BILATERAL BOUNDARY STABILITY IN CONSERVED DYNAMICAL SYSTEMS

The Conservation Constraint as Unified Obstruction to Interior Singularity and Exterior Escape

Bilateral_Boundary_Stability_Gaconnet_2026.pdf

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Don L. Gaconnet

LifePillar Institute for Recursive Sciences

ORCID: 0009-0001-6174-8384

https://osf.io/j5836/files/e2yrm DOI: 10.13140/RG.2.2.31777.06249

January 2026

Keywords: Navier-Stokes regularity, Collatz conjecture, conservation laws, singularity formation, blow-up, dynamical systems, boundary stability, Millennium Prize Problems, turbulence, discrete dynamics

ABSTRACT

Two long-standing problems in mathematics—the Navier-Stokes existence and smoothness problem and the Collatz conjecture—appear unrelated: one concerns continuous fluid dynamics, the other discrete integer iteration. This paper demonstrates they are structurally coupled through a common conservation constraint. We prove the Bilateral Stability

Theorem: for any dynamical system satisfying the conservation constraint ∮ε dt = 0 (where ε represents the exchange differential across a system boundary), the boundary is bilaterally stable—it cannot be breached from inside (singularity formation) or escaped from outside (infinite divergence). Interior singularity and exterior escape are proven to be dual violations of the same conservation law. Applied to Navier-Stokes, the theorem implies that smooth solutions persist globally if and only if the conservation constraint holds—blow-up requires conservation violation. Applied to Collatz-type iteration, the theorem implies all trajectories remain bounded and eventually converge if the discrete conservation constraint holds—escape requires the same violation. The structural coupling generates cross-domain predictions: if singularity is possible in one domain, escape should be possible in the other, and vice versa. Falsification conditions are specified.

1. INTRODUCTION

1.1 Two Problems, One Structure?

The Navier-Stokes existence and smoothness problem asks whether smooth solutions to the three-dimensional incompressible Navier-Stokes equations persist globally or can develop finite-time singularities—points where velocity becomes unbounded (Fefferman, 2006). The problem has resisted solution for over a century and remains one of seven Millennium Prize Problems.

The Collatz conjecture proposes that for any positive integer n, the iteration n → n/2 (if even) or n → 3n+1 (if odd) eventually reaches 1 (Lagarias, 2010). Despite its elementary statement, the conjecture has resisted proof for nearly ninety years. Erdős remarked that 'mathematics is not yet ready for such problems.'

These problems appear unrelated. One is continuous, the other discrete. One concerns fluid physics, the other number theory. One asks about interior blow-up, the other about exterior escape.

This paper demonstrates they are structurally coupled: both test whether a conservation constraint prevents boundary failure, and the failures they test are dual to each other.

1.2 The Conservation Constraint

Consider a dynamical system with an interior domain, an exterior domain, and a boundary between them. Let ε denote the exchange differential—the net flow across the boundary at any instant:

ε = Φ_in - Φ_out

where Φ_in is inward flux and Φ_out is outward flux.

The conservation constraint requires that over any complete cycle, total exchange integrates to zero:

∮ε dt = 0

This permits local imbalance (ε > 0 or ε < 0 at any instant) but prohibits global imbalance over closed paths. What accumulates must dissipate. What flows out must return.

1.3 The Bilateral Claim

Our central claim: the conservation constraint prevents both interior breach and exterior escape simultaneously. These failures are not independent—they are dual violations of the same law.

Interior Breach (Singularity): Unbounded concentration at an interior point. The boundary develops a 'hole' through which interior structure collapses. In Navier-Stokes, this would be finite-time blow-up.

Exterior Escape (Divergence): Unbounded extension without return. A trajectory extends to infinity without cycling back. In Collatz, this would be a sequence that never reaches 1.

We prove that if ∮ε dt = 0 holds, both failures are impossible. If either occurs, ∮ε dt ≠ 0. The conservation constraint is a unified obstruction to boundary failure from either direction.

1.4 Plan of the Paper

Section 2 develops the formal framework. Section 3 proves the Bilateral Stability Theorem. Section 4 applies the theorem to Navier-Stokes. Section 5 applies it to Collatz-type dynamics. Section 6 establishes the structural coupling and its predictions. Section 7 specifies falsification conditions. Section 8 concludes.

2. FORMAL FRAMEWORK

2.1 System Specification

Definition 2.1 (Bounded Dynamical System): A bounded dynamical system S consists of:

(i) Interior domain D_int: a bounded region with internal dynamics

(ii) Exterior domain D_ext: an unbounded region with external dynamics

(iii) Boundary ∂D: the interface between D_int and D_ext

(iv) State function Ψ: S × T → ℝⁿ mapping system configurations over time

(v) Exchange differential ε: T → ℝ measuring net flow across ∂D

2.2 Conservation Constraint

Definition 2.2 (Conservation Constraint): A bounded dynamical system satisfies the conservation constraint if for all closed paths C in state-time space:

∮_C ε dt = 0

Equivalently: the total exchange over any complete cycle integrates to zero. Accumulation phases (ε > 0) exactly balance dissipation phases (ε < 0).

Proposition 2.1: The conservation constraint is equivalent to the existence of a potential function Φ such that ε = -dΦ/dt.

Proof: Standard result from conservative dynamics. If ε = -dΦ/dt, then ∮ε dt = -∮dΦ = -[Φ]_cycle = 0 for any closed path. Conversely, if ∮ε dt = 0 for all closed paths, then ε is an exact differential and a potential exists. □

2.3 Stability Definitions

Definition 2.3 (Interior Stability): The interior D_int is stable if no finite-time singularity forms—no point x ∈ D_int has ||Ψ(x,t)|| → ∞ as t → T for finite T.

Definition 2.4 (Exterior Stability): The exterior D_ext is stable if all trajectories remain bounded—no trajectory γ ⊂ D_ext has ||γ(t)|| → ∞ without returning to bounded regions.

Definition 2.5 (Bilateral Stability): A bounded dynamical system is bilaterally stable if both D_int is interior stable and D_ext is exterior stable.

2.4 Failure Modes

Definition 2.6 (Interior Breach): Interior breach occurs when there exists x* ∈ D_int and T* < ∞ such that ||Ψ(x*,t)|| → ∞ as t → T*. The interior 'blows up' in finite time.

Definition 2.7 (Exterior Escape): Exterior escape occurs when there exists trajectory γ ⊂ D_ext such that ||γ(t)|| → ∞ as t → ∞ without γ entering any bounded region. The trajectory 'escapes to infinity.'

Bilateral stability means: no interior breach AND no exterior escape.

3. THE BILATERAL STABILITY THEOREM

3.1 Statement

Theorem 3.1 (Bilateral Stability): If a bounded dynamical system S satisfies the conservation constraint ∮ε dt = 0, then S is bilaterally stable: neither interior breach nor exterior escape can occur.

3.2 Proof

Lemma 3.1 (No Interior Breach): If ∮ε dt = 0, then interior breach cannot occur.

Proof: Suppose interior breach occurs at point x* at time T*. Then ||Ψ(x*,t)|| → ∞ as t → T*.

For ||Ψ(x*,t)|| to become unbounded, the state at x* must accumulate without bound. This requires sustained net inflow to x*: ε(x*) > 0 persistently as t → T*.

Consider a small region R containing x* and a time interval [T* - δ, T*]. The total exchange into R over this interval is:

∫_{T*-δ}^{T*} ε dt

For blow-up to occur, this integral must be unbounded (infinite accumulation in finite time).

But the conservation constraint requires ∮ε dt = 0 over any complete cycle. If accumulation occurs without corresponding dissipation, the integral over a cycle containing [T* - δ, T*] cannot close to zero.

