic regulation through the dynamics of response to a standardised stimulus rather than through resting level, and the Study of Physical Resilience and Aging has developed that proposal into an empirical programme (Walston et al., 2023). Bandeen-Roche and colleagues (2025) analysed multi-system stimulus-response data in the SPRING pilot — Holter time series, cortisol response to adrenocorticotropic hormone stimulation, and repeated diurnal salivary cortisol — and derived dynamic component scores intended to capture adaptive capacity across systems. Related work has applied provocative testing and orthostatic challenge in community-dwelling older adults toward the same end.

That programme and the present construct agree on the measurement condition and differ on the aggregation rule. Both hold that reserve is visible under provocation rather than at rest, and both measure response dynamics rather than levels. Three differences follow. The quantity extracted here is a ratio of two fitted rate constants, restoration against degradation, in the same instrument and identical units, rather than a component score derived across heterogeneous measures. Domains are combined by a minimum rather than by a composite or factor structure, on the claim that non-substitutable domains admit no averaging. And the window is the acute challenge-and-recovery cycle, seconds to minutes, whereas the resilience programme is anchored to major clinical stressors and to recovery trajectories over months.

The novelty claimed is therefore narrow. Not that provocation is the correct measurement condition, which that literature established. Not that multi-system dynamics carry information about reserve, which it has demonstrated. The claim is that the quantity to be minimised across domains is a matched-instrument rate asymmetry, and that the minimum rather than a composite is the governing figure.

Search statement. A structured prior-art search was conducted in PubMed on 26 August 2026 using ten Title/Abstract query strings combining terms for rate-based reserve measurement, minimum and limiting-factor rules, multi-domain physiological capacity, and stimulus-response characterisation of resilience. The full strings are given in Supplementary Table S1. Counts were read from the PubMed result header and independently confirmed against the NCBI E-utilities esearch interface on the same day. No date, language, or article-type filter was applied, and counts are not deduplicated across strings. Returned counts ranged from 1 to 1,423, and are given by string below. The five intersections closest to the present construct returned 7, 9, 5, 2 and 1 records (24 Title/Abstract records; no overlapping PMIDs). All 24 were screened. None reports loss and recovery rates measured across physiological domains and reduced to a governing minimum. The nearest named constructs in those intersections are Convertino’s compensatory reserve (a single integrated waveform index) and Ferrucci’s aerobic resilience (a capacity ratio inside the mitochondrial and energetic domain), both addressed above. The remaining records are lexical collisions: epidemiological incidence rate ratios attached to a frailty label called physiological reserve; heart-rate asymmetry of RR intervals; protein-domain architecture; and industrial loss and recovery rates. No prior report was identified of a multi-domain minimum of matched-instrument gain-to-loss rate ratios under controlled acute challenge. Counts change as MEDLINE grows; the strings, not the integers, are the reproducible object.

#	Construct intersection	Hits
1	Rate ratio × reserve × multidomain	7
2	Law of the minimum × capacity	127
3	Weakest link / governing domain	41
4	Rate asymmetry × reserve	9
5	Recovery rate × individual differences	5
6	Physiological resilience × kinetics	51
7	Stimulus-response × reserve	19
8	Multisystem × bottleneck	2
9	Compensatory reserve × multidomain	1
10	Functional reserve × measurement	1,423

Strings 1, 4, 5, 8 and 9 are the sparse intersections closest to the construct; string 10 is a breadth check on measurement language around functional reserve and is not a novelty claim. The search was limited to one database, to PubMed-indexed material, and to title and abstract fields. Grey literature, non-indexed monographs, and pre-1966 sources were not searched systematically.

Novelty claim (modest form). This article proposes a multi-domain kinetic operationalization of functional reserve in which the smallest restoration-to-degradation rate ratio identifies the governing domain under a standardized acute challenge.

7. Falsification

The construct is falsified if, under measurement satisfying section 2:

ρ_min has no more predictive value than the mean of the eligible point estimates (ρ̂_d).

The governing domain is unstable within persons across repeated equivalent challenges.

ρ_min carries no relation to the recovery interval required for net adaptation.

8. Limitations

The construct is untested. The acute-window boundary rule remains to be validated across laboratories. τ is a free parameter whose influence must be reported. Standardised challenge protocols are not specified here. The prior-art search was limited to a single database and to title and abstract screening of the 24 records at the five sparse intersections; grey literature and non-indexed sources were not searched systematically.

Declarations