Viability Space Decomposition: A geometric partition of survival outcomes in single- and multi-agent systems

Connor McShaffrey; Randall D. Beer

arxiv: 2605.16753 · v1 · pith:IDCMOTKXnew · submitted 2026-05-16 · 🧬 q-bio.QM · math.DS

Viability Space Decomposition: A geometric partition of survival outcomes in single- and multi-agent systems

Connor McShaffrey , Randall D. Beer This is my paper

Pith reviewed 2026-05-19 19:50 UTC · model grok-4.3

classification 🧬 q-bio.QM math.DS

keywords viability space decompositionmortality manifoldsordering manifoldscollapse manifoldsviability portraitsurvival outcomesODE modelsmulti-agent systems

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The pith

Viability space decomposition partitions state space into regions of qualitatively similar survival outcomes using mortality, ordering, and collapse manifolds.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Models of living agents often use ordinary differential equations constrained to bounded viability regions, where leaving the region means death. Traditional tools like attractors and separatrices cannot fully describe the global space of behaviors because death is an absorbing outcome outside the viable set. The paper develops viability space decomposition to introduce mortality manifolds that bound viable regions, ordering manifolds that sequence distinct viability boundaries, and collapse manifolds that handle interactions in multi-agent cases. These elements together produce a viability portrait that classifies the entire state space by survival character. The approach is applied to three example models to show how it reveals complete global dynamics for subcellular networks, individual cells, and coupled cell systems.

Core claim

Viability space decomposition supplies a geometric partition of the state space for viability-constrained ODE models by identifying mortality manifolds that separate viable from non-viable states, ordering manifolds that rank successive viability limits, and collapse manifolds that resolve multi-agent interactions, thereby yielding a complete viability portrait of qualitatively distinct survival outcomes.

What carries the argument

Viability space decomposition, which uses mortality, ordering, and collapse manifolds to produce a viability portrait that partitions state space according to survival outcomes.

Load-bearing premise

The introduced manifolds can be identified and combined to achieve a complete decomposition of all survival outcomes for the full class of viability-constrained ODE models without leaving regions unclassified.

What would settle it

A viability-constrained ODE model in which some open regions of state space exhibit mixed or unclassified survival outcomes after all mortality, ordering, and collapse manifolds have been located.

Figures

Figures reproduced from arXiv: 2605.16753 by Connor McShaffrey, Randall D. Beer.

**Figure 2.** Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. In the network diagrams, normal arrows represent [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗

**Figure 6.** Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗

**Figure 7.** Figure 7: FIG. 7. Combining all of the elements from Fig. 6 gives us [PITH_FULL_IMAGE:figures/full_fig_p016_7.png] view at source ↗

**Figure 8.** Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p018_8.png] view at source ↗

**Figure 9.** Figure 9: (b). This reveals the same peak values for the [PITH_FULL_IMAGE:figures/full_fig_p021_9.png] view at source ↗

**Figure 10.** Figure 10: FIG. 10. The complete numerical exploration of the behaving cell model, comprising one million simulations per two-dimensional [PITH_FULL_IMAGE:figures/full_fig_p022_10.png] view at source ↗

read the original abstract

What determines whether an organism or collective will survive under particular conditions? This question is asked across the life sciences when determining adaptive fit, developing efficacious treatments for diseases, and assessing the risks posed by ecological shifts. To aid their investigations, researchers employ models of agents which must respect particular constraints to remain alive. By constraining the dynamics of these agents to bounded viability regions, these models form a class of extended dynamical systems where transient dynamics can lead to death, making traditional attractors and separatrices insufficient for characterizing the global space of possible behaviors. To remedy this, we develop viability space decomposition, an analysis framework for ordinary differential equation models of agents with viability constraints. We first introduce the general theory, revealing how several new classes of manifolds (mortality, ordering, and collapse) permit a complete decomposition of state space into regions of qualitatively similar survival outcomes: a viability portrait. We then demonstrate the method by completely analyzing the global behavior of three models: a subcellular network, a behaving cell with the same physiology, and two coupled cell networks. Finally, we finish by discussing how the framework scales and future directions for its development and application.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a geometric decomposition of viability spaces in constrained ODE models using mortality, ordering, and collapse manifolds, with concrete demos on three low-dimensional cases, but the completeness claim looks tied to those specific setups.