More precisely: Let C be a cycle through the accumulation phase. If ∫_{accumulation} ε dt → ∞, then for ∮_C ε dt = 0, we need ∫_{dissipation} ε dt → -∞. But dissipation is bounded below by the system's total content—you cannot dissipate more than exists.

Therefore unbounded accumulation violates conservation. Interior breach implies ∮ε dt ≠ 0. Contrapositive: ∮ε dt = 0 implies no interior breach. □

Lemma 3.2 (No Exterior Escape): If ∮ε dt = 0, then exterior escape cannot occur.

Proof: Suppose exterior escape occurs: trajectory γ has ||γ(t)|| → ∞ as t → ∞ without returning.

For ||γ(t)|| → ∞, the trajectory must persistently move outward. This requires sustained net outflow: ε < 0 along γ persistently.

Consider the path integral along γ:

∫_γ ε dt < 0 (persistent outflow)

For ∮ε dt = 0 over any closed path, outflow along γ must be balanced by inflow along the return path. But if γ never returns (escape), there is no return path. The cycle cannot close.

Formally: escape implies existence of trajectory with ∫_γ ε dt < 0 and no compensating path. This means ∮ε dt ≠ 0 for any cycle containing γ—which is impossible if the system satisfies conservation.

Therefore escape violates conservation. Exterior escape implies ∮ε dt ≠ 0. Contrapositive: ∮ε dt = 0 implies no exterior escape. □

Theorem 3.1 Proof: By Lemma 3.1, conservation implies no interior breach. By Lemma 3.2, conservation implies no exterior escape. Therefore conservation implies bilateral stability. □

3.3 The Duality

Theorem 3.2 (Duality of Failures): Interior breach and exterior escape are dual violations of the conservation constraint.

Proof:

Interior breach requires: ∫ε dt → +∞ (unbounded accumulation, ε > 0 dominates)

Exterior escape requires: ∫ε dt → -∞ (unbounded dissipation, ε < 0 dominates)

These are sign-opposite violations of the same constraint ∮ε dt = 0.

If conservation holds: neither +∞ nor -∞ accumulation is possible. Both failures blocked.

If conservation fails: the failure has a sign. Positive failure (∫ε dt > 0) enables interior breach.

Negative failure (∫ε dt < 0) enables exterior escape.

The failures are dual: they lie on opposite sides of the conservation constraint. □

Corollary 3.1 (Structural Coupling): If interior breach is possible in a system, then exterior escape is possible in a structurally coupled system with reversed exchange sign, and vice versa.

4. APPLICATION TO NAVIER-STOKES

4.1 The Navier-Stokes Equations

The incompressible Navier-Stokes equations describe fluid flow:

∂u/∂t + (u · ∇)u = -∇p + ν∇²u

∇ · u = 0

where u is velocity, p is pressure, and ν is kinematic viscosity.

The regularity question: given smooth initial data u(x,0) with finite energy, does the solution remain smooth for all time, or can finite-time singularity (blow-up) develop?

4.2 Energy Conservation

The Navier-Stokes equations admit an energy identity. For smooth solutions:

d/dt ∫|u|² dx = -2ν ∫|∇u|² dx

Energy decreases due to viscous dissipation. For finite-energy initial data, total energy remains bounded:

∫|u(t)|² dx ≤ ∫|u(0)|² dx

This is a global bound on total energy. The regularity question concerns whether local concentrations can become unbounded despite global bounds.

4.3 The Conservation Mapping

Proposition 4.1: The Navier-Stokes energy identity corresponds to the conservation constraint ∮ε dt = 0 when ε represents local energy exchange.

Derivation: Define the local exchange differential ε(x,t) as the rate of energy transfer at point x:

ε(x,t) = (u · ∇)u · u + u · ∇p - ν|∇u|²

The first term represents convective transfer; the second, pressure work; the third, viscous dissipation.

For incompressible flow (∇ · u = 0), the pressure term integrates to zero over the domain. The convective term redistributes energy without creating or destroying it. Only viscous dissipation removes energy.

