> 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 09J — Eight-Chart Quadratic Floor on B01
### From 4D Boundary Cover to Eight Polyhedral Sign Charts: a Second-Order Lower Bound over the Entire Local B01 Box
### Eight-Chart Quadratic Floor on B01: Reducing the Full Local Velocity Box to Polyhedral Quadratic Programs

**Document ID:** `LESR-LUC-09J-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 / B01`  
**Prerequisites:** LESR Paper 07–09I  
**Status:** `FULL-B01 SECOND-ORDER CHART FLOOR / RESEARCH KKT ENUMERATION / FINITE-RHO REMAINDER OPEN`

---

# Abstract

Paper 09I has completed, at research-certificate level:

$$
\boxed{
\text{the radial responsibility of the local B01 primary box}.
}
$$

That is, for:

$$
\rho\in[10^{-4},1.4\times10^{-4}],
$$

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

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

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

all admissible:

$$
r\ge s
$$

can be quotiented, by local secant + radial convexity, back to:

$$
r=s.
$$

B01's remaining core problem is therefore the boundary slice:

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

09I attempted to cover all of B01 finitely using chart-free Hausdorff area continuity directly. The first layer:

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

of cells could close some peripheral regions, but many cells remained because the global Hausdorff bound was excessively sensitive to rotation:

$$
\rho
$$

.

This paper does not continue refining the $\rho$ grid.

Instead it returns to the first-order topology classification already completed by Paper 09E.

09E has proved that the primary neighboring charts are completely determined by:

$$
\operatorname{sgn}E_T,
\qquad
\operatorname{sgn}E_L,
\qquad
\operatorname{sgn}E_R
$$

these three sign bits.

There are therefore exactly:

$$
\boxed{
2^3=8
}
$$

open first-order charts.

For any chart:

$$
\sigma
\in
\{-1,+1\}^3,
$$

This paper builds an 18-piece one-sided positive-$\rho$ chart from canonical support formulas and directly uses a second-order Jet:

$$
A_\sigma(\rho;u,v,w)
=
A_0
+
\rho\Phi_1^\sigma(u,v,w)
+
\rho^2\Psi_2^\sigma(u,v,w)
+
O(\rho^3)
$$

to extract:

$$
\boxed{
\Psi_2^\sigma.
}
$$

Since:

- the first-order motion of body centers is affine in $(u,v,w)$;
- first-order transition speeds are affine;
- second-order transition speeds are quadratic;
- the area primitive is a quadratic form plus trigonometric endpoint expansion;

each:

$$
\Psi_2^\sigma(u,v,w)
$$

is exactly a:

$$
\boxed{
\text{quadratic polynomial in three variables}.
}
$$

B01's second-order completeness therefore no longer needs 4D dense sampling.

It becomes eight finite quadratic programs:

$$
\boxed{
\min
\Psi_2^\sigma(u,v,w)
}
$$

subject to：

1. B01 velocity box；
2. three sign inequalities:
   $$
   \sigma_T E_T\ge0,
   $$
   $$
   \sigma_L E_L\ge0,
   $$
   $$
   \sigma_R E_R\ge0.
   $$

Each feasible region is a compact polytope.

For the global minimum of a quadratic function on a compact polytope, every minimizer lies at a relative-interior stationary point of some face, or on a lower-dimensional face.

It is therefore sufficient to enumerate KKT systems with:

$$
\boxed{
0,1,2,3
}
$$

independent active constraints.

This paper performs complete active-face enumeration for all eight charts.

The chartwise minima are:

| Sign chart $(E_T,E_L,E_R)$ | $\inf\Psi_2^\sigma$ |
|---|---:|
| $(-,-,-)$ | $0.1941331878\ldots$ |
| $(-,-,+)$ | $0.1771421602\ldots$ |
| $(-,+,-)$ | $0.1082369861\ldots$ |
| $(-,+,+)$ | $\boxed{0.1060833695\ldots}$ |
| $(+,-,-)$ | $0.3088556046\ldots$ |
| $(+,-,+)$ | $0.3438822109\ldots$ |
| $(+,+,-)$ | $0.2570779006\ldots$ |
| $(+,+,+)$ | $0.3003930106\ldots$ |

The entire local B01 velocity box therefore has:

$$
\boxed{
\Psi_2
\ge
0.1060833695\ldots
>
0.1.
}
$$

The minimum does not lie at the first-order critical velocity of Paper 09B/09C.

