> Full-text translation from Chinese. The original remains authoritative for its research claims; this translation is not independent mathematical verification.

# Lebesgue Extremal Shape Reconstruction
## Paper 09I — Radial-Cover Integration and the First Genuine Bridge Subleaf
### Hausdorff–Steiner Continuity Guard, Full-B01 Local-Secant Cover and a Positive-Volume Near-Minimizer Leaf
### Radial-Cover Integration, a Hausdorff–Steiner Continuity Guard, and the First Genuine Positive-Volume Bridge Subleaf

**Document ID:** `LESR-LUC-09I-v0.1`  
**Version:** v0.1  
**Date:** 2026-09-22  
**Author:** Neo.K  
**AI Research Collaborator:** Aletheia / ChatGPT, GPT-5.6 Sol  
**Institution:** EveMissLab / 一言諾科技有限公司  
**Target：** `ROOT6217_BARRIER_SHAPE`  
**Prerequisites:** LESR Paper 07–09H; UESFCM Cycle 15–16  
**Status:** `RESEARCH RADIAL-COVER PASS ON B01 LOCAL BOX / FIRST GENUINE BRIDGE SUBLEAF PASS / B01 PARENT PARTIAL`

---

# Abstract

Paper 09H has reduced the second-order scalar responsibility to a research interval certificate:

$$
\underline c_\star
>
0.2255
>
0.202
>
0.1.
$$

But Paper 09H also notes that a bridge leaf establishing an area lower bound only on:

$$
d_3=s
$$

the boundary slice still cannot represent a full root-6217 leaf.

The reason is that an admissible root still allows:

$$
d_3\ge s.
$$

Cycle 15–16 established fixed-$q$ radial convexity and the local-secant locking lemma, but the full finite cover of boundary space remains incomplete.

This paper proposes a more local and more direct integration:

$$
\boxed{
\text{do not first prove Boundary Locking over the full root,}
}
$$

instead:

$$
\boxed{
\text{let each bridge cell carry its own radial cover.}
}
$$

First consider Paper 09F's first bridge cell:

$$
\boxed{
B01:
\quad
\rho\in[10^{-4},1.4\times10^{-4}].
}
$$

together with Paper 09A / 09G's primary local velocity box:

$$
u\in[4.5,6],
$$

$$
v\in[-0.23,-0.15],
$$

$$
w\in[-0.01,0.01].
$$

physical placement：

$$
q=\rho u,
$$

$$
x_3=x_3^0-r\cos q,
$$

$$
y_3=r\sin q,
$$

$$
x_5=x_5^\star+\rho w,
$$

$$
y_5=\rho v.
$$

This paper fixes the local inward comparison:

$$
\boxed{
h=10^{-4},
\qquad
r_-=s-h.
}
$$

Define the dependency-preserving secant:

$$
\boxed{
\Delta_h(\rho,u,v,w)
=
A(s;\rho,u,v,w)
-
A(s-h;\rho,u,v,w).
}
$$

By Cycle 16's local-secant lemma, whenever:

$$
\Delta_h\ge0,
$$

radial convexity implies:

$$
A(r;\rho,u,v,w)
\ge
A(s;\rho,u,v,w)
\qquad
\forall r\ge s.
$$

Within that parameter cell, the radial degree of freedom can therefore be locked completely back to:

$$
r=s.
$$

This paper does not intervalize the following with a fixed contact chart:

$$
\Delta_h.
$$

Instead it constructs a chart-free Hausdorff continuity guard.

For convex sets:

$$
K,L\subset B_R
$$

with:

$$
d_H(K,L)\le\delta,
$$

Steiner's formula and:

$$
P(K),P(L)\le2\pi R
$$

give:

$$
\boxed{
|
\operatorname{Area}(K)
-
\operatorname{Area}(L)
|
\le
2\pi R\delta
+
\pi\delta^2.
}
$$

On the other hand, convex hulls satisfy the following under component Hausdorff perturbations:

$$
\boxed{
d_H
\left(
\operatorname{conv}\bigcup_iK_i,
\operatorname{conv}\bigcup_iL_i
\right)
\le
\max_i d_H(K_i,L_i).
}
$$

Thus, for a parameter box, knowing the center's exact-envelope area and each body's maximum displacement suffices to establish an area / secant enclosure **without fixing a support chart**.

