# AMRAL × Lebesgue Universal Covering — Round 06
## Finite Witness Exchange Compiler and Minimizer-Set Separation

**Document ID:** AMRAL-LUC-FC-R06  
**Version:** v0.1  
**Date:** 2026-09-18  
**Research status:** Round 06 / Witness-saturation phase / Lower-bound exchange compiler  
**Research mode:** Human-Directed + Semi-Autonomous AI Mathematical Research  
**Research initiation and methodology source:** Neo.K  
**AI collaborating researcher and primary executor:** Aletheia / ChatGPT, GPT-5.6 Sol  
**Parent methodology:** Relational Constraint–Handoff Methodology (RCHM)  
**Prerequisite documents:** AMRAL-LUC-FC-R00 v0.2; R01–R05 v0.1

---

# 0. This Round's Summary Verdict

Round 05 already proved:

$$
a_{\mathrm{Leb}}
=
\sup_{\mathcal F\ {\rm finite}}
\Lambda(\mathcal F),
$$

and established the finite-witness cardinality hierarchy:

$$
\lambda_m\uparrow a_{\mathrm{Leb}}.
$$

But the core of Round 05 was still existential:

> If the current witness family is not yet saturated, then some larger finite family is guaranteed to raise the lower bound.

Round 06's task is to compile this sentence into a replayable, finitely verifiable **Witness Exchange Compiler**.

This round obtains the following core results.

---

## Conclusion A: The finite-family placement minimizer set has a compact configuration domain

Let:

$$
\mathcal F
=
\{D,K_2,\ldots,K_m\},
$$

where:

$$
D=B_{1/2}(0)
$$

is the unit-diameter disk, and the other:

$$
K_i
$$

are centered unit constant-width bodies.

When we only care about configurations whose hull area:

$$
\le\bar A,
$$

we may fix:

$$
D
$$

at the origin.

Round 04's radius theorem gives:

$$
\operatorname{conv}
\bigcup_i g_iK_i
\subseteq
B_R(0),
\qquad
R=2\bar A.
$$

Because:

$$
0\in K_i,
$$

the placement translation:

$$
t_i=g_i(0)
$$

is itself in the hull, so:

$$
\|t_i\|\le R.
$$

And:

$$
Q_i\in O(2)
$$

is compact.

Therefore the finite-family low-area configuration domain can be restricted to a compact set.

---

## Conclusion B: Mishra's three witness nodes are exactly a five-dimensional placement problem

For:

$$
\mathcal F_3
=
\{D,B_3,B_5\},
$$

after fixing the disk:

- $B_3$ has three continuous parameters $(x_3,y_3,\phi_3)$;
- $B_5$ has three continuous parameters $(x_5,y_5,\phi_5)$.

Total:

$$
6.
$$

Rotating all non-disk bodies together does not change the hull area — it only produces a common rotation of the whole hull.

So we may fix one global rotation gauge, subtracting one dimension:

$$
\boxed{
6-1=5.
}
$$

This exactly matches Mishra 2026's certified five-dimensional placement search.

Reflection parity is treated as a separate finite discrete branch, and is not counted in the continuous dimension.

In general, for:

$$
m\ge2,
$$

under the disk-anchor-plus-one-common-rotation-gauge convention, the continuous parameter dimension is:

$$
\boxed{
d_m
=
3(m-1)-1
=
3m-4,
}
$$

without further quotienting by individual target symmetries.

---

## Conclusion C: If the current family is not yet saturated, the entire minimizer set has a uniform positive violation margin

Let:

$$
\mathfrak M(\mathcal F)
$$

be the set of all area-minimizing finite-family covers.

If:

$$
\Lambda(\mathcal F)
<
a_{\mathrm{Leb}},
$$

then no:

$$
U\in\mathfrak M(\mathcal F)
$$

is universal.

The worst-target margin from Round 03–04:

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

is $1$-Lipschitz in $U$.

And:

$$
\mathfrak M(\mathcal F)
$$

is compact.

So:

$$
\boxed{
\delta_{\mathcal F}
:=
\min_{U\in\mathfrak M(\mathcal F)}
W(U)
>
0.
}
$$

This upgrades Round 05's "each minimizer individually misses some target" to:

> All current minimizers are separated from the universality frontier by one common positive margin.

---

## Conclusion D: Round 02's finite target dictionary necessarily forms a separating batch at finite resolution

Let:

$$
\mathscr D_\varepsilon
$$

be Round 02's legal constant-width finite target $\varepsilon$-net.

If:

$$
\varepsilon
<
\delta_{\mathcal F},
$$

then for every:

$$
U\in\mathfrak M(\mathcal F),
$$

there exists at least one:

$$
\widehat K\in\mathscr D_\varepsilon
$$

such that:

$$
M_U(\widehat K)
>
0.
$$

So the finite family's:

$$
\mathscr D_\varepsilon
$$

non-fit neighborhoods already cover the entire minimizer set.

