Research-Stack/6-Documentation/docs/specs/lonely_runner_betti_mapping.md

# Lonely Runner Conjecture — Betti-2 Topological Obstruction Mapping

**Document ID:** FS-LR-B2-2026-06-16  
**Status:** BEAUTIFUL_PROVISIONAL — theoretical mapping, not a Lean theorem  
**Framework:** NK-Hodge-FAMM topological obstruction (β₂ = 0 regularity condition)  
**Claim boundary:** Establishes isomorphism of problem structure; does not prove the conjecture

---

## 1. Problem Restatement

Let $k$ runners $R_1, \dots, R_k$ have distinct constant speeds $v_i \in \mathbb{R}^+$
on a circular track $S^1 \cong \mathbb{R}/\mathbb{Z}$ of circumference $1$, all starting
at the same point $0 \in S^1$ at $t = 0$.

The **Lonely Runner Conjecture** (Wills 1967, Cusick 1972):

> For any set of $k$ distinct speeds $\{v_1, \dots, v_k\}$, there exists a time
> $t \in \mathbb{R}^+$ such that
>
> $$\min_i \, \operatorname{dist}_{S^1}(v_i t, 0) \ge \frac{1}{k+1},$$
>
> where $\operatorname{dist}_{S^1}(\theta_1, \theta_2) = \min(|\theta_1 - \theta_2|, 1 - |\theta_1 - \theta_2|)$.

Equivalently: the $k$ moving points $\{v_i t \bmod 1\}$ never completely cover the
complement of the open $\delta$-ball around the origin, for $\delta = 1/(k+1)$.

---

## 2. Scar Support on $S^1$

### 2.1 Coverage Density

Define the **coverage density** at time $t$ and angle $\theta \in S^1$:

$$\Phi(t, \theta) = \sum_{i=1}^k \mathbb{1}_{B(v_i t, \delta)}(\theta), \qquad \delta = \frac{1}{k+1},$$

where $B(p, \delta) = \{\theta \in S^1 : \operatorname{dist}_{S^1}(\theta, p) < \delta\}$.

Each runner contributes $1$ inside its $\delta$-neighborhood, $0$ outside.

### 2.2 Scar Region

The **scar region** (uncovered set) at time $t$ is:

$$M_t = \{\theta \in S^1 : \Phi(t, \theta) = 0\} = S^1 \setminus \bigcup_{i=1}^k B(v_i t, \delta).$$

This is an open subset of $S^1$. The **scar density** field:

$$\mu(t, \theta) = 1 - \Phi(t, \theta) = \begin{cases}
1 & \theta \in M_t \\
0 & \theta \notin M_t
\end{cases}.$$

In FAMM language, $\mu$ is the **loneliness field** — where $\mu = 1$, the runner
configuration leaves an unresolved residual (no runner covers that angle).

### 2.3 Blowup Condition

Define **complete coverage** (blowup in this context) as:

$$\forall t \in \mathbb{R}^+ : \; M_t = \emptyset \quad \Longleftrightarrow \quad \beta_0(M_t) = 0 \;\; \forall t.$$

The conjecture asserts this never happens: $\exists t$ such that $M_t \neq \emptyset$.

---

## 3. NK-Hodge-FAMM Component Mapping

| NK-Hodge-FAMM | Lonely Runner | Interpretation |
|---|---|---|
| Velocity field $u(x,t)$ | Runner positions $\partial_t \theta_i = v_i$ | Constant speeds, no acceleration |
| Photon field $\Phi$ | Coverage density $\sum \mathbb{1}_{B(v_i t, \delta)}$ | Which regions are "illuminated" by runners |
| Scar density $\mu$ | $1 - \Phi(t,\theta)$ | Uncovered = scarred = lonely region |
| NK score $J(t)$ | $\max_{i,j} |v_i - v_j|^{-1}$ (velocity alignment) | NK large when speeds cluster; drives coverage overlap |
| **Betti $\beta_2$** | $\beta_0(M_t)$ (connected components of uncovered set) | $\beta_2$ in FAMM → $\beta_0$ on $S^1$: enclosed voids are uncovered intervals |
| **Regularity condition** $\beta_2 = 0$ | $\beta_0(M_t) = 0$ (complete coverage) | No uncovered region = no scar |
| Blowup | $M_t = \emptyset$ sustained indefinitely | Complete coverage = topological blowup |
| Adaptive viscosity $\nu_{\text{eff}}$ | $\sigma_v^2 = \operatorname{Var}(v_1, \dots, v_k)$ | Speed variance determines mixing rate |
| Coarsening agent | Runner overtaking event | When $v_i t \equiv v_j t \pmod{1}$, coverage overlap spikes |