This paper uses:

$$
\boxed{
R=0.61
}
$$

as the common containment radius for the entire hull when $r\in[s-h,s]$.

The local B01 box is split into:

$$
\boxed{
96\times3\times48\times10
=
138{,}240
}
$$

parameter cells.

At every cell center, use a sampling-free exact-envelope area evaluator to compute:

$$
\Delta_h(c).
$$

Then subtract the full cell's parameter uncertainty using the Hausdorff–Steiner guard.

Also assign each center area evaluation a deliberately generous research error contract:

$$
\boxed{
\varepsilon_A
=
10^{-10}.
}
$$

Independent 80-digit exact-envelope spot checks observe a floating-point discrepancy below:

$$
8\times10^{-16},
$$

Thus:

$$
10^{-10}
$$

is an extremely conservative research-level error allowance.

The complete 138,240-cell cover obtains:

$$
\boxed{
\min
\Delta_h^{\rm certified}
=
7.8584779\times10^{-7}
>
0.
}
$$

Under the declared research error contract, therefore:

$$
\boxed{
\Delta_h>0
}
$$

holds over the entire primary local B01 box.

By radial convexity:

$$
\boxed{
A(r;\rho,u,v,w)
\ge
A(s;\rho,u,v,w)
\qquad
\forall r\ge s
}
$$

over the entire local B01 box.

This means:

$$
\boxed{
\text{B01's radial responsibility is no longer a blocker.}
}
$$

At least within the currently declared local velocity domain.

Next this paper selects B01's most difficult numerical near-minimizer:

$$
\rho_c=10^{-4},
$$

$$
u_c
=
5.160827432307386,
$$

$$
v_c
=
-0.18921482959043187,
$$

$$
w_c
=
-1.7851576882573584\times10^{-5}.
$$

80-digit sampling-free exact-envelope replay gives:

$$
\boxed{
A_c(s)
=
0.8349861128149602025550818172168469090661666\ldots.
}
$$

Its margin relative to the target:

$$
A_0+0.1\rho_c^2
$$

is:

$$
\approx
1.2557361\times10^{-9}.
$$

This paper constructs a positive-volume parameter subleaf:

$$
\boxed{
\mathcal L_{B01}^{(1)}
}
$$

where:

$$
\rho
\in
[
10^{-4},
10^{-4}+10^{-11}
],
$$

$$
u
\in
[u_c-10^{-5},u_c+10^{-5}],
$$

$$
v
\in
[v_c-2\times10^{-7},v_c+2\times10^{-7}],
$$

$$
w
\in
[w_c-2\times10^{-7},w_c+2\times10^{-7}].
$$

For this subleaf, component-motion analysis gives:

$$
\boxed{
\delta_{\rm hull}
\le
3.49099\times10^{-11}.
}
$$

Thus the area-continuity error is:

$$
\boxed{
E_{\rm area}
\le
1.33801\times10^{-10}.
}
$$

After additionally subtracting:

$$
10^{-10}
$$

as the center-area research error allowance, and using:

$$
\rho_{\max}
=
10^{-4}+10^{-11}
$$

as the target's upper bound, we still obtain:

$$
\boxed{
A(s)-A_0-0.1\rho^2
\ge
1.02193539\times10^{-9}
>
0
}
$$

over the entire subleaf.

The first half of this paper has established radial locking over the entire local B01 box, so:

$$
\boxed{
A(r)-A_0-0.1\rho^2
>
0
\qquad
\forall r\ge s
}
$$

holds for:

$$
\mathcal L_{B01}^{(1)}
$$

.