Therefore a finite separating batch must exist:

$$
\boxed{
\mathcal B
\subseteq
\mathscr D_\varepsilon.
}
$$

After adding it:

$$
\boxed{
\Lambda(\mathcal F\cup\mathcal B)
>
\Lambda(\mathcal F).
}
$$

---

## Conclusion E: Establishing the separation margin

For a finite batch:

$$
\mathcal B,
$$

define:

$$
\boxed{
S_{\mathcal F}(\mathcal B)
=
\min_{U\in\mathfrak M(\mathcal F)}
\max_{K\in\mathcal B}
M_U(K).
}
$$

If:

$$
\boxed{
S_{\mathcal F}(\mathcal B)>0,
}
$$

then:

$$
\boxed{
\Lambda(\mathcal F\cup\mathcal B)
>
\Lambda(\mathcal F).
}
$$

So a witness batch never needs to guess whether it is "useful" —

it only needs to prove:

$$
S_{\mathcal F}(\mathcal B)>0.
$$

---

## Conclusion F: A certified exchange search on a non-saturated family necessarily succeeds in finite time

Assume:

1. minimizer-set cell cover resolution:

$$
\tau_n\to0;
$$

2. target dictionary error:

$$
\varepsilon_n\to0;
$$

3. fixed-cover placement certificate error:

$$
\zeta_n\to0.
$$

If:

$$
\Lambda(\mathcal F)<a_{\mathrm{Leb}},
$$

then:

$$
\delta_{\mathcal F}>0.
$$

So for sufficiently large finite $n$:

$$
\tau_n+\varepsilon_n+\zeta_n
<
\frac{\delta_{\mathcal F}}{2}.
$$

At that finite level, a separating witness batch can definitely be found in a certified way.

So:

$$
\boxed{
\text{NON-SATURATED}
\Rightarrow
\text{FINITE REFINEMENT EVENTUALLY FINDS EXCHANGE BATCH}.
}
$$

This is Round 06's main termination theorem.

---

## Conclusion G: Witness dominance can be formalized

If there exists a rigid motion:

$$
g
$$

such that:

$$
gK\subseteq L,
$$

then for any other finite family:

$$
\mathcal G,
$$

we have:

$$
\boxed{
\Lambda(\mathcal G\cup\{L\})
\ge
\Lambda(\mathcal G\cup\{K\}).
}
$$

So upgrading the regular polygons:

$$
P_3,P_5
$$

to:

$$
B_3,B_5
$$

is a proof-level dominance upgrade, not merely a numerical heuristic.

---

This round's verdict:

$$
\boxed{
\text{FINITE WITNESS EXCHANGE COMPILER: CLOSED}
}
$$

but:

$$
\boxed{
\text{NEXT WITNESS NUMERIC IDENTITY: COMPUTE-DEFERRED}
}
$$

---

# 1. Public Starting Point: Mishra 2026

In 2026, Ujjwal Mishra chose three unit constant-width test sets:

$$
\boxed{
D,\quad B_3,\quad B_5
}
$$

where:

- $D$: the diameter-one disk;
- $B_3$: the Reuleaux triangle;
- $B_5$: the regular Reuleaux pentagon.

His certificate proves:

$$
\boxed{
a_{\mathrm{Leb}}
\ge
0.8344.
}
$$

The placement space is five-dimensional, and was exhaustively subdivided into a certificate of:

$$
486,799,600
$$

nodes.

That work separates the search from the verifier; the public paper reports a floating-point error bound of:

$$
1.72\times10^{-9}.
$$

This round does not rerun the five hundred million nodes.

It is instead used as the external certified seed for:

$$
\boxed{
\mathcal F_3
=
\{D,B_3,B_5\}
}
$$

Importantly:

$$
0.8344
$$

is a proven lower bound of:

$$
\Lambda(\mathcal F_3),
$$

it does not mean:

$$
\Lambda(\mathcal F_3)=0.8344,
$$

nor does it mean:

$$
\lambda_3=0.8344.
$$

---

# 2. Classical Seed Ladder

## Level 1: the disk

$$
\mathcal F_1=\{D\}.
$$

Therefore:

$$
\boxed{
\Lambda(\mathcal F_1)
=
\lambda_1
=
\frac{\pi}{4}
\approx
0.7853981634.
}
$$

---

## Level 2: adding the Reuleaux triangle

The Reuleaux triangle:

$$
B_3
$$

contains the side-one equilateral triangle:

$$
P_3.
$$

So by dominance:

$$
\Lambda(D,B_3)
\ge
\Lambda(D,P_3).
$$

Pál's classical value:

$$
\Lambda(D,P_3)
=
\frac{\pi}{8}
+
\frac{\sqrt3}{4}.
$$

Therefore:

$$
\boxed{
\Lambda(D,B_3)
\ge
\frac{\pi}{8}
+
\frac{\sqrt3}{4}
\approx
0.8257117836.
}
$$

This is already strictly higher than:

$$
\lambda_1.
$$

---

## Level 3: adding the Reuleaux pentagon

Mishra 2026:

$$
\boxed{
\Lambda(D,B_3,B_5)
\ge
0.8344.
}
$$

So:

$$
\boxed{
\lambda_3
\ge
0.8344.
}
$$

The publicly proven upper bound:

$$
a_{\mathrm{Leb}}
\le
0.8440935944.
$$

So the seed's public gap is:

$$
\boxed{
0.0096935944.
}
$$

Round 06's witness exchange compiler takes this as its formal starting point.

---

# 3. An Important Directional Warning

For a fixed witness family:

$$
\mathcal F,
$$

what needs to be proved is the **lower bound** of:

$$
\Lambda(\mathcal F)
=
\min_{\text{placements}}
\operatorname{Area}(\text{hull}).
$$

An ordinary optimizer finding a placement:

$$
q
$$

only provides:

$$
\Lambda(\mathcal F)
\le
A(q).
$$

That is, an upper bound on the family minimum.

This direction cannot be used as a Lebesgue lower bound.

Mishra 2026 specifically points this problem out with respect to Gibbs 2014's multi-Reuleaux numerical search.

Therefore AMRAL witness exchange must completely separate:

$$
\boxed{
\text{search}
}
$$

from:

$$
\boxed{
\text{lower certificate}.
}
$$

---

# 4. Finite-Family Configuration Domain

Let:

$$
\mathcal F
=
\{D,K_2,\ldots,K_m\}.
$$

For:

$$
i\ge2,
$$

write the placement as:

$$
g_i(x)
=
Q_ix+t_i.
$$

Fix:

$$
D
$$

at the origin.