It occurs in the chart:

$$
\boxed{
(-,+,+)
}
$$

on the box face:

$$
\boxed{
u=6,
\qquad
v=-0.15,
}
$$

with:

$$
\boxed{
w
=
-0.0043290860412957563\ldots.
}
$$

At this point:

$$
E_T<0,
\qquad
E_L>0,
\qquad
E_R>0.
$$

The winning chart's quadratic polynomial is:

$$
\begin{aligned}
\Psi_2^{(-,+,+)}
={}&
0.02794365170082480318
-0.00610709314035057786\,u \\
&+
0.52831149053377130555\,v
+
0.41030844970918459524\,w \\
&-
0.00038801325227352718\,u^2
+
8.42348018991951288801\,v^2 \\
&+
10.13141844912428934197\,w^2
-
0.02073227314117337278\,uv \\
&-
0.00441000000000000000\,uw
+
1.97419256890616049061\,vw.
\end{aligned}
$$

On the:

$$
u=6,
\qquad
v=-0.15
$$

face, taking the stationary point in $w$ gives the above:

$$
w_\star^{B01}.
$$

Substitute back:

$$
\boxed{
\Psi_{2,\min}^{B01}
=
0.1060833695262153239987\ldots.
}
$$

Paper 09J therefore obtains a statement broader than 09D's critical-cone result:

$$
\boxed{
\text{positive second-order growth occurs not only on the first-order critical cone;}
}
$$

rather:

$$
\boxed{
\text{the chartwise second-order floor of the entire local B01 velocity box exceeds }0.1.
}
$$

B01's finite-$\rho$ responsibility therefore now consists only of:

$$
\boxed{
\text{uniform higher-order remainder}.
}
$$

This paper also builds a third-order Jet diagnostic.

Across the eight charts:

$$
\Psi_3^\sigma
$$

are all cubic polynomials.

Numerical polytope search observes:

$$
\boxed{
|\Psi_3|
<
0.42
}
$$

over the current B01 domain.

This paper does not promote:

$$
0.42
$$

to an interval theorem.

Instead it sets a very generous future certificate target:

$$
\boxed{
|\Psi_3|
\le1.
}
$$

To retain a larger safety margin for finite-$\rho$ closure, this paper also adopts a conservative quadratic floor:

$$
\boxed{
\Psi_2\ge0.1055.
}
$$

At:

$$
\rho_{\max}
=
1.4\times10^{-4}
$$

if a future proof establishes:

$$
|\Psi_3|\le1
$$

and the fourth-and-higher remainder satisfies:

$$
\boxed{
|R_4|
\le
250000\,\rho^4,
}
$$

then:

$$
\begin{aligned}
A-A_0-\rho\Phi_1
&\ge
0.1055\rho^2
-
\rho^3
-
250000\rho^4 \\
&=
\rho^2
\left(
0.1055
-\rho
-250000\rho^2
\right).
\end{aligned}
$$

For:

$$
\rho\le1.4\times10^{-4}
$$

we have:

$$
0.1055
-
0.00014
-
250000(1.4\times10^{-4})^2
=
\boxed{
0.10046.
}
$$

Thus:

$$
\boxed{
A
\ge
A_0
+
\rho\Phi_1
+
0.10046\rho^2.
}
$$

Also, since Paper 09E gives:

$$
\Phi_1\ge0,
$$

this implies:

$$
\boxed{
A
>
A_0
+
0.1\rho^2
}
$$

over all of B01.

In other words, the current B01 parent status has been further compressed from:

```text
4D FINITE COVER OPEN
```

to:

```text
RADIAL RESPONSIBILITY:
    RESEARCH CLOSED

FIRST-ORDER CHART COMPLETENESS:
    ANALYTICALLY CLOSED

SECOND-ORDER FULL-BOX FLOOR:
    RESEARCH CLOSED

FINITE-RHO REMAINDER:
    OPEN
```

The required remainder bounds are extremely generous:

$$
|\Psi_3|\le1
$$

and:

$$
|R_4|/\rho^4\le250000.
$$

09J therefore does not silently promote the B01 parent to `CLOSED`.

But it reduces B01's final mathematical responsibility from:

$$
\boxed{
\text{four-dimensional area finite cover}
}
$$

to:

$$
\boxed{
\text{a uniform higher-order Taylor enclosure}.
}
$$

---

# 0. Current Definition of B01

$$
\rho
\in
[10^{-4},1.4\times10^{-4}],
$$

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

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

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

Paper 09I has quotiented the:

$$
r\ge s
$$

radial responsibility back to:

$$
r=s
$$

over the declared local B01 box.

---

# 1. Primary residuals

Following Paper 09E:

$$
E_T
=
s u\beta_{5T}
-
\beta_{3T}(\gamma_T+v),
$$

$$
E_L
=
-s\sin\theta_\star L_\alpha u
+
L_\theta
(
\gamma_L+\sin\theta_\star v-\cos\theta_\star w
),
$$

$$
E_R
=
-s\sin\theta_\star L_\alpha u
+
L_\theta
(
\gamma_L+\sin\theta_\star v+\cos\theta_\star w
).
$$

---

# 2. Eight sign charts

Each:

$$
\sigma
=
(\sigma_T,\sigma_L,\sigma_R)
\in
\{-1,+1\}^3
$$

defines the polytope:

$$
\boxed{
P_\sigma
=
B_{\rm vel}
\cap
\{
\sigma_TE_T\ge0,
\sigma_LE_L\ge0,
\sigma_RE_R\ge0
\}.
}
$$

---

# 3. Why exactly eight charts suffice

Paper 09E has paired the six triple-support normals into three mirror / cyclic pairs.

For each signed residual:

- a positive sign selects one member of the pair;
- a negative sign selects the mirror/cyclic member.

The generic positive-$\rho$ primary chart is therefore determined only by three signs.

---

# 4. 18-piece chart compiler

For each:

$$
\sigma
$$

the companion generator fixes the corresponding 18-piece winner sequence.

Transition boundaries are generated by:

- body intrinsic boundaries；
- adjacent pairwise support equalities；

.

All roots are expanded using an implicit Jet:

$$
\theta_j
=
\theta_j^0
+
\tau_j\rho
+
\eta_j\rho^2
+
O(\rho^3).
$$

---

# 5. Area expansion

For each chart:

$$
A_\sigma
=
\sum_j
\left[
\mathcal F_j(\theta_{j+1})
-
\mathcal F_j(\theta_j)
\right].
$$

Directly extract the coefficients:

$$
A_\sigma
=
A_0
+
\rho\Phi_1^\sigma
+
\rho^2\Psi_2^\sigma
+
O(\rho^3).
$$

---

# 6. $\Psi_2^\sigma$ is quadratic

Each:

$$
\Psi_2^\sigma
$$

is directly reconstructed by the companion generator as:

$$
\boxed{
c
+
\ell^Tx
+
\frac12x^THx,
\qquad
x=(u,v,w).
}
$$

This is not finite-$\rho$ fitting.

---

# 7. Coefficient extraction

Since the function is known to be quadratic,

only the analytic Jet evaluator's values at:

$$
0,
\quad
\pm e_1,
\quad
\pm e_2,
\quad
\pm e_3,
\quad
e_1+e_2,
\quad
e_1+e_3,
\quad
e_2+e_3
$$

are needed.

For example:

$$
\ell_u
=
\frac{
q(e_1)-q(-e_1)
}{2},
$$

$$
q_{uu}
=
\frac{
q(e_1)+q(-e_1)
}{2}
-q(0),
$$

$$
q_{uv}
=
q(e_1+e_2)
-q(e_1)
-q(e_2)
+q(0).
$$

Coefficient generation is therefore finite analytic reconstruction.

---

# 8. Quadratic global minimum on a polytope

For a compact polytope:

$$
P
$$

and a quadratic function:

$$
q,
$$

every global minimizer:

1. is stationary in the interior;
2. or stationary in the relative interior of some codimension-1 face;
3. or lies on a lower-dimensional face.

In three dimensions, enumeration of at most three independent active constraints suffices.

---

# 9. B01 constraints

Each chart has:

- $u$ lower / upper；
- $v$ lower / upper；
- $w$ lower / upper；
- three residual-sign inequalities.

Total:

$$
9
$$

linear inequalities.

The KKT compiler enumerates:

$$
\sum_{k=0}^3
\binom9k
=
130
$$

active-set patterns / chart,

then removes rank-deficient and infeasible candidates.

---

# 10. Complete active-face ledger

All eight charts execute the same procedure.

The companion JSON retains:

- quadratic coefficients；
- Hessian eigenvalues；
- winning active set；
- candidate point；
- sign residuals；
- direct analytic-Jet replay values.

---

# 11. Per-chart minima

Result:

$$
\boxed{
\begin{array}{c|c}
\sigma & \inf_{P_\sigma}\Psi_2^\sigma\\
\hline
(-,-,-)&0.1941331878\ldots\\
(-,-,+)&0.1771421602\ldots\\
(-,+,-)&0.1082369861\ldots\\
(-,+,+)&0.1060833695\ldots\\
(+,-,-)&0.3088556046\ldots\\
(+,-,+)&0.3438822109\ldots\\
(+,+,-)&0.2570779006\ldots\\
(+,+,+)&0.3003930106\ldots
\end{array}
}
$$

---

# 12. Global B01 second-order floor

Thus:

$$
\boxed{
\inf_{\sigma}
\inf_{P_\sigma}
\Psi_2^\sigma
=
0.1060833695262153\ldots.
}
$$

---

# 13. Winning chart

winning sign tuple：

$$
\boxed{
(-,+,+).
}
$$

Therefore:

$$
E_T<0,
$$

$$
E_L>0,
$$

$$
E_R>0.
$$

---

# 14. Winning active face

KKT active set：

$$
\boxed{
u=6,
\qquad
v=-0.15.
}
$$

$w$ is not on the box boundary.

It is determined by face stationarity.

---

# 15. Closed-form $w$

On the winning quadratic:

$$
\partial_w\Psi_2=0.
$$

Thus:

$$
\boxed{
w_\star
=
-\frac{
\ell_w
+
q_{uw}u
+
q_{vw}v
}{
2q_{ww}
}.
}
$$

Substitute:

$$
u=6,
\qquad
v=-0.15
$$

to obtain:

$$
\boxed{
w_\star
=
-0.0043290860412957563477567259\ldots.
}
$$

---

# 16. Winning value

Substitute again:

$$
\boxed{
\Psi_2
=
0.1060833695262153239987374696\ldots.
}
$$

---

# 17. Why this is stronger than 09D

09D addresses the second epi-derivative on the:

$$
\Phi_1=0
$$

critical cone,

obtaining approximately:

$$
0.2255863.
$$

09J allows all neighboring charts with:

$$
\Phi_1>0
$$

.

It therefore seeks the full chartwise second-coefficient floor over the entire velocity box.

The minimum can naturally be lower than the critical-cone value.

But it remains:

$$
\boxed{
>0.1.
}
$$

---

# 18. Numerical finite-$\rho$ consistency

Direct exact-envelope counter-search found, near B01's upper end, a normalized remainder of:

$$
\frac{
A-A_0-\rho\Phi_1
}{
\rho^2
}
\approx
0.1060773.
$$

Compared with the asymptotic value:

$$
0.10608337
$$

the difference is only approximately:

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

This indicates that the finite-$\rho$ correction is very small.

Status：

`NUMERICAL DIAGNOSTIC`

---

# 19. Third-order Jet

The companion diagnostic extends to:

$$
A
=
A_0
+
\rho\Phi_1
+
\rho^2\Psi_2
+
\rho^3\Psi_3
+
O(\rho^4).
$$

For each chart:

$$
\Psi_3^\sigma
$$

is a cubic polynomial.

---

# 20. Cubic diagnostic

The current polytope counter-search observes:

$$
\boxed{
\max
|\Psi_3|
<
0.42.
}
$$

The largest observed value is approximately:

$$
0.4135
$$

occurring near the:

$$
(+,-,-)
$$

chart.

No theorem is claimed.

---

# 21. Deliberately broad cubic target

The next interval compiler need prove only:

$$
\boxed{
|\Psi_3|
\le1.
}
$$

Relative to the observed:

$$
<0.42
$$

this has a margin greater than a factor of 2.

---

# 22. Conservative second-order floor

Although research KKT gives:

$$
0.10608336\ldots,
$$

the finite-proof contract uses only:

$$
\boxed{
\Psi_2\ge0.1055.
}
$$

retaining more than:

$$
5.8\times10^{-4}
$$

of coefficient slack as the arithmetic audit budget.

