> 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 09C — Secondary Sliver Asymptotics and the Explicit Second Epi-Derivative
### Five Potential Second-Order Slivers, Convex Piecewise-Linear $\Phi_2$ and Finite LP Certification
### Five Potential Second-Scale Slivers, a Convex Piecewise-Linear $\Phi_2$, and Reduction to a Finite LP Certificate

**Document ID:** `LESR-LUC-09C-v0.1`  
**Version:** v0.1  
**Date:** 2026-09-21  
**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 07–09B; BCODR-LUC-07  
**Status:** `SECOND-SCALE ANALYTIC REDUCTION / EXPLICIT PHI2 CANDIDATE / INTERVAL AUDIT OPEN`

---

# Abstract

Paper 09B advanced the primary critical velocity on root-6217's positive-rotation topology wall from numerical fitting to analytic selection:

$$
\boxed{
(u_\star,v_\star,w_\star)
=
(
5.157126245268359,
-0.18922818990664067,
0
).
}
$$

and obtained:

$$
\boxed{
\Phi_1^{\rm primary}
=
\gamma_w w,
\qquad
\gamma_w
=
0.1838996392372776.
}
$$

Once the primary collapse equations force:

$$
w_\star=0
$$

the actual remaining responsibility is the second epi-derivative.

Paper 09A v0.2 observed three extremely thin positive-$\rho$ intervals and retained:

$$
O(\rho^2)
$$

as a working hypothesis. However, the first Cycle-12 positive sample used for that observation did not lie exactly at the primary critical velocity subsequently derived analytically in Paper-09B.

This paper repeats sampling-free support-envelope asymptotics on the exact primary critical branch, obtaining a more precise structure:

$$
\boxed{
\text{not a fixed collection of three secondary slivers,}
}
$$

but:

$$
\boxed{
\text{a finite secondary chart family of five potential }O(\rho^2)\text{ slivers.}
}
$$

These five potential slivers are located at:

1. near $\theta_\star$: the $B_5$ arc;
2. near $\pi/2$: the disk;
3. near $\pi-\theta_\star$: the $B_5$ corner;
4. near $\pi+\theta_\star$: the $B_5$ corner;
5. near $2\pi-\theta_\star$: the $B_5$ arc.

Introduce secondary corrections:

$$
\boxed{
u(\rho)=u_\star+\rho u_2,
}
$$

$$
\boxed{
v(\rho)=v_\star+\rho v_2,
}
$$

$$
\boxed{
w(\rho)=\rho w_2,
}
$$

equivalently:

$$
q
=
\rho u_\star+\rho^2u_2,
$$

$$
y_5
=
\rho v_\star+\rho^2v_2,
$$

$$
x_5
=
x_5^\star+\rho^2w_2.
$$

Under this nested blow-up, each secondary transition angle has:

$$
\theta_{ij}(\rho)
=
\theta_0
+
\rho\tau
+
\rho^2\eta_{ij}
+
o(\rho^2),
$$

and every signed sliver width becomes:

$$
\boxed{
\delta_j
=
\eta_j^+
-
\eta_j^-,
}
$$

an affine function of:

$$
(u_2,v_2,w_2)
$$

.

High-precision asymptotic replay gives three independent signed widths; reflection / cyclic covariance supplies the remaining two as opposite-sign pairs. The second-order area functional ultimately becomes:

$$
\boxed{
\Phi_2(u_2,v_2,w_2)
=
c_0
+
\gamma_w w_2
+
\lambda_S
\left(
|\delta_+|
+
|\delta_-|
\right)
+
\lambda_T
[\delta_T]_+.
}
$$

where:

$$
[x]_+
=
\max\{x,0\},
$$

with nominal constants:

$$
\boxed{
c_0
\approx
0.258433713,
}
$$

$$
\boxed{
\lambda_S
\approx
0.00719758120,
}
$$

$$
\boxed{
\lambda_T
\approx
0.02821424418.
}
$$

Thus $\Phi_2$ is no longer an unknown Hessian, but a:

$$
\boxed{
\textbf{convex piecewise-linear finite-dimensional function}.
}
$$

Its minimization reduces to a small linear program.