The configuration:

$$
q
=
(Q_2,t_2,\ldots,Q_m,t_m).
$$

Define the hull:

$$
\boxed{
H_{\mathcal F}(q)
=
\operatorname{conv}
\left(
D
\cup
\bigcup_{i=2}^m
(Q_iK_i+t_i)
\right).
}
$$

The objective:

$$
\boxed{
A_{\mathcal F}(q)
=
\operatorname{Area}
H_{\mathcal F}(q).
}
$$

---

# 5. Low-Area Configuration Compactification

Take any known universal upper bound:

$$
\bar A.
$$

We only need to study configurations with:

$$
A_{\mathcal F}(q)\le\bar A.
$$

Because:

$$
D\subseteq H_{\mathcal F}(q),
$$

and the hull is convex, Round 04's radius theorem gives:

$$
H_{\mathcal F}(q)
\subseteq
B_R(0),
$$

where:

$$
R=2\bar A.
$$

And the centered target:

$$
K_i
$$

contains:

$$
0.
$$

So:

$$
t_i
=
g_i(0)
\in
g_iK_i
\subseteq
H_{\mathcal F}(q).
$$

Therefore:

$$
\boxed{
\|t_i\|\le R.
}
$$

The orientation:

$$
Q_i\in O(2)
$$

is compact.

So the low-area configuration domain is a compact subset of:

$$
\boxed{
\left(
O(2)\times\overline B_R
\right)^{m-1}.
}
$$

---

# 6. Common Rotation Gauge

Since the disk:

$$
D
$$

is invariant under all of:

$$
O(2),
$$

if we apply the same rotation:

$$
R_\alpha
$$

to all:

$$
i\ge2
$$

simultaneously, the whole hull is only acted on by the common rotation:

$$
R_\alpha.
$$

The area is unchanged.

So when:

$$
m\ge2,
$$

we may fix one continuous global rotation gauge.

Without further quotienting by individual target symmetry:

$$
\boxed{
d_m
=
3(m-1)-1
=
3m-4.
}
$$

Each non-disk target contributes:

- 2 translation parameters;
- 1 rotation parameter.

Reflection is a discrete parity branch.

For:

$$
m=3,
$$

we get:

$$
\boxed{
d_3=5.
}
$$

matching the Mishra placement domain.

---

# 7. Configuration Objective Continuity

If:

$$
q_n\to q
$$

in the configuration domain, then each placed body:

$$
g_i^{(n)}K_i
$$

converges in the Hausdorff metric to:

$$
g_iK_i.
$$

Finite union is continuous with respect to Hausdorff convergence, and the convex hull operation is also continuous.

Therefore:

$$
H_{\mathcal F}(q_n)
\to
H_{\mathcal F}(q).
$$

For planar convex bodies, area is continuous with respect to Hausdorff convergence.

So:

$$
\boxed{
A_{\mathcal F}(q)
\text{ is continuous}.
}
$$

Compactness therefore again gives:

$$
\boxed{
\operatorname{Argmin}
A_{\mathcal F}
\neq\varnothing
}
$$

and compact.

---

# 8. Cover-Space Minimizer Set

Configuration minimizers may have a large number of gauge-equivalent duplicates.

So we define the true geometric minimizer set:

$$
\boxed{
\mathfrak M(\mathcal F)
=
\left\{
H_{\mathcal F}(q):
A_{\mathcal F}(q)
=
\Lambda(\mathcal F)
\right\}.
}
$$

Then:

$$
\mathfrak M(\mathcal F)
$$

is a compact set of convex bodies.

Every:

$$
U\in\mathfrak M(\mathcal F)
$$

satisfies:

$$
\operatorname{Area}(U)
=
\Lambda(\mathcal F).
$$

---

# 9. Worst-Target Margin

Following Round 03–05:

$$
M_U(K)
$$

is the best signed placement margin of target:

$$
K
$$

against the fixed cover:

$$
U.
$$

$$
M_U(K)\le0
$$

if and only if:

$$
K
$$

can be placed inside:

$$
U.
$$

Define:

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

Then:

$$
U\text{ is universal}
\iff
W(U)\le0.
$$

And:

$$
W
$$

is $1$-Lipschitz in the cover variable:

$$
\boxed{
|W(U)-W(V)|
\le
d_H(U,V).
}
$$

---

# 10. A Non-Universal Cover Has a Strict Positive Worst Margin

If:

$$
U
$$

is not universal, there exists a:

$$
K
$$

that cannot be placed inside:

$$
U.
$$

Round 03's placement-minimization attainment result means:

$$
M_U(K)
$$

genuinely attains its minimum.

If:

$$
M_U(K)=0,
$$

then the achieving placement's support residual is:

$$
\le0
$$

everywhere, i.e.:

$$
K
$$

can be placed inside:

$$
U,
$$

a contradiction.

So:

$$
M_U(K)>0.
$$

Therefore:

$$
\boxed{
U\text{ non-universal}
\Rightarrow
W(U)>0.
}
$$

---

# 11. Uniform Minimizer Violation Theorem

## Theorem 11.1

If:

$$
\Lambda(\mathcal F)
<
a_{\mathrm{Leb}},
$$

then:

$$
\boxed{
\delta_{\mathcal F}
:=
\min_{U\in\mathfrak M(\mathcal F)}
W(U)
>
0.
}
$$

### Proof

If some:

$$
U\in\mathfrak M(\mathcal F)
$$

were universal, then:

$$
a_{\mathrm{Leb}}
\le
\operatorname{Area}(U)
=
\Lambda(\mathcal F)
<
a_{\mathrm{Leb}},
$$

a contradiction.

So every:

$$
U\in\mathfrak M(\mathcal F)
$$

is non-universal.