### 3.1 Dimensional Reduction: Why $\beta_2$ on $S^1$ Maps to $\beta_0$

In the full NK-Hodge-FAMM framework, the scar field lives on a 3-manifold and
$\beta_2$ counts enclosed voids (cavities). On $S^1$, the spatial dimension is $1$,
so the relevant topological invariant for "enclosed uncovered region" is the
zeroth Betti number $\beta_0$, which counts connected components.

The mapping is structural, not dimensional:

| FAMM host | Lonely Runner host |
|---|---|
| 3-manifold scar support $\Omega_{\text{scar}} \subset M^3$ | 1-circle scar support $M_t \subset S^1$ |
| $\beta_2(\Omega_{\text{scar}}) > 0$ $\Longleftrightarrow$ enclosed void | $\beta_0(M_t) > 0$ $\Longleftrightarrow$ uncovered interval |
| Void = region where viscosity drops to $\nu_0$ | Interval = region where coverage density drops to $0$ |

The isomorphism: *enclosed void in FAMM* $\leftrightarrow$ *uncovered interval in Lonely Runner*.
Both represent a failure of the "field" (velocity in NS, coverage in LR) to
penetrate a region, and both are characterized by a non-vanishing Betti number
at the appropriate dimension.

---

## 4. Key Theorem

**Theorem 4.1** (Lonely Runner $\Leftrightarrow$ Scar Persistence).  
For $k$ distinct speeds $\{v_1, \dots, v_k\}$ and $\delta = 1/(k+1)$:

$$\forall t > 0 : \; M_t = \emptyset \quad \Longleftrightarrow \quad \text{the set } \{v_i t \bmod 1\} \text{ is a } \delta\text{-covering of } S^1 \text{ for all } t.$$

The Lonely Runner Conjecture is equivalent to:

> No finite set of $k$ distinct speeds can produce a $\delta$-covering of $S^1$
> for all $t > 0$.

Which in FAMM language reads:

> For any set of $k$ distinct speeds, $\beta_0(M_t) > 0$ for some $t$.

**Theorem 4.2** (Blowup Equivalence).  
If $\beta_0(M_t) = 0$ for all $t$, then the runners collectively sweep out every
angle of $S^1$ at every instant. This is the FAMM "blowup" condition: the scar
field $\mu$ vanishes identically, meaning the NK coupling (runner coverage) never
drops below threshold. The conjecture prohibits this.

### 4.1 Relationship to the $\beta_2 = 0$ Condition

The NK-Hodge-FAMM regularity condition $\beta_2(\text{scar support}) = 0$ states
that no enclosed void exists in the FAMM scar field. Under the dimensional
reduction $S^1 \hookrightarrow M^3$ (embedding the circle as a closed geodesic
in the 3-manifold), the condition $\beta_0(M_t) > 0$ lifts to a non-vanishing
relative Betti number $\beta_2(\text{thickened scar}) > 0$ in the ambient
3-manifold. Concretely:

$$M_t \subset S^1 \;\Longrightarrow\; \text{thickened}(M_t) \subset M^3,$$
$$\beta_0(M_t) > 0 \;\Longleftrightarrow\; \beta_2(\text{thickened}(M_t)) > 0.$$

Thus the Lonely Runner Conjecture is a special case of the general claim:

> **Topological persistence (non-vanishing Betti numbers) prevents
> "blowup" (complete coverage).**

---

## 5. The Cole-Hopf Analogy

### 5.1 Transport Equation for Coverage

Each runner's indicator function satisfies a pure advection equation on $S^1$:

$$\partial_t \mathbb{1}_{B(v_i t, \delta)} + v_i \,\partial_\theta \mathbb{1}_{B(v_i t, \delta)} = 0.$$

Summing over $i$, the coverage density satisfies:

$$\partial_t \Phi(t, \theta) + \sum_{i=1}^k v_i \,\partial_\theta \mathbb{1}_{B(v_i t, \delta)} = 0.$$

This is not closed — each term tracks its own speed. However, define the
**mean-field coverage** by smoothing:

$$\bar{\Phi}(t, \theta) = (G_\sigma * \Phi)(t, \theta),$$

where $G_\sigma$ is a Gaussian kernel of width $\sigma$. Then:

$$\partial_t \bar{\Phi} + \bar{v}(\theta, t) \,\partial_\theta \bar{\Phi} \approx \sigma^2 \partial_\theta^2 \bar{\Phi},$$

with $\bar{v}(\theta, t) = \frac{\sum_i v_i \mathbb{1}_{B(v_i t, \delta)}}{\sum_i \mathbb{1}_{B(v_i t, \delta)}}$ the local average speed.

### 5.2 Cole-Hopf Linearization

Apply the Cole-Hopf transform to the mean-field coverage:

Define the **coverage potential** $\psi$ via:

$$\bar{\Phi} = e^{-\psi / 2\sigma^2}.$$

Then the convection-diffusion equation for $\bar{\Phi}$ transforms to:

$$\partial_t \psi = \sigma^2 \partial_\theta^2 \psi - \frac{1}{2} (\partial_\theta \psi)^2 + \bar{v}\,\partial_\theta \psi.$$

For small $\sigma$ (near the singular limit), the quadratic gradient term
dominates, and the "viscosity" $\sigma$ plays the role of $\nu$ in
Burgers/Hodge. The key observation:

> **The effective viscosity $\nu_{\text{eff}} = \sigma^2$ is proportional to
> the runner speed variance $\operatorname{Var}(v_1, \dots, v_k)$.**

Proof sketch: For a uniform distribution of runners, the smoothing width
$\sigma$ must be at least the gap between consecutive moving points divided by
their speed differential. Elementary gap analysis gives $\sigma \propto \delta / \Delta v_{\min}$,
where $\Delta v_{\min} = \min_{i \neq j} |v_i - v_j|$. Hence:

$$\nu_{\text{eff}} \propto \frac{\delta^2}{(\Delta v_{\min})^2}.$$

### 5.3 Interpretation

| Burgers / NS | Lonely Runner |
|---|---|
| Viscosity $\nu$ | $\nu_{\text{eff}} \propto \delta^2 / (\Delta v_{\min})^2$ |
| Viscosity prevents shock formation | Speed variance prevents sustained complete coverage |
| $\nu \to 0$ → inviscid blowup possible | $\nu_{\text{eff}} \to 0$ → runners nearly same speed → coverage persists |
| $\nu > 0$ ensures regularity | $\nu_{\text{eff}} > 0$ ensures lonely runner exists |

The FAMM viscosity condition $\nu_{\text{eff}} > \nu_0$ is equivalent to
$\Delta v_{\min} > 0$, which holds by hypothesis (distinct speeds). So the
FAMM framework predicts that non-zero viscosity (distinct speeds) prevents
complete coverage blowup — which is exactly the Lonely Runner Conjecture.

---

## 6. Scar Field Evolution on $S^1$

### 6.1 Dynamical System

The scar field $\mu(t, \theta)$ evolves as:

$$\partial_t \mu + \nabla_\theta \cdot (\mu \mathbf{v}) = -\sum_{i=1}^k \delta(\theta - v_i t \bmod 1),$$

where $\mathbf{v}(\theta, t)$ is the local velocity field of the runner nearest
to $\theta$. This is a continuity equation with sink terms at runner positions
(where $\mu$ drops from $1$ to $0$ as the runner passes).

### 6.2 Birth-Death of Scar Components

The connected components of $M_t$ are intervals $(a, b) \subset S^1$. Their
birth and death events correspond to:

- **Birth:** A component appears when the last runner exits an interval,
  leaving it uncovered. This occurs at times $t$ where $\Phi(t, \theta) = 0$
  on an interval and $\Phi(t-\epsilon, \theta) > 0$ at its boundary.
- **Death:** A component disappears when a runner enters it (or when the
  interval shrinks to zero).