read the letter

The main thing to know is that McShaffrey and Beer introduce viability space decomposition to partition state space in viability-constrained ODE models into regions with similar survival outcomes. They define three new manifold classes—mortality, ordering, and collapse—that together with the viable regions produce what they call a viability portrait. This moves past standard attractors and separatrices because death can occur outside the bounds. They then apply the method to a subcellular network, a single behaving cell, and two coupled cells, showing how the decomposition maps global behaviors in each case. Those worked examples are the clearest part of the contribution and make the framework feel usable for models where agents or collectives must stay inside viability regions to survive. The approach could help with questions about adaptive fit or risk in biological systems. The soft spot is the generality. The demonstrations stay in low-dimensional systems with relatively simple viability boundaries, and the abstract does not include an explicit theorem that these manifolds can always be located to exhaust the state space without gaps or overlaps in higher-dimensional or topologically messier models. If the construction depends on smoothness or low codimension that is not stated, the complete decomposition may not hold for arbitrary viability-constrained ODEs. The paper would be stronger with either a proof of scope or a clearer discussion of when the method might leave trajectories unclassified. This is aimed at researchers working on dynamical systems in biology who already use constrained agent models. A reader looking for new tools to characterize survival outcomes globally would get practical value from the examples. I would send it to peer review. The core framework and the three analyses are developed enough to merit referee time, even if revisions are needed on the theoretical reach.

Referee Report

1 major / 1 minor

Summary. The paper develops viability space decomposition, an analysis framework for ODE models of agents subject to viability constraints. It introduces mortality, ordering, and collapse manifolds that are claimed to enable a complete geometric partition of state space into regions of qualitatively similar survival outcomes (a viability portrait). The framework is demonstrated via full global analysis of three models: a subcellular network, a single behaving cell, and two coupled cells.

Significance. If the completeness claim holds, the work supplies a systematic geometric tool for characterizing global behavior in viability-constrained dynamical systems, where traditional attractors and separatrices are insufficient. The three explicit demonstrations provide concrete, reproducible examples of the method in action and could support falsifiable predictions about survival regions in biological models.

major comments (1)

[General theory] General theory (prior to the demonstrations): the manuscript asserts that the new manifolds permit a complete decomposition for the full class of viability-constrained ODE models, yet no explicit theorem or proof is supplied that guarantees the manifolds can always be identified and that their union (together with viability regions) exhausts state space without unclassified trajectories or overlaps. The demonstrations are restricted to three low-dimensional cases with relatively simple viability boundaries; this leaves the generality claim load-bearing but unproven for higher-dimensional or topologically complex systems.

minor comments (1)

[Abstract] The abstract refers to 'several new classes of manifolds' without giving their defining properties or construction; a brief definitional sentence would improve immediate readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their positive evaluation of the work's significance and for the constructive major comment. We address the concern regarding the general theory below.

read point-by-point responses

Referee: [General theory] General theory (prior to the demonstrations): the manuscript asserts that the new manifolds permit a complete decomposition for the full class of viability-constrained ODE models, yet no explicit theorem or proof is supplied that guarantees the manifolds can always be identified and that their union (together with viability regions) exhausts state space without unclassified trajectories or overlaps. The demonstrations are restricted to three low-dimensional cases with relatively simple viability boundaries; this leaves the generality claim load-bearing but unproven for higher-dimensional or topologically complex systems.

Authors: We agree that a formal theorem would strengthen the presentation of the general theory. In the revised manuscript we will add an explicit theorem stating that, under standard assumptions (Lipschitz continuity of the vector field and compactness of the viability region), the mortality, ordering, and collapse manifolds together with the viability regions form a partition of the state space with no unclassified trajectories or overlaps. The proof will proceed by showing that every initial condition has a unique forward orbit that either remains in the viability region for all time or intersects the viability boundary at a first time, after which the orbit is classified by the type of boundary intersection (mortality, ordering, or collapse). We will also include a brief discussion of the coordinate-free, dimension-independent nature of the construction, noting that while the three demonstrations are low-dimensional for computational clarity, the definitions and partition property extend directly to higher-dimensional and topologically more complex viability sets. We will add a remark on practical identification challenges in high dimensions without altering the theoretical completeness claim. revision: yes

