Appendix D: Compendium of Formulated Problems

The following problem set turns the framework into a graduated research program: resource-bounded distinction $\to$ attention topology $\to$ non-reciprocal mechanics $\to$ spectral, control-theoretic, and topological analysis.

Part I: Non-Reciprocal Active Mechanics and Physical Systems

Problem 1: The Non-Reciprocal Triplet Engine

Three identical point particles of mass $m$ move on a frictionless line from $x_1(0)=0$, $x_2(0)=a$, and $x_3(0)=b$, where $0<a<b/2$. Each particle selects its unique farthest neighbor and obeys $F_i=G\sum_{j\neq i}A_{ij}(x)(x_j-x_i)$.

  x = 0               x = a                               x = b
  (1) ─────────────── (2) ─────────────────────────────── (3)
   │                   │                                   │
   └───────◄───────────┴───────────────►───────────────────┘

At the initial state, $q_1=3$, $q_2=3$, and $q_3=1$. Hence $F_{\mathrm{tot}}=G(b-a)$, so

Before the first switch, particles 1 and 3 form a mutual pair. With $\omega=\sqrt{2G/m}$ and $\Omega=\sqrt{G/m}$,

The first topological switch is the first positive solution of

At the switch, $x_2(t^*)=(x_1(t^*)+x_3(t^*))/2=b/2$. Therefore

independent of $G$ and $m$. The coupling and mass set the clock, while the switch configuration is determined by the topology skeleton.

Problem 2: The Asymmetric Binary Propellor

For two particles with drag $-\gamma\mathbf v_i$, separation $\mathbf r=\mathbf x_2-\mathbf x_1$, and pair forces $G(1+\alpha)\mathbf r$ and $-G(1-\alpha)\mathbf r$,

            F_21 = -G(1-α)r
       (1) ◄─────────────── (2)
        └─────────────────►
           F_12 = G(1+α)r

Integrating from zero to infinity gives

The non-reciprocal work channel is

which is exactly the center-of-mass drag dissipation.

Problem 3: Column-Imbalance Drift and Representation Sinks

Let $A$ be row-stochastic. Followers $i\ge2$ attend to leader 1 with weight $\alpha$, while leader 1 attends to particle 2 with weight $\beta$. Then $\boldsymbol\delta=A^T\mathbf1-\mathbf1$ has components

           ┌────────── (Weight α) ──────────┐
           ▼                                │
      (Leader 1) ───► (Follower 2)     (Followers 3...N)
           │   (Weight β)  ▲                │
           └───────────────┴────────────────┘

The center-of-mass equation is

The leader's sink/source threshold is exact:

The sign of the actual drift remains configuration-dependent through $\boldsymbol\delta^T\mathbf x$; the threshold is not, by itself, a universal theorem about the direction of self-propulsion.

Part II: Geometry, Stability, and Pregeometric Bounds

Problem 4: The Graph Stability Radius Theorem

For a configuration with unique farthest neighbors, let $\eta_i$ be the gap between the largest and second-largest distances and let $\eta_{\min}=\min_i\eta_i>0$. Since each pairwise distance changes by less than $2\delta$ under $\|\mathbf y_i-\mathbf x_i\|<\delta$,

This gives a quantitative stability radius for the functional digraph.

Problem 5: Pregeometric Scale Invariance and the Identity Boundary

An undifferentiated pregeometric state with no internal ruler is invariant under $x\mapsto\lambda x$ for $\lambda>0$ until the first operational distinction breaks the symmetry. If the boundary projection maps Absent, Latent, Stalled, and Unresolved antecedents to the same accessible state $D_1$, those antecedents are observationally unidentifiable from the operational history of $D_1$ alone.

Part III: The Open Research Program

Problem 6: Spectral Classification of Functional Laplacians

For a binary row-stochastic functional graph $A_q$ whose cycles have length two, analyze $L_q=I-A_q$. Prove the zero-eigenvalue multiplicity bound associated with connected components and determine the real parts of the remaining modes.

Problem 7: Zeno-Free Dwell Time under Hysteresis

Under the switching rule $d_{ik}-d_{ij}>h$, bounded velocities $\|\mathbf v_i\|\le v_{\max}$ imply a positive dwell-time bound of the form

Establish the hypotheses under which this excludes Zeno execution and gives global well-posedness.

Problem 8: Existence of Active Limit Cycles

For the nondimensional switched system, derive the closed-orbit balance

Use it to obtain necessary bounds for stable non-conservative limit cycles.

Problem 9: Effective Graph Complexity and Persistent Homology

Combine softmax witness selection with a capacity cutoff and characterize the effective graph. Bound the Betti numbers $\beta_p(K_W)=\dim H_p(K_W)$ in terms of body count and ambient dimension.

Problem 10: The Dimensionless Phase Diagram

Map the parameter space $(N,\hat\varepsilon,\zeta)$ across frozen-topology, smooth-relaxation, Filippov-sliding, active-limit-cycle, and chaotic-switching regimes. The transition boundaries must be stated with their hypotheses; labels such as $\zeta\gg1$ or $N\gg1$ are scaling regimes, not exact phase boundaries.

Master Index

The resulting research arc is: