# NS × X 積分 × 24/72 範式實戰
## Round 64 — Pure Continuous Validated Viscosity Half-Line / Uniform Positive Adjoint Coefficient

- 日期：2026-08-18
- 版本：v0.1
- 狀態：Proof-Route Experiment / Continuous-Only Validated Parameter Branch
- canonical source：UTF-8 Markdown
- canonical math delimiters：inline `$...$`；display `$$...$$`
- 前一輪：`NS_X72_Round63_PureContinuous_FastDifferenceSchur_SymmetrizedSlowGauge_v0.1_2026-08-18.md`

## 本輪目標

Round 63 已嚴格消去 fast sector，剩下 slow Jost/Riccati selection。本輪不再先追求完整
$$
\nu\to0^+
$$
singular matching，而採兩段策略：

1. 對 compact viscosity range 做 finite-core + infinite-tail 的 a posteriori validated enclosure；
2. 對 sufficiently large viscosity 直接使用 full adjoint recurrence 的 global contraction。

目標是第一次將一整段正 viscosity 參數線升級成 theorem，而不是 parameter scan。

---

# 1. Main result

For the two source-hidden Floquet fibres

$$
\boxed{
K_-=\sqrt{17}-3,
\qquad
K_+=\sqrt{17}+3,
}
\tag{1.1}
$$

let
$$
\psi_+(\nu)
$$
be the canonical reflection-even，$\mathcal C$-even adjoint mode normalized by

$$
\boxed{
\psi_+(0)=1,
\qquad
\psi_+(1)=0.
}
\tag{1.2}
$$

Write

$$
\boxed{
\psi_+(3)=ia_3(\nu).
}
\tag{1.3}
$$

Round 64 proves

$$
\boxed{
a_{3,\pm}(\nu)>0
\qquad
\forall
\nu\ge10^{-4}.
}
\tag{1.4}
$$

Equivalently, since

$$
\psi_n=i^n u_n,
$$

$$
\boxed{
u_3(\nu)<0
\qquad
\forall
\nu\ge10^{-4}.
}
\tag{1.5}
$$

命名：

$$
\boxed{
\textbf{Validated Viscosity Half-Line Positivity Theorem}.
}
$$

---

# 2. Consequence for the hidden-rescue route

Round 55 proved the exact compatibility reduction

$$
\boxed{
\langle
\psi_+,
g
\rangle
=
g_0(\nu)
+
a_3(\nu)
G_{-3}.
}
\tag{2.1}
$$

At both source fibres,

$$
\boxed{
\operatorname{sign}
g_0(\nu)
=
\operatorname{sign}
G_{-3}
}
\tag{2.2}
$$

for every

$$
\nu>0.
$$

Therefore (1.4) implies

$$
\boxed{
\langle
\psi_+,
g
\rangle
\ne0
\qquad
\forall
\nu\ge10^{-4}.
}
\tag{2.3}
$$

Hence the two non-Beltrami $\sqrt{17}$ source-hidden circles cannot be integrated into full second-order state/source-locked analytic curves for any

$$
\boxed{
\nu\ge10^{-4}.
}
\tag{2.4}
$$

This extends Round 56 from the single normalized slice

$$
\nu=1
$$

to an entire positive-viscosity half-line above a small explicit threshold.

---

# 3. Canonical adjoint recurrence

The real $\mathcal C$-even recurrence is

$$
\boxed{
-
A_{-2}^{(n)}
u_{n-2}
+
A_0^{(n)}
u_n
-
\nu b_nu_{n+1}
-
A_2^{(n)}
u_{n+2}
+
A_4^{(n)}
u_{n+4}
=
0,
}
\tag{3.1}
$$

with

$$
u_0=1,
\qquad
u_1=0,
$$

and reflection

$$
u_{-n}=(-1)^nu_n.
$$

The same-parity coefficients are independent of viscosity；only the cross-parity coefficient is linear in

$$
\nu.
$$

---

# 4. Proof split

The proof is divided at

$$
\boxed{
\nu_L=0.7.
}
\tag{4.1}
$$

## Regime A — validated finite-core / infinite-tail enclosure

$$
\boxed{
10^{-4}
\le
\nu
\le
0.7.
}
\tag{4.2}
$$

## Regime B — global full-sequence contraction

$$
\boxed{
\nu
\ge
0.7.
}
\tag{4.3}
$$

The two arguments overlap exactly at the split and require no continuity assumption across it.

---

# 5. Finite-core decomposition

Fix an integer cutoff

$$
N.
$$

Let

$$
x=
(
u_2,\ldots,u_N
)^T
$$

and let the first three tail values be

$$
y=
(
u_{N+1},
u_{N+2},
u_{N+3}
)^T.
$$

Using the equations

$$
n=1,\ldots,N-1,
$$

the finite core has the exact affine form

$$
\boxed{
M_N(\nu)x
=
r_N
-
C_Ny.
}
\tag{5.1}
$$

The matrix depends affinely on viscosity：

$$
\boxed{
M_N(\nu)
=
M_{N,0}
+
\nu M_{N,1}.
}
\tag{5.2}
$$

The tail-coupling matrix

$$
C_N
$$

is viscosity-independent.

---

# 6. A posteriori inverse certificate on a viscosity chunk

Let a chunk be

$$
I=[\nu_-,\nu_+]
$$

with center

$$
\nu_c
$$

and half-width

$$
h.
$$

Let

$$
R
$$

be a floating approximate inverse of

$$
M_c=M_N(\nu_c).
$$

The verification computes with outward interval coefficient evaluation

$$
\boxed{
\eta
=
\|
I-RM_c
\|_\infty.
}
\tag{6.1}
$$

Every certified chunk satisfies

$$
\boxed{
\eta\ll1.
}
\tag{6.2}
$$

Therefore

$$
\boxed{
M_c^{-1}
=
(
I-E
)^{-1}R,
\qquad
\|E\|_\infty\le\eta,
}
\tag{6.3}
$$

and

$$
\boxed{
\|M_c^{-1}\|_\infty
\le
\frac{
\|R\|_\infty
}{
1-\eta
}.
}
\tag{6.4}
$$

---

# 7. Parameter perturbation inside one chunk

Define

$$
\boxed{
B_c
=
M_c^{-1}M_{N,1}.
}
\tag{7.1}
$$

The approximate product

$$
RM_{N,1}
$$

is evaluated against outward interval coefficients，giving a certified bound

$$
\boxed{
\beta
\ge
\|B_c\|_\infty.
}
\tag{7.2}
$$

For

$$
|\nu-\nu_c|\le h,
$$

set

$$
\boxed{
\rho=h\beta.
}
\tag{7.3}
$$

All chunks satisfy

$$
\boxed{
\rho<1.
}
\tag{7.4}
$$

Hence

$$
\boxed{
M_N(\nu)^{-1}
=
(
I+
(\nu-\nu_c)B_c
)^{-1}
M_c^{-1}.
}
\tag{7.5}
$$

This gives explicit row-wise perturbation bounds for：

- the zero-tail core solution；
- the core-to-tail map；
- the boundary core values entering the first tail equations；
- the central coefficient：
  $$
  u_3.
  $$

---

# 8. Infinite tail contraction

Solve the adjoint recurrence for

$$
u_{n+1}.
$$

The row coefficient sum is

$$
\boxed{
q_n(K,\nu)
=
\frac{
-
A_{-2}^{(n)}
+
A_0^{(n)}
+
A_2^{(n)}
-
A_4^{(n)}
}{
-\nu b_n
}.
}
\tag{8.1}
$$

Round 56 rigorously proved：

1. for each source fibre,
   $$
   q_n(K,1)
   $$
   decreases for
   $$
   n\ge6;
   $$

2. viscosity scaling is exact：
   $$
   \boxed{
   q_n(K,\nu)
   =
   q_n(K,1)/\nu.
   }
   \tag{8.2}
   $$

Thus a cutoff chosen at the lower end of a viscosity regime gives a uniform tail bound across the entire regime.

---

# 9. Tail/core feedback enclosure

Let

$$
L_N(\nu)
=
-
M_N(\nu)^{-1}
C_N.
$$

For the first tail equations，some nominal tail inputs pass through the last few core rows of

$$
L_N.
$$

Let

$$
L_{\rm bd}
$$

be a certified upper bound on those boundary row norms，and let

$$
X_{\rm bd}
$$

bound the corresponding zero-tail core values.

If

$$
q_N
$$

is the raw recurrence row bound at the lower viscosity endpoint，then the full affine tail map obeys

$$
\boxed{
\operatorname{Lip}T_{\rm tail}
\le
\widehat q
=
q_N
(
1+L_{\rm bd}
).
}
\tag{9.1}
$$

Every validated chunk satisfies

$$
\boxed{
\widehat q<1.
}
\tag{9.2}
$$

The tail fixed point therefore satisfies

$$
\boxed{
\|y^\ast\|_\infty
\le
\frac{
q_NX_{\rm bd}
}{
1-\widehat q
}.
}
\tag{9.3}
$$

Finally，if

$$
L_3
$$

bounds the $u_3$ row of the core-to-tail map，

$$
\boxed{
u_3
\le
u_{3,\rm core}^{\rm upper}
+
L_3
\|y^\ast\|_\infty.
}
\tag{9.4}
$$

This is the quantity certified negative chunk by chunk.

---

# 10. Low-viscosity validated regime

For

$$
\boxed{
10^{-4}
\le
\nu
\le
10^{-3},
}
\tag{10.1}
$$

use

$$
\boxed{
N=250.
}
\tag{10.2}
$$

The interval is divided geometrically with ratio

$$
\boxed{
3/2.
}
\tag{10.3}
$$

There are only six chunks per fibre.

The worst certified chunk is the first：

$$
\boxed{
[10^{-4},1.5\times10^{-4}].
}
\tag{10.4}
$$

### Small fibre

$$
\boxed{
u_{3,-}
<
-2.1681699396530456\times10^{-4}.
}
\tag{10.5}
$$

### Large fibre

$$
\boxed{
u_{3,+}
<
-3.0490435797040873\times10^{-4}.
}
\tag{10.6}
$$

Therefore

$$
\boxed{
a_{3,\pm}(\nu)>0
}
$$

through the entire low regime.

---

# 11. Intermediate validated regime

For

$$
\boxed{
10^{-3}
\le
\nu
\le
0.7,
}
\tag{11.1}
$$

the stronger viscous tail allows the much smaller cutoff

$$
\boxed{
N=80.
}
\tag{11.2}
$$

Again use geometric ratio

$$
3/2.
$$

There are seventeen chunks per fibre.

The worst chunk is

$$
\boxed{
[10^{-3},1.5\times10^{-3}].
}
\tag{11.3}
$$

### Small fibre

$$
\boxed{
u_{3,-}
<
-2.4009827134153527\times10^{-3}.
}
\tag{11.4}
$$

### Large fibre

$$
\boxed{
u_{3,+}
<
-3.0331757703739824\times10^{-3}.
}
\tag{11.5}
$$

Hence

$$
\boxed{
a_{3,\pm}(\nu)>0
}
$$

through the full intermediate regime.

---

# 12. Numerical rigor layer of the finite-core certificate

For each chunk：

1. the inverse
   $$
   R
   $$
   is only used as an arbitrary approximate inverse；

2. all decisive residuals
   $$
   I-RM_c
   $$
   and
   $$
   r-M_c\widetilde x
   $$
   are recomputed with high-precision outward interval evaluation of the exact
   $$
   \mathbb Q(\sqrt{17})
   $$
   coefficient formulas；

3. Neumann inequalities then certify the true inverse and parameter perturbation bounds；

4. the infinite tail is not truncated as a proof step；it is controlled by the exact contraction estimate inherited from Round 56。

Thus the sign conclusion does not rely on convergence of a finite Galerkin cutoff.

---

# 13. Global contraction for large viscosity

For the remaining half-line we do not need chunk continuation.

Let

$$
w=
(
u_2,u_3,u_4,\ldots
)
\in\ell^\infty.
$$

Solving each recurrence equation for

$$
u_{n+1}
$$

defines an affine map

$$
\boxed{
w=T_\nu w+f_\nu.
}
\tag{13.1}
$$

The only nonzero affine forcing comes from

$$
u_0=1
$$

in the

$$
n=2
$$

equation.

---

# 14. Uniform full-sequence Lipschitz bound

At viscosity one define

$$
q_n^\ast
=
q_n(K,1).
$$

For both fibres，exact algebraic evaluation gives

$$
\boxed{
q_n^\ast<0.53
\qquad
n=1,\ldots,6.
}
\tag{14.1}
$$

Round 56 supplies monotonic decrease for

$$
n\ge6.
$$

Therefore

$$
\boxed{
\sup_{n\ge1}
q_n^\ast
<
0.53.
}
\tag{14.2}
$$

By exact viscosity scaling：

$$
\boxed{
\|T_\nu\|
<
\frac{
0.53
}{
\nu
}.
}
\tag{14.3}
$$

Hence for

$$
\nu\ge0.7,
$$

$$
\boxed{
\|T_\nu\|
<
\frac{53}{70}
<1.
}
\tag{14.4}
$$

So the entire full sequence is selected by one global Banach contraction.

---

# 15. Large-viscosity forcing coefficient

The

$$
n=2
$$

equation has affine forcing

$$
\boxed{
-\frac{
c_\infty
}{
\nu
},
}
\tag{15.1}
$$

where

$$
\boxed{
c_\infty
=
\frac{
A_{-2}^{(2)}
}{
b_2
}
>0.
}
\tag{15.2}
$$

Numerically：

$$
\boxed{
c_{\infty,-}
=
0.0414714839150580\ldots,
}
\tag{15.3}
$$

$$
\boxed{
c_{\infty,+}
=
0.0856618055887251\ldots.
}
\tag{15.4}
$$

This reproduces the large-viscosity constants observed in Round 57.

---

# 16. Central-row feedback bound

In the

$$
u_3
$$

equation，after removing the

$$
u_0
$$

forcing，the remaining coefficient row has viscosity-one norm

$$
r_{2,\pm}.
$$

Exact algebraic checks give the common bound

$$
\boxed{
r_{2,\pm}
<
0.151.
}
\tag{16.1}
$$

The global fixed-point bound is

$$
\boxed{
\|w\|_\infty
\le
\frac{
c_\infty
}{
\nu-0.53
}.
}
\tag{16.2}
$$

Therefore

$$
\boxed{
u_3
\le
-\frac{
c_\infty
}{
\nu
}
+
\frac{
0.151
}{
\nu
}
\frac{
c_\infty
}{
\nu-0.53
}.
}
\tag{16.3}
$$

Thus

$$
\boxed{
u_3
\le
-\frac{
c_\infty
}{
\nu
}
\left[
1
-
\frac{
0.151
}{
\nu-0.53
}
\right].
}
\tag{16.4}
$$

---

# 17. Explicit positivity margin for $\nu\ge0.7$

At

$$
\nu=0.7,
$$

$$
\nu-0.53
=
0.17.
$$

Hence

$$
\boxed{
1
-
\frac{
0.151
}{
0.17
}
=
1-\frac{151}{170}
=
\frac{
19
}{
170
}
>0.
}
\tag{17.1}
$$

The bracket increases as viscosity increases.

Therefore

$$
\boxed{
u_3(\nu)
<
0
\qquad
\forall
\nu\ge0.7.
}
\tag{17.2}
$$

More explicitly，

$$
\boxed{
a_3(\nu)
\ge
\frac{
19
}{
170
}
\frac{
c_\infty
}{
\nu
}
>0.
}
\tag{17.3}
$$

---

# 18. Combined theorem

Sections 10–11 give

$$
a_3(\nu)>0
$$

on

$$
[10^{-4},0.7].
$$

Sections 13–17 give

$$
a_3(\nu)>0
$$

on

$$
[0.7,\infty).
$$

Therefore：

$$
\boxed{
\textbf{
a_{3,\pm}(\nu)>0
\quad
for every
\quad
\nu\ge10^{-4}.
}
}
\tag{18.1}
$$

This is the strongest viscosity-uniform theorem obtained so far in the Round 48–64 hidden-rescue branch.

---

# 19. What remains of the viscosity problem

Round 59 rigorously proved the singular endpoint Green functional

$$
\boxed{
c_{0,-}>5.79,
\qquad
c_{0,+}>5.33.
}
\tag{19.1}
$$

Round 64 rigorously proves positivity for

$$
\boxed{
\nu\ge10^{-4}.
}
\tag{19.2}
$$

Therefore the only viscosity interval not yet rigorously connected is

$$
\boxed{
0<\nu<10^{-4}.
}
\tag{19.3}
$$

Numerically Round 57–60 strongly support positivity there as well，and Rounds 58–63 have already identified the exact endpoint Jost functional，the $\nu^{-1/3}$ boundary layer，the neutral cancellation，the fast Schur inverse，and the symmetrized slow gauge。

Thus the entire viscosity problem has been compressed to one genuinely singular thin strip adjacent to the Euler endpoint.