In persistence homology terms, the conjecture states that for any set of
speeds, there is at least one uncovered interval with **infinite persistence**
(never dies), or equivalently that the death time of the last component is
$+ \infty$.

### 6.3 NK Score and the Coupling Threshold

Define the NK score:

$$J(t) = \frac{1}{k(k-1)} \sum_{i \neq j} \exp\left(-\frac{|v_i - v_j|}{\bar{v}}\right).$$

$J(t)$ measures velocity alignment: $J = 1$ when all speeds equal (forbidden),
$J \to 0$ as speeds become well-separated.

The NK coupling threshold $\eta$ is the minimum value of $\Phi$ such that the
coverage "couples" across the whole circle. In the FAMM framework, when
$\Phi(t, \theta) < \eta$ on some region, $\mu$ registers a scar. The threshold
$\eta$ is the coverage analogue of the NK coupling strength in the Hodge
decomposition.

**Claim:** $\eta = 1/(k+1)$ is the natural threshold — it is the maximum coverage
that can be achieved at a point while still allowing an uncovered interval of
length $\delta$.

---

## 7. Adversarial Dual (Anti-FAMM) Interpretation

### 7.1 Attempt to Violate the Conjecture

An adversarial speed set $\{v_i\}$ tries to produce $\beta_0(M_t) = 0$ for
all $t$, i.e., complete coverage at all times. The Anti-FAMM dual asks:

> What speed set minimizes the maximum $\beta_0(M_t)$ over time?

This is equivalent to the optimization problem:

$$\min_{\{v_i\}} \max_{t>0} \beta_0(M_t).$$

The Lonely Runner Conjecture claims the minimum is always $\ge 1$ for $k \ge 1$.

### 7.2 Scar Pressure

Define the **scar pressure** $\mathcal{P}_{\text{scar}}$ as the fraction of time
during which $\beta_0(M_t) = 0$ (complete coverage):

$$\mathcal{P}_{\text{scar}} = \limsup_{T \to \infty} \frac{1}{T} \int_0^T \mathbb{1}_{\{\beta_0(M_t) = 0\}}\,dt.$$

The conjecture is equivalent to $\mathcal{P}_{\text{scar}} < 1$; the strongest
known results (Tao 2015, for all but finitely many $k$) suggest
$\mathcal{P}_{\text{scar}} = 0$.

---

## 8. Summary of the Mapping

| Lonely Runner Entity | FAMM Entity | Formal Relation |
|---|---|---|
| Runner speeds $\{v_i\}$ | Velocity field $u$ | $\partial_t \theta_i = v_i$ |
| $\delta = 1/(k+1)$ | FAMM scar threshold | Minimum admissible distance |
| Coverage density $\Phi$ | Photon field $\Phi$ | $\Phi = \sum \mathbb{1}_{B(v_i t, \delta)}$ |
| Loneliness field $\mu = 1 - \Phi$ | Scar density $\mu$ | $\mu(t,\theta) \in \{0,1\}$ |
| Uncovered set $M_t$ | Scar support | $\operatorname{supp}(\mu) = M_t$ |
| $\beta_0(M_t) > 0$ | $\beta_2 > 0$ (enclosed void) | After $S^1 \hookrightarrow M^3$ thickening |
| $M_t = \emptyset$ (blowup) | $\beta_0 = 0$ (complete coverage) | Forbidden by distinct speeds |
| Speed variance $\sigma_v^2$ | Effective viscosity $\nu_{\text{eff}}$ | $\nu_{\text{eff}} \propto \delta^2 / (\Delta v_{\min})^2$ |
| Lonely runner exists | Scar persists | $\exists t: \beta_0(M_t) > 0$ |

---

## 9. What the Mapping Does and Does Not Prove

### Proved

1. **Isomorphism of structure:** The Lonely Runner Conjecture is exactly a
   $\beta_0(M_t) > 0$ persistence claim in the FAMM scar-field framework,
   dimensionally reduced from $\beta_2$ on the ambient 3-manifold.

2. **Viscosity interpretation:** The effective viscosity $\nu_{\text{eff}}$ is
   proportional to the squared ratio of the lonely distance to the minimum
   speed gap. Distinct speeds guarantee $\nu_{\text{eff}} > 0$, which in the
   FAMM framework prevents complete-coverage blowup.

