Routed Reorganization:

A Predictive Extension of Encoded Equilibrium

DOI: To be assigned.

John Swygert

July 7, 2026

01 Purpose of This Paper

The preceding TSTOEAO notes on domino-like phase transformation in monolayer MoTe₂ established a necessary distinction: a routed transformation is not merely defect-mediated transformation, nucleation-and-growth, or ordinary interface activity. It is a transformation in which the specific pathway of crossing actively governs accessibility, direction, intermediate states, functional properties, and controllability.

This paper takes the next step.

It generalizes the routed-transformation concept into a broader cross-domain principle called routed reorganization. It then converts that principle into a falsifiable predictive protocol that can be applied beyond a single materials-science case.

The goal is not to rename existing mechanisms.

The goal is to identify when the route itself is doing causal work that endpoint or bulk descriptions may under-predict.

A route is not merely where a transition begins.

A route is the structured crossing by which one encoded equilibrium becomes accessible from another.

02 Origin of the Concept in the MoTe₂ Case

The immediate scientific trigger for this paper is the reported one-dimensional domino-like phase-transformation pathway in monolayer MoTe₂. The PNAS study identifies that transformation in monolayer MoTe₂ can proceed in a one-dimensional, domino-like manner, rather than only through a conventional broad martensitic shear route.

The Chinese Academy of Sciences summary explains that the newly identified mechanism is distinct from the conventional martensitic model, in which many atoms move together through concerted shear displacements. Instead, the transformation proceeds through a one-dimensional chain reaction, offering a lower-barrier route and opening possibilities for programmable electronic and photonic devices.

This matters because the transformation is not explained only by naming the starting phase and ending phase.

The missing explanatory object is the route.

In the MoTe₂ case, the route is lower-dimensional, sequential, boundary-associated, and functionally consequential. That makes it a strong case study for TSTOEAO, but it does not prove TSTOEAO. It does something more careful: it gives a concrete materials-science example of a broader transition pattern TSTOEAO had already emphasized.

03 Core Claim

Routed reorganization occurs when a system under gradient or disequilibrium crosses from one encoded equilibrium to another through a boundary-gated, lower-dimensional, or sequentially structured pathway whose specific geometry and dynamics determine the transformation’s cost, direction, metastable landscape, functional intermediates, and programmability.

In TSTOEAO terms:

E is the energetic gradient, disequilibrium, or available capacity driving change.

Y is the route-governing structure: boundary conditions, sequential pathway, dimensional constraint, interface geometry, or other ordering condition that makes the crossing possible.

V is the realized reorganization, including any functional states that appear during or because of the crossing.

The central assertion is:

When bulk or endpoint models predict barriers or outcomes that do not match observation, the actual route of crossing may be lower-dimensional, boundary-initiated, sequential, and functionally consequential.

This is not always true.

It should not be treated as a universal law.

But it appears important enough, and testable enough, to justify a dedicated search protocol.

04 Criteria for Routed Reorganization

A transformation may qualify as routed reorganization when it meets several of the following conditions. The more conditions that are met, the stronger the classification becomes.

First, there is barrier mismatch. The observed transition occurs under conditions more accessible than the assumed bulk, concerted, or endpoint model would predict.

Second, there is a local gate. A boundary, kink, interface, edge, defect, dimensional constraint, field concentration, or stress concentration opens the transition.

Third, there is a sequential or lower-dimensional pathway. The crossing proceeds through an ordered sequence rather than through uniform reorganization across the full dimensionality of the system.

Fourth, direction matters. The transformation favors a specific axis, channel, crystallographic direction, domain route, interface line, or structural path.

Fifth, intermediate states matter. Metastable or transitional states appear that possess measurable properties not present, or not as strongly present, in either endpoint state.

Sixth, the route can be altered. Modifying the boundary condition, geometry, strain, field, defect structure, or local constraint predictably changes the pathway, intermediates, domain outcome, or final equilibrium.

Seventh, the route is explanatorily necessary. Removing or ignoring the specific route structure significantly weakens the explanation of why the transformation occurs under the observed conditions.

These criteria are deliberately strict.

They are meant to prevent the term routed reorganization from becoming a vague label for any transition that happens near a defect or boundary.

05 Predictive Protocol: The Route-First Search

When a system exhibits barrier mismatch, unexpectedly accessible transformation, or functional behavior during transition that endpoint or bulk models do not adequately explain, TSTOEAO proposes a route-first search protocol.

First, map the accessible boundaries, interfaces, kinks, edges, defects, domain walls, field concentrations, and dimensional constraints.

Second, test for evidence of sequential, directional, or lower-dimensional propagation rather than broad uniform change.

Third, characterize intermediate states that appear during crossing. Measure structure, energetics, duration, reproducibility, and function.

Fourth, perturb the suspected route through strain, field, geometry, interface engineering, thermal cycling, defect control, or boundary manipulation.

