# AMRAL × Lebesgue Universal Covering — Round 07
## Mishra-Seed Minimizer Atlas and Threshold-Conditioned Witness Lifting

**Document ID:** AMRAL-LUC-FC-R07  
**Version:** v0.1  
**Date:** 2026-09-18  
**Research status:** Round 07 / Witness-saturation phase / Minimizer-atlas 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–R06 v0.1  

---

# 0. Round Summary and Verdict

Round 06 already established:

$$
\boxed{
\text{NON-SATURATED}
\Rightarrow
\text{finite refinement eventually finds a separating witness batch}.
}
$$

But what was still missing was a practically usable minimizer atlas, and a computational architecture for how to avoid dimensional explosion as the number of witnesses grows.

Round 07 takes Mishra 2026's already-proven three-witness family:

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

as its seed, and completes the following work.

---

## Conclusion A: The Official Five-Dimensional Seed and Domain Have Been Successfully Interfaced

The normalized placement coordinates can be written as:

$$
q
=
(x_3,y_3,\phi_5,x_5,y_5).
$$

where:

$$
\phi_5
\in
[0,2\pi/5).
$$

For the certified target:

$$
T=0.8344,
$$

the official sharper a-priori domain gives:

$$
|t_3|
\le
0.192366411526,
$$

$$
|t_5|
\le
0.195890714644.
$$

For this round's proposed next lower-bound milestone:

$$
\boxed{
T_1=0.8350,
}
$$

the same analytic domain formula gives:

$$
\boxed{
|t_3|
\le
0.194856180909,
}
$$

$$
\boxed{
|t_5|
\le
0.197820670401.
}
$$

---

## Conclusion B: The Official Best-Exhibited Seed Falls Almost on a Near-Collinear / Near-Co-Oriented Branch

The official rigorous ceiling log, for:

$$
D+B_3+B_5
$$

gives the exhibited placement:

$$
x_3
=
-0.01215883209899,
$$

$$
y_3
\approx
-3.71\times10^{-8},
$$

$$
\phi_5
=
1.256661190387,
$$

$$
x_5
=
0.02299660691706,
$$

$$
y_5
\approx
-3.90\times10^{-6}.
$$

Since:

$$
B_5
$$

has $5$-fold rotational symmetry,

$$
2\pi/5
=
1.2566370614359\ldots
$$

the normalized angle is:

$$
\boxed{
\phi_5^{\mathrm{mod}}
\approx
2.4128951\times10^{-5}.
}
$$

That is:

- the centers of $B_3$ and $B_5$ are almost on the same horizontal line;
- $B_5$ is almost back at the symmetry-zero orientation;
- the three-body best-exhibited arrangement shows a clear low-dimensional structural signal.

The official exact-kernel area:

$$
0.834780945912,
$$

rigorous outer area:

$$
0.834781190917.
$$

This is only an upper witness for the family minimum, not a lower certificate.

---

## Conclusion C: Constructing the Threshold-Conditioned Minimizer Atlas

For the base family:

$$
\mathcal F
$$

and threshold:

$$
T,
$$

define:

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

The only base configurations that truly need a new witness added are:

$$
\boxed{
\mathcal Q_{<T}
=
\{q:A_{\mathcal F}(q)<T\}.
}
$$

If a base placement itself already satisfies:

$$
A_{\mathcal F}(q)\ge T,
$$

then adding any witness can only make the hull area larger, so there is no need to lift the new witness's three placement parameters.

Therefore:

$$
\boxed{
\text{extra witness dimensions only need activation on the base sublevel atlas}.
}
$$

---

## Conclusion D: The Threshold-Conditioned Witness Lifting Theorem

If the base configuration space is covered by finitely many cells:

$$
C_1,\ldots,C_N
$$

and for each cell, either:

- base lower bound:

  $$
  L_{\mathcal F}(C_j)\ge T;
  $$

- or there exists a witness:

  $$
  K_j
  $$

  such that:

  $$
  \inf_{q\in C_j}
  \inf_{g\in E(2)}
  \operatorname{Area}
  \left(
  \operatorname{conv}
  (H_{\mathcal F}(q)\cup gK_j)
  \right)
  \ge T,
  $$

then for the batch:

$$
\mathcal B
=
\{K_j\},
$$

we have:

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

This means witness lifting can be **cell-local**, rather than a global Cartesian-product expansion.

---

## Conclusion E: The Witness-Switching Lower Bound Theorem

For a fixed base hull:

$$
H,
$$

define the single-witness lift functional:

$$
\boxed{
J_K(H)
=
\min_{g\in E(2)}
\operatorname{Area}
\left(
\operatorname{conv}(H\cup gK)
\right).
}
$$

Then for a finite batch:

$$
\mathcal B,
$$

we have:

$$
\boxed{
\Lambda(\mathcal F\cup\mathcal B)
\ge
\min_{q_{\mathcal F}}
\max_{K\in\mathcal B}
J_K(H_{\mathcal F}(q_{\mathcal F})).
}
$$

So different base cells can be closed by different witnesses.

Adding $r$ new witnesses does not require raising the certificate directly to a single joint box problem of dimension:

$$
d_{\mathcal F}+3r.
$$

One can instead keep a:

$$
d_{\mathcal F}
$$

-dimensional base atlas, and then choose, for each cell, one:

$$
3\text{-D witness lift subproblem}.
$$

---

## Conclusion F: For $\mathcal F_3+B_7$, Round 07 Proposes Its First Concrete Milestone

The official scaling data, for:

$$
D+B_3+B_5+B_7
$$

reports a numerical optimum of:

$$
\boxed{
0.836494901
}
$$

But this is only a search ceiling, not a rigorous lower bound.

The official brute-force scaling, for direct 8-D certification up to:

$$
T=0.835
$$

extrapolates to about:

$$
5.527\times10^{12}
$$

boxes, about:

$$
5635.8
$$

days at the measured throughput.

This is not a mathematical impossibility — only an empirical cost extrapolation of the existing brute-force implementation.

Round 07 therefore selects:

$$
\boxed{
T_1=0.8350
}
$$

as the first threshold-conditioned lifting milestone.

---

## Conclusion G: A Small SEARCH-ONLY Probe Provisionally Ranks $B_7$ First Among Regular-Reuleaux Candidates

Fixed near the official exhibited $\mathcal F_3$ seed, this round performs a non-proof exploration using a boundary-sampled hull plus a numerical optimizer:

| New witness | Minimum hull at fixed seed (SEARCH-ONLY) |
|---|---:|
| $B_7$ | about $0.83713$ |
| $B_9$ | about $0.83563$ |
| $B_{11}$ | about $0.83613$ |
| $B_{13}$ | about $0.83597$ |

So on this **single base representative**:

$$
B_7
$$

is strongest.

But the official full four-body search's numerical optimum:

$$
0.836494901
$$

is lower than the fixed-base $B_7$ lift of about $0.83713$.

This precisely demonstrates:

> once a new witness is added, the old base family reconfigures; one cannot test only a single old minimizer point.

So Round 07's minimizer atlas is necessary.

---

This round's verdict:

$$
\boxed{
\text{MINIMIZER-ATLAS SCHEMA: CLOSED}
}
$$

$$
\boxed{
\text{CONDITIONAL WITNESS LIFTING: CLOSED}
}
$$

$$
\boxed{
\text{FIRST ACTUAL }0.8350\text{ CERTIFICATE: COMPUTE-DEFERRED}
}
$$

---

# 1. External Data Audit

This round directly checks against:

## Mishra 2026 Paper

**Ujjwal Mishra, _Curves of constant width and Lebesgue's covering problem_, arXiv:2608.30538.**

Confirmed:

- test sets:

  $$
  D,B_3,B_5;
  $$

- the five-dimensional normalized placement domain;
- certified lower bound:

  $$
  0.8344;
  $$

- certificate:

  $$
  486,799,600
  $$

  nodes;

- independent verifier;
- worst leaf slack:

  $$
  1.240900\times10^{-5};
  $$

- rigorous floating error:

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

---

## Official Source Repository

Repository:

`Ujjwal238/universal-cover-problem`

proof-path files checked:

- `src/domain.py`
- `src/geom.py`
- `src/verifyB.py`
- `src/ceilings.py`
- `src/phase1.py`
- `src/phase2.py`
- `src/certgen.py`
- `src/certemitB.py`
- `logs/domain.log`
- `logs/ceilings.log`
- `logs/scaling.log`
- `logs/phase4_B.log`

This round does not treat the repo's search output as a new theorem; it is used only to pin down the externally certified seed, parameterization, computational cost, and search-prior.

---

# 2. Base family normalization

seed family:

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

Fix:

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

Use a global rotation gauge to fix:

$$
B_3
$$

orientation.

remaining continuous parameters:

$$
\boxed{
q
=
(x_3,y_3,\phi_5,x_5,y_5).
}
$$

where:

$$
\phi_5
\in
[0,2\pi/5).
$$

The reflection branch, because regular Reuleaux bodies are mirror-symmetric, does not add an extra continuous dimension.

---

# 3. Threshold-dependent translation domain

The official `domain.py` defines:

$$
f(d)
=
\frac14
\left(
\pi-\arccos\frac{1}{2d}
\right)
+
\frac12
\sqrt{d^2-\frac14},
\qquad
d\ge\frac12.
$$

Let:

$$
d_\star(T)
$$

satisfy:

$$
f(d_\star)=T.
$$

For the regular Reuleaux $n$-gon corner circumradius:

$$
R_n
=
\frac{1}
{
2\sin\left(
\frac{\pi(n-1)}{2n}
\right)
},
$$

translation radius bound:

$$
\boxed{
t_n(T)
=
-R_n\cos\frac{\pi}{n}
+
\sqrt{
R_n^2\cos^2\frac{\pi}{n}
-
R_n^2
+
d_\star(T)^2
}.
}
$$

---

# 4. Domain values

## At $T=0.8344$

$$
d_\star
\approx
0.693830648713.
$$

$$
\boxed{
t_3
\approx
0.192366411526,
}
$$

$$
\boxed{
t_5
\approx
0.195890714644.
}
$$

---

## At proposed milestone $T_1=0.8350$

$$
\boxed{
t_3(T_1)
\approx
0.194856180909,
}
$$

$$
\boxed{
t_5(T_1)
\approx
0.197820670401.
}
$$

The domain grows slightly larger as the target increases, which is logical:

the higher the threshold, the fewer translations can be excluded by a single-point / disk area estimate alone.

---

# 5. Information Limits of the Mishra Certificate

The headline certificate uses:

$$
1\text{ bit/node}
$$

DFS preorder:

- `1`: split;
- `0`: pruned leaf.

The boxes themselves are not stored.

The verifier regenerates every box from the root plus the deterministic split rule.

So the original certificate is suited to:

$$
\boxed{
\text{verify all boxes were pruned at a fixed threshold}
}
$$

but does not directly provide:

$$
\boxed{
\text{near-minimizer atlas}.
}
$$

So Round 07 cannot claim:

> minimizer cells have already been read off from the official certificate.

The actual approach is:

> modify / bypass the original traversal, add a `retain-near-threshold` channel, and then generate the atlas.

---

# 6. Threshold-conditioned sublevel atlas

For the base family:

$$
\mathcal F,
$$

configuration objective:

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

For threshold:

$$
T,
$$

define the true sublevel region:

$$
\boxed{
\mathcal Q_{<T}
=
\{q:A_{\mathcal F}(q)<T\}.
}
$$

We do not need to know exactly:

$$
\mathcal Q_{<T}
$$

's boundary.

We only need a finite cell cover:

$$
\boxed{
\mathcal Q_{<T}
\subseteq
\bigcup_{j=1}^{N_T}C_j.
}
$$

---

# 7. Atlas emitter

For base box:

$$
C
$$

there is already the official erosion lower bound:

$$
L_{\mathcal F}(C)
\le
A_{\mathcal F}(q)
\qquad
\forall q\in C.
$$

So the traversal:

```text
visit base box C

if L_F(C) >= T:
    label BASE-PRUNED
    stop

else if max_effective_width(C) <= h_atlas:
    label ATLAS-RETAINED
    emit C
    stop

else:
    split C
```

The output:

$$
\{C_j\}
$$

is guaranteed to cover:

$$
\mathcal Q_{<T}.
$$

because any point that truly satisfies:

$$
A_{\mathcal F}(q)<T
$$

can never fall inside a cell that was pruned for having:

$$
L_{\mathcal F}(C)\ge T.
$$

---

# 8. The Difference Between the Near-Minimizer Atlas and the Exact Minimizer Atlas

Round 07 does not require first solving exactly for:

$$
\operatorname{Argmin}A_{\mathcal F}.
$$

For pushing forward a new lower bound:

$$
T
$$

what is actually relevant is the entire sublevel region where:

$$
\boxed{
A_{\mathcal F}(q)<T
}
$$

holds.

So:

$$
\boxed{
\text{Threshold Atlas}
}
$$

is more direct than:

$$
\boxed{
\text{Exact Minimizer Atlas}.
}
$$

If the goal is only:

$$
T=0.8350,
$$

there is no need to first pin down the minimizer near:

$$
0.8347809\ldots
$$

to machine precision.

It suffices to cover the entirety of the base configurations with:

$$
\boxed{
A_{\mathcal F}<0.8350
}
$$

This is the first major computational compression.

---

# 9. Single-witness lift functional

For base hull:

$$
H
$$

and a new legal target:

$$
K,
$$

define:

$$
\boxed{
J_K(H)
=
\min_{g\in E(2)}
\operatorname{Area}
\left(
\operatorname{conv}(H\cup gK)
\right).
}
$$

It answers:

> once the base hull is fixed, when the new witness $K$ is packed in as area-efficiently as possible, how much area is required at minimum?

Clearly:

$$
J_K(H)
\ge
\operatorname{Area}(H).
$$

---

# 10. Threshold-Conditioned Witness Lifting theorem

## Theorem 10.1

Suppose the base configuration domain is covered by finitely many cells:

$$
C_1,\ldots,C_N
$$

For each cell:

$$
C_j,
$$

assume at least one of the following two conditions holds.

### Type A: base closed

$$
\boxed{
L_{\mathcal F}(C_j)\ge T.
}
$$

### Type B: witness lifted

There exists:

$$
K_j
$$

such that:

$$
\boxed{
\inf_{q\in C_j}
J_{K_j}
\left(
H_{\mathcal F}(q)
\right)
\ge T.
}
$$

Let:

$$
\mathcal B
=
\{K_j:\ C_j\text{ Type B}\}.
$$

Then:

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

### Proof

Take any full-family placement.

Its base restriction corresponds to some:

$$
q\in C_j.
$$

If:

$$
C_j
$$

is Type A, then the base hull is already:

$$
\ge T.
$$

The full hull contains the base hull, so:

$$
\ge T.
$$

If it is Type B, the full hull contains:

$$
\operatorname{conv}
\left(
H_{\mathcal F}(q)\cup gK_j
\right)
$$

for the:

$$
gK_j
$$

appearing in that full placement.

By the cell-lift assumption, no matter how:

$$
g
$$

is chosen, the area is always:

$$
\ge T.
$$

Hence the full hull:

$$
\ge T.
$$

This holds for all full placements, therefore:

$$
\Lambda(\mathcal F\cup\mathcal B)\ge T.
$$

Q.E.D.

---

# 11. Witness-switching lower theorem

For a base configuration:

$$
q,
$$

let:

$$
H(q)=H_{\mathcal F}(q).
$$

For a finite witness batch:

$$
\mathcal B,
$$

## Theorem 11.1

$$
\boxed{
\Lambda(\mathcal F\cup\mathcal B)
\ge
\min_q
\max_{K\in\mathcal B}
J_K(H(q)).
}
$$

### Proof

Fix the base:

$$
q.
$$

the full placement is:

$$
\{g_K\}_{K\in\mathcal B}.
$$

the full hull area:

$$
A_{\mathrm{full}}
$$

for any:

$$
K
$$

always satisfies:

$$
A_{\mathrm{full}}
\ge
\operatorname{Area}
\left(
\operatorname{conv}(H(q)\cup g_KK)
\right).
$$

So:

$$
A_{\mathrm{full}}
\ge
\max_{K\in\mathcal B}
A_K(q,g_K).
$$

Minimizing independently over each:

$$
g_K
$$

$$
\min_{\{g_K\}}
\max_K
A_K(q,g_K)
=
\max_K
\min_{g_K}
A_K(q,g_K)
=
\max_K
J_K(H(q)).
$$

Finally, taking the minimum over:

$$
q.
$$

Q.E.D.

---

# 12. Why This Theorem Matters

Naively adding:

$$
r
$$

witnesses:

$$
K_1,\ldots,K_r
$$

would take the placement's continuous dimension:

$$
d_{\mathcal F}
$$

and turn it directly into:

$$
d_{\mathcal F}+3r,
$$

which, even after subtracting any possible symmetry gauge, still explodes rapidly.

The witness-switching theorem changes this to:

$$
\boxed{
d_{\mathcal F}
\text{-D base atlas}
+
\text{cell-local }3\text{-D witness lifts}.
}
$$

Different cells can use different:

$$
K.
$$

So the batch's cardinality does not multiply directly into the local placement dimension.

---

# 13. Lazy dimension activation

Round 07 calls this strategy:

$$
\boxed{
\text{Lazy Witness Dimension Activation}.
}
$$

For a base cell:

1. First compute:

   $$
   L_{\mathcal F}(C).
   $$

2. If:

   $$
   L_{\mathcal F}(C)\ge T,
   $$

   do not create any new witness dimensions.

3. Only cells with:

   $$
   L_{\mathcal F}(C)<T
   $$

   activate the:

   $$
   (\phi_K,x_K,y_K)
   $$

   three-dimensional witness placement search.

4. If one witness has already closed the cell, no other witnesses are tested.

This is exact branch-and-bound logic, not heuristic pruning.

---

# 14. The Erosion Bound Extends Directly to the Lifted Witness

Mishra's core lemma:

> if, within a placement box, the maximum displacement of every Reuleaux corner is $\delta$, then the fixed core obtained by eroding each defining unit disk's radius from $1$ to $1-\delta$ is contained in every placement in that box.

So base cell:

$$
C
$$

and the new witness placement box:

$$
B_K
$$

can directly form common cores:

$$
E_3(C),
\qquad
E_5(C),
\qquad
E_K(B_K).
$$

then:

$$
\boxed{
\operatorname{Area}
\operatorname{conv}
\left(
D,
E_3,
E_5,
E_K
\right)
}
$$

is a rigorous lower bound on the hull area over the whole of:

$$
C\times B_K.
$$

So Conditional Lifting does not need to invent a new curved-body lower bound.

It only needs to turn the official erosion machinery into:

$$
\boxed{
\text{base cell persistent}
+
\text{one lazily activated witness box}.
}
$$

---

# 15. Cell-switching verifier

For atlas cell:

$$
C_j
$$

and witness pool:

$$
\mathcal P
=
\{K^{(1)},\ldots,K^{(s)}\},
$$

try in sequence:

$$
K^{(a)}.
$$

If its full three-dimensional placement domain can be certified by a finite erosion certificate showing:

$$
\inf_{q\in C_j,g}
\operatorname{Area}
\operatorname{conv}
(H(q)\cup gK^{(a)})
\ge T,
$$

then record:

`CELL-CLOSED-BY = K^(a)`

and stop processing that cell.

If no single witness closes the cell:

- subdivide the base cell;
- or expand the witness pool;
- there is no need to immediately combine two witnesses into a joint 6-D search.

because after subdivision, different single witnesses may separately close the children.

---

# 16. Batch witness certificate is local-switchable

Suppose the atlas has:

$$
1000
$$

hard cells.

Among these:

- $B_7$ closes $600$;
- some asymmetric Fourier target closes $250$;
- $B_9$ closes the remaining $150$.

Then the global family:

$$
\boxed{
\mathcal F_3
\cup
\{B_7,K_{\mathrm{asym}},B_9\}
}
$$

can be certified, by Theorem 10.1, to give:

$$
\Lambda\ge T.
$$

It does not require the same witness to close all cells.

This is the area-threshold version of Round 06's separating-batch theory.

---

# 17. Robust cell-lift transfer theorem

Sometimes, instead of running erosion directly over the whole of:

$$
C\times B_K
$$

we instead have a base representative hull:

$$
H_C
$$

with:

$$
d_H
\left(
H(q),H_C
\right)
\le
\tau
\qquad
\forall q\in C.
$$

In this case, a robust transfer can be established using area continuity.

---

## Theorem 17.1

Let:

$$
D=B_{1/2}(0)
\subset H(q),H_C.
$$

If:

$$
J_K(H_C)
>
T
+
\omega_T(\tau),
$$

where:

$$
\boxed{
\omega_T(\tau)
=
4\pi T\tau
+
\pi\tau^2,
}
$$

then:

$$
\boxed{
J_K(H(q))\ge T
\qquad
\forall q\in C.
}
$$

### Proof

Suppose for contradiction that some:

$$
q\in C
$$

satisfies:

$$
J_K(H(q))<T.
$$

Then there exists a placement:

$$
g
$$

such that:

$$
C_q
=
\operatorname{conv}(H(q)\cup gK)
$$

has area:

$$
<T.
$$

Since:

$$
D\subset C_q,
$$

the Round 04 radius theorem gives:

$$
C_q\subset B_{2T}(0)
$$

once anchored at a common point.

So:

$$
\operatorname{Per}(C_q)
\le
4\pi T.
$$

Also:

$$
H_C
\subset
H(q)+\tau B.
$$

So:

$$
\operatorname{conv}(H_C\cup gK)
\subset
C_q+\tau B.
$$

Steiner formula:

$$
\operatorname{Area}
(C_q+\tau B)
\le
\operatorname{Area}(C_q)
+
4\pi T\tau
+
\pi\tau^2.
$$

Hence:

$$
J_K(H_C)
<
T+\omega_T(\tau),
$$

a contradiction.

Q.E.D.

---

# 18. Atlas cell record schema

Every:

$$
C_j
$$

stores at least:

```text
cell_id
parent_id
depth
parameter_intervals:
    x3
    y3
    phi5
    x5
    y5

base_lower_area
base_center_area (SEARCH-ONLY unless rigorous upper)
effective_widths
erosion_delta_R3
erosion_delta_R5
R3_core_nonempty
R5_core_nonempty

hull_support_interval_hash
hausdorff_cell_radius (if available)

status:
    BASE-PRUNED
    ATLAS-RETAINED
    LIFT-ACTIVE
    CELL-CLOSED
    COMPUTE-DEFERRED

closed_by_witness
lift_lower_bound
threshold
margin
provenance_hash
```

This is the minimum cell schema for a future cloud/local crystal.

---

# 19. Root atlas for $T_1=0.8350$

This round proposes the first improvement milestone:

$$
\boxed{
T_1=0.8350.
}
$$

This would raise the published lower bound:

$$
0.8344
$$

by:

$$
6\times10^{-4}.
$$

base normalized domain:

$$
\phi_5\in[0,2\pi/5).
$$

translation radial cut:

$$
|t_3|
\le
0.194856180909,
$$

$$
|t_5|
\le
0.197820670401.
$$

The implementation can first use enclosing square boxes:

$$
x_3,y_3
\in
[-t_3,t_3],
$$

$$
x_5,y_5
\in
[-t_5,t_5],
$$

and then use a radial a-priori test to prune the square corners.

---

# 20. Official best exhibited seed

official `logs/ceilings.log`:

$$
q^\dagger
=
(
-0.01215883209899,
-3.7081986\times10^{-8},
1.256661190387,
0.02299660691706,
-3.8969815\times10^{-6}
).
$$

modulo:

$$
2\pi/5,
$$

written as:

$$
\boxed{
q^\dagger_{\mathrm{norm}}
\approx
(
-0.01215883210,
0,
2.4128951\times10^{-5},
0.02299660692,
0
).
}
$$

exact-kernel family area:

$$
\boxed{
0.834780945912.
}
$$

rigorous outer polygon:

$$
\boxed{
0.834781190917.
}
$$

So, relative to:

$$
T_1=0.8350,
$$

the base headroom is only about:

$$
\boxed{
2.19\times10^{-4}.
}
$$

This means:

$$
T_1
$$

only needs to handle the sublevel region very close to the base family's ceiling.

---

# 21. Official full 8-D search prior

official `logs/scaling.log`, for:

$$
D+B_3+B_5+B_7
$$

reports the numerical optimum:

$$
\boxed{
0.836494901.
}
$$

This number is numerical evidence for the search ceiling:

$$
\Lambda(D,B_3,B_5,B_7)
\le
0.836494901
$$

It is not a lower bound.