---

# 23. Fourth-and-higher remainder target

Let:

$$
R_4
=
A
-
A_0
-
\rho\Phi_1
-
\rho^2\Psi_2
-
\rho^3\Psi_3.
$$

This paper proposes a very generous acceptance bound:

$$
\boxed{
|R_4|
\le
250000\,\rho^4.
}
$$

---

# 24. Why $250000$ is enough

At:

$$
\rho_{\max}
=
1.4\times10^{-4},
$$

if:

$$
\Psi_2\ge0.1055,
$$

$$
|\Psi_3|\le1,
$$

$$
|R_4|\le250000\rho^4,
$$

then:

$$
\frac{
A-A_0-\rho\Phi_1
}{
\rho^2
}
\ge
0.1055
-
0.00014
-
250000(1.4\times10^{-4})^2.
$$

The right-hand side is:

$$
\boxed{
0.10046.
}
$$

---

# 25. Composition with first order

Paper 09E：

$$
\Phi_1\ge0.
$$

Thus:

$$
A-A_0
\ge
0.10046\rho^2.
$$

Hence:

$$
\boxed{
A
>
A_0+0.1\rho^2.
}
$$

---

# 26. Composition with radial closure

Paper 09I's research radial cover gives:

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

for all：

$$
r\ge s
$$

inside B01 local box。

Thus, once the higher-order remainder is certified,

the B01 parent can be closed over the full root.

---

# 27. Current B01 status

```text
RADIAL DEGREE:
    RESEARCH CLOSED

FIRST-ORDER CHART ATLAS:
    ANALYTICALLY CLOSED

SECOND-ORDER 8-CHART COMPILER:
    PASS

SECOND-ORDER FULL-BOX FLOOR:
    ~0.1060833695

CONSERVATIVE FLOOR TARGET:
    0.1055

THIRD-ORDER OBSERVED MAX ABS:
    <0.42

THIRD-ORDER CERTIFICATE TARGET:
    <=1

FOURTH+ REMAINDER TARGET:
    <=250000*rho^4

FINITE-RHO B01:
    OPEN

B01 PARENT:
    PARTIAL
```

---

# 28. What was removed

Before 09J，B01 looked like：

$$
\boxed{
\text{4D adaptive finite-cover problem}.
}
$$

After 09J：

$$
\boxed{
\text{one broad uniform Taylor remainder problem}.
}
$$

---

# 29. Why this is a real compression

A 4D cover near the critical tube needs very fine:

$$
\rho
$$

resolution, because the raw Hausdorff rotation bound ignores first-order cancellation.

09J directly encodes cancellation into:

$$
\Phi_1+\rho\Psi_2.
$$

It therefore no longer substitutes a geometric-displacement upper bound for area calculus.

---

# 30. Next node

The next round should directly undertake:

$$
\boxed{
\textbf{LESR Paper 09K — Uniform Cubic Bound and Fourth-Order B01 Closure}.
}
$$

It requires only:

1. intervalize the eight cubic polynomials:
   $$
   \Psi_3^\sigma;
   $$
2. prove:
   $$
   |\Psi_3|\le1;
   $$
3. use a fourth-order Jet / Taylor model to prove:
   $$
   |R_4|\le250000\rho^4;
   $$
4. compose 09E + 09I + 09J；
5. if all PASS:
   $$
   \boxed{
   B01:\ PARTIAL\to CLOSED.
   }
   $$

---

# 31. Conclusion

Paper 09I has removed B01's radial freedom.

Paper 09J removes a further, larger computational burden:

$$
\boxed{
\text{B01 need not be exhausted using extremely fine 4D boxes.}
}
$$

Paper 09E's finite-chart atlas allows direct extraction of eight chartwise quadratic forms.

Complete active-face enumeration gives:

$$
\boxed{
\Psi_2^{B01}
\ge
0.1060833695262153\ldots
>
0.1.
}
$$

B01 therefore has only higher-order small terms remaining.

The required higher-order bounds are very generous:

$$
|\Psi_3|\le1,
$$

$$
|R_4|\le250000\rho^4.
$$

This indicates that B01 is already very close to actual finite-radius closure.

---

# Canonical Source Declaration

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

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

This paper's quadratic coefficients are generated by an analytic Jet chart compiler; active-face/KKT enumeration is currently a research numerical certificate, and the publication arithmetic audit remains open.

This paper does not claim that the B01 parent is closed, that the root-6217 positive-rotation gate is closed, or that the global Lebesgue universal covering problem is solved.

===END===