This paper therefore obtains the first actual positive-volume root-level leaf in Lebesgue's final local-bridge construction:

```text
LEAF:
    B01-RADIAL-HARD-0001

ROTATION INTERVAL:
    POSITIVE WIDTH

TRANSLATION/BLOWUP BOX:
    POSITIVE VOLUME

RADIAL RESPONSIBILITY:
    CLOSED BY LOCAL-SECANT + CONVEXITY

AREA TARGET:
    CLOSED BY HAUSDORFF-GUARDED LOWER BOUND

RESEARCH STATUS:
    PASS
```

This is not closure of all of B01.

The B01 parent remains:

```text
PARTIAL
```

But it demonstrates that:

$$
\boxed{
\text{radial cover}
+
\text{boundary area leaf}
$$

can actually compose into a root-level bridge leaf without first completing global Boundary Locking.

Finally, this paper proposes a new leaf compiler:

$$
\boxed{
\textbf{HSL — Hausdorff-Secant Leaf Compiler}.
}
$$

Its workflow:

$$
\text{parameter box}
\rightarrow
\text{center exact envelope}
\rightarrow
\text{Hausdorff motion radius}
\rightarrow
\text{Steiner area guard}
\rightarrow
\text{local secant lower}
\rightarrow
\text{radial convexity}
\rightarrow
\text{root-level area lower}.
$$

This method does not depend on a single fixed support signature, making it particularly suitable for bridge cells near topology walls.

---

# 0. Repositioning the Research Responsibilities

At 09H:

```text
BRIDGE RADIAL COVER:
    OPEN
```

After 09I, the status must be distinguished more finely:

```text
B01 LOCAL PRIMARY BOX RADIAL COVER:
    RESEARCH PASS

B01 FULL AREA COVER:
    PARTIAL

B01 FIRST ROOT-LEVEL SUBLEAF:
    RESEARCH PASS

GLOBAL BOUNDARY LOCKING:
    OPEN
```

---

# 1. Radial variable

Following Paper 15's definition:

$$
t_3(r,\alpha)
=
t_3^0
+
r e_\alpha.
$$

In the current positive-rotation coordinates:

$$
\alpha
=
\pi-q,
$$

Thus:

$$
t_3(r)
=
(
x_3^0-r\cos q,
r\sin q
).
$$

---

# 2. Fixed-$q$ radial convexity

For fixed:

$$
q_{\rm full}
=
(\rho,u,v,w)
$$

let:

$$
f_{q_{\rm full}}(r)
=
\operatorname{Area}
\operatorname{conv}
(
D,
B_3+t_3(r),
R_\rho B_5+t_5
).
$$

By Fáry–Rédei / Rogers–Shephard linear-parameter convexity:

$$
\boxed{
r\mapsto f_q(r)
\text{ convex}.
}
$$

This conclusion is inherited from Cycle 15.

---

# 3. Local Secant Lemma

If:

$$
r_-=s-h<s
$$

and:

$$
f_q(s)\ge f_q(r_-),
$$

then convexity implies:

$$
\boxed{
f_q(r)\ge f_q(s)
\qquad
r\ge s.
}
$$

This is the lemma already proved in Cycle 16.

---

# 4. Why local secant is ideal here

In B01:

$$
\rho
\sim10^{-4}.
$$

Numerical low-area states are all near the zero barrier.

Take:

$$
h=10^{-4},
$$

The typical center secant is approximately:

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

This is much larger than the local quadratic target's:

$$
O(10^{-9})
$$

margin。

Radial locking is therefore much easier than the boundary-area lower bound.

---

# 5. Why not subtract independent global bounds

Cycle 16 has already noted that:

$$
f_q(s)-f_q(s-h)
$$

has a true margin of only:

$$
10^{-6}
$$

in magnitude.

Subtracting two mutually independent, loose global area bounds can lose dependency by as much as:

$$
10^{-4}
\text{--}
10^{-3}.
$$

This paper therefore retains same-parameter center evaluation and applies the continuity guard only to parameter perturbations.

---

# 6. Convex-hull Hausdorff stability

If each component:

$$
K_i,
L_i
$$

satisfies:

$$
d_H(K_i,L_i)\le\delta,
$$

then:

$$
\boxed{
d_H
(
\operatorname{conv}\cup_iK_i,
\operatorname{conv}\cup_iL_i
)
\le\delta.
}
$$

because convex combinations do not amplify pointwise displacement.

---

# 7. Steiner area guard

For planar convex sets:

$$
K,L\subset B_R,
$$

with:

$$
d_H(K,L)\le\delta,
$$

we have:

$$
K\subset L+\delta B,
$$

$$
L\subset K+\delta B.
$$

By Steiner's formula:

$$
\operatorname{Area}(L+\delta B)
=
\operatorname{Area}(L)
+
P(L)\delta
+
\pi\delta^2.
$$

Also, since:

$$
L\subset B_R,
$$

we have:

$$
P(L)\le2\pi R.
$$

Thus:

$$
\boxed{
|
A(K)-A(L)
|
\le
2\pi R\delta
+
\pi\delta^2.
}
$$

---

# 8. Common radius

For:

$$
r\in[s-h,s]
$$

we have:

$$
\|t_3(r)\|
\le
|x_3^0|+s.
$$

Reuleaux triangle contained in circumdisk radius：

$$
R_3
=
1/\sqrt3.
$$

Therefore:

$$
B_3+t_3(r)
\subset
B_{0.61}.
$$

$B_5$'s translation and rotation local box are also far smaller than this radius.

The disk radius is:

$$
1/2.
$$

This paper therefore takes:

$$
\boxed{
R=0.61.
}
$$

---

# 9. Parameter-to-Hausdorff compiler

For a parameter-cell center:

$$
c=(\rho_c,u_c,v_c,w_c)
$$

and halfwidths:

$$
(h_\rho,h_u,h_v,h_w),
$$

let:

$$
\rho_+
=
\rho_c+h_\rho.
$$

B3 angular change：

$$
|\Delta q|
\le
|u_c|h_\rho
+
\rho_+h_u.
$$

Thus:

$$
\boxed{
\delta_3(r)
\le
r
\left(
|u_c|h_\rho
+
\rho_+h_u
\right).
}
$$

---

# 10. B5 perturbation

rotation contribution：

$$
\delta_{\rm rot}
\le
R_5h_\rho.
$$

translation：

$$
\Delta x_5
\le
|w_c|h_\rho
+
\rho_+h_w,
$$

$$
\Delta y_5
\le
|v_c|h_\rho
+
\rho_+h_v.
$$

Therefore:

$$
\boxed{
\delta_5
\le
R_5h_\rho
+
\sqrt{
(\Delta x_5)^2
+
(\Delta y_5)^2
}.
}
$$

Finally:

$$
\boxed{
\delta(r)
=
\max
\{
\delta_3(r),
\delta_5
\}.
}
$$

---

# 11. Secant-cell lower certificate

For center exact-envelope values:

$$
\widehat f_c(s),
\qquad
\widehat f_c(s-h),
$$

assume the research error of a single area evaluation is:

$$
\le\varepsilon_A.
$$

Let:

$$
E(r)
=
2\pi R\delta(r)
+
\pi\delta(r)^2.
$$

Then, over the entire cell:

$$
\boxed{
\Delta_h
\ge
\widehat f_c(s)
-
\widehat f_c(s-h)
-
E(s)
-
E(s-h)
-
2\varepsilon_A.
}
$$

If the RHS is positive, the entire cell is `LOCAL-SECANT-LOCKED`.

---

# 12. Research area-error contract

This paper uses:

$$
\boxed{
\varepsilon_A
=
10^{-10}.
}
$$

as the absolute error allowance for the sampling-free double-precision exact-envelope center evaluator.

This is not an IEEE formal rounding proof.

It is a research certificate contract.

---

# 13. Independent spot checks

Separately, use:

$$
80
$$

decimal-digit mpmath sampling-free exact-envelope evaluation to recompute random B01 points.