Circularity Check

0 steps flagged

No significant circularity; framework introduces independent geometric constructs

full rationale

The paper defines viability space decomposition by introducing new manifold classes (mortality, ordering, collapse) as part of a general theory for viability-constrained ODEs, then applies them to three specific low-dimensional models. No step reduces a claimed prediction or completeness result to a fitted parameter, self-citation chain, or definitional renaming; the central decomposition is constructed explicitly from the stated geometric partitions rather than presupposing the target classification. Demonstrations remain illustrative rather than load-bearing for the general claim, and the derivation stays self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

Abstract-only review limits visibility into specific free parameters or axioms; the core assumption is that viability-constrained ODEs require new manifold types beyond traditional attractors.

axioms (1)

domain assumption Models of agents with viability constraints form extended dynamical systems where transient dynamics can lead to death and traditional attractors/separatrices are insufficient.
Directly stated in the abstract as the motivation and class of systems addressed.

invented entities (1)

mortality manifolds, ordering manifolds, collapse manifolds no independent evidence
purpose: To enable complete decomposition of state space into regions of similar survival outcomes.
New classes introduced by the framework as described in the abstract.

pith-pipeline@v0.9.0 · 5735 in / 1333 out tokens · 41908 ms · 2026-05-19T19:50:40.914193+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

mortality manifolds M = {ϕt(m) : t ≤ 0} ... ordering manifolds O ... collapse manifolds C
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

viability portrait ... hybrid automaton with guard conditions Gα

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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This makes the set of molecules autocatalytic

Model specification The single-cell physiology is composed of two molecu- lar species M = {M1, M2} which reciprocally catalyze each other and are synthesized from one of two food molecules F = {F1, F2} respectively. This makes the set of molecules autocatalytic. Fig. 4(a) depicts this cir- cuit with normal arrows representing material transfor- mation, ro...

work page
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Numerical exploration To approximate the possible existential outcomes in the single-cell physiology, we uniformly sample a million initial conditions in the range [Mi] ∈ [0, 20]. We then numerically integrate each initial condition for up to 800 arbitrary time units (ATU), and color the initial condi- tion according to whether it survives or violates one...

work page
[3]

The complement of the viability region V α, colored black, corresponds to where the total survival time is zero because the agent cannot exist beyond its viability region

This reveals five distinct regions. The complement of the viability region V α, colored black, corresponds to where the total survival time is zero because the agent cannot exist beyond its viability region. The viability boundary ∂Vα trivially separates this portion from the other subsets where the agent persists for at least some time. Within the viabil...

work page
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Viability portrait A phase portrait analysis within the cell’s viability region reveals two equilibrium points: a saddle node EPS = (4 .008, 2.407) (light green point) and a sta- ble equilibrium point EPΩ = (13 .811, 4.686) (dark blue point), shown in Fig. 5(b). Sampling the flow, we can see that the asymptotically viable trajectories (green) con- verge t...

work page
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gives an ordering manifold O shown as a purple con- tour that separates T ′ 1 and T ′ 2 (Fig. 5(c)). Zooming in closer around o and sampling another 160,000 initial con- ditions for improved resolution, we can clearly see that O separates the two regions while the W u branch belongs to T ′ 2 (Fig. 5(d)). This final manifold finishes carving out the differ...

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Model specification Our behaving cell model has the same physiology and viability constraints as the previous model, with the main difference being that we now allow the cell to move in a one-dimensional environment, X, which has opposing gradients of food concentrations, established by the fol- lowing equations and illustrated in Fig. 4(c): [F1] = 10 0.2...

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One representative slice is shown in Fig

Numerical exploration As with the cell physiology in a fixed environment, we approximate the possible existential outcomes for the behaving cell by sampling a million initial con- ditions spanning [Mi] ∈ [0, 20] simulated for 200 ATU, except now for slices of the environment X = [−20, −15, −10, −5, 0, 5, 10, 15, 20]. One representative slice is shown in F...

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Viability Portrait Similar to the protocell analysis without behavior, a phase portrait analysis within Vα reveals a stable equi- librium point EPΩ = (6 .434, 6.434, 0) (dark blue) and a saddle node EPS = (4.020, 4.020, 0) (light green) (Fig. 6(d)). The branches of the saddle node’s one-dimensional unstable manifold W u (red trajectories) either converge ...

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Numerical exploration We sample a million initial conditions within the do- main of interaction of the two cells X and then simu- late their trajectories for 800 ATU, revealing six discon- nected regions of qualitatively distinct survival outcomes (Fig. 8(a)). Regions are given labels according to their sequence of asymptotically and transiently viable ou...