The nominal LP minimizer lies at:

$$
\boxed{
\delta_+
=
\delta_-
=
\delta_T
=
0,
}
$$

corresponding to:

$$
\boxed{
(u_2^\star,v_2^\star,w_2^\star)
\approx
(
37.01279,
0.133549,
-0.178616
).
}
$$

and gives:

$$
\boxed{
c_\star^{\rm nominal}
=
\inf\Phi_2
\approx
0.22558634.
}
$$

Relative to Paper-08's certificate target:

$$
0.1,
$$

the nominal margin is:

$$
\boxed{
0.22558634-0.1
\approx
0.12558634.
}
$$

This paper does not claim that this nominal coefficient is a publication-level theorem. The actual remaining work has been reduced to:

1. interval-enclose secondary transition coefficients；
2. interval-enclose $c_0,\gamma_w,\lambda_S,\lambda_T$；
3. obtain an outward-rounded lower certificate for the finite LP;
4. establish a uniform higher-order remainder for $0<\rho\le\rho_0$;
5. connect to Paper-09B's full multi-chart $\Phi_1$ nonnegativity certificate.

The local quadratic gate has therefore been reduced from an unknown multiscale Hessian to:

$$
\boxed{
\text{finite interval LP}
+
\text{uniform remainder}.
}
$$

---

# 0. Recenter: What Exactly Did 09B Leave?

Paper 09B has established:

$$
\mathcal C_1
=
\{
(u_\star,v_\star,0)
\}
$$

as the first-order critical set within the reconstructed primary topology family.

Its numerical values:

$$
u_\star
=
5.157126245268359,
$$

$$
v_\star
=
-0.18922818990664067.
$$

and:

$$
w_\star=0.
$$

The primary first-area derivative is:

$$
\boxed{
\Phi_1^{\rm primary}
=
0.1838996392372776\,w.
}
$$

On the exact primary collapse branch, therefore:

$$
\boxed{
\Phi_1=0.
}
$$

What remains is:

$$
\boxed{
\Phi_2.
}
$$

---

# 1. Why Not Compute Second Order Directly Along Fixed $(u_\star,v_\star,0)$?

Taking only:

$$
u=u_\star,
\qquad
v=v_\star,
\qquad
w=0,
$$

produces merely one particular second-order path.

The actual epi-derivative must allow sequences:

$$
(u(\rho),v(\rho),w(\rho))
\rightarrow
(u_\star,v_\star,0).
$$

The most natural secondary corrections are therefore:

$$
u(\rho)
=
u_\star+\rho u_2,
$$

$$
v(\rho)
=
v_\star+\rho v_2,
$$

$$
w(\rho)
=
\rho w_2.
$$

Returning to physical variables:

$$
\boxed{
q
=
\rho u_\star
+
\rho^2u_2,
}
$$

$$
\boxed{
y_5
=
\rho v_\star
+
\rho^2v_2,
}
$$

$$
\boxed{
x_5
=
x_5^\star
+
\rho^2w_2.
}
$$

These are:

$$
\boxed{
\textbf{secondary blow-up coordinates}.
}
$$

---

# 2. Secondary Expansion of the $B_3$ Centre

From:

$$
\alpha_3
=
\pi-q
$$

and:

$$
q
=
\rho u_\star+\rho^2u_2
$$

we obtain:

$$
y_3
=
s\sin q.
$$

Thus:

$$
\boxed{
y_3
=
s u_\star\rho
+
s u_2\rho^2
+
O(\rho^3).
}
$$

On the other hand:

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

hence:

$$
\boxed{
x_3
=
x_3^\star
+
\frac12s u_\star^2\rho^2
+
O(\rho^3).
}
$$

Notably:

$$
u_2
$$

does not enter the second-order term of $x_3$, but does enter that of $y_3$.

---

# 3. $B_5$ secondary expansion

$B_5$'s rotation parameter itself remains:

$$
\rho.
$$

Its translation is:

$$
x_5
=
x_5^\star+\rho^2w_2,
$$

$$
y_5
=
\rho v_\star+\rho^2v_2.
$$

The second-scale variables:

$$
(u_2,v_2,w_2)
$$

therefore modify only the primary branch's second-order placement.

---

# 4. A General Secondary Transition Equation

Consider two support pieces:

$$
i,
\qquad
j
$$

intersecting at the zero-wall normal:

$$
\theta_0
$$

.

Define:

$$
F_{ij}(\theta,\rho)
=
h_i(\theta;\rho)
-
h_j(\theta;\rho).
$$

Primary collapse already gives:

$$
F_{ij}(\theta_0,0)=0.
$$

and selects a common first-order speed:

$$
\tau
$$

such that:

$$
F_{ij}
(
\theta_0+\rho\tau,
\rho
)
=
O(\rho^2).
$$

Now write:

$$
\boxed{
\theta_{ij}(\rho)
=
\theta_0
+
\rho\tau
+
\rho^2\eta_{ij}
+
o(\rho^2).
}
$$

The second-order Taylor coefficient then has the form:

$$
\boxed{
F_{ij}^{(2)}
=
b_{ij}\eta_{ij}
+
c_{ij}(u_2,v_2,w_2).
}
$$

where:

$$
b_{ij}
=
\partial_\theta
F_{ij}
(\theta_0,0),
$$

Here:

$$
c_{ij}
$$

is affine in the secondary variables.

Therefore:

$$
\boxed{
\eta_{ij}
=
-
\frac{
c_{ij}(u_2,v_2,w_2)
}{
b_{ij}
}.
}
$$

All second-scale transition coefficients are therefore affine.

---

# 5. Signed sliver width

If three pieces:

$$
A,
\quad
B,
\quad
C
$$

collapse at the primary scale to the same:

$$
(\theta_0,\tau),
$$

Define:

$$
\eta_{AB},
\qquad
\eta_{BC}.
$$

the middle piece:

$$
B
$$

has signed second-scale width:

$$
\boxed{
\delta_B
=
\eta_{BC}
-
\eta_{AB}.
}
$$

If:

$$
\delta_B>0,
$$

then:

$$
B
$$

forms an actual:

$$
O(\rho^2)
$$

winner interval：

$$
\boxed{
\Delta\theta_B
=
\rho^2\delta_B
+
o(\rho^2).
}
$$

If:

$$
\delta_B<0,
$$

the two pairwise roots occur in reversed order,

Thus:

$$
B
$$

does not form an interval on the upper support envelope.

The actual interval width is therefore:

$$
\boxed{
\Delta\theta_B
=
\rho^2[\delta_B]_+
+
o(\rho^2).
}
$$

---

# 6. Why Must 09A's Three Thin Slivers Be Corrected?

Paper 09A v0.2 uses the first Cycle-12 positive sample:

$$
\rho
\approx
0.0010069207.
$$

The effective primary velocity of that numerical optimizer sample is approximately:

$$
u
\approx
5.1967,
$$

not the value later derived analytically in Paper 09B:

$$
u_\star
\approx
5.157126.
$$

What 09A observed at the time:

1. top disk sliver；
2. left $B_5$ corner；
3. right $B_5$ corner；

is:

$$
\boxed{
\text{the secondary chart of a near-critical sample,}
}
$$

not:

$$
\boxed{
\text{the unique secondary chart at the exact primary critical centre.}
}
$$

---

# 7. Exact-Critical Replay Finds Two Previously Hidden Slivers

Repeating the sampling-free support envelope at:

$$
(u_\star,v_\star,0)
$$

:

besides the top-disk sliver,

we also find:

1. near $\theta_\star$: an extremely thin $B_5$ arc;
2. near $2\pi-\theta_\star$: an extremely thin $B_5$ arc.

By contrast:

$$
\pi-\theta_\star,
\qquad
\pi+\theta_\star
$$

the $B_5$ corner intervals near these angles do not enter the envelope at the secondary centre.

The actual classification therefore concerns not three fixed intervals,

but:

$$
\boxed{
5\text{ potential secondary slivers}.
}
$$

---

# 8. Five Potential Slivers

Define:

## $S_1$ — $\theta_\star$ wall

winner candidate：

$$
B_5\text{ arc }v_3.
$$

local ordering：

$$
B_3\text{ corner }v_0
\rightarrow
B_5\text{ arc }v_3
\rightarrow
D.
$$

## $S_2$ — top wall

winner candidate：

$$
D.
$$

local ordering：

$$
B_5\text{ corner }v_1
\rightarrow
D
\rightarrow
B_3\text{ arc }v_2.
$$

## $S_3$ — $\pi-\theta_\star$ wall

winner candidate：

$$
B_5\text{ corner }v_2.
$$

local ordering：

$$
B_3\text{ arc }v_0
\rightarrow
B_5\text{ corner }v_2
\rightarrow
D.
$$

## $S_4$ — $\pi+\theta_\star$ wall

winner candidate：

$$
B_5\text{ corner }v_3.
$$

local ordering：

$$
D
\rightarrow
B_5\text{ corner }v_3
\rightarrow
B_3\text{ arc }v_0.
$$

## $S_5$ — $2\pi-\theta_\star$ wall

winner candidate：

$$
B_5\text{ arc }v_2.
$$

local ordering：

$$
D
\rightarrow
B_5\text{ arc }v_2
\rightarrow
B_3\text{ corner }v_0.
$$

---

# 9. Exact-center observed scaling

Along the exact-primary path with:

$$
u_2=v_2=w_2=0
$$

high-precision replay shows:

## $\theta_\star$ $B_5$ arc

$$
\frac{
\Delta\theta_{S_1}
}{
\rho^2
}
\rightarrow
2.72111\ldots
$$

## top disk

$$
\frac{
\Delta\theta_{S_2}
}{
\rho^2
}
\rightarrow
1.78711\ldots
$$

## left $B_5$ corner

signed width：

$$
\delta_{S_3}
\approx
-2.72111,
$$

Thus:

$$
\boxed{
S_3\text{ absent at the exact secondary center}.
}
$$

## right $B_5$ corner

Likewise absent.

## end $B_5$ arc

$$
\frac{
\Delta\theta_{S_5}
}{
\rho^2
}
\rightarrow
2.72111\ldots
$$

The actual upper-envelope secondary structure at the exact primary centre is therefore:

$$
\boxed{
S_1
+
S_2
+
S_5.
}
$$

---

# 10. Reflection / cyclic pairing

Second-scale replay shows two signed pairs:

$$
S_1
\leftrightarrow
S_4,
$$

and:

$$
S_3
\leftrightarrow
S_5.
$$

For each pair,

when the signed width on one side is positive,

the other side's is negative,

and conversely.

Two independent signed variables therefore suffice:

$$
\boxed{
\delta_+
}
$$

and:

$$
\boxed{
\delta_-.
}
$$

The actual total width contribution of each pair is represented by:

$$
|\delta_+|,
\qquad
|\delta_-|
$$

.

The top disk separately has:

$$
\boxed{
\delta_T.
}
$$

---

# 11. Secondary signed-width affine model

High-precision nested-blow-up replay gives nominal affine asymptotics:

$$
\boxed{
\delta_+
\approx
2.721112
-
0.02682772\,u_2
+
7.435229\,v_2
+
15.234440\,w_2.
}
$$

The other independent pair can be written as:

$$
\boxed{
\delta_-
\approx
-2.721112
-
0.02682772\,u_2
+
7.435229\,v_2
-
15.234440\,w_2.
}
$$

top wall：

$$
\boxed{
\delta_T
\approx
1.787108
-
0.02848341\,u_2
-
5.487547\,v_2.
}
$$

In the second-order limit, the coefficient of:

$$
w_2
$$

in:

$$
\delta_T
$$

vanishes.

---

# 12. These Three Affine Forms Replace the Ambiguity of $\delta_1,\ldots,\delta_k$

Paper 09A v0.2 could write only:

$$
\delta_1,\ldots,\delta_k.
$$

Paper 09C now compresses secondary topology to:

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

that is:

$$
\boxed{
3\text{ signed second-scale coordinates}
}
$$

control:

$$
\boxed{
5\text{ potential slivers}.
}
$$

This is a finite secondary atlas.

---

# 13. Area functional

For piecewise support:

$$
h(\theta),
$$

the area is:

$$
\boxed{
A
=
\frac12
\int_0^{2\pi}
\left(
h^2
-
(h')^2
\right)d\theta.
}
$$

Define:

$$
\boxed{
f
=
\frac12
\left(
h^2-(h')^2
\right).
}
$$

If a transition has:

$$
\theta_j
=
\theta_j^0
+
\rho\tau_j
+
\rho^2\eta_j
+
\cdots,
$$

the second-scale endpoint contribution is:

$$
\boxed{
\eta_j
\left(
f_{\rm left}^0
-
f_{\rm right}^0
\right).
}
$$

---

# 14. Primary 15-piece second coefficient

First ignore all second-scale third-body slivers,

letting each latent triple wall transition directly between its two primary pieces at the secondary scale.

This gives the primary 15-piece second coefficient:

$$
\boxed{
\Phi_2^{(15)}
=
c_0
+
\gamma_w w_2.
}
$$

where:

$$
\boxed{
c_0
\approx
0.258433713,
}
$$

and:

$$
\boxed{
\gamma_w
=
0.1838996392372776.
}
$$

Most importantly:

$$
u_2,
\qquad
v_2
$$

again cancel from the primary interior second coefficient.

They affect the following only through secondary topology widths:

$$
\Phi_2.
$$

---

# 15. Why Does $\gamma_w$ Reappear?

Paper 09B has:

$$
\Phi_1^{\rm primary}
=
\gamma_w w.
$$

Now:

$$
w
=
\rho w_2.
$$

Thus:

$$
\rho\Phi_1
=
\rho\gamma_w(\rho w_2)
=
\rho^2\gamma_w w_2.
$$

The second epi-derivative therefore naturally includes:

$$
\boxed{
\gamma_w w_2.
}
$$

This is not a new coefficient.

It is the lift of Paper-09B's first derivative through the secondary correction.

---

# 16. Area Correction from Inserting a Secondary Sliver

Consider:

$$
A
\rightarrow
B
\rightarrow
C.
$$

Let:

$$
\eta_{AB},
\qquad
\eta_{BC}
$$

be the two second-scale transition locations,

and let:

$$
\eta_{AC}
$$

be the direct transition when $B$ is omitted.

actual endpoint contribution：

$$
\eta_{AB}
(f_A-f_B)
+
\eta_{BC}
(f_B-f_C).
$$

primary direct contribution：

$$
\eta_{AC}
(f_A-f_C).
$$

The difference reduces to:

$$
\boxed{
\lambda_B
[
\delta_B
]_+,
}
$$

where:

$$
\delta_B
=
\eta_{BC}
-
\eta_{AB}.
$$

---

# 17. Secondary sliver area coefficient

Let the angular support slopes be:

$$
s_1
=
h_A'-h_B',
$$

$$
s_2
=
h_B'-h_C'.
$$

From:

$$
F_{AC}
=
F_{AB}
+
F_{BC}
$$

we obtain the direct-root relation.

Finally:

$$
\boxed{
\lambda_B
=
(f_B-f_C)
-
\frac{s_2}{s_1+s_2}
(f_A-f_C).
}
$$

This is determined entirely by zero-wall geometry.

---

# 18. Side-wall coefficient

For the four:

$$
\theta_\star,
\quad
\pi-\theta_\star,
\quad
\pi+\theta_\star,
\quad
2\pi-\theta_\star
$$

latent triple walls,

nominal zero-wall evaluation gives the same:

$$
\boxed{
\lambda_S
\approx
0.00719758120147
>
0.
}
$$

---

# 19. Top-wall coefficient

At the wall:

$$
\pi/2
$$

,

we obtain:

$$
\boxed{
\lambda_T
\approx
0.02821424417983
>
0.
}
$$

All actual second-scale slivers generated therefore provide an area:

$$
\boxed{
\text{nonnegative correction}.
}
$$

---

# 20. Explicit $\Phi_2$

Combine the primary 15-piece term with all potential slivers:

$$
\boxed{
\Phi_2(u_2,v_2,w_2)
=
c_0
+
\gamma_w w_2
+
\lambda_S
\left(
|\delta_+|
+
|\delta_-|
\right)
+
\lambda_T
[\delta_T]_+.
}
$$

This is Paper 09C's core reduction.

---

# 21. Why an Absolute Value?

Only one member of each side pair can become a positive-width upper-envelope sliver at the second scale.

If:

$$
\delta_+>0,
$$

one side of the pair appears.

If:

$$
\delta_+<0,
$$

the corresponding sliver at the paired wall appears.

The total side contribution is therefore:

$$
\lambda_S
\left(
[\delta_+]_+
+
[-\delta_+]_+
\right)
=
\lambda_S|\delta_+|.
$$

Likewise for:

$$
\delta_-.
$$

---

# 22. Why Is the Top a Positive Part Rather Than an Absolute Value?

The middle piece at the top wall is the disk.

If:

$$
\delta_T>0,
$$

the disk forms an actual secondary interval.

If:

$$
\delta_T<0,
$$

the disk does not form an interval,

and the two primary pieces transition directly.

Therefore:

$$
\boxed{
\lambda_T[\delta_T]_+.
}
$$

---

# 23. $\Phi_2$ Is Convex Piecewise-Linear

Because:

$$
c_0+\gamma_w w_2
$$

is affine,

and:

$$
|\delta_+|,
\qquad
|\delta_-|,
\qquad
[\delta_T]_+
$$

are all convex piecewise-linear,

Thus:

$$
\boxed{
\Phi_2
\text{ is convex piecewise-linear in }
(u_2,v_2,w_2).
}
$$

there is no need to guess a Hessian further.

---

# 24. Second-Order Minimization Reduces to an LP

Introduce auxiliary variables:

$$
t_+,
\quad
t_-,
\quad
t_T
$$

satisfying:

$$
t_+
\ge
\delta_+,
$$

$$
t_+
\ge
-\delta_+,
$$

$$
t_-
\ge
\delta_-,
$$

$$
t_-
\ge
-\delta_-,
$$

$$
t_T
\ge
\delta_T,
$$

$$
t_T
\ge
0.
$$

Minimize:

$$
\boxed{
c_0
+
\gamma_w w_2
+
\lambda_S(t_++t_-)
+
\lambda_Tt_T.
}
$$

This is a small LP with:

$$
\boxed{
3+3
}
$$

variables.

---

# 25. Nominal LP minimizer

Solving this LP at the current high-precision coefficient centre,

the minimum lies at:

$$
\boxed{
\delta_+
=
\delta_-
=
\delta_T
=
0.
}
$$

The three second-scale signed gaps therefore collapse simultaneously.

The solution gives:

$$
\boxed{
u_2^\star
\approx
37.01279,
}
$$

$$
\boxed{
v_2^\star
\approx
0.133549,
}
$$

$$
\boxed{
w_2^\star
\approx
-0.178616.
}
$$

---

# 26. Why Is $u_2^\star\approx37$ Not a Large Primary Drift?

The actual primary variable is:

$$
u(\rho)
=
u_\star
+
\rho u_2.
$$

For example, when:

$$
\rho=10^{-4}
$$

:

the secondary correction is only:

$$
\rho u_2
\approx
0.0037.
$$

As:

$$
\rho\downarrow0,
$$

we still have:

$$
u(\rho)
\rightarrow
u_\star.
$$

This is therefore an admissible epi-derivative sequence.

---

# 27. Nominal minimum coefficient

At the secondary minimizer,

all sliver penalties collapse simultaneously at order:

$$
\rho^2
$$

.

Therefore:

$$
\Phi_2^{\min}
=
c_0
+
\gamma_w w_2^\star.
$$

Substitute:

$$
c_0
\approx
0.258433713,
$$

$$
w_2^\star
\approx
-0.178616,
$$

to obtain:

$$
\boxed{
c_\star^{\rm nominal}
\approx
0.22558634.
}
$$

---

# 28. Relationship with Paper-09A's Numerical $\approx0.22$

Paper 09A v0.2 observed from the positive profile:

$$
\frac{
m(\rho)-m(0)
}{
\rho^2
}
\approx
0.22.
$$

Paper 09C now obtains:

$$
\boxed{
0.22558634
}
$$

as the explicit second-scale LP's nominal minimum.

The two agree at the level of:

$$
\boxed{
\text{the same quadratic-growth regime}.
}
$$

This suggests that 09A's numerical branch is very likely already approaching the actual secondary optimum.

---

# 29. Fresh finite-$\rho$ replay

Using the secondary LP centre:

$$
(u_2^\star,v_2^\star,w_2^\star)
$$

rebuild the exact support envelope,

obtaining:

$$
\rho=10^{-4}:
\quad
\frac{A(\rho)-A_0}{\rho^2}
\approx
0.2255848,
$$

$$
\rho=5\times10^{-5}:
\quad
\approx
0.2255856,
$$

$$
\rho=2\times10^{-5}:
\quad
\approx
0.2255860,
$$

$$
\rho=10^{-5}:
\quad
\approx
0.2255862.
$$

The numerical sequence therefore converges stably toward:

$$
\boxed{
0.2255863\ldots
}
$$

.

Status：

`HIGH-PRECISION NUMERICAL REPLAY / NOT INTERVAL PROOF`

---

# 30. Certificate target margin

Paper-08 / 09A target：

$$
\boxed{
\Phi_2
\ge
0.1.
}
$$

nominal result：

$$
\boxed{
c_\star
\approx
0.22558634.
}
$$

The margin is therefore:

$$
\boxed{
m_{\Phi_2}
\approx
0.12558634.
}
$$

This margin is very large relative to the current numerical uncertainty of the coefficients.

The local gate is therefore no longer:

$$
\boxed{
\text{barely positive second variation}.
}
$$

but:

$$
\boxed{
\text{a finite LP with substantial nominal safety margin}.
}
$$

---

# 31. What Does Paper 09C Actually Complete?

This paper does not claim that:

$$
\boxed{
\Phi_2\ge0.1
}
$$

is publication-certified.

What this paper completes is the reduction from:

$$
\boxed{
\text{unknown multiscale second-order problem}
}
$$

to:

$$
\boxed{
\text{explicit finite convex LP}.
}
$$

This is structural closure.

---

# 32. CERT-09C.1 — Five-Sliver Exhaustion

Interval support equations must prove that:

for sufficiently small:

$$
\rho>0
$$

and within the critical neighbourhood of the primary-velocity box,

all:

$$
o(\rho)
$$

winner intervals can come only from the five latent walls in this paper.

There must be no sixth hidden secondary sliver.

---

# 33. CERT-09C.2 — Secondary Affine Widths

For:

$$
\delta_+,
\qquad
\delta_-,
\qquad
\delta_T
$$

construct an interval affine enclosure:

$$
\delta_j
\in
a_j
+
b_{ju}u_2
+
b_{jv}v_2
+
b_{jw}w_2
+
[-\varepsilon_j,\varepsilon_j].
$$

---

# 34. CERT-09C.3 — Area Coefficients

Interval-enclose：

$$
c_0,
$$

$$
\gamma_w,
$$

$$
\lambda_S,
$$

$$
\lambda_T.
$$

and prove:

$$
\boxed{
\lambda_S>0,
\qquad
\lambda_T>0.
}
$$

---

# 35. CERT-09C.4 — Interval LP

Using outward-rounded coefficient boxes,

compute a rigorous lower bound:

$$
\boxed{
\underline c_\star
>
0.1.
}
$$

Since nominally:

$$
c_\star
\approx
0.22558634,
$$

substantial interval uncertainty is admissible.

---

# 36. CERT-09C.5 — Uniform remainder

The second-order epi coefficient itself controls only:

$$
\rho\downarrow0.
$$

To promote this to Paper-08's finite local gate:

$$
0\le\rho\le0.003,
$$

a remainder is still needed:

$$
A(\rho)
=
A_0
+
\rho^2\Phi_2
+
R_3(\rho).
$$

Prove:

$$
\boxed{
|R_3(\rho)|
\le
C_3\rho^3
}
$$

or, more generally:

$$
o(\rho^2)
$$

uniform bound。

---

# 37. A Local Radius Split Can Reduce Remainder Pressure

The second-order expansion need not be extended directly to:

$$
0.003.
$$

We can select:

$$
0<\rho\le\rho_{\rm micro},
$$

and use the 09C interval LP + remainder.

Then, for:

$$
\rho_{\rm micro}
\le\rho\le0.003
$$

use a finite interval-envelope certificate.

Therefore:

$$
\boxed{
\text{asymptotic calculus}
+
\text{finite interval bridge}
}
$$

is usually more robust than a single Taylor bound.

---

# 38. CERT-09B.4 Still Cannot Be Skipped

Paper 09C performs second-scale analysis on:

$$
\mathcal C_1
$$

.

Paper-09B's following certificate still must be completed:

$$
\boxed{
\text{full neighboring first-order chart nonnegativity}.
}
$$

Every first-order direction not satisfying:

$$
E_T=E_L=E_R=0
$$

:

must either:

1. enter an explicitly declared neighbouring chart;
2. or produce strictly positive $\Phi_1$.

---

# 39. The Local Proof Stack Is Therefore Now Finite

The complete local stack:

$$
\boxed{
\text{09B first-order chart isolation}
}
$$

$$
+
$$

$$
\boxed{
\text{09C five-sliver exhaustion}
}
$$

$$
+
$$

$$
\boxed{
\text{09C interval LP lower bound}
}
$$

$$
+
$$

$$
\boxed{
\text{uniform finite-}\rho\text{ remainder / bridge}.
}
$$

There is no new infinite topology family.

---

# 40. Why Is the Secondary Chart Count Still Finite?

The zero barrier has only finitely many support pieces.

Secondary intervals can be generated only by:

- pairwise support equalities；
- intrinsic piece boundaries；
- latent triple walls；

.

Paper 09C has reduced the second-scale candidates in the critical neighbourhood to:

$$
\boxed{
5.
}
$$

The signed widths are further reduced to:

$$
\boxed{
3.
}
$$

Topological complexity has therefore been made highly finite.

---

# 41. Relationship Between 09C and BCODR-LUC-07

BCODR-LUC-07 has clarified:

$$
\boxed{
\text{IMBG meeting point}
\neq
\text{shape minimizer}.
}
$$

09C therefore performs second-scale analysis along:

$$
\boxed{
\text{Paper-09B outward critical branch}
}
$$

.

BCODR supplies only:

$$
\boxed{
\text{this outward branch also participates in the surviving root-131 certificate interaction}.
}
$$

The two points are not conflated.

---

# 42. Root-131 Local Candidate Alignment Can Still Run Independently Locally

CERT-07.1–07.8 still belong to the:

$$
\boxed{
\text{candidate / germ / mirror alignment lane}.
}
$$

They do not block 09C's derivation of the second epi-derivative.

Only if support-signature alignment later reports PASS

can:

$$
\text{Cycle-17 candidate}
$$

and:

$$
\text{09B/09C analytic branch}
$$

be promoted to one structural shape spine.

---

# 43. The Tail Gate Is Unchanged

Paper 09C treats only the local positive-rotation gate on:

$$
0\le\rho\le0.003
$$

.

For:

$$
0.003
\le
\rho
\le
\pi/120
$$

the finite-certificate / tight-reference tail architecture still applies.

---

# 44. The Global Lebesgue Theorem Status Does Not Automatically Change with 09C's Nominal LP

Explicitly:

$$
\boxed{
a_{\rm Leb}\ge0.835
}
$$

is still not established by this paper alone.

09C addresses:

$$
\boxed{
\text{root-6217 constrained barrier local rotation obligation}.
}
$$

AMRAL's global finite-closure residuals still require separate completion.

---

# 45. Implications of 09C for the Final Stress Test

Once the 09C interval certificate is complete,

the root-6217 local shape atlas will no longer treat the neighbourhood of:

$$
\rho=0
$$

as merely looking numerically minimal.