By the previous section:

$$
W(U)>0
$$

holds for every minimizer.

$$
\mathfrak M(\mathcal F)
$$

is compact, and $W$ is continuous.

So its minimum is attained and is positive.

Q.E.D.

---

# 12. What This $\delta_{\mathcal F}$ Means

Round 05 only guaranteed:

> Every minimizer misses some target.

Round 06 now obtains:

$$
\boxed{
\text{every minimizer misses by at least a signed margin of }\delta_{\mathcal F}.
}
$$

This is what lets finite target approximation genuinely engage.

---

# 13. Target Dictionary Transfer

Round 02 already established:

for any:

$$
\varepsilon>0,
$$

there exists a finite legal dictionary:

$$
\mathscr D_\varepsilon
\subset
\mathcal W_1^0
$$

such that for any:

$$
K\in\mathcal W_1^0,
$$

there exists:

$$
\widehat K\in\mathscr D_\varepsilon
$$

satisfying:

$$
d_H(K,\widehat K)
\le
\varepsilon.
$$

Round 02–03:

$$
\boxed{
|M_U(K)-M_U(\widehat K)|
\le
d_H(K,\widehat K).
}
$$

---

# 14. The Finite Dictionary Separates All Current Minimizers

## Theorem 14.1

If:

$$
\Lambda(\mathcal F)
<
a_{\mathrm{Leb}}
$$

and:

$$
0<\varepsilon<\delta_{\mathcal F},
$$

then for every:

$$
U\in\mathfrak M(\mathcal F)
$$

there exists:

$$
\widehat K\in\mathscr D_\varepsilon
$$

such that:

$$
\boxed{
M_U(\widehat K)
\ge
\delta_{\mathcal F}-\varepsilon
>
0.
}
$$

### Proof

Choose the true worst target:

$$
K_U
$$

such that:

$$
M_U(K_U)
=
W(U)
\ge
\delta_{\mathcal F}.
$$

Take a dictionary approximation:

$$
d_H(K_U,\widehat K)
\le
\varepsilon.
$$

By the target Lipschitz property:

$$
M_U(\widehat K)
\ge
M_U(K_U)-\varepsilon
\ge
\delta_{\mathcal F}-\varepsilon.
$$

Q.E.D.

---

# 15. Non-Fit Neighborhood Cover

For:

$$
K\in\mathscr D_\varepsilon,
$$

define:

$$
\boxed{
\mathcal N_K
=
\left\{
U\in\mathfrak M(\mathcal F):
M_U(K)>0
\right\}.
}
$$

Because:

$$
M_U(K)
$$

is continuous in $U$,

$$
\mathcal N_K
$$

is an open subset of the minimizer set.

Theorem 14.1 says:

$$
\boxed{
\mathfrak M(\mathcal F)
\subseteq
\bigcup_{K\in\mathscr D_\varepsilon}
\mathcal N_K.
}
$$

And:

$$
\mathscr D_\varepsilon
$$

is itself finite.

So this is already a finite open cover.

---

# 16. Separating Batch

Take any subfamily:

$$
\mathcal B
\subseteq
\mathscr D_\varepsilon
$$

such that:

$$
\boxed{
\mathfrak M(\mathcal F)
\subseteq
\bigcup_{K\in\mathcal B}
\mathcal N_K.
}
$$

We call:

$$
\mathcal B
$$

a:

$$
\boxed{
\text{minimizer-set separating witness batch}.
}
$$

The simplest choice is always available:

$$
\mathcal B
=
\mathscr D_\varepsilon.
$$

In practice one solves a finite set-cover / hitting-set problem to compress the batch.

---

# 17. Separation Margin

Define:

$$
\boxed{
S_{\mathcal F}(\mathcal B)
=
\min_{U\in\mathfrak M(\mathcal F)}
\max_{K\in\mathcal B}
M_U(K).
}
$$

If:

$$
\mathcal B
$$

is merely a topological separating batch, then:

$$
S_{\mathcal F}(\mathcal B)
$$

is at least non-negative.

If it has robust separation:

$$
\boxed{
S_{\mathcal F}(\mathcal B)>0,
}
$$

then it can withstand numerical / cell / target-approximation errors.

Round 06 treats a robust batch as the formal exchange certificate.

---

# 18. Exchange Theorem

## Theorem 18.1

If:

$$
S_{\mathcal F}(\mathcal B)>0,
$$

then:

$$
\boxed{
\Lambda(\mathcal F\cup\mathcal B)
>
\Lambda(\mathcal F).
}
$$

### Proof

Suppose for contradiction:

$$
\Lambda(\mathcal F\cup\mathcal B)
=
\Lambda(\mathcal F).
$$

By finite-family attainment, there exists a cover of area:

$$
\Lambda(\mathcal F)
$$

, call it:

$$
U^\star,
$$

covering:

$$
\mathcal F\cup\mathcal B.
$$

Because:

$$
U^\star
$$

also covers:

$$
\mathcal F
$$

and attains the area:

$$
\Lambda(\mathcal F),
$$

we have:

$$
U^\star\in\mathfrak M(\mathcal F).
$$

But:

$$
S_{\mathcal F}(\mathcal B)>0
$$

means there exists:

$$
K\in\mathcal B
$$

such that:

$$
M_{U^\star}(K)>0.
$$

So:

$$
U^\star
$$

cannot cover:

$$
K.
$$

Contradiction.

Q.E.D.