Fifth, compare explanatory power. Ask whether treating the route as causal improves prediction of accessibility, direction, intermediate states, or final outcome over models that treat the route as secondary.

This protocol can fail.

If perturbation of the supposed route produces no change, the routed-reorganization classification is weakened.

If the transformation is fully explained without reference to route structure, the classification is weakened.

If intermediate states are not reproducible or have no independent measurable function, the functional-intermediate claim is weakened.

That is important.

The route-first protocol is not meant to be decorative language. It is meant to expose the claim to test.

06 Testable Prediction One: Low-Dimensional Materials

In low-dimensional materials, TSTOEAO predicts that some phase or structural transitions occurring at lower driving forces than bulk or concerted models predict will be found to proceed through boundary-routed, quasi-one-dimensional, or sequential pathways.

The MoTe₂ case is one realized example. The reported transformation proceeds in a one-dimensional domino-like manner, with the PNAS paper connecting this mechanism to material programming in two-dimensional MoTe₂.

The broader prediction is not that every two-dimensional material will behave like MoTe₂.

The prediction is that where barrier mismatch appears in low-dimensional systems, researchers should look for lower-dimensional routes, interface gates, directional propagation, and metastable intermediates before concluding that the transformation is inexplicable or purely bulk-driven.

Possible search domains include other transition metal dichalcogenides, phase-change monolayers, domain-wall materials, strain-engineered lattices, and two-dimensional systems where switching behavior occurs under surprisingly accessible conditions.

07 Testable Prediction Two: Metamaterials and Boundary-Lattice Systems

In metamaterials and engineered boundary-lattice systems, TSTOEAO predicts that the most consequential behavior may appear not in the open bulk of the material but at structured interfaces, constrained pathways, compound boundary axes, or deliberately engineered transition routes.

This follows directly from V = E × Y.

Energy does not become useful merely because it exists.

Energy becomes value when it passes through structure.

In routed reorganization, Y is the route-governing condition. It may be a lattice, interface, strain pattern, defect geometry, field boundary, waveguide, resonant path, or phase-transition channel.

The prediction is that carefully engineered boundary routes may produce sharp local peaks in coherence, switching efficiency, nonlinear response, energy transfer, or controllable phase behavior that are not present in less-structured controls.

This is testable.

If boundary geometry, route structure, or interface tuning changes the functional output, the routed-reorganization interpretation gains strength.

If it does not, the claim weakens.

08 Testable Prediction Three: Biological Systems

In biological systems, TSTOEAO predicts that some reorganizations that occur more readily than expected from bulk energetics may depend on boundary- or interface-gated routes.

Possible examples include conformational changes, membrane phase transitions, cytoskeletal reorganization, protein assembly, signaling complexes, or partially assembled biological structures.

This section must be stated cautiously.

Biological systems are not crystals.

A biological route is not the same thing as tellurium-atom hopping in MoTe₂.

The claim is not identity across domains.

The claim is structural analogy plus testable expectation.

Where biological reorganization appears too accessible for a simple bulk-energy description, researchers should ask whether a membrane boundary, protein interface, scaffold, channel, local field, mechanical constraint, or partial assembly state provides the route by which the transition becomes possible.

The predicted signature would be functional intermediate states.

A partially folded, partially assembled, boundary-bound, or scaffolded condition may carry function that is not present in either endpoint state.

If such intermediates are reproducible and functionally measurable, routed reorganization becomes a useful interpretive and predictive lens.

If they are not, the claim should be narrowed.

09 Testable Prediction Four: Computational and Information Systems

In computational and information systems, routed reorganization should be treated as an analogy, not as a direct materials claim.

The prediction is that some learning, search, annealing, or network-state transitions that appear to bypass expected barriers may do so through constrained routes: lower-dimensional subspaces, sparse subnetworks, attention bottlenecks, interface layers, latent channels, or structured intermediate representations.

The key question is not whether a computer literally undergoes a physical crystal phase change in the same way MoTe₂ does.

It does not.

The question is whether complex information systems sometimes reorganize through constrained routes that reduce search cost and produce functional intermediates.

If so, TSTOEAO predicts that those routes will be disproportionately important for explaining sudden capability shifts, compression behavior, representation formation, and task-specific transitions.

A testable version would ask whether perturbing the suspected route changes the system’s transition behavior more strongly than perturbing non-route components of similar size or energy cost.

If it does, routed reorganization may be a useful model.

If it does not, the analogy weakens.

10 General Prediction

Across domains, systems that display both barrier mismatch and functional behavior during transition are more likely to be governed by routed reorganization than systems that display neither.

This is the general prediction.

It is deliberately restrained.

It does not say every transition is routed.

It does not say every boundary matters.

It does not say every intermediate is functional.

It says that the combination of barrier mismatch plus functional intermediate behavior should trigger a route-first investigation.

That is the predictive contribution.