For the naive full 8-D certification target:

$$
0.8350,
$$

the official empirical scaling extrapolation is:

$$
\boxed{
\sim5.527\times10^{12}
\text{ boxes}
}
$$

and:

$$
\boxed{
\sim5635.8\text{ days}
}
$$

at measured throughput.

Round 07's atlas-lift strategy exists precisely to avoid:

$$
\boxed{
\text{pre-expanding B7's three dimensions across all base configurations}.
}
$$

---

# 22. SEARCH-ONLY fixed-seed probe

This round uses:

- the official exhibited base placement;
- sampled Reuleaux boundaries;
- a numerical convex hull;
- differential evolution;

to perform only candidate ranking.

This **is not a certificate**.

The reproduced base inner-sampled hull is about:

$$
0.8347793.
$$

close to the official exact value:

$$
0.834780945912
$$

but the sampled hull is an inner approximation, so it cannot be used as a rigorous upper/lower theorem.

---

# 23. Regular Reuleaux candidate ranking

On the fixed base seed:

## $B_7$

search-only:

$$
\boxed{
J_{B_7}(H^\dagger)
\approx
0.83713.
}
$$

## $B_9$

$$
\approx
0.83563.
$$

## $B_{11}$

$$
\approx
0.83613.
$$

## $B_{13}$

$$
\approx
0.83597.
$$

So, on this seed:

$$
\boxed{
B_7
}
$$

's lift is strongest.

status:

`SEARCH-PRIOR ONLY`

---

# 24. Why the Fixed-Seed Value $0.83713$ Is Not a Four-Body Lower Bound

The official full four-body numerical ceiling:

$$
0.836494901
$$

is lower than:

$$
0.83713.
$$

Reason:

Once:

$$
B_7
$$

is added:

$$
B_3,B_5
$$

can also move again.

So:

$$
\boxed{
\min_{q_3}
J_{B_7}(H_{\mathcal F_3}(q_3))
<
J_{B_7}(H_{\mathcal F_3}(q^\dagger)).
}
$$

This gap is itself a validation of Round 07's minimizer-atlas approach:

> the old best point is not a sufficient base state; the entire threshold sublevel region must be covered.

---

# 25. $B_7$'s Current Status

Currently it can be labeled:

`SEARCH-PRIOR`

+

`EXTERNAL-NUMERICAL-FAMILY-CEILING`

It cannot yet be labeled:

`NEW-SEPARATOR-CERTIFIED`

because it has not yet been proven that:

$$
\inf_{q\in\mathcal Q_{<0.835}}
J_{B_7}(H(q))
\ge
0.835.
$$

Round 07 only turns this statement into the next computable exact gate.

---

# 26. New exact gate R07-G1

## Gate R07-G1

Prove:

$$
\boxed{
\forall q\in\mathcal Q_{<0.8350},
\quad
J_{B_7}(H_{\mathcal F_3}(q))
\ge
0.8350.
}
$$

If this holds:

$$
\boxed{
\Lambda(D,B_3,B_5,B_7)
\ge
0.8350.
}
$$

therefore:

$$
\boxed{
a_{\mathrm{Leb}}
\ge
0.8350.
}
$$

This would be a genuine new strict improvement relative to:

$$
0.8344.
$$

Currently:

`COMPUTE-DEFERRED`

---

# 27. A More General Batch Gate

If:

$$
B_7
$$

cannot close all atlas cells by itself, that does not mean failure.

One can find a finite:

$$
\mathcal B
=
\{B_7,K_2,\ldots,K_r\}
$$

such that:

$$
\boxed{
\forall q\in\mathcal Q_{<T},
\quad
\max_{K\in\mathcal B}
J_K(H(q))
\ge T.
}
$$

then the witness-switching theorem gives:

$$
\boxed{
\Lambda(\mathcal F_3\cup\mathcal B)\ge T.
}
$$

Different atlas cells can be closed by different:

$$
K.
$$

This is exactly the area-threshold upgrade of Round 06's separating batch.

---

# 28. The Witness Pool Must Not Contain Only Regular Reuleaux Polygons

The first-priority pool may contain:

$$
B_7,B_9,B_{11},B_{13}.
$$

But Round 02 already told us that the entire constant-width target domain can be finitely approximated.

So subsequent rounds must add:

- low odd Fourier-mode bodies;
- asymmetric constant-width bodies;
- a curvature-density dictionary;
- near-extreme candidates of the $r(\theta)\in\{0,1\}$ type;
- regular Reuleaux polygons.

If only:

$$
B_{2k+1},
$$

is run, this may produce representation bias.

---

# 29. Atlas refinement policy

For cell:

$$
C
$$

one can compute:

- base lower:

  $$
  L_{\mathcal F}(C);
  $$

- base center search area;
- current best witness-lift lower;
- cell width;
- erosion slack.

Suggested priority:

$$
\boxed{
\text{priority}(C)
=
(T-L_{\mathcal F}(C))
\times
\text{uncertainty}(C)
\times
\text{witness-failure score}.
}
$$

This is only a scheduling heuristic.

Correctness does not depend on priority.

