# AMRAL × Lebesgue Universal Covering — Round 01
## Canonical Support-Function + Constant-Width Handoff Compiler

**Document ID:** AMRAL-LUC-FC-R01  
**Version:** v0.1  
**Date:** 2026-09-18  
**Research Status:** Round 01 / Formal Attack / Structural-Closure Advance  
**Research Mode:** Human-Directed + Semi-Autonomous AI Mathematical Research  
**Research Initiator & Methodology Source:** Neo.K  
**AI Collaborating Researcher & Primary Executor:** Aletheia / ChatGPT, GPT-5.6 Sol  
**Parent Methodology:** Relational Constraint–Handoff Methodology (RCHM)  
**Parent Protocol:** AMRAL-LUC-FC-R00 v0.2  

---

# 0. Round Summary Verdict

Round 01's task is not to search for new numerical bounds, but to answer the first exact gate specified by Round 00:

$$
\boxed{
\text{original Lebesgue target domain}
\longrightarrow
\text{constant-width body}
\longrightarrow
\text{support function}
}
$$

namely, whether this handoff can be completely fixed without swapping quantifiers, without losing degenerate cases, and without relying on numerical approximation.

This round produces six main results.

1. **Exact reduction of the original test domain is complete.**  
   For a convex universal cover $U$, covering every planar set of diameter at most one is equivalent to covering every planar convex body of constant width one.

2. **The $d<1$ gap in prior research is closed.**  
   The old six-way Q3 mainly addressed diameter-one sets. This round fills in the completion + dilation for $0<d<1$, as well as the degenerate case $d=0$.

3. **The support function becomes an exact containment handoff.**  
   For compact convex sets, containment, directed Hausdorff distance, and rigid-motion placement can all be written exactly as support-function inequalities.

4. **The constant-width body acquires a new canonical shape variable.**  
   Beyond the support function $h$, this round introduces a translation-invariant curvature-radius density:

   $$
   r(\theta)
   =
   h(\theta)+h''(\theta).
   $$

   The translation-equivalence classes of unit constant-width bodies correspond to an explicit bounded function domain.

5. **The outer worst case is upgraded from $\sup$ to $\max$.**  
   Once the Steiner point is fixed, the family of unit constant-width bodies is compact under the Hausdorff topology, and containment skew is $1$-Lipschitz with respect to the target shape, so a worst-case target genuinely exists.

6. **Under a fixed orientation, the translation optimum admits an active certificate of at most three directions.**  
   For a fixed rotation/reflection $Q$, the optimal translation is a two-dimensional convex minimax; by subgradient calculus + Carathéodory, it can be witnessed by at most three active directions.

Round 01's overall verdict is therefore:

$$
\boxed{
\text{SUPPORT-HANDOFF: CLOSED}
}
$$

$$
\boxed{
\text{CONSTANT-WIDTH REDUCTION: CLOSED}
}
$$

$$
\boxed{
\text{SHAPE-FINITE-COMPILER: OPEN}
}
$$

This round does not claim to solve the Lebesgue universal covering problem, nor does it produce any new rigorous numerical bound.

---

# 1. Research Inputs and Inheritance from Prior Work

This round primarily inherits four existing lines of research.

## 1.1 Round 00

Round 00 has already designated:

$$
\text{support function}
$$

as the primary handoff interface, and requires Round 01 to reconstruct the exact theorem, hypotheses, and scope of the constant-width reduction.

## 1.2 The 2026-07 Six-Way Parallel Verification

The old six-way study obtained:

- Q1: the $1$-Lipschitz property of containment skew with respect to the candidate body;
- Q2: the support-function formulation;
- Q3: completion-driven constant-width reduction;
- Q4: Lipschitz sandwiching for finite angular/rotational sampling;
- Q5: predictive $SE(2)$ search;
- Q6: a finite-family removable-region criterion.

This round does not simply take these as fully proved, but reconstructs and repositions them item by item.

## 1.3 The One-Dimensional Metric-Kernel Study

The one-dimensional index and distance kernel:

$$
D_K(s,t)
=
\|q_K(s)-q_K(t)\|
$$

is retained as a representation-fidelity auditor.

This round's proof layer instead prioritizes the support function, because the original Lebesgue problem itself requires a convex universal cover, and the support function is exact for convex containment.

## 1.4 External Mathematical Interfaces

This round relies on the following known external structures as a foundation:

- the standard definition of a Lebesgue universal cover takes convex planar sets;
- every bounded Euclidean set has a same-diameter completion;
- in the Euclidean plane, a diametrically complete body is equivalent to a constant-width body;
- the support function of a constant-width-one convex body satisfies:

$$
h(\theta)+h(\theta+\pi)=1;
$$

- for a planar constant-width-one body:

$$
0
\le
h''(\theta)+h(\theta)
\le
1
$$

holds almost everywhere.

These are all established results in convex geometry, not new theorems claimed by this round.

---

# 2. The Exact Scope of the Original Problem

Let:

$$
\mathcal A_{\le1}
=
\left\{
A\subset\mathbb R^2:
A\neq\varnothing,\ 
\operatorname{diam}(A)\le1
\right\}.
$$

Let $\mathcal U$ be the set of nonempty planar compact convex sets.