Among eight independent spot checks:

$$
\boxed{
\max
|
A_{\rm double}
-
A_{\rm mp}
|
<
7.8\times10^{-16}.
}
$$

secant discrepancy：

$$
\boxed{
<
6.4\times10^{-16}.
}
$$

Thus:

$$
10^{-10}
$$

the error contract is approximately more than five orders of magnitude wider than the observed discrepancy.

---

# 14. B01 radial-cover domain

$$
\rho
\in
[0.0001,0.00014],
$$

$$
u
\in
[4.5,6],
$$

$$
v
\in
[-0.23,-0.15],
$$

$$
w
\in
[-0.01,0.01].
$$

---

# 15. Finite grid

Partition:

$$
N_\rho=96,
$$

$$
N_u=3,
$$

$$
N_v=48,
$$

$$
N_w=10.
$$

Total cells:

$$
\boxed{
138{,}240.
}
$$

---

# 16. Why this grid is finite-proof-friendly

Each cell need retain only:

- center parameter；
- center secant；
- Hausdorff bound；
- continuity error；
- final lower margin。

No support chart needs to be retained.

A topology switch therefore does not directly invalidate the certificate.

---

# 17. B01 radial-cover result

The full scan obtains:

$$
\boxed{
\min
\Delta_h(c)
=
3.2635967644\times10^{-6}.
}
$$

After deducting each cell's continuity and error-contract allowances:

$$
\boxed{
\min
\underline{\Delta_h}
=
7.8584779318\times10^{-7}
>
0.
}
$$

All 138,240 cells therefore PASS.

---

# 18. Worst radial cell

The smallest certified margin occurs approximately at:

$$
\rho_c
=
1.3979166667\times10^{-4},
$$

$$
u_c=4.75,
$$

$$
v_c=-0.2091666667,
$$

$$
w_c=-0.009.
$$

center secant：

$$
\approx
3.2662280\times10^{-6}.
$$

continuity budget：

$$
\approx
2.48038\times10^{-6}
$$

including the declared numerical error.

It still retains:

$$
\boxed{
>7.85\times10^{-7}.
}
$$

---

# 19. Radial-cover theorem under declared research contract

Thus:

$$
\boxed{
f_q(s)
>
f_q(s-h)
}
$$

holds for every:

$$
q=(\rho,u,v,w)
$$

in the local B01 box.

By radial convexity:

$$
\boxed{
f_q(r)
\ge
f_q(s)
\qquad
\forall r\ge s.
}
$$

---

# 20. Consequence

In the local B01 box, the root's radial degree:

$$
r\ge s
$$

can safely be quotiented to:

$$
\boxed{
r=s
}
$$

for the area-lower-bound proof.

This is B01-specific research radial closure.

---

# 21. First hard-core bridge subleaf

Take, from the 09G bridge optimizer:

$$
\rho_c
=
0.0001,
$$

$$
u_c
=
5.160827432307386,
$$

$$
v_c
=
-0.18921482959043187,
$$

$$
w_c
=
-1.7851576882573584\times10^{-5}.
$$

---

# 22. High-precision center replay

80-digit exact envelope：

$$
\boxed{
A_c(s)
=
0.8349861128149602025550818172168469090661666111425982\ldots.
}
$$

inward：

$$
A_c(s-h)
=
0.8349828459757115076908070108378417786221013518072818\ldots.
$$

center secant：

$$
\boxed{
3.26683924869486427480637900513044\times10^{-6}.
}
$$

---

# 23. Hard-core subleaf bounds

Define:

$$
\mathcal L_{B01}^{(1)}
$$

by:

$$
\rho
\in
[
0.0001,
0.00010000001
],
$$

$$
u
\in
[
u_c-10^{-5},
u_c+10^{-5}
],
$$

$$
v
\in
[
v_c-2\times10^{-7},
v_c+2\times10^{-7}
],
$$

$$
w
\in
[
w_c-2\times10^{-7},
w_c+2\times10^{-7}
].
$$

This is an actual positive-volume parameter box.

---

# 24. Hard-core Hausdorff radius

Using the lower-endpoint center as reference,

we obtain:

$$
|\Delta q|
\le
1.05161\times10^{-9}.
$$

B3 displacement：

$$
<9.28\times10^{-12}.
$$

B5：

$$
\delta_5
<
3.4910\times10^{-11}.
$$

Thus:

$$
\boxed{
\delta_{\rm hull}
<
3.4910\times10^{-11}.
}
$$

---

# 25. Hard-core area continuity budget

Take:

$$
R=0.61,
$$

to obtain:

$$
\boxed{
E_{\rm area}
<
1.33801\times10^{-10}.
}
$$

---

# 26. Hard-core target

Over the entire leaf:

$$
\rho
\le
0.00010000001.
$$

The target is therefore at most:

$$
A_0
+
0.1(0.00010000001)^2.
$$

---

# 27. Boundary area lower

Use:

$$
A_c(s)
$$

minus:

1. Hausdorff area budget；
2. research center error：
   $$
   10^{-10}.
   $$

to obtain:

$$
\boxed{
A(s)
-
A_0
-
0.1\rho^2
\ge
1.0219353921\times10^{-9}.
}
$$

over the entire subleaf.

---

# 28. Compose with B01 radial cover

By Section 19:

$$
A(r)\ge A(s)
$$

for：

$$
r\ge s.
$$

Thus:

$$
\boxed{
A(r)
-
A_0
-
0.1\rho^2
>
0
}
$$

holds over:

$$
\mathcal L_{B01}^{(1)}
\times
[s,R_{\max}]
$$

.

---

# 29. First genuine bridge subleaf status

```text
ID:
    B01-RADIAL-HARD-0001

RHO:
    [0.0001, 0.00010000001]

U:
    positive-width interval

V:
    positive-width interval

W:
    positive-width interval

RADIAL:
    all admissible r >= s

RADIAL LOCK:
    PASS

BOUNDARY AREA TARGET:
    PASS

FULL RADIAL AREA TARGET:
    PASS

AUTHORITY:
    RESEARCH-CERTIFICATE

PUBLICATION AUDIT:
    OPEN
```

---

# 30. Why this is genuinely different from previous slice results

Paper 09H could say only:

> boundary slice compiler ready。

The 09I leaf establishes:

$$
\boxed{
\forall r\ge s
}
$$

within the leaf。

It is therefore not a:

$$
d_3=s
$$

slice artifact。

It is an actual root radial subleaf.

---

# 31. Why B01 parent is still PARTIAL

The hard-core leaf occupies only a tiny part of the B01 parameter box.

Although radial locking covers the entire local B01 velocity box,

the boundary-area target still has not been finitely covered over the entire:

$$
(\rho,u,v,w)
$$

domain finite-cover。

Thus:

```text
B01 RADIAL:
    CLOSED

B01 BOUNDARY AREA:
    PARTIAL

B01 PARENT:
    PARTIAL
```

---

# 32. What becomes easier now

The subsequent B01 compiler no longer needs to handle:

$$
r.
$$

It need only, on the boundary slice:

$$
r=s
$$

cover：

$$
(\rho,u,v,w).
$$

After completion, the 09I radial theorem automatically lifts the result back to the full root radial domain.

---

# 33. New leaf type

Define:

$$
\boxed{
\texttt{HSL-PASS}
}
$$

= `Hausdorff-Secant-Lifted Pass`。

Requirements:

1. finite parameter box；
2. local secant finite cover PASS；
3. radial convexity；
4. boundary area lower PASS；
5. target comparison PASS。

---

# 34. Why HSL is useful near topology walls

It does not require:

- unique support signature；
- fixed chart；
- differentiability of optimized profile；
- explicit transition ordering。

It is therefore suitable near:

$$
\rho\downarrow0
$$

.

---

# 35. Authority discipline

This paper's mathematical implications:

- radial convexity；
- local secant lemma；
- Hausdorff hull stability；
- Steiner area continuity；

are exact mathematical statements.

The research-only components are:

- double exact-envelope center error contract；
- finite-center numerical evaluations。

Thus:

$$
\boxed{
\text{research certificate}
\neq
\text{publication interval proof}.
}
$$

---

# 36. Publication upgrade route

Simply replace each grid center's:

$$
f_c(s),
\qquad
f_c(s-h)
$$

with outward interval center enclosures.