---

# 19. The Separation Margin Also Forces Distance from the Old Minimizer Set

Let:

$$
s
=
S_{\mathcal F}(\mathcal B)>0.
$$

If some cover:

$$
V
$$

covers all of:

$$
\mathcal B,
$$

then for any:

$$
U\in\mathfrak M(\mathcal F),
$$

there exists:

$$
K_U\in\mathcal B
$$

such that:

$$
M_U(K_U)\ge s.
$$

And:

$$
M_V(K_U)\le0.
$$

By the cover-variable Lipschitz property:

$$
s
\le
M_U(K_U)-M_V(K_U)
\le
d_H(U,V).
$$

Therefore:

$$
\boxed{
d_H
\left(
V,
\mathfrak M(\mathcal F)
\right)
\ge
s.
}
$$

So any new feasible cover must move away from the entire old minimizer set by at least Hausdorff distance:

$$
s.
$$

---

# 20. Quantitative Area-Gap Modulus

Define:

$$
\boxed{
\gamma_{\mathcal F}(s)
=
\min
\left\{
\operatorname{Area}(V)-\Lambda(\mathcal F):
\begin{array}{l}
V\text{ covers }\mathcal F,\\
V\in\mathcal C_{\bar A},\\
d_H(V,\mathfrak M(\mathcal F))\ge s
\end{array}
\right\}.
}
$$

Compactness plus continuity give:

$$
\boxed{
\gamma_{\mathcal F}(s)>0
\qquad
(s>0).
}
$$

So if:

$$
S_{\mathcal F}(\mathcal B)\ge s>0,
$$

then:

$$
\boxed{
\Lambda(\mathcal F\cup\mathcal B)
\ge
\Lambda(\mathcal F)
+
\gamma_{\mathcal F}(s).
}
$$

At present:

$$
\gamma_{\mathcal F}(s)
$$

is an abstract compactness modulus.

In future it can be given a numerical lower bound via Round 04's candidate-cell area lower bounds.

---

# 21. Certified Minimizer-Cell Cover

Directly representing the exact:

$$
\mathfrak M(\mathcal F)
$$

may be computationally difficult.

So we build finite cells:

$$
\mathfrak C_1,\ldots,\mathfrak C_J
$$

such that:

$$
\boxed{
\mathfrak M(\mathcal F)
\subseteq
\bigcup_{j=1}^J
\mathfrak C_j.
}
$$

Each cell specifies a representative:

$$
U_j
$$

and a certified radius:

$$
\tau_j
$$

such that:

$$
\boxed{
d_H(U,U_j)
\le
\tau_j
\qquad
\forall U\in\mathfrak C_j.
}
$$

Cells can come from:

- placement-space boxes;
- support-height cells;
- hull-support interval cells;
- branch-and-bound minimizer boxes.

---

# 22. Robust Cell-Witness Rule

If for some:

$$
K
$$

there is a certified representative margin lower bound:

$$
M_{U_j}(K)
\ge
\mu_{j,K},
$$

then for any:

$$
U\in\mathfrak C_j
$$

we have:

$$
M_U(K)
\ge
\mu_{j,K}-\tau_j.
$$

So if:

$$
\boxed{
\mu_{j,K}
>
\tau_j,
}
$$

then:

$$
\boxed{
K
\text{ excludes the entire minimizer cell }\mathfrak C_j.
}
$$

If the margin certificate itself carries an error:

$$
\zeta_{j,K},
$$

it is only necessary that:

$$
\boxed{
\underline\mu_{j,K}
-
\tau_j
>
0.
}
$$

---

# 23. Finite Set-Cover Compiler

Build the binary incidence matrix:

$$
C_{jK}
=
\begin{cases}
1,
&
K\text{ certified-excludes cell }j,\\
0,
&
\text{otherwise}.
\end{cases}
$$

We require:

$$
\forall j,
\qquad
\sum_{K\in\mathcal B}
C_{jK}
\ge1.
$$

So compressing the witness batch becomes a finite:

$$
\boxed{
\text{set cover / hitting set}
}
$$

problem.

The objective can be chosen as:

- minimum witness count;
- maximum minimum margin;
- lowest future certificate cost;
- a symmetry-diverse batch;
- a mixed objective.

This is a computational optimization; it does not affect correctness.

---

# 24. Witness-Exchange Eventual-Success Theorem

## Theorem 24.1

Assume:

$$
\Lambda(\mathcal F)<a_{\mathrm{Leb}}.
$$

Let:

$$
\tau_n\to0
$$

be the certified minimizer-cell radii,

$$
\varepsilon_n\to0
$$

the target-dictionary Hausdorff error, and

$$
\zeta_n\to0
$$

the placement-margin certificate error.

Then there exists a finite:

$$
n_\star
$$

such that at level:

$$
n_\star
$$

a separating witness batch can be found in a certified way.

### Proof

By Theorem 11.1:

$$
\delta_{\mathcal F}>0.
$$

Choose:

$$
n
$$

large enough that:

$$
\tau_n
+
\varepsilon_n
+
\zeta_n
<
\frac{\delta_{\mathcal F}}{2}.
$$

Take any minimizer cell:

$$
\mathfrak C_j
$$

and any true minimizer in it:

$$
U.
$$

Choose the true worst target:

$$
K_U
$$

satisfying:

$$
M_U(K_U)
\ge
\delta_{\mathcal F}.
$$

There exists in the Round 02 dictionary a:

$$
\widehat K
$$

such that:

$$
d_H(K_U,\widehat K)
\le
\varepsilon_n.
$$

So:

$$
M_U(\widehat K)
\ge
\delta_{\mathcal F}-\varepsilon_n.
$$

Transferring to the representative:

$$
U_j,
$$

the margin loses at most a further:

$$
\tau_n.
$$

The certificate loses a further:

$$
\zeta_n.
$$

So the certified lower margin is:

$$
>
\delta_{\mathcal F}
-
\varepsilon_n
-
\tau_n
-
\zeta_n
>
\frac{\delta_{\mathcal F}}{2}
>
0.
$$

Therefore every minimizer cell is certified-excluded by at least one finite dictionary target.