---

# 30. Atlas lineage

Every child cell must retain:

```text
parent_id
split_axis
split_point
child_index
```

and retain:

```text
base_bound_version
geometry_kernel_hash
target_threshold
witness_pool_version
```

This allows a second AI to independently replay:

- which cells were base-pruned;
- which cells needed a lift;
- which witness closed which cell;
- which cells are not yet closed.

---

# 31. Atlas status taxonomy

## `BASE-PRUNED`

$$
L_{\mathcal F}(C)\ge T.
$$

## `LIFT-CLOSED`

Some:

$$
K
$$

has already proven:

$$
\inf_{q\in C}J_K(H(q))\ge T.
$$

## `MULTI-WITNESS-CLOSED`

Closed by a witness-switching batch.

## `REFINE`

No certificate yet; continue splitting.

## `SEARCH-HIT`

Only a numerical witness candidate exists; not yet proven.

## `COMPUTE-DEFERRED`

Requires heavy local search.

## `BOUNDARY-ANOMALY`

Some erosion / core / interval assumption fails and must be handled separately.

---

# 32. Local compute specification for $T_1=0.8350$

## Phase A: Base atlas

input:

$$
\mathcal F_3=D+B_3+B_5.
$$

threshold:

$$
T_1=0.8350.
$$

root:

$$
q=(x_3,y_3,\phi_5,x_5,y_5).
$$

uses:

- official exact geometry kernel;
- official erosion lower bound;
- sharper domain;
- deterministic split weights.

output:

$$
\mathcal A_{0.835}
$$

finite retained cells.

---

## Phase B: B7 lazy lift

For each retained base cell:

1. attach:

   $$
   (\phi_7,x_7,y_7);
   $$

2. symmetry:

   $$
   \phi_7\in[0,2\pi/7);
   $$

3. use threshold-dependent translation bound for $B_7$;
4. erosion base bodies and $B_7$ over joint cell;
5. prune if hull-core area:

   $$
   \ge0.8350;
   $$

6. if all $B_7$ placement boxes close:

   `CELL-CLOSED-BY-B7`

7. otherwise retain failing subcells for alternative witness scan.

---

# 33. Why this is not the same as naive 8-D certification

naive:

```text
root 8-D box
split all eight variables as needed
```

Round 07:

```text
5-D base traversal
    |
    +-- base lower >= T  -> close, never create B7 dimensions
    |
    +-- base lower < T   -> retain atlas cell
                              |
                              +-- activate local 3-D B7 search
```

So:

$$
\boxed{
\text{new dimensions exist only on the hard base sublevel region}.
}
$$

The magnitude of the computational savings is unknown and must be measured empirically.

This round does not claim that the 5635.8 days will necessarily be brought down to an acceptable time.

---

# 34. Exact certificate format for conditional lift

Every base leaf:

```text
base_cell_id
base_box_intervals
base_lower_bound
base_status
```

If lifted:

```text
witness_id
witness_root_domain
lift_certificate_stream_hash
lift_nodes
lift_leaves
lift_worst_slack
lift_error_bound
threshold
verdict
```

global verifier:

1. replay base atlas;
2. every leaf must either be:
   - `BASE-PRUNED`, or
   - have at least one valid lift certificate;
3. verify all child coverage;
4. verify no unclassified leaf;
5. aggregate:

   $$
   \Lambda\ge T.
   $$

This verifier can be kept separate from the search code.

---

# 35. Batch-switch certificate format

If there are multiple witnesses:

```text
cell_id -> witness_id
```

this is a finite assignment.

Every witness_id points to:

$$
\text{cell-local 3-D lift certificate}.
$$

Global correctness only requires:

$$
\boxed{
\forall\text{ retained base cell},
\quad
\exists\text{ one valid witness lift certificate}.
}
$$

It does not require that every:

$$
K\in\mathcal B
$$

be computed in every cell.

This can substantially reduce the amount of certificate data.

---

# 36. Relation to RCHM saturation

Round 05:

$$
\text{finite witness saturation}
$$

asks whether the lower family can reach the truth.

Round 06:

$$
\text{exchange compiler}
$$

guarantees that, when not yet saturated, a batch will eventually be found.

Round 07 now turns batch exchange into:

$$
\boxed{
\text{threshold sublevel atlas}
+
\text{cell-local witness lift}
}
$$

So the RCHM residual gate moves from:

$$
\text{abstract witness discovery}
$$

and compresses further into:

$$
\boxed{
\text{finite atlas coverage problem}.
}
$$

---

# 37. Current witness ancestry

## $D$

`REUSED / ANCHOR`

## $B_3$

`DOMINANCE-UPGRADE`

relative to the classical:

$$
P_3.
$$

## $B_5$

`DOMINANCE-UPGRADE`

relative to the classical:

$$
P_5.
$$

## $B_7$

Currently:

`SEARCH-PRIOR`

If R07-G1 succeeds:

`NEW-SEPARATOR-CERTIFIED`

## $B_9,B_{11},B_{13}$

Currently:

`SEARCH-PRIOR`

not automatically upgraded by a fixed-seed numerical lift.

---

# 38. COMPUTE-DEFERRED

## C07-1: Base sublevel atlas $T=0.8350$

Mandatory.