Call $U\in\mathcal U$ a Lebesgue universal cover if:

$$
\forall A\in\mathcal A_{\le1},
\quad
\exists g\in E(2),
\quad
gA\subseteq U.
$$

The Lebesgue constant is:

$$
a_{\mathrm{Leb}}
=
\inf
\left\{
\operatorname{Area}(U):
U\in\mathcal U,\ 
U\text{ universal}
\right\}.
$$

Every reduction in this round preserves the quantifier order:

$$
\boxed{
\forall A\ \exists g
}
$$

This must not be rewritten as:

$$
\exists g\ \forall A.
$$

---

# 3. Gate 1: From an Arbitrary Set to a Compact Convex Set

## Theorem 3.1: Closed Convexification Preserves Diameter

For any nonempty bounded $A\subset\mathbb R^2$:

$$
\operatorname{diam}
\left(
\overline{\operatorname{conv}}A
\right)
=
\operatorname{diam}(A).
$$

### Proof

First:

$$
A
\subseteq
\overline{\operatorname{conv}}A,
$$

so:

$$
\operatorname{diam}(A)
\le
\operatorname{diam}
\left(
\overline{\operatorname{conv}}A
\right).
$$

On the other hand, for any:

$$
x
=
\sum_i\alpha_i a_i,
\qquad
y
=
\sum_j\beta_j b_j
$$

belonging to $\operatorname{conv}A$, where:

$$
\alpha_i,\beta_j\ge0,
\qquad
\sum_i\alpha_i
=
\sum_j\beta_j
=
1,
$$

we have:

$$
x-y
=
\sum_{i,j}
\alpha_i\beta_j(a_i-b_j).
$$

Hence:

$$
\|x-y\|
\le
\sum_{i,j}
\alpha_i\beta_j
\|a_i-b_j\|
\le
\operatorname{diam}(A).
$$

Therefore:

$$
\operatorname{diam}(\operatorname{conv}A)
\le
\operatorname{diam}(A).
$$

Taking the closure does not change the diameter's supremum, so the conclusion holds.

Q.E.D.

---

## Corollary 3.2: The Original Problem Can Be Restricted to Compact Convex Targets

If $U$ is convex, and:

$$
g\overline{\operatorname{conv}}A
\subseteq U,
$$

then naturally:

$$
gA
\subseteq U.
$$

Conversely, compact convex sets of diameter at most one are already a subfamily of $\mathcal A_{\le1}$.

Therefore:

$$
\boxed{
U\text{ universal for }\mathcal A_{\le1}
}
$$

if and only if:

$$
\boxed{
U\text{ covers every nonempty compact convex }K
\text{ with }\operatorname{diam}(K)\le1.
}
$$

### RCHM Verdict

This step eliminates:

- non-closed sets;
- non-convex topological detail;
- arbitrary holes;
- arbitrary internal fragmentation.

But it does not eliminate:

- shape complexity;
- rotation;
- translation;
- the continuum of directions.

Recorded as:

`HANDOFF-A: PASS`

---

# 4. Gate 2: From a Compact Convex Target to a Constant-Width-One Target

Let:

$$
\mathcal W_1
=
\left\{
W\subset\mathbb R^2:
W\text{ compact convex body of constant width }1
\right\}.
$$

This section proves:

$$
\boxed{
\text{testing all }\operatorname{diam}\le1
\iff
\text{testing only }\mathcal W_1.
}
$$

---

## Theorem 4.1: Unit Constant-Width Completion Reduction

For any nonempty compact convex $K\subset\mathbb R^2$, if:

$$
\operatorname{diam}(K)\le1,
$$

then there exists $W\in\mathcal W_1$ such that:

$$
K\subseteq W.
$$

### Proof

Let:

$$
d
=
\operatorname{diam}(K).
$$

We split into three cases.

### Case 1: $d=1$

By the Euclidean completion theorem, there exists a diametrically complete $C$ such that:

$$
K\subseteq C,
\qquad
\operatorname{diam}(C)=1.
$$

In the Euclidean plane, a complete body is equivalent to a constant-width body, so:

$$
C\in\mathcal W_1.
$$

Take:

$$
W=C.
$$

### Case 2: $0<d<1$

Again by the completion theorem, there exists a constant-width-$d$ convex body $C_d$ such that:

$$
K\subseteq C_d.
$$

Take any:

$$
c\in C_d.
$$

Define:

$$
W
=
c+\frac{1}{d}(C_d-c).
$$

Since:

$$
\frac1d>1
$$

and $C_d$ is convex with $c\in C_d$, we have:

$$
C_d
\subseteq
W.
$$

Therefore:

$$
K
\subseteq
W.
$$

Under this dilation, every length is multiplied by $1/d$, so $W$'s constant width is:

$$
\frac1d\cdot d
=
1.
$$

Hence:

$$
W\in\mathcal W_1.
$$

### Case 3: $d=0$

In this case $K$ is a single point.

Any disk of radius $1/2$ containing that point is a constant-width-one body.

So the conclusion holds.

Q.E.D.

---

## Theorem 4.2: Exact Constant-Width Reduction of the Lebesgue Test Domain

For any compact convex $U$, the following two statements are equivalent:

1.

$$
\forall K
\text{ compact convex},
\quad
\operatorname{diam}(K)\le1
\Rightarrow
\exists g\in E(2):
gK\subseteq U;
$$

2.

$$
\forall W\in\mathcal W_1,
\quad
\exists g\in E(2):
gW\subseteq U.
$$

### Proof

$1\Rightarrow2$:

Every $W\in\mathcal W_1$ has:

$$
\operatorname{diam}(W)=1,
$$

so it is a member of the first family.

$2\Rightarrow1$:

Take any $K$ of diameter at most one.

By Theorem 4.1, there exists:

$$
W\in\mathcal W_1
$$

such that:

$$
K\subseteq W.
$$

By hypothesis, there exists $g$ such that:

$$
gW\subseteq U.
$$

Hence:

$$
gK
\subseteq
gW
\subseteq
U.
$$

Q.E.D.

---

## Round 01's Verdict on the Old Q3

The core direction of the old six-way Q3 is correct.

But the original formulation focused mainly on "diameter one."

This round fills in:

- $0<d<1$;
- $d=0$;
- exact quantifier equivalence.

The status is therefore updated to:

`Q3: REBUILD-PASS`

This reduction can now enter the main proof graph.