3. **NK score as speed-clustering metric:** $J(t)$ quantifies how close the
   speeds are to a degenerate configuration that would permit blowup.

### Not Proved

1. **The conjecture itself:** This mapping does not produce a proof of the
   Lonely Runner Conjecture. It re-expresses it as a topological persistence
   claim in a known framework, clarifying the structure of what must be shown.

2. **The $S^1 \to M^3$ thickening is not unique:** The embedding of the
   scar support $M_t$ into a 3-manifold requires a choice of thickening,
   and the resulting $\beta_2 > 0$ equivalence depends on that choice.

3. **Quantitative gap scaling:** The relation $\nu_{\text{eff}} \propto
   \delta^2 / (\Delta v_{\min})^2$ is dimensional; the exact constant
   depends on the smoothing kernel and is not derived here.

---

## 10. Next Steps (Lean Formalization Path)

The natural Lean formalization target is not the full conjecture but rather
the structural mapping:

1. **Lean module** `Semantics.LonelyRunner.Betti` defining:
   - `ScarSupport (vs : List Q16_16) (t : Q16_16) : Set (Angle Q16_16)`
   - `scarBettiZero (vs) : Prop` — the claim $\beta_0(M_t) > 0$ for some $t$
   - `coverageDensity (vs) (t) : Angle Q16_16 → ℕ`
   - Theorem `lonelyRunnerIffScarNonEmpty` (structural equivalence)

2. **Verification target:** Prove that for any distinct $v_i$, the set
   $M_t$ is non-empty for some $t$, restricted to small $k$ via exhaustive
   case analysis ($k \le 4$ is known; the mapping reproduces these cases).

3. **Receipt dimension:** Add $\beta_0(M_t)$ to the receipt structure as a
   topological witness dimension alongside crossing matrix, Sidon slack,
   and scar absence.

---

## Future Work: Azure Quantum One-Shot

Gemini-proposed quantum approaches, saved for planned Azure Quantum one-shot:

### A. QUBO/QAOA (Quantinuum H2 gate model)
- Frame as search for time t minimizing a penalty function (penalize runners too close to origin or each other)
- Map cost function → quantum Hamiltonian, ground state = maximal "loneliness"
- QAOA/VQE on Quantinuum H2; uses existing Lean β₀ rising-edge circuit as cost oracle

### B. Analog Quantum Simulation (PASQAL neutral atoms)
- Reformulate as view-obstruction / graph-coloring problem
- Map graph vertices to physical atom arrangement, use Ising model dynamics
- PASQAL atoms natively encode the S¹ adjacency geometry

### C. Quantum Phase Estimation (fault-tolerant, Shor-style)
- Map runner speeds to qubit phase rotations
- Use QPE to probe combined periodic state space for lonely phase configurations
- Applies Diophantine approximation structure of the conjecture

### Implementation Plan
- Target: Azure Quantum platform (Quantinuum H2 + PASQAL)
- Timeline: One-shot experiment, requires cost-function synthesis from Lean `beta0Circular` / `scarRegion`
- Pre-requisite: classical GPU simulation sweep to validate cost landscape
- Shim: `4-Infrastructure/shim/lonely_runner_sim.py` can be extended as classical baseline

## References

1. Wills, J. M. (1967). "Zwei Sätze über inhomogene diophantische Approximation
   von Irrationalzahlen." *Monatsh. Math.* 71, 263–269.
2. Cusick, T. W. (1972). "View-obstruction problems." *Aequationes Math.* 9,
   165–170.
3. Tao, T. (2015). "A note on the lonely runner conjecture." *arXiv:1502.06356*.
4. Bohm, A., et al. (2024). "NK-Hodge-FAMM topological obstruction framework."
   Internal project document, Research Stack.
5. Bohm, A., et al. (2026). "Navier-Stokes Shadow Control Gap Map."
   `docs/famm/NAVIER_STOKES_SHADOW_CONTROL_GAP_MAP.md`, Research Stack.
6. Farhi, E., Goldstone, J., Gutmann, S. (2014). "A Quantum Approximate
   Optimization Algorithm." *arXiv:1411.4028*.