Over any closed region and time cycle, the conservation constraint ∮ε dt = 0 holds: energy is redistributed, not created.

4.4 Application of Bilateral Stability

Theorem 4.1: If Navier-Stokes solutions satisfy the conservation constraint (energy identity), then finite-time blow-up cannot occur.

Proof: By Theorem 3.1 (Bilateral Stability), if ∮ε dt = 0, then no interior breach occurs. Finite-time blow-up IS interior breach: ||u(x*,t)|| → ∞ as t → T* for some x* and finite T*. Therefore conservation implies no blow-up. □

Interpretation: Blow-up would require unbounded local energy concentration. But the energy identity constrains total energy and the conservation structure prevents unbounded local accumulation from the global bound. The energy cannot concentrate infinitely at a point while remaining globally bounded—this would violate the conservation structure.

4.5 Relation to Existing Results

This result connects to known regularity conditions:

• Leray (1934) established global weak solutions but left regularity open

• Serrin (1962) and subsequent work identified sufficient conditions for regularity in terms of velocity bounds

• Tao (2016) constructed blow-up for an averaged Navier-Stokes equation, demonstrating

that the specific structure of NS, not just its form, matters

The bilateral stability theorem identifies the conservation structure as the critical factor: blow-up requires violation of conservation, and the Navier-Stokes conservation structure may be too rigid to permit such violation.

5. APPLICATION TO COLLATZ-TYPE DYNAMICS

5.1 The Collatz Conjecture

The Collatz function C: ℕ → ℕ is defined by:

C(n) = n/2 if n is even

C(n) = 3n + 1 if n is odd

The conjecture: for all n ≥ 1, repeated iteration of C eventually reaches 1.

Equivalently: no trajectory escapes to infinity, and no non-trivial cycle exists other than {1, 4, 2}.

5.2 Discrete Conservation

Collatz iteration involves discrete rather than continuous dynamics. We define a discrete analogue of the conservation constraint.

Definition 5.1: For Collatz iteration, define the exchange differential at step k as:

ε_k = log(C(n_k)) - log(n_k) = log(C(n_k)/n_k)

This measures the multiplicative change at each step. For even n: ε = log(1/2) = -log 2 < 0 (contraction). For odd n: ε = log((3n+1)/n) ≈ log 3 > 0 for large n (expansion).

Definition 5.2: A Collatz trajectory satisfies discrete conservation if the sum over any cycle is zero:

Σ_cycle ε_k = 0

5.3 The Density Argument

Proposition 5.1: For almost all n, long Collatz trajectories satisfy Σε_k < 0.

Heuristic derivation: Even numbers halve (ε = -log 2 ≈ -0.693). Odd numbers nearly triple (ε ≈ log 3 ≈ 1.099). But odd numbers produce even numbers (3n+1 is even), guaranteeing at least one halving per odd step.

Average behavior: Starting from random n, about half the steps are 'halving-only' (even preceded by even) and half are 'triple-then-halve' (odd followed by mandatory even). Net average:

⟨ε⟩ ≈ (1/2)(-log 2) + (1/2)(log 3 - log 2) = (1/2)(log 3 - 2log 2) = (1/2)log(3/4) < 0

The expected exchange is negative: trajectories contract on average.

5.4 Application of Bilateral Stability

Theorem 5.1: If Collatz trajectories satisfy the discrete conservation constraint (no net escape), then no trajectory escapes to infinity.

Proof: By Theorem 3.1 (Bilateral Stability), if the conservation constraint holds, no exterior escape occurs. Escape to infinity IS exterior escape: ||n_k|| → ∞ as k → ∞. Therefore conservation implies no escape. □

Proposition 5.2: A divergent Collatz trajectory (if one exists) must violate the density argument—it must have Σε_k → +∞.

Proof: For n_k → ∞, we need log(n_k) → ∞. Since log(n_k) = log(n_0) + Σ_{j=0}^{k-1} ε_j, divergence requires Σε_j → +∞. But the density argument shows typical trajectories have Σε_j → -∞. A divergent trajectory must be atypical—it must find a path that systematically avoids the contractive steps. □

5.5 The Structural Question

The Collatz conjecture reduces to: does a path exist that systematically achieves Σε_k → +∞?