Output retained cell count, depth distribution, support/hull intervals.

---

## C07-2: B7 conditional lift

Priority.

If all cells close:

immediately proceed to the full independent verifier.

---

## C07-3: Unclosed-cell target scan

Run only against $B_7$'s failed cells:

- $B_9$;
- $B_{11}$;
- $B_{13}$;
- the Round 02 Fourier dictionary.

---

## C07-4: Batch set cover

Build:

$$
\text{atlas cells}\times\text{witnesses}
$$

incidence.

Solve for a minimum / high-margin set cover.

---

## C07-5: Independent verifier

The search emitter and verifier must:

- not share prune decisions;
- share only mathematical constants / definitions;
- the verifier reconstructs boxes;
- the verifier checks that every base leaf is legally classified;
- lifted certificate hashes are fixed.

---

# 39. Small exploratory code in package

This round's package contains:

`AMRAL_LUC_FC_Round_07_search_prior.py`

Purpose:

1. reconstruct the regular Reuleaux boundary representation;
2. use the official exhibited $\mathcal F_3$ seed;
3. for:

   $$
   B_7,B_9,B_{11},B_{13}
   $$

   perform a fixed-base numerical placement search;
4. output a

   `SEARCH-ONLY`

   ranking.

This program:

- does not use interval arithmetic;
- its convex hull comes from boundary sampling;
- does not constitute a lower certificate;
- is only for witness ordering.

---

# 40. Assigned topic for Round 08

## AMRAL-LUC-FC-R08
### Conditional-Lift Certificate Compiler for the $0.8350$ Milestone

If the local end has not yet returned the heavy compute, Round 08 can still begin with:

1. exact lifted-box erosion formulas;
2. $B_7$'s threshold-dependent domain;
3. a base/lift split scheduler;
4. certificate stream grammar;
5. verifier invariants;
6. a fallback multi-witness branch.

If the local end already has an atlas:

begin directly with:

$$
\boxed{
\mathcal A_{0.835}
\to
B_7\text{ cell closure}.
}
$$

---

# 41. Reproducibility checklist

## External seed source

`VERIFIED`

## Official five-dimensional parameterization

`VERIFIED`

## Sharper translation domain

`VERIFIED`

## Official exhibited near-minimizer seed

`VERIFIED`

## Threshold-conditioned atlas theorem

`PROVED`

## Witness-switching theorem

`PROVED`

## Lazy dimension activation

`PROVED AS CERTIFICATE LOGIC`

## Robust cell-lift transfer

`PROVED`

## Regular Reuleaux ranking

`SEARCH-ONLY`

## $B_7$ as certified next witness

`OPEN`

## $0.8350$ lower bound

`OPEN / COMPUTE-DEFERRED`

---

# 42. Shortest Handoff Conclusion

Round 07's core contribution is not that a fourth certified witness has already been found.

What has actually been completed is:

$$
\boxed{
\text{5-D base family sublevel atlas}
\to
\text{cell-local 3-D witness lifting}
}
$$

as well as:

$$
\boxed{
\Lambda(\mathcal F\cup\mathcal B)
\ge
\min_q
\max_{K\in\mathcal B}
J_K(H_{\mathcal F}(q)).
}
$$

So a witness batch no longer equals a direct sum of dimensions.

For the first actual milestone:

$$
\boxed{
T_1=0.8350,
}
$$

the next exact gate has become:

$$
\boxed{
\forall q\in\mathcal Q_{<0.835},
\quad
J_{B_7}(H(q))
\ge0.835
\ ?
}
$$

If $B_7$ cannot complete this alone:

use a finite witness-switching batch.

This is a strategic shift from "finding the fourth shape" to "covering the entire near-minimizer atlas."

---

# References

1. U. Mishra, *Curves of constant width and Lebesgue's covering problem*, arXiv:2608.30538, 2026.
2. U. Mishra, official source repository `Ujjwal238/universal-cover-problem`, 2026.
3. P. Gibbs, *An Upper Bound for Lebesgue's Covering Problem*, arXiv:1810.10089, 2018.
4. P. Gibbs, *A New Slant on Lebesgue's Universal Covering Problem*, arXiv:1401.8217, 2014.
5. S. Zeng, *An exact hierarchy for Lebesgue's universal covering constant and a certified 0.834 lower bound*, arXiv:2609.01284, 2026.
6. Neo.K + Aletheia, *AMRAL × Lebesgue Universal Covering — Round 00–06*, 2026-09-18.

---

# 43. Declaration

This round does not claim:

- that $B_7$ has already rigorously raised the lower bound to $0.8350$;
- that the fixed-seed numerical ranking is a theorem;
- that the official four-body numerical optimum is a rigorous lower bound;
- that atlas-lift is necessarily faster than the full 8-D search by any particular amount.

What this round has actually accomplished is:

$$
\boxed{
\text{minimizer-set exchange}
\to
\text{threshold-conditioned atlas lifting}.
}
$$

Large-scale atlas and lift-certificate computation is left as `COMPUTE-DEFERRED`; computations that have not yet been executed are not written up as proofs.