---

# 5. Gate 3: The Support-Function Containment Compiler

For a compact convex set $K$, define the support function:

$$
h_K(u)
=
\max_{x\in K}
x\cdot u,
\qquad
u\in S^1.
$$

Write:

$$
u_\theta
=
(\cos\theta,\sin\theta),
$$

and abbreviate:

$$
h_K(\theta)
=
h_K(u_\theta).
$$

---

## Theorem 5.1: Convex Containment Is Exactly Equivalent to the Support Function

For compact convex $A,U$:

$$
A\subseteq U
$$

if and only if:

$$
h_A(u)
\le
h_U(u)
\qquad
\forall u\in S^1.
$$

### Proof

If:

$$
A\subseteq U,
$$

then for every $u$:

$$
\max_{x\in A}x\cdot u
\le
\max_{y\in U}y\cdot u.
$$

Conversely, use the half-space reconstruction of a convex body:

$$
U
=
\bigcap_{u\in S^1}
\left\{
x:
x\cdot u
\le
h_U(u)
\right\}.
$$

If $h_A\le h_U$, then every $x\in A$ simultaneously satisfies all the supporting half-space inequalities, so:

$$
x\in U.
$$

Q.E.D.

---

## Theorem 5.2: Support Function of a Rigid Motion

Let:

$$
g(x)
=
Qx+t,
\qquad
Q\in O(2),
\quad
t\in\mathbb R^2.
$$

Then:

$$
h_{gK}(u)
=
h_K(Q^Tu)+t\cdot u.
$$

Therefore:

$$
gK\subseteq U
$$

if and only if:

$$
h_K(Q^Tu)
+t\cdot u
\le
h_U(u)
\qquad
\forall u\in S^1.
$$

This is the canonical handoff for every exact containment certificate that follows.

---

# 6. Directed Hausdorff Distance and Exact Slack

Define the directed containment distance:

$$
d_+(A,U)
=
\sup_{x\in A}
\operatorname{dist}(x,U).
$$

For convex compact $A,U$:

$$
A
\subseteq
U+\varepsilon B
$$

if and only if:

$$
h_A(u)
\le
h_U(u)+\varepsilon
\qquad
\forall u\in S^1.
$$

Hence:

$$
\boxed{
d_+(A,U)
=
\max
\left\{
0,
\max_{u\in S^1}
\left(
h_A(u)-h_U(u)
\right)
\right\}.
}
$$

For a placement $g=(Q,t)$:

$$
d_+(gK,U)
=
\max
\left\{
0,
\max_{u\in S^1}
\left[
h_K(Q^Tu)
+t\cdot u
-h_U(u)
\right]
\right\}.
$$

The old six-way Q2 can therefore be upgraded to:

`Q2: PASS / PRIMARY HANDOFF`

---

# 7. The Exact Placement Functional

For fixed $K,U$, define:

$$
\Psi_U(K)
=
\inf_{Q\in O(2),\ t\in\mathbb R^2}
d_+(QK+t,U).
$$

Then:

$$
\Psi_U(K)=0
$$

if and only if there exists a rigid motion such that:

$$
gK\subseteq U.
$$

The Lebesgue universal condition can therefore be written as:

$$
\forall K\in\mathcal W_1,
\quad
\Psi_U(K)=0.
$$

Later we will prove that the outer worst-case value is in fact attained.

---

# 8. The Constant-Width Support Compiler

For $W\in\mathcal W_1$, its support function satisfies:

$$
\boxed{
h(\theta)+h(\theta+\pi)=1.
}
$$

This condition is independent of translation, because the translation term:

$$
t\cdot u_\theta
$$

cancels with:

$$
t\cdot u_{\theta+\pi}
=
-t\cdot u_\theta
$$

---

# 9. The Steiner Gauge and Fourier Structure

The Fourier expansion of the support function is written as:

$$
h(\theta)
=
a_0
+
\sum_{n=1}^{\infty}
\left(
a_n\cos n\theta
+
b_n\sin n\theta
\right).
$$

The unit-constant-width condition:

$$
h(\theta)+h(\theta+\pi)=1
$$

gives:

$$
a_0
=
\frac12,
$$

and all even modes vanish:

$$
a_{2m}
=
b_{2m}
=
0
\qquad
m\ge1.
$$

The $n=1$ mode is exactly the translation degree of freedom.

The Steiner point is:

$$
s(W)
=
(a_1,b_1).
$$

Choosing:

$$
s(W)=0
$$

as the canonical translation gauge then gives:

$$
a_1=b_1=0.
$$

So the support function of a centered constant-width body reduces to just:

$$
\boxed{
h(\theta)
=
\frac12
+
\sum_{\substack{n\ge3\\n\ \mathrm{odd}}}
\left(
a_n\cos n\theta
+
b_n\sin n\theta
\right).
}
$$

This step is precisely what eliminates:

$$
\boxed{
\text{target-shape translation gauge}.
}
$$

Note that this is not the placement translation.

The $t$ in the placement still exists, because the target body must still be moved into the candidate cover.

---

# 10. A New Canonical Variable: The Curvature-Radius Density

For the support function of a unit constant-width body, define:

$$
\boxed{
r(\theta)
=
h''(\theta)+h(\theta).
}
$$

For a general non-smooth constant-width body, $h$ is still enough to make the above expression meaningful almost everywhere; more generally it can first be understood in the sense of distributions or the surface-area measure.

Established constant-width convex geometry gives:

$$
0
\le
r(\theta)
\le
1
$$

almost everywhere.

Differentiating:

$$
h(\theta)+h(\theta+\pi)=1
$$

twice gives:

$$
h''(\theta)+h''(\theta+\pi)=0.
$$

Hence:

$$
\boxed{
r(\theta)+r(\theta+\pi)=1
}
$$

almost everywhere.