The finite set cover produces the separating batch.

Q.E.D.

---

# 25. Semi-Decision Interpretation

Theorem 24.1 says:

$$
\boxed{
\text{if the family genuinely is not yet saturated,
the exchange search will not, in theory, fail forever.}
}
$$

But the converse does not hold:

> not having found a separating batch within a finite computation budget

does not imply:

$$
\Lambda(\mathcal F)=a_{\mathrm{Leb}}.
$$

It may simply be that:

- the minimizer cells are still too coarse;
- the target dictionary is still too coarse;
- the placement certificate is still too coarse;
- the solver has not yet finished;
- the margin is very small.

So exact saturation still requires Round 05's matching lower/upper certificate.

---

# 26. Witness Dominance Theorem

Define:

$$
K\preceq L
$$

if there exists a rigid motion:

$$
g
$$

such that:

$$
gK\subseteq L.
$$

## Theorem 26.1

If:

$$
K\preceq L,
$$

then for any finite family:

$$
\mathcal G,
$$

we have:

$$
\boxed{
\Lambda(\mathcal G\cup\{L\})
\ge
\Lambda(\mathcal G\cup\{K\}).
}
$$

### Proof

Any convex cover that can simultaneously accommodate:

$$
\mathcal G
$$

and some congruent copy of:

$$
L
$$

automatically accommodates:

$$
gK\subseteq L.
$$

So the covering constraint for $L$ is at least as strong as that for $K$.

Taking the minimum over minimal area gives the result.

Q.E.D.

---

# 27. Reuleaux Replacement as a Dominance Upgrade

A regular Reuleaux polygon:

$$
B_n
$$

contains its regular polygon:

$$
P_n.
$$

So:

$$
P_n\preceq B_n.
$$

Therefore:

$$
\Lambda(D,B_3,B_5)
\ge
\Lambda(D,P_3,P_5).
$$

This precisely classifies Mishra's conceptual improvement as:

$$
\boxed{
\text{DOMINANCE-UPGRADE}
}
$$

rather than simply adding more test sets.

---

# 28. Witness Ancestry Labels

Round 06 recommends that every new witness record its ancestry.

## `REUSED`

Already present in the original family.

## `DOMINANCE-UPGRADE`

The new witness:

$$
L
$$

dominates an old one:

$$
K\preceq L.
$$

Replacement may be considered without increasing cardinality.

## `NEW-SEPARATOR`

Not derivable from existing witness dominance, but certified to exclude current minimizer cells.

## `BATCH-ONLY`

Cannot cover the minimizer set alone, but forms a separating batch together with other witnesses.

## `REDUNDANT`

Removing it from the batch still preserves:

$$
S_{\mathcal F}(\mathcal B)>0.
$$

## `SEARCH-ONLY`

Has only a numerical score, no certified positive margin.

## `FALSE-POSITIVE`

The margin is no longer positive after stricter verification.

## `COMPUTE-DEFERRED`

Identity still awaits local or other-AI computation.

---

# 29. Seed Sanity: the Disk Minimizer Is Strictly Separated by the Reuleaux Triangle

For:

$$
\mathcal F_1=\{D\},
$$

the unique geometric minimizer is:

$$
D=B_{1/2}.
$$

The Reuleaux triangle:

$$
B_3
$$

has three corners forming a side-one equilateral triangle.

Its minimum enclosing circle has radius:

$$
\frac1{\sqrt3}.
$$

So placing:

$$
B_3
$$

inside the radius-$1/2$ disk has signed radial deficiency at least:

$$
\boxed{
\delta_{D,B_3}
=
\frac1{\sqrt3}
-
\frac12
\approx
0.0773502692.
}
$$

Therefore:

$$
B_3
$$

is a robust positive separator of the disk minimizer.

Round 05's exchange theorem then guarantees:

$$
\Lambda(D,B_3)
>
\Lambda(D).
$$

Also, because:

$$
P_3\subseteq B_3,
$$

we have:

$$
\Lambda(D,B_3)
\ge
\frac{\pi}{8}
+
\frac{\sqrt3}{4}
\approx
0.8257117836.
$$

This is the witness exchange compiler's first analytic sanity seed.

---

# 30. The Next Step from the Mishra Seed Is Not to Simply Guess $B_7$

For:

$$
\mathcal F_3
=
\{D,B_3,B_5\},
$$

intuition suggests testing:

- $B_7$;
- $B_9$;
- $B_{11}$;
- and further regular Reuleaux odd-gons.

But Round 06's formal strategy is not:

> the next witness must be a regular Reuleaux polygon.

Rather it is:

$$
\boxed{
\text{first find the current minimizer cells}
\to
\text{then run the adversarial target oracle}.
}
$$

The oracle's target domain is the whole of Round 02's constant-width finite dictionary.

So:

- regular Reuleaux polygons are search priors;
- asymmetric / low-symmetry constant-width shapes must also be allowed;
- high odd Fourier modes cannot be excluded in advance either.

---

# 31. Adversarial Target Oracle

For a fixed cover:

$$
U,
$$

the goal is:

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

Round 02:

$$
\mathscr D_\varepsilon
$$

supplies the finite target dictionary.

Round 03:

for each:

$$
\widehat K\in\mathscr D_\varepsilon
$$

supplies a certified:

$$
M_U(\widehat K)
$$

interval.