Instead, it will have:

$$
\boxed{
\text{first-order critical cone}
+
\text{second-order finite topology atlas}
+
\text{positive certified LP}.
}
$$

Only then is it suitable for freezing as part of the final adversarial geometry atlas.

---

# 46. Adversarial interpretation

Future 3D / SOB / flexible stress testing should attack not:

$$
\boxed{
\Phi_2^{\rm nominal}.
}
$$

but:

> Does an admissible geometric mechanism exist outside this five-sliver atlas?

If a sixth secondary class is found,

09C closure immediately reopens.

This is fully consistent with Adversarial Geometric Closure.

---

# 47. Current status

```text
PRIMARY CRITICAL VELOCITY:
    ANALYTICALLY DERIVED

PRIMARY PHI1:
    gamma_w * w

PRIMARY CRITICAL SET:
    SINGLETON INSIDE RECONSTRUCTED FAMILY

SECONDARY PHYSICAL CORRECTIONS:
    (u2, v2, w2)

POTENTIAL SECONDARY SLIVERS:
    5

INDEPENDENT SIGNED WIDTHS:
    3

SECONDARY WIDTH MODEL:
    AFFINE / NUMERICALLY REPLAYED

PRIMARY-15 PHI2:
    c0 + gamma_w*w2

FULL PHI2:
    CONVEX PIECEWISE LINEAR

NOMINAL LP MIN:
    ~0.22558634

TARGET:
    0.1

NOMINAL MARGIN:
    ~0.12558634

INTERVAL SECONDARY LP:
    OPEN

UNIFORM REMAINDER:
    OPEN

FULL FIRST-ORDER MULTICHART CERT:
    OPEN

GLOBAL LEBESGUE LOWER BOUND:
    NOT CERTIFIED BY THIS PAPER
```

---

# 48. Next analytic / certification node

The next node need not invent new local calculus.

It should directly undertake:

$$
\boxed{
\textbf{LESR Paper 09D — Interval Secondary LP Certificate and Uniform Local Remainder}.
}
$$

Goals:

1. interval-isolate five-sliver atlas；
2. certify three affine signed widths；
3. certify $\lambda_S,\lambda_T>0$；
4. solve interval LP；
5. prove：
   $$
   \underline c_\star>0.1;
   $$
6. establish a uniform micro-radius remainder;
7. connect a finite interval bridge to:
   $$
   \rho=0.003.
   $$

---

# 49. Local Handoff Priorities

Local secondary verification can independently recompute:

1. zero barrier support constants；
2. Paper-09B $u_\star,v_\star,w_\star$；
3. five latent wall pairwise roots；
4. $\rho^{-2}$ sliver-width convergence；
5. affine coefficient fitting / symbolic differentiation；
6. primary 15-piece $c_0$；
7. $\lambda_S,\lambda_T$；
8. LP center；
9. exact-envelope finite-$\rho$ convergence；
10. intervalized version。

Do not treat this paper's nominal JSON directly as the final proof certificate.

---

# 50. Conclusion

Paper 09A found:

$$
\boxed{
\text{ordinary Hessian picture is wrong}.
}
$$

Paper 09B found:

$$
\boxed{
\text{primary topology selects a unique first-order critical velocity}.
}
$$

Paper 09C now further obtains:

$$
\boxed{
\text{secondary topology is finite and polyhedral}.
}
$$

The final second epi-derivative is no longer an unknown complex function, but:

$$
\boxed{
\Phi_2
=
c_0
+
\gamma_w w_2
+
\lambda_S
(
|\delta_+|+|\delta_-|
)
+
\lambda_T[\delta_T]_+.
}
$$

where:

$$
\delta_+,
\qquad
\delta_-,
\qquad
\delta_T
$$

are all affine forms of:

$$
(u_2,v_2,w_2)
$$

.

Therefore:

$$
\boxed{
\text{second-order local geometry}
\rightarrow
\text{finite convex LP}.
}
$$

nominal minimum：

$$
\boxed{
c_\star
\approx
0.22558634
>
0.1.
}
$$

This is not a publication theorem,

but reduces the final local analytic obstruction from:

$$
\boxed{
\text{multi-scale topology uncertainty}
}
$$

to:

$$
\boxed{
\text{interval coefficient audit}
+
\text{tiny LP}
+
\text{uniform remainder}.
}
$$

The root-6217 local positive-rotation gate has therefore genuinely entered the:

$$
\boxed{
\textbf{certificate engineering phase}.
}
$$

---

# Canonical Source Declaration

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

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

This paper strictly distinguishes:

- analytically derived primary structure；
- high-precision second-scale numerical replay；
- explicit finite $\Phi_2$ reduction；
- nominal LP solution；
- publication-level interval obligations。

This paper does not claim that the global Lebesgue universal covering problem is solved.

===END===