The bilateral stability theorem rephrases this as: does a conservation-violating trajectory exist?

If the Collatz iteration preserves a hidden conservation structure—if the seeming chaos conceals a closure principle—then escape is impossible. The conjecture becomes: prove the conservation structure holds for all trajectories.

6. STRUCTURAL COUPLING AND CROSS-DOMAIN PREDICTIONS

6.1 The Coupling

Theorems 4.1 and 5.1 reveal structural coupling between Navier-Stokes regularity and Collatz convergence: both are tests of bilateral stability under conservation.

• Navier-Stokes tests interior stability: can the boundary be breached from inside?

• Collatz tests exterior stability: can the boundary be escaped from outside?

• Both are prevented by the same conservation constraint: ∮ε dt = 0

6.2 Cross-Domain Predictions

The coupling generates testable predictions:

Prediction 6.1 (Symmetry of Failure): If blow-up is proven possible for Navier-Stokes under some conditions, then escape should be possible for a Collatz-type system with analogous conditions.

Specifically: Tao (2016) constructed blow-up for an averaged Navier-Stokes equation that breaks the fine structure of the original. By duality, a 'averaged Collatz' breaking the fine structure should permit escape.

Prediction 6.2 (Symmetry of Regularity): If all Collatz trajectories provably converge, the proof mechanism should transfer to Navier-Stokes regularity.

Specifically: if Collatz convergence is proven via a conservation structure, the analogous structure should establish Navier-Stokes regularity.

Prediction 6.3 (Coupled Counterexamples): A counterexample to Collatz (divergent trajectory) and a counterexample to Navier-Stokes (blow-up) should be structurally related—both violating the conservation constraint in dual ways.

6.3 Why the Problems Are Hard

The structural coupling explains why both problems have resisted solution:

1. They are not independent problems admitting separate solutions. They are dual manifestations of a single constraint.

2. The conservation structure may be too subtle for direct verification. Proving ∮ε dt = 0 for all trajectories/flows requires global analysis.

3. The failures, if they exist, would be highly atypical—conservation-violating paths in a conservation-respecting landscape.

4. The problems test the limits of mathematical analysis: when does local dynamics guarantee global conservation?

6.4 The Unified Claim

Summary: Navier-Stokes regularity and Collatz convergence are not two problems but two aspects of one problem: does the conservation constraint ensure bilateral boundary stability in continuous and discrete domains?

If yes: both problems are resolved (smooth solutions persist; all trajectories converge).

If no: both admit counterexamples (blow-up and escape are possible).

If mixed: the structural coupling is wrong—which would falsify the bilateral stability framework.

7. FALSIFICATION CONDITIONS

7.1 Framework Predictions

The framework generates specific falsifiable predictions:

F1 (Conservation Implies Regularity): If a fluid system satisfies energy conservation (energy identity holds), then no finite-time blow-up occurs.

Falsification: Demonstrate blow-up in a system provably satisfying conservation. (Note: Tao's averaged NS blow-up occurs in a system that explicitly breaks conservation structure.)

F2 (Conservation Implies Convergence): If an iterative system satisfies discrete conservation (net exchange is zero), then no trajectory escapes.

Falsification: Demonstrate escape in a system provably satisfying discrete conservation.

F3 (Duality of Failures): Interior breach (blow-up) and exterior escape (divergence) are sign-opposite violations of conservation.

Falsification: Demonstrate blow-up with ∫ε dt < 0 or escape with ∫ε dt > 0, contradicting the sign assignments.

F4 (Structural Coupling): Navier-Stokes and Collatz are coupled through the conservation constraint.

Falsification: Resolve one problem while the other remains open, with no transfer of proof mechanism—demonstrating the problems are structurally independent.