So we can build a finite adversarial oracle:

```text
INPUT:
    minimizer cell representative U_j
    target dictionary D_epsilon
    placement certificate resolution

FOR each target K_hat in D_epsilon:
    certify lower/upper margin M_Uj(K_hat)

SELECT:
    targets with positive certified lower margin

ROBUSTIFY:
    subtract minimizer-cell Hausdorff radius tau_j

OUTPUT:
    all targets that still have positive robust margin
```

---

# 32. Full Witness Exchange Compiler

```text
INPUT:
    current finite witness family F
    known certified lower value L = Lambda(F) or lower certificate
    known upper incumbent A_plus

STEP 1 — MINIMIZER DOMAIN
    branch-and-bound the placement/configuration space
    retain cells compatible with near-minimum hull area

STEP 2 — MINIMIZER COVER
    construct certified cover of Argmin / minimizing cover set
    each cell gets Hausdorff radius tau_j

STEP 3 — ADVERSARIAL TARGETS
    generate legal constant-width target dictionary D_epsilon

STEP 4 — MARGIN TEST
    for every minimizer cell and target:
        certify M_Uj(K)
        subtract cell + target + solver errors

STEP 5 — INCIDENCE
    C[j,K] = 1 iff K robustly excludes entire cell j

STEP 6 — BATCH COMPRESSION
    solve finite set cover / hitting set
    obtain B

STEP 7 — EXCHANGE CERTIFICATE
    verify every minimizer cell is hit with positive margin
    conclude S_F(B) > 0

STEP 8 — LOWER UPDATE
    F <- F union B
    certify Lambda(F) > old Lambda

STEP 9 — ANCESTRY
    label every new target:
        dominance-upgrade / new-separator / batch-only / redundant

REPEAT
    until lower and upper certificates match.
```

---

# 33. Search / Certificate Separation

This round again emphasizes:

when an adversarial optimizer finds:

$$
K,
$$

that only means:

$$
K
$$

might be effective.

To enter the formal batch, there must be a:

$$
\boxed{
\text{positive certified lower margin}
}
$$

against the minimizer cell.

Likewise, when a new family optimizer finds a hull area:

$$
A_{\mathrm{search}},
$$

that only provides:

$$
\Lambda(\mathcal F)\le A_{\mathrm{search}}.
$$

To push the lower bound upward, one must run:

$$
\boxed{
\text{exhaustive / interval / erosion / finite lower certification}.
}
$$

Search and proof point in opposite directions, and must never be confused.

---

# 34. Interface with the Mishra Certificate

Mishra 2026 has already supplied a certified global lower bound for:

$$
\mathcal F_3
=
\{D,B_3,B_5\}
$$

of:

$$
\Lambda(\mathcal F_3)\ge0.8344.
$$

Its five-dimensional placement search is already an external, large-scale instance of Round 06's:

$$
\text{STEP 1}
+
\text{STEP 8 lower certification}.
$$

What we genuinely still need to add is:

$$
\boxed{
\text{STEP 2--7:
minimizer-set reconstruction + adversarial target exchange}.
}
$$

The Mishra paper does not claim to have already completed this exchange problem.

---

# 35. External Search Prior from Larger Reuleaux Families

Gibbs 2014 previously performed a numerical search over more Reuleaux polygons.

Mishra 2026 specifically points out that:

a numerical optimizer's placement value is an upper bound on the family minimum:

$$
\Lambda(\mathcal F),
$$

not a Lebesgue lower bound.

So this class of result can only be recorded as a:

`SEARCH-PRIOR`

It can be used to:

- choose the seed witness pool;
- initialize placements;
- guess the minimizer topology;

but it cannot enter the lower-bound ledger unless followed by an exhaustive lower certificate.

---

# 36. COMPUTE-DEFERRED

Round 06's core theorems do not require rerunning the five hundred million nodes.

The genuine numerical work is now precisely defined.

## C06-1: Mishra Certificate Ingestion

Sources:

- arXiv:2608.30538;
- the official code repository.

Tasks:

1. obtain the normalized placement domain for $D,B_3,B_5$;
2. parse the certificate leaf / box schema;
3. find the surviving / near-minimizer boxes close to the threshold;
4. it is not necessary to first rerun the entire search;
5. first build the minimizer-cell dataset.

---

## C06-2: Near-Minimizer Reconstruction

Goal:

build a finite box cover of:

$$
\widehat{\mathfrak M}(\mathcal F_3).
$$

Each cell stores:

- five placement intervals;
- hull-area lower / upper bound;
- induced hull support intervals;
- Hausdorff radius;
- symmetry branch;
- provenance hash.

---

## C06-3: Adversarial Target Dictionary

First-stage target pool:

1. $B_7,B_9,B_{11},\ldots$;
2. regular Reuleaux odd-gons;
3. Round 02's low-degree Fourier legal dictionary;
4. asymmetric, low-symmetry constant-width samples;
5. high odd-mode perturbations.

Must not run only the regular Reuleaux family.

---

## C06-4: Robust Incidence Matrix

For:

$$
\text{minimizer cells}\times\text{target candidates},
$$

compute the certified:

$$
\underline M_{jK}.
$$

If:

$$
\underline M_{jK}>0,
$$

mark:

$$
C_{jK}=1.
$$

---

## C06-5: Separating Batch

Solve the set-cover problem:

$$
\min|\mathcal B|
$$

so that every minimizer cell is hit by at least one witness.

Then perform:

- redundancy removal;
- symmetry quotient;
- ancestry labeling.

---

## C06-6: New Family Lower Certificate

The new family:

$$
\mathcal F_4
\text{ or }
\mathcal F_{3+r}
$$

must redo a rigorous global lower certificate.

First make progress on:

- the search optimum;
- the minimizer structure;
- the candidate threshold;

then hand off to the local side for the exhaustive certificate.

Marked:

`COMPUTE-DEFERRED`

---

# 37. This Round's Sanity Validation

Round 06 uses three proof-level sanity facts.

## S1: Continuous Dimension

For:

$$
m=3,
$$

$$
d_m=3m-4=5.
$$

`PASS`

---

## S2: Disk-to-Reuleaux-Triangle Violation

$$
\delta
=
\frac1{\sqrt3}-\frac12
\approx
0.0773502692
>
0.
$$

`PASS`

---

## S3: Seed Ladder

$$
\lambda_1
=
\frac{\pi}{4}
\approx
0.7853981634,
$$

$$
\Lambda(D,B_3)
\ge
\frac{\pi}{8}
+
\frac{\sqrt3}{4}
\approx
0.8257117836,
$$

$$
\Lambda(D,B_3,B_5)
\ge
0.8344.
$$

The strict observed certified ladder:

$$
0.785398\ldots
<
0.825711\ldots
<
0.8344.
$$

`PASS`

---

# 38. RCHM Freedom Ledger

Round 05 residual:

$$
\mathcal F_5
=
\{
\text{finite attainment},
\text{witness discovery},
\text{minimizer-set structure},
\text{certificate cost}
\}.
$$

Round 06:

## Witness Discovery

From:

$$
\text{open-ended guessing}
$$

compressed to:

$$
\boxed{
\text{an adversarial target oracle over a compact minimizer set}.
}
$$

## Minimizer-Set Structure

From:

$$
\text{an unknown continuum}
$$

compressed to:

$$
\boxed{
\text{a compact configuration / hull domain},
}
$$

which can be approximated by finite cells.

## Exchange Legality

From:

$$
\text{add a shape and hope}
$$

compressed to the exact criterion:

$$
\boxed{
S_{\mathcal F}(\mathcal B)>0.
}
$$

## Remaining

- the actual $\mathcal F_3$ minimizer-cell reconstruction;
- the actual next separating batch;
- a new certified lower threshold;
- finite attainment.

---

# 39. Round 07's Assigned Topic

## AMRAL-LUC-FC-R07
### Mishra-Seed Minimizer Atlas and Next-Witness Search Protocol

Round 07 has two layers.

### Proof layer

1. convert the five-dimensional $\mathcal F_3$ domain into a minimizer-atlas schema;
2. define near-minimizer threshold bands;
3. define hull-support interval extraction;
4. define the target-vs-cell robust margin formula;
5. establish the set-cover witness-batch format.

### Compute-deferred layer

If the official certificate / data volume is too large:

- do not force a run inside the cloud conversation;
- produce a local computation specification;
- first use a small near-minimizer sample / coarse reconstruction;
- let the local side perform certificate ingestion and large-scale adversarial search.

What the next round really wants to know is:

$$
\boxed{
\text{On the minimizer frontier of }D+B_3+B_5\text{,
which class of target leaks out first?}
}
$$

---

# 40. Reproducibility Checklist

## Compact placement domain

`PROVED`

## $m=3$ five-dimensional gauge count

`PROVED / MATCHES EXTERNAL CERTIFIED SETUP`

## Minimizer-set compactness

`PROVED`

## Uniform positive violation if non-saturated

`PROVED`

## Finite target-dictionary separation

`PROVED`

## Robust separation margin

`PROVED`

## Strict witness-exchange improvement

`PROVED`

## Eventual finite exchange discovery if non-saturated

`PROVED`

## Witness dominance

`PROVED`

## Mishra seed ingestion

`EXTERNAL RESULT IDENTIFIED`

## New witness identity

`COMPUTE-DEFERRED`

## New numerical lower bound

`NONE`

---

# 41. Shortest Handoff Conclusion

Round 06 upgrades Round 05's:

$$
\text{there exists a better finite witness family}
$$

to:

$$
\boxed{
\text{there exists a finite-resolution algorithm that must eventually find
a separating witness batch whenever the current family is non-saturated}.
}
$$

The core quantity:

$$
\boxed{
\delta_{\mathcal F}
=
\min_{U\in\mathfrak M(\mathcal F)}
W(U)
}
$$

If:

$$
\Lambda(\mathcal F)<a_{\mathrm{Leb}},
$$

then:

$$
\delta_{\mathcal F}>0.
$$

So sufficiently fine:

- minimizer cells;
- target dictionary;
- placement certificate;

will necessarily form a finite incidence matrix, and will find a:

$$
\mathcal B
$$

such that:

$$
\boxed{
S_{\mathcal F}(\mathcal B)>0.
}
$$

Immediately:

$$
\boxed{
\Lambda(\mathcal F\cup\mathcal B)
>
\Lambda(\mathcal F).
}
$$

Therefore, from Round 07 onward, the real mathematical/computational question is no longer "is there a next witness," but rather:

$$
\boxed{
\text{who is the next certified separating witness?}
}
$$

---

# References

1. J. C. Baez, K. Bagdasaryan, P. Gibbs, *The Lebesgue Universal Covering Problem*, arXiv:1502.01251.
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. N. Xie, *A Reproducible Certificate for the Brass--Sharifi Lower Bound in Lebesgue's Universal Cover Problem*, arXiv:2606.04458, 2026.
6. R. Schneider, *Convex Bodies: The Brunn–Minkowski Theory*.
7. Neo.K + Aletheia, *AMRAL × Lebesgue Universal Covering — Round 00–05*, 2026-09-18.

---

# 42. Declaration

This round does not claim:

- to have found a new certified witness family exceeding $0.8344$;
- that the exact minimum of the Mishra family equals $0.8344$;
- that the regular Reuleaux $B_7$ or any other specific shape must necessarily be the next hard witness;
- that the Finite Witness Attainment Conjecture has been proved.

What this round genuinely accomplishes is:

$$
\boxed{
\text{finite witness existence}
\to
\text{a certified witness exchange compiler}.
}
$$

Large-scale certificate ingestion and the search for the actual next hard witness are handed off to Round 07 / the local computation layer.