---

# 11. The First-Moment Conditions

Since:

$$
r=h+h'',
$$

integrating the $2\pi$-periodic function by parts gives:

$$
\int_0^{2\pi}
r(\theta)\cos\theta\,d\theta
=
0,
$$

and:

$$
\int_0^{2\pi}
r(\theta)\sin\theta\,d\theta
=
0.
$$

These two conditions can also be understood as the surface-area balance condition of a closed convex curve.

We therefore define the admissible curvature-density domain:

$$
\boxed{
\mathcal R
=
\left\{
r\in L^\infty(S^1):
\begin{array}{l}
0\le r\le1\ \text{a.e.},\\
r(\theta+\pi)=1-r(\theta)\ \text{a.e.},\\
\int_0^{2\pi}r(\theta)\cos\theta\,d\theta=0,\\
\int_0^{2\pi}r(\theta)\sin\theta\,d\theta=0
\end{array}
\right\}.
}
$$

This is the most important new common domain introduced in this round.

---

# 12. Theorem: The Exact Correspondence Between Constant-Width Translation Classes and $\mathcal R$

## Theorem 12.1

The translation-equivalence classes of unit constant-width planar convex bodies correspond bijectively to $\mathcal R$.

More precisely, the map:

$$
[W]_{\mathrm{trans}}
\longmapsto
r_W
=
h_W+h_W''
$$

is a bijection:

$$
\boxed{
\mathcal W_1/\operatorname{Trans}(\mathbb R^2)
\cong
\mathcal R.
}
$$

### Proof: well-defined

Translation $W\mapsto W+t$ sends:

$$
h(\theta)
\mapsto
h(\theta)
+t_x\cos\theta
+t_y\sin\theta.
$$

And:

$$
(D^2+1)\cos\theta
=
0,
$$

$$
(D^2+1)\sin\theta
=
0.
$$

So:

$$
h+h''
$$

is invariant under translation.

Hence $r$ is well defined on the translation class.

---

### Proof: injective modulo translation

If two support functions $h_1,h_2$ have the same $r$:

$$
h_1+h_1''
=
h_2+h_2'',
$$

then:

$$
f
=
h_1-h_2
$$

satisfies:

$$
f''+f=0.
$$

Hence:

$$
f(\theta)
=
a\cos\theta+b\sin\theta.
$$

This is exactly the support-function difference produced by a translation $t=(a,b)$.

So the two convex bodies differ only by a translation.

---

### Proof: surjective

Take any:

$$
r\in\mathcal R.
$$

Consider the periodic equation:

$$
h''+h=r.
$$

The periodic kernel of the operator:

$$
D^2+1
$$

is:

$$
\operatorname{span}
\{
\cos\theta,\sin\theta
\}.
$$

The two first-moment conditions of $\mathcal R$ guarantee precisely that $r$ is orthogonal to this kernel, so a periodic solution exists.

After fixing the gauge:

$$
a_1=b_1=0,
$$

the solution is unique.

Since:

$$
r\in L^\infty,
$$

we may take:

$$
h\in W^{2,\infty}(S^1)
\subset
C^{1,1}(S^1).
$$

And since:

$$
h''+h=r\ge0
$$

almost everywhere, the standard support-function criterion yields some convex body $W$.

Next let:

$$
w(\theta)
=
h(\theta)+h(\theta+\pi).
$$

We have:

$$
w''+w
=
r(\theta)+r(\theta+\pi)
=
1.
$$

On the other hand $w$ is naturally $\pi$-periodic.

The general $2\pi$-periodic solution of the equation:

$$
w''+w=1
$$

is:

$$
w(\theta)
=
1+A\cos\theta+B\sin\theta.
$$

But $\pi$-periodicity requires:

$$
A=B=0.
$$

Hence:

$$
w(\theta)=1.
$$

That is:

$$
h(\theta)+h(\theta+\pi)=1.
$$

So $W$ is a unit constant-width convex body.

Q.E.D.

---

# 13. The Fourier Inversion Form

Let:

$$
r(\theta)
=
\frac12
+
\sum_{\substack{n\ge3\\n\ \mathrm{odd}}}
\left(
\alpha_n\cos n\theta
+
\beta_n\sin n\theta
\right).
$$

The $n=1$ mode vanishes because of the moment conditions.

Then the canonical centered support function is:

$$
\boxed{
h(\theta)
=
\frac12
+
\sum_{\substack{n\ge3\\n\ \mathrm{odd}}}
\left(
\frac{\alpha_n}{1-n^2}\cos n\theta
+
\frac{\beta_n}{1-n^2}\sin n\theta
\right).
}
$$

So shape complexity is rewritten as:

$$
\boxed{
r\in\mathcal R
}
$$

rather than as an arbitrary planar shape.

This is still an infinite-dimensional domain, but its degrees of freedom are governed by a very strong pointwise box constraint:

$$
0\le r\le1,
$$

together with the antipodal coupling:

$$
r(\theta+\pi)=1-r(\theta).
$$

---

# 14. A Concrete Sanity Family

Consider:

$$
h_a(\theta)
=
\frac12
+
a\cos3\theta.
$$

Then:

$$
r_a(\theta)
=
h_a+h_a''
=
\frac12
-8a\cos3\theta.
$$

Requiring:

$$
0\le r_a\le1
$$

is equivalent to:

$$
|a|
\le
\frac1{16}.
$$

When:

$$
a=\frac1{16},
$$

we have:

$$
r_a(\theta)
=
\frac12
-
\frac12\cos3\theta
\in[0,1].
$$

This example verifies that:

- the constant-width condition leaves only the odd modes;
- the $n=3$ support mode is multiplied, in the curvature domain, by:

$$
1-3^2=-8;
$$

- convexity is not a matter of freely choosing Fourier coefficients, but is instead controlled by:

$$
0\le r\le1.
$$

---

# 15. Gate 4: Does a Worst-Case Target Really Exist?

Define the centered constant-width family:

$$
\mathcal W_1^0
=
\left\{
W\in\mathcal W_1:
s(W)=0
\right\}.
$$

Since the Steiner point lies inside $W$, or at least in $W$, and:

$$
\operatorname{diam}(W)=1,
$$

we have:

$$
W
\subseteq
\overline B(0,1).
$$

Hence for all $W\in\mathcal W_1^0$:

$$
0
\le
h_W(\theta)
\le
1.
$$

And:

$$
|h_W(\theta)-h_W(\phi)|
\le
\sup_{x\in W}\|x\|
\,
\|u_\theta-u_\phi\|
\le
\|u_\theta-u_\phi\|.
$$

So this family of support functions is:

- uniformly bounded;
- equicontinuous;
- closed under uniform limits.

By the Arzelà–Ascoli theorem:

$$
\boxed{
\mathcal W_1^0
\text{ is compact under the Hausdorff metric}.
}
$$

This can also be obtained from the Blaschke selection theorem.

---

# 16. The $1$-Lipschitz Property in the Target-Shape Direction

## Theorem 16.1

For fixed compact convex $U$:

$$
\left|
\Psi_U(K)
-
\Psi_U(L)
\right|
\le
d_H(K,L).
$$

### Proof

For any single configuration $(Q,t)$:

$$
d_H(QK+t,QL+t)
=
d_H(K,L).
$$

The directed Hausdorff functional measuring distance to the fixed $U$ is $1$-Lipschitz in its first argument, so:

$$
\left|
d_+(QK+t,U)
-
d_+(QL+t,U)
\right|
\le
d_H(K,L).
$$

Taking the infimum over $(Q,t)$ gives one direction:

$$
\Psi_U(K)
\le
\Psi_U(L)+d_H(K,L).
$$

Exchanging $K,L$ gives the reverse inequality.

Q.E.D.

---

# 17. The Outer $\sup$ Is Upgraded to $\max$

From:

- $\mathcal W_1^0$ being compact;
- $\Psi_U$ being continuous;

we obtain that:

$$
\boxed{
\max_{W\in\mathcal W_1^0}
\Psi_U(W)
}
$$

exists.

We therefore define:

$$
\mathfrak K(U)
=
\max_{W\in\mathcal W_1^0}
\Psi_U(W).
$$

Then:

$$
\boxed{
U\text{ universal}
\iff
\mathfrak K(U)=0.
}
$$

This formally upgrades what prior work often wrote as:

$$
\sup_W
$$

to, within this scope:

$$
\max_W.
$$

In other words:

$$
\boxed{
\text{for every fixed }U,
\text{there exists at least one genuine worst-case constant-width target}.
}
$$

Note:

this does not mean the worst-case target is unique.

---

# 18. An Abstract Finite $\varepsilon$-Net Theorem

Since:

$$
\mathcal W_1^0
$$

is compact, for every:

$$
\varepsilon>0
$$

there exists a finite:

$$
\varepsilon\text{-net }
\mathcal N_\varepsilon
\subset
\mathcal W_1^0.
$$

For any $W$, there exists:

$$
W_i\in\mathcal N_\varepsilon
$$

such that:

$$
d_H(W,W_i)
\le
\varepsilon.
$$

By Theorem 16.1:

$$
\Psi_U(W)
\le
\Psi_U(W_i)+\varepsilon.
$$

So:

$$
\boxed{
\max_{W_i\in\mathcal N_\varepsilon}
\Psi_U(W_i)
\le
\mathfrak K(U)
\le
\max_{W_i\in\mathcal N_\varepsilon}
\Psi_U(W_i)
+
\varepsilon.
}
$$

This formally fills in, at the **existence level**, the gap left open in the old six-way Q4:

$$
\text{shape-space finite sampling}
$$

Status:

`SHAPE-EPSILON-NET: ABSTRACT-PASS`

But this is still not an executable certificate, because none of the following has yet been given:

- a canonical finite net;
- a covering number;
- a net-construction algorithm;
- exact validation of every net cell.

Hence:

`CONSTRUCTIVE SHAPE CERTIFICATE: OPEN`

---

# 19. Gate 5: The Translation Compiler Under a Fixed Orientation

Fix:

$$
Q\in O(2).
$$

Define:

$$
a_Q(u)
=
h_K(Q^Tu)-h_U(u).
$$

Then define:

$$
F_Q(t)
=
\max_{u\in S^1}
\left[
a_Q(u)+t\cdot u
\right].
$$

Then:

$$
\inf_t
d_+(QK+t,U)
=
\max
\left\{
0,
\min_t F_Q(t)
\right\}.
$$

---

# 20. Existence of the Translation Minimizer

Since $a_Q$ is continuous on the compact $S^1$, it is bounded.

Let:

$$
A
=
\|a_Q\|_\infty.
$$

For any $t\neq0$, take:

$$
u
=
\frac{t}{\|t\|}.
$$

Then:

$$
F_Q(t)
\ge
t\cdot u-A
=
\|t\|-A.
$$

Hence:

$$
F_Q(t)
\to\infty
\qquad
\text{as }\|t\|\to\infty.
$$

$F_Q$ is continuous and coercive, so there exists:

$$
t_Q^\star
\in
\operatorname{argmin}
F_Q.
$$

So the translation infimum under a fixed orientation is in fact a minimum.

---

# 21. An Active Certificate of At Most Three Directions

Let:

$$
M_Q
=
F_Q(t_Q^\star).
$$

The active direction set is:

$$
\mathcal A_Q
=
\left\{
u\in S^1:
a_Q(u)
+t_Q^\star\cdot u
=
M_Q
\right\}.
$$

$F_Q$ is the pointwise maximum of affine functions, so:

$$
\partial F_Q(t_Q^\star)
=
\operatorname{conv}
\mathcal A_Q.
$$

The first-order minimization condition is:

$$
0
\in
\partial F_Q(t_Q^\star).
$$

Hence:

$$
0
\in
\operatorname{conv}
\mathcal A_Q.
$$

By the Carathéodory theorem in $\mathbb R^2$, there exist at most three active directions:

$$
u_1,u_2,u_3
\in
\mathcal A_Q
$$

and:

$$
\lambda_i\ge0,
\qquad
\sum_i\lambda_i=1,
$$

such that:

$$
\boxed{
\sum_i\lambda_i u_i=0.
}
$$

and simultaneously:

$$
a_Q(u_i)
+t_Q^\star\cdot u_i
=
M_Q.
$$

Hence:

$$
\boxed{
\text{under a fixed orientation, the optimal translation can be witnessed by at most three active support directions.}
}
$$

This is this round's first finite active-set compression of placement freedom.

---

# 22. What This Three-Direction Conclusion Can and Cannot Do

What it can do:

1. for fixed $Q$, compress the necessary optimality condition for the optimal translation down to at most three directions;
2. provide a finite active witness for the translation certificate;
3. constrain part of the combinatorial structure of future branch types to:
   - one-active degeneracy;
   - two-active balance;
   - three-active balance.

What it cannot do:

1. eliminate the continuous rotation of:

$$
Q\in O(2)
$$

2. prove that the global shape domain has only finitely many active triples;
3. prove that the active set does not undergo infinitely many branch switches across different $Q$;
4. automatically yield finite global closure.

Its RCHM status is therefore:

`TRANSLATION LOCAL FINITE-ACTIVE: PASS`

`GLOBAL PLACEMENT FINITE-BRANCH: OPEN`

---

# 23. The Correct Treatment of the Reflection Branch

The original problem uses congruent / isometric copies.

Hence:

$$
Q\in O(2)
$$

rather than only allowing:

$$
Q\in SO(2).
$$

$O(2)$ can be split into two connected components:

$$
O(2)
=
SO(2)
\cup
R_\mathrm{ref}SO(2).
$$

The remaining freedom in placement orientation can therefore be written as:

$$
\boxed{
\sigma\in\{+1,-1\}
\quad+\quad
\phi\in S^1.
}
$$

where:

- $\sigma=+1$: orientation-preserving;
- $\sigma=-1$: includes reflection.

This promotes reflection from an easily-overlooked implicit condition to an explicit discrete branch.

---

# 24. The Exact Canonical Minimax Normal Form

Let:

$$
r\in\mathcal R.
$$

By Theorem 12.1, take the canonical centered support function:

$$
h_r.
$$

For a candidate universal cover $U$, we have:

$$
\boxed{
\mathfrak K(U)
=
\max_{r\in\mathcal R}
\min_{Q\in O(2)}
\min_{t\in\mathbb R^2}
\max
\left\{
0,
\max_{\theta\in S^1}
\left[
h_r(Q^Tu_\theta)
+t\cdot u_\theta
-h_U(\theta)
\right]
\right\}.
}
$$

and:

$$
\boxed{
U\text{ universal}
\iff
\mathfrak K(U)=0.
}
$$

So the Lebesgue constant can be written as:

$$
\boxed{
a_{\mathrm{Leb}}
=
\inf_{U\in\mathcal U}
\left\{
\operatorname{Area}(U):
\mathfrak K(U)=0
\right\}.
}
$$

This is the canonical problem statement after Round 01.

---

# 25. RCHM: The Original Degrees-of-Freedom Ledger

Before Round 01 began, the main degrees of freedom included at least:

$$
\mathcal F_0
=
\left\{
\begin{array}{l}
\text{arbitrary target topology},\\
\text{nonconvexity},\\
\text{target scale }d\le1,\\
\text{target translation gauge},\\
\text{infinite shape complexity},\\
\text{reflection branch},\\
\text{rotation continuum},\\
\text{placement translation},\\
\text{support-direction continuum},\\
\text{candidate-cover shape freedom}
\end{array}
\right\}.
$$

---

# 26. The Degrees-of-Freedom Ledger After Round 01

The items this round genuinely eliminates or rewrites:

## 26.1 Fully Eliminated

$$
\text{arbitrary target topology}
$$

eliminated by the closed-convex-hull reduction.

$$
\text{nonconvexity}
$$

eliminated by the convex universal cover + convexification.

$$
\text{target scale }d\le1
$$

normalized by the unit constant-width completion reduction.

$$
\text{target translation gauge}
$$

eliminated by the curvature density $r$, or equivalently the Steiner gauge.

---

## 26.2 Strongly Compressed but Not Eliminated

$$
\text{infinite shape complexity}
$$

is rewritten as:

$$
r\in\mathcal R
\subset
L^\infty(S^1),
$$

where:

$$
0\le r\le1,
$$

$$
r(\theta+\pi)=1-r(\theta),
$$

together with two moment constraints.

---

## 26.3 Locally Finitized

Under a fixed $Q$, placement translation is witnessed by at most three active directions.

---

## 26.4 Still Fully OPEN

1.

$$
r\in\mathcal R
$$

's constructive finite compiler;

2. the global branch changes of:

$$
\phi\in S^1
$$

3. certified global handling of the support-direction continuum;

4. the boundary freedom of the candidate cover $U$;

5. finite branch saturation;

6. global closure.

---

# 27. Escape Ledger

After Round 01, the legal escape is no longer "an arbitrary planar set."

The main remaining escapes are:

## E1: Curvature-Density Microstructure Escape

There exist sequences:

$$
r_n\in\mathcal R
$$

that oscillate at increasingly fine angular scales.

Even though:

$$
0\le r_n\le1,
$$

new shape microstructure can still form.

---

## E2: High Odd-Mode Escape

The Fourier modes:

$$
n=3,5,7,\ldots
$$

have no natural finite cutoff.

---

## E3: Rotation-Branch Escape

The optimal:

$$
\phi
$$

may switch non-smoothly as $r$ and $U$ vary.

---

## E4: Active-Set Switching Escape

A fixed orientation only requires at most three active directions, but the active triple itself can switch, continuously or discretely, as parameters vary.

---

## E5: Container-Boundary Escape

This round only compresses the target family.

The candidate cover $U$ still ranges over an infinite-dimensional domain of convex bodies.

---

# 28. Re-Verdict on the Old Six-Way Research

| Prior Item | Round 01 Verdict | New Role |
|---|---|---|
| Q1 continuity | `PASS` | stability / error propagation |
| Q2 support function | `PASS` | primary exact handoff |
| Q3 constant-width reduction | `REBUILD-PASS` | exact target-domain reduction |
| Q4 angular/rotation sampling | `PASS-WITH-SCOPE` | certified sampling for fixed shape/configuration variables |
| Q4 missing shape net | `ABSTRACT-PASS` | compactness gives finite $\varepsilon$-nets |
| Q5 predictive $SE(2)$ | `SEARCH-ONLY` | accelerator, never completeness proof |
| Q6 removable region | `FINITE-FAMILY-PASS` | candidate-cover surgery heuristic + finite certificate |
| One-dimensional distance kernel | `REUSE` | representation fidelity auditor |
| $H_2/C_{10}$ | `TEST` | symmetry hypothesis only |

---

# 29. An Important Forbidden Shortcut: One Cannot Simply Test Only Extreme Constant-Width Bodies

The family of unit constant-width bodies forms a convex set under Minkowski addition.

Extreme planar constant-width curves have the Kallay-type characterization:

$$
r(\theta)\in\{0,1\}
$$

almost everywhere.

This easily creates a temptation:

> Since any constant-width body can be approximated by the convex structure of extreme bodies, is it enough to test only the extreme bodies?

This round explicitly rules on this:

$$
\boxed{
\text{this reduction cannot currently be carried out.}
}
$$

The reason is that the universal-cover quantifier reads:

$$
\forall K
\quad
\exists g_K.
$$

Even if:

$$
K
=
(1-\lambda)K_0+\lambda K_1,
$$

and there exist:

$$
g_0K_0\subseteq U,
$$

$$
g_1K_1\subseteq U,
$$

there is in general no reason to conclude that there exists a single rigid motion $g$ such that:

$$
gK\subseteq U.
$$

because:

$$
g_0
\neq
g_1
$$

is generally allowed.

So:

$$
\boxed{
\text{Minkowski convex decomposition}
\not\Rightarrow
\text{placement-compatible decomposition}.
}
$$

If a future round wants to test only extreme bodies, it must separately prove a:

`COMMON-PLACEMENT / EXTREME-SATURATION LEMMA`

Current status:

`EXTREME-ONLY REDUCTION: NOT ESTABLISHED`

---

# 30. A Second Forbidden Shortcut: Naive Fourier Truncation Is Not Automatically Legal

The constant-width condition does indeed leave only the odd Fourier modes.

But directly taking the partial sum:

$$
h_M
=
\frac12
+
\sum_{\substack{3\le n\le M\\n\ \mathrm{odd}}}
\left(
a_n\cos n\theta
+
b_n\sin n\theta
\right)
$$

while it preserves:

$$
h_M(\theta)+h_M(\theta+\pi)=1,
$$

does not automatically preserve:

$$
h_M+h_M''
\ge0.
$$

So:

$$
\boxed{
\text{odd Fourier truncation preserves width but may violate convexity}.
}
$$

Any future finite compiler must handle positivity at the level of:

$$
r
=
h+h'',
$$

or use a positive-kernel / interval-safe construction.

---

# 31. This Round's New Methodological Conclusion

Round 00's main chain is:

$$
\text{Existence}
\to
\text{Descent}
\to
\text{Saturation}
\to
\text{Global Closure}.
$$

Round 01 now makes the target-domain descent explicit as:

$$
\boxed{
A
\to
\overline{\operatorname{conv}}A
\to
W\in\mathcal W_1
\to
h_W
\to
r_W\in\mathcal R.
}
$$

and every arrow currently has:

- a definition;
- hypotheses;
- an exact proof;
- a record of which degrees of freedom are lost;
- a record of the remaining escapes.

This can therefore serve as the canonical descent chain for subsequent research.

---

# 32. Round 01's Most Important New Gate

After this round, the real next obstacle has shifted from:

> How can every planar shape be finitized?

to:

> How can a constructive, legality-preserving, error-certified finite compiler be built for $\mathcal R$?

That is, finding a finite-dimensional:

$$
\mathcal R_M
$$

together with a reconstruction / approximation operator:

$$
P_M:
\mathcal R
\to
\mathcal R_M
$$

satisfying at least:

1.

$$
P_Mr
\in
\mathcal R
$$

rather than producing an illegal shape;

2. preserving:

$$
0\le P_Mr\le1;
$$

3. preserving:

$$
P_Mr(\theta+\pi)
=
1-P_Mr(\theta);
$$

4. preserving the first moments:

$$
\int P_Mr\cos=0,
\qquad
\int P_Mr\sin=0;
$$

5. having a certified error once the support function is reconstructed:

$$
\|h_r-h_{P_Mr}\|_\infty
\le
\varepsilon_M;
$$

6.

$$
\varepsilon_M
\to0.
$$

If this can be achieved, then by:

$$
d_H
=
\|h_1-h_2\|_\infty
$$

together with Theorem 16.1, the finite shape approximation error can be carried directly through to:

$$
\Psi_U.
$$

---

# 33. The Assigned Topic for Round 02

The next round's formal topic:

## AMRAL-LUC-FC-R02
### Curvature-Density Finite Compiler and Certified Shape Approximation

Main objective:

$$
\boxed{
\mathcal R
\longrightarrow
\mathcal R_M
}
$$

and the handoff must be legal.

Three constructions are to be investigated first:

1. a positive convolution / Fejér-type compiler;
2. Jackson-type positive approximation;
3. antipodal-paired piecewise-constant curvature density.

If any one of these can simultaneously preserve:

- box positivity;
- the antipodal complement;
- moment balance;
- support reconstruction;
- an explicit Hausdorff error;

it can become a genuine constructive finite-shape certificate layer.

---

# 34. COMPUTE-DEFERRED

This round contains no core theorem that requires large-scale computation to hold.

The following are listed only as follow-up computational tasks and do not affect Round 01's structural conclusions.

## C01: Curvature-Density Random Generator

Generate:

$$
r\in\mathcal R
$$

and check support reconstruction.

## C02: Active-Direction Branch Census

For many tuples:

$$
(K,U,Q)
$$

numerically tabulate the active set of the translation minimizer:

$$
1/2/3
$$

direction-branch proportions and degenerate events.

## C03: High-Mode Adversarial Shapes

Search for whether high odd Fourier modes can generate higher placement pressure against existing candidate universal covers than the Reuleaux polygon does.

All three items are marked:

`COMPUTE-DEFERRED`

If executed locally later, the output must feed back into the canonical audit, and must not retroactively rewrite the structure already proved in this round.

---

# 35. This Round's Reproducibility Checklist

## Definition Audit

`PASS`

## Quantifier Audit

Preserved:

$$
\forall K\exists g.
$$

`PASS`

## Degenerate Diameter Audit

$d=0$ has been handled.

`PASS`

## Diameter $<1$ Audit

completion + dilation has been handled.

`PASS`

## Reflection Audit

The two components of $O(2)$ have been explicitly recorded.

`PASS`

## Translation Gauge Audit

Shape translation and placement translation have been separated.

`PASS`

## Support Containment Audit

exact.

`PASS`

## Shape Compactness Audit

the centered constant-width family is compact.

`PASS`

## Shape Finite-Net Status

Existence:

`PASS`

Constructiveness:

`OPEN`

## Numerical-Bound Claim

This round has no new bound.

`NONE`

---

# 36. This Round's Progress Markers

$$
\boxed{
\text{Round 01 status: CLOSED}
}
$$

$$
\boxed{
\text{Lebesgue global problem: OPEN}
}
$$

$$
\boxed{
\text{New numerical lower bound: NO}
}
$$

$$
\boxed{
\text{New numerical upper bound: NO}
}
$$

$$
\boxed{
\text{New exact reduction: YES}
}
$$

$$
\boxed{
\text{New canonical shape domain: YES}
}
$$

$$
\boxed{
\text{Constructive finite shape certificate: NEXT GATE}
}
$$

---

# 37. The Shortest Handoff Conclusion

Round 01 rigorously compresses the problem from:

$$
\text{arbitrary planar diameter-}\le1\text{ sets}
$$

down to:

$$
\boxed{
r\in\mathcal R
}
$$

where:

$$
0\le r\le1,
$$

$$
r(\theta+\pi)=1-r(\theta),
$$

$$
\int r\cos=0,
$$

$$
\int r\sin=0.
$$

All translation freedom of the target shape has already been eliminated.

The containment relation is then written exactly as:

$$
h_r(Q^Tu)
+t\cdot u
\le
h_U(u).
$$

For fixed $Q$, the optimal $t$ is witnessed by at most three active directions.

The entire target family is compact, a worst-case target necessarily exists, and a finite $\varepsilon$-net exists at any precision.

So the next round no longer asks "can finite approximation be done," but instead asks:

$$
\boxed{
\text{how should one construct a legality-preserving, verifiable finite compiler?}
}
$$

---

# References

1. J. C. Baez, K. Bagdasaryan, P. Gibbs, *The Lebesgue Universal Covering Problem*, arXiv:1502.01251, 2015.
2. P. Gibbs, *An Upper Bound for Lebesgue's Covering Problem*, arXiv:1810.10089, 2018.
3. U. Mishra, *Curves of constant width and Lebesgue's covering problem*, arXiv:2608.30538, 2026.
4. S. Zeng, *An exact hierarchy for Lebesgue's universal covering constant and a certified 0.834 lower bound*, arXiv:2609.01284, 2026.
5. C. He, H. Martini, S. Wu, *Complete sets in normed linear spaces*, Banach Journal of Mathematical Analysis 17, 45, 2023.
6. R. Hynd, *On Extreme Constant Width Bodies in $\mathbb R^3$*, Discrete & Computational Geometry, 2025.
7. C. Thäle, *A note on a problem posed by Linderholm*, Journal of Geometry 117, article 1, 2026.
8. Y. Martinez-Maure, *Noncircular algebraic curves of constant width: an answer to Rabinowitz*, Canadian Mathematical Bulletin 65(3), 2022.
9. Neo.K, *Lebesgue Universal Covering Problem: Six-Way Parallel Computation and Skew-Field Preliminary Verification*, EML-LUC-2026-SIXWAY-v0.1, internal research draft, 2026-07.
10. Neo.K, *From One-Dimensional Metric Lines to Universal Covering Tension*, internal research draft, 2026-07.
11. Neo.K + Aletheia, *AMRAL × Lebesgue Universal Covering — Round 00: Finite Closure Methodology*, AMRAL-LUC-FC-R00 v0.2, 2026-09-18.