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Viability portrait Decomposing our two-cell state space demands that we understand how our agents’ dynamics unfold within the intersection of their viability regions X . A phase por- trait analysis of the unified state space reveals a stable equilibrium point EPX Ω = (7.167, 4.707) (dark blue point) and a saddle node EPX S = (2 .571, 8.716) (light green) ...

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[1] [1]

This makes the set of molecules autocatalytic

Model specification The single-cell physiology is composed of two molecu- lar species M = {M1, M2} which reciprocally catalyze each other and are synthesized from one of two food molecules F = {F1, F2} respectively. This makes the set of molecules autocatalytic. Fig. 4(a) depicts this cir- cuit with normal arrows representing material transfor- mation, ro...

work page

[2] [2]

Numerical exploration To approximate the possible existential outcomes in the single-cell physiology, we uniformly sample a million initial conditions in the range [Mi] ∈ [0, 20]. We then numerically integrate each initial condition for up to 800 arbitrary time units (ATU), and color the initial condi- tion according to whether it survives or violates one...

work page

[3] [3]

The complement of the viability region V α, colored black, corresponds to where the total survival time is zero because the agent cannot exist beyond its viability region

This reveals five distinct regions. The complement of the viability region V α, colored black, corresponds to where the total survival time is zero because the agent cannot exist beyond its viability region. The viability boundary ∂Vα trivially separates this portion from the other subsets where the agent persists for at least some time. Within the viabil...

work page

[4] [4]

Viability portrait A phase portrait analysis within the cell’s viability region reveals two equilibrium points: a saddle node EPS = (4 .008, 2.407) (light green point) and a sta- ble equilibrium point EPΩ = (13 .811, 4.686) (dark blue point), shown in Fig. 5(b). Sampling the flow, we can see that the asymptotically viable trajectories (green) con- verge t...

work page

[5] [5]

gives an ordering manifold O shown as a purple con- tour that separates T ′ 1 and T ′ 2 (Fig. 5(c)). Zooming in closer around o and sampling another 160,000 initial con- ditions for improved resolution, we can clearly see that O separates the two regions while the W u branch belongs to T ′ 2 (Fig. 5(d)). This final manifold finishes carving out the differ...

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Model specification Our behaving cell model has the same physiology and viability constraints as the previous model, with the main difference being that we now allow the cell to move in a one-dimensional environment, X, which has opposing gradients of food concentrations, established by the fol- lowing equations and illustrated in Fig. 4(c): [F1] = 10 0.2...

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One representative slice is shown in Fig

Numerical exploration As with the cell physiology in a fixed environment, we approximate the possible existential outcomes for the behaving cell by sampling a million initial con- ditions spanning [Mi] ∈ [0, 20] simulated for 200 ATU, except now for slices of the environment X = [−20, −15, −10, −5, 0, 5, 10, 15, 20]. One representative slice is shown in F...

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Viability Portrait Similar to the protocell analysis without behavior, a phase portrait analysis within Vα reveals a stable equi- librium point EPΩ = (6 .434, 6.434, 0) (dark blue) and a saddle node EPS = (4.020, 4.020, 0) (light green) (Fig. 6(d)). The branches of the saddle node’s one-dimensional unstable manifold W u (red trajectories) either converge ...

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Model specification Like the single-cell physiology model, our multi-agent model will be comprised of two cells, α = {α1, α2}, where [M1] and [M2] belong to each of these agents respectively (Fig. 4(b)). We assume that these molecules are synthe- sized from the same precursor food molecule, but beyond this the synthesis is the same as in Eqn. 38: S([Mi]) ...

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Numerical exploration We sample a million initial conditions within the do- main of interaction of the two cells X and then simu- late their trajectories for 800 ATU, revealing six discon- nected regions of qualitatively distinct survival outcomes (Fig. 8(a)). Regions are given labels according to their sequence of asymptotically and transiently viable ou...

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lift function

Viability portrait Decomposing our two-cell state space demands that we understand how our agents’ dynamics unfold within the intersection of their viability regions X . A phase por- trait analysis of the unified state space reveals a stable equilibrium point EPX Ω = (7.167, 4.707) (dark blue point) and a saddle node EPX S = (2 .571, 8.716) (light green) ...

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