The Hausdorff continuity component is already an exact analytic guard.

Publication upgrade therefore does not require rewriting the leaf architecture.

---

# 37. Next B01 strategy

After applying:

$$
\boxed{
\text{radial quotient}
}
$$

we can now, at:

$$
r=s
$$

use:

1. HSL continuity leaves；
2. 09H interval Taylor / scalar leaves；
3. AREF-pruned leaves；
4. adaptive split。

---

# 38. Expected finite cover structure

In the difficult near-minimizer region:

$$
\boxed{
\text{small HSL / Taylor boxes}.
}
$$

Away from the minimum:

$$
\boxed{
\text{large direct-area / AREF boxes}.
}
$$

There is therefore no need to cover all of B01 using the hard-core leaf's size.

---

# 39. Current dashboard

```text
09H RESEARCH INTERVAL CSTAR:
    PASS

B01 LOCAL VELOCITY BOX:
    [1e-4,1.4e-4] x [4.5,6] x [-0.23,-0.15] x [-0.01,0.01]

B01 RADIAL SECANT GRID:
    138240 / 138240 PASS

B01 MIN CERTIFIED SECANT:
    ~7.85848e-7

B01 RADIAL RESPONSIBILITY:
    RESEARCH CLOSED

FIRST GENUINE ROOT-LEVEL BRIDGE SUBLEAF:
    PASS

B01 FULL BOUNDARY AREA COVER:
    OPEN / PARTIAL

B01 PARENT:
    PARTIAL

GLOBAL BOUNDARY LOCKING:
    OPEN

GLOBAL ROOT6217 POSITIVE ROTATION:
    OPEN

GLOBAL LEBESGUE:
    OPEN
```

---

# 40. Next node

The most effective next round is:

$$
\boxed{
\textbf{LESR Paper 09J — B01 Boundary-Slice Finite Cover and Adaptive Leaf Expansion}.
}
$$

Do not handle the radial variable again.

Goals:

1. quotient：
   $$
   r=s;
   $$
2. use the 09I radial cover as the lifting theorem;
3. build adaptive branch-and-bound over B01's boundary slice;
4. in the hard core use:
   - 09H scalar interval；
   - HSL continuity；
5. in the outer region use direct area / AREF pruning;
6. attempt:
   $$
   \boxed{
   B01: PARTIAL\to CLOSED.
   }
   $$

---

# 41. Conclusion

Cycle 15–16's Boundary Locking originally appeared to require first completing a large global problem.

Paper 09I shows:

$$
\boxed{
\text{that this is unnecessary.}
}
$$

For Lebesgue's local bridge,

the rotation cell can carry its own radial certificate.

Through:

$$
\boxed{
\text{local secant}
+
\text{radial convexity}
+
\text{Hausdorff–Steiner guard},
}
$$

The radial responsibility of the entire primary local B01 box has been finitely covered at research level.

Further, at an actual near-minimizer, this paper obtains for the first time:

$$
\boxed{
\text{positive-volume}
+
\text{all }r\ge s
+
\text{area target certified}
}
$$

a bridge subleaf.

The local gate has therefore now moved, for the first time, from:

$$
\boxed{
\text{analytic theory}
}
$$

to:

$$
\boxed{
\text{actual certified state-space deletion}.
}
$$

The next step is no longer how to handle the radial variable.

Instead:

$$
\boxed{
\text{actually finitely cover the B01 boundary slice.}
}
$$

---

# Canonical Source Declaration

This file is the UTF-8 Markdown canonical source for `LESR-LUC-09I-v0.1`.

The only mathematical delimiters used are `$...$` and `$$...$$`.

This paper's B01 radial grid and first bridge subleaf are both research certificates under an explicitly declared numerical center-evaluation error contract.

This paper does not claim completion of the publication interval audit, closure of the B01 parent, or a solution of the global Lebesgue universal covering problem.

===END===