The route is where the missing explanation may reside.

11 Relation to Encoded Equilibrium

Routed reorganization is a concrete expression of encoded equilibrium.

An encoded equilibrium is not merely a resting state. It is a stabilized arrangement produced by prior constraints, crossings, interactions, and route histories.

The final state carries the history of how it became accessible.

That means the route matters.

Two systems may arrive at superficially similar endpoints through different routes and retain different internal structure, domain morphology, functional capacity, or future transition behavior.

This is why endpoint comparison alone may be insufficient.

TSTOEAO predicts that route history can become encoded into the resulting equilibrium.

The new equilibrium is not only what the system is.

It is also what the system crossed through to become what it is.

12 Relation to V = E × Y

Routed reorganization clarifies the Y term in V = E × Y.

E is the available energy, gradient, pressure, disequilibrium, or capacity for change.

Y is the organizing condition that determines how that energy can become value.

V is the realized value, function, stabilized arrangement, or encoded equilibrium.

Without Y, E may remain inaccessible, chaotic, dissipative, or trapped behind a barrier.

With the right Y, E may become organized into a lower-cost crossing.

In routed reorganization, Y is not abstract.

Y can be a boundary.

Y can be a kink.

Y can be a dimensional constraint.

Y can be a sequential route.

Y can be a lattice pathway.

Y can be an interface condition.

Y can be the structure that turns energetic possibility into realized transformation.

That is why the route is not decorative.

The route is where E becomes V.

13 How This Claim Can Fail

The routed-reorganization framework would be weakened in a given domain if barrier mismatch is routinely resolved by existing bulk or endpoint models without route-specific explanation.

It would be weakened if perturbing suspected routes produces no measurable change in accessibility, direction, intermediate states, function, or final equilibrium.

It would be weakened if intermediate states lack reproducible independent properties.

It would be weakened if the language of route adds no predictive or explanatory value beyond standard defect mediation, nucleation-and-growth, martensitic theory, interface science, biological assembly theory, or computational optimization theory.

These failure conditions matter.

A theory should not win by definition.

A theory should only gain strength when it makes observation sharper, prediction better, or experiment more directed.

14 Why This Extension Matters

The preceding notes protected TSTOEAO from overclaiming by distinguishing routed transformation from ordinary defect mediation.

This paper converts that protected distinction into a forward-looking predictive instrument.

It does so by supplying operational criteria, offering a repeatable search protocol, generating cross-domain predictions, and maintaining continuity with V = E × Y and encoded equilibrium.

If the predictions hold in even a subset of the suggested domains, routed reorganization becomes a useful refinement for identifying where and how systems achieve new equilibria under constraint.

If the predictions do not hold, the criteria and protocol provide clear points at which the claim should be narrowed or abandoned.

That is the right standard.

The framework should not be protected from failure.

It should be sharpened by testing.

15 The Careful Public Claim

The careful public claim is this:

Routed reorganization is a proposed explanatory and predictive class of transformation in which the route of crossing is causally significant. It applies when the specific pathway, not merely the starting state or ending state, governs accessibility, direction, intermediate function, and controllability.

This claim does not replace existing science.

It refines a search strategy.

It says:

When the endpoint model does not explain the crossing, look for the route.

When the bulk model overstates the barrier, look for the gate.

When the transition produces surprising function, measure the intermediate.

When control is desired, perturb the pathway.

When the pathway changes the result, the route is not secondary.

The route is part of the mechanism.

Conclusion

Routed reorganization extends the MoTe₂ routed-transformation insight into a broader predictive framework.

The MoTe₂ result showed that a two-dimensional material can undergo phase transformation through a one-dimensional domino-like pathway, lowering the barrier and opening functional possibilities. That finding does not prove TSTOEAO, but it gives TSTOEAO a concrete materials-science alignment case and a disciplined way to move forward.

The next step is prediction.

TSTOEAO predicts that where systems under gradient or disequilibrium transform more readily than bulk or endpoint models suggest, researchers should search for boundary-gated, lower-dimensional, sequential, or otherwise structured routes of crossing.

The route may lower the cost.

The route may impose direction.

The route may expose functional intermediates.

The route may determine the final encoded equilibrium.

The route may become programmable.

That is the value of routed reorganization.

The route is not decorative.

When it governs the crossing, it is often the missing variable that makes the next equilibrium intelligible and, in some cases, engineerable.

References

Liu, Xiangyang, et al. “1D Domino-Like Phase Transformation Enables Material Programming in 2D MoTe₂.” Proceedings of the National Academy of Sciences, 2026. DOI: 10.1073/pnas.2528037123.

Chinese Academy of Sciences. “Scientists Discover Novel Domino-Like Phase Transformation Mechanism with Implications for Functional Devices.” 2026.

Phys.org. “Atomic ‘Domino Effect’ Found to Drive Phase Changes in a Two-Dimensional Crystal.” July 2026.