7.2 Experimental Approaches

Numerical experiments can test framework predictions:

• Compute ∮ε dt for near-singular Navier-Stokes flows; verify conservation violation precedes blow-up

• Compute Σε_k for extremely long Collatz trajectories; verify contraction dominates

• Search for Collatz-like iterations that provably escape; verify they violate discrete conservation

• Construct NS-like equations with broken conservation; verify blow-up becomes possible

7.3 What Would Not Falsify the Framework

The following would not constitute falsification:

• Proof of Navier-Stokes regularity by methods not invoking conservation (compatible, not conflicting)

• Proof of Collatz by methods not invoking conservation (compatible, not conflicting)

• Numerical evidence of near-singular behavior (not actual singularity)

• Extremely long Collatz trajectories (not actual escape)

Falsification requires demonstrated violation of the framework's specific predictions, not mere alternative approaches.

8. CONCLUSION

We have established the Bilateral Stability Theorem: the conservation constraint ∮ε dt = 0 prevents both interior breach (singularity formation) and exterior escape (infinite divergence). These failures are dual violations of a single law—sign-opposite deviations from conservation.

Applied to Navier-Stokes, the theorem implies that blow-up requires conservation violation. The energy identity establishes conservation for smooth solutions; the question reduces to whether solutions remain smooth enough to satisfy the identity.

Applied to Collatz, the theorem implies that escape requires systematically violating the density argument. Almost all trajectories contract on average; escape requires finding paths that persistently expand—paths that violate the hidden conservation structure.

The structural coupling between these problems is the central insight. They are not independent puzzles but dual aspects of bilateral boundary stability. Resolution of one should illuminate the other.

The framework does not solve either problem. It reframes them: from 'can singularity form?' to 'can conservation be violated?' and from 'can trajectories escape?' to 'can conservation be violated from the other direction?' The unified question is whether the conservation constraint holds universally in each domain.

The boundary holds—or it fails in dual ways. The mathematics will determine which.

REFERENCES

Constantin, P., & Foias, C. (1988). Navier-Stokes Equations. University of Chicago Press.

Fefferman, C. L. (2006). Existence and smoothness of the Navier-Stokes equation. In The Millennium Prize Problems (pp. 57-67). Clay Mathematics Institute.

Gaconnet, D. L. (2025). The Echo-Excess Principle: Substrate Law of Generative Existence. LifePillar Institute.

Gaconnet, D. L. (2025). Cognitive Field Dynamics: A Unified Theory. LifePillar Institute.

Gaconnet, D. L. (2026). The Triadic Minimum for Non-Equilibrium Steady States. LifePillar Institute.

Lagarias, J. C. (Ed.). (2010). The Ultimate Challenge: The 3x+1 Problem. American Mathematical Society.

Leray, J. (1934). Sur le mouvement d'un liquide visqueux emplissant l'espace. Acta Mathematica, 63(1), 193-248.

Serrin, J. (1962). On the interior regularity of weak solutions of the Navier-Stokes equations. Archive for Rational Mechanics and Analysis, 9(1), 187-195.

Tao, T. (2016). Finite time blowup for an averaged three-dimensional Navier-Stokes equation. Journal of the American Mathematical Society, 29(3), 601-674.

Terras, R. (1976). A stopping time problem on the positive integers. Acta Arithmetica, 30, 241-252.

DOCUMENT CONTROL

Title: Bilateral Boundary Stability in Conserved Dynamical Systems: The Conservation Constraint as Unified Obstruction to Interior Singularity and Exterior Escape

Author: Don L. Gaconnet

Institution: LifePillar Institute for Recursive Sciences

ORCID: 0009-0001-6174-8384

Date: January 2026

Classification: Mathematical Physics / Dynamical Systems / Millennium Prize Problems

Keywords: Navier-Stokes regularity, Collatz conjecture, conservation laws, singularity formation, blow-up, bilateral stability, boundary stability, Millennium Prize Problems

© 2026 Don L. Gaconnet. All Rights Reserved.