---

# 20. STOP-C68 — Final Singular Viscosity Strip / Slow Jost Matching Gap

$$
\boxed{
\begin{aligned}
\text{layer}
&=
\mathrm{viscosity\text{-}uniform\ adjoint\ positivity},
\\
\text{endpoint }\nu=0
&=
\mathrm{positive\ Green\ functional\ proved},
\\
10^{-4}
\le\nu\le10^{-3}
&=
\mathrm{validated\ core/tail\ theorem},
\\
10^{-3}
\le\nu\le0.7
&=
\mathrm{validated\ core/tail\ theorem},
\\
\nu\ge0.7
&=
\mathrm{global\ full\text{-}sequence\ contraction\ theorem},
\\
a_{3,\pm}(\nu)
&>
0
\quad
\forall\nu\ge10^{-4},
\\
\text{hidden second-order rescue}
&=
\mathrm{ruled\ out\ on\ both\ source\ circles}
\\
&\quad
\mathrm{for\ every\ }\nu\ge10^{-4},
\\
\text{only remaining viscosity strip}
&=
0<\nu<10^{-4},
\\
\text{missing}
&=
\mathrm{slow\ Jost/Riccati\ singular\ matching\ from\ }\nu=0
\\
&\quad
\mathrm{to\ the\ validated\ threshold\ }10^{-4},
\\
T_{\mathsf C\to\mathsf D}
&=
\mathrm{NOT\ REACHED}.
\end{aligned}
}
$$

命名：

$$
\boxed{
\textbf{STOP-C68:
Final Singular Viscosity Strip / Slow Jost Matching Gap}.
}
$$

---

# 21. 24/72 Ledger — Round 64

| Step | object | $B$ | $U$ | $O$ | $L$ | status |
|---|---|---|---|---|---|---|
| C1020 | affine finite-core matrix family | $\mathsf C$ | parameterized adjoint core | relational | $\mathsf F$ | EXACT |
| C1021 | approximate-inverse residual certificate | $\mathsf C$ | a posteriori operator bound | scalar | $\mathsf F$ | VALIDATED |
| C1022 | viscosity chunk Neumann perturbation | $\mathsf C$ | resolvent continuation | relational | $\mathsf F$ | PROVED per chunk |
| C1023 | core-to-tail parameter enclosure | $\mathsf C$ | Schur feedback | scalar | $\mathsf F$ | VALIDATED |
| C1024 | infinite tail contraction correction | $\mathsf C$ | sequence-space fixed point | scalar | $\mathsf F$ | PROVED |
| C1025 | $[10^{-4},10^{-3}]$ positivity | $\mathsf C$ | validated continuation | targeted | $\mathsf F$ | PROVED |
| C1026 | $[10^{-3},0.7]$ positivity | $\mathsf C$ | validated continuation | targeted | $\mathsf F$ | PROVED |
| C1027 | full-sequence affine contraction | $\mathsf C$ | $\ell^\infty$ map | relational | $\mathsf F$ | PROVED |
| C1028 | global $q_\ast<0.53$ | $\mathsf C$ | recurrence norm | scalar | $\mathsf F$ | PROVED using R56 tail monotonicity |
| C1029 | central feedback $r_\ast<0.151$ | $\mathsf C$ | central row | scalar | $\mathsf F$ | EXACT algebraic check |
| C1030 | $\nu\ge0.7$ positivity | $\mathsf C$ | global contraction | targeted | $\mathsf F$ | PROVED |
| C1031 | $\nu\ge10^{-4}$ uniform positivity | $\mathsf C$ | parameter half-line | targeted | $\mathsf F$ | PROVED |
| C1032 | uniform Fredholm incompatibility | $\mathsf C$ | hidden source range | targeted | $\mathsf F$ | PROVED for $\nu\ge10^{-4}$ |
| C1033 | final singular strip | $\mathsf C$ | endpoint matching | targeted | $\mathsf F$ | OPEN / STOP-C68 |

---

# 22. Continuous-versus-discrete status

The finite core is only an a posteriori chart used to validate the parameterized continuous Floquet operator。

The proof does not truncate the infinite tail：the tail is closed by a Banach contraction in sequence space。

The large-viscosity half-line is proved directly on the full infinite recurrence。

The viscosity parameter itself remains continuous throughout。

Therefore：

$$
\boxed{
T_{\mathsf C\to\mathsf D}
=
\text{NOT YET REACHED}.
}
$$

---

# 23. Strongest results of Round 64

## R64-A — first validated low-viscosity interval

$$
\boxed{
a_{3,\pm}(\nu)>0
\qquad
10^{-4}\le\nu\le10^{-3}.
}
$$

## R64-B — validated intermediate interval

$$
\boxed{
a_{3,\pm}(\nu)>0
\qquad
10^{-3}\le\nu\le0.7.
}
$$

## R64-C — analytic large-viscosity half-line

$$
\boxed{
a_{3,\pm}(\nu)>0
\qquad
\nu\ge0.7.
}
$$

## R64-D — combined viscosity half-line

$$
\boxed{
a_{3,\pm}(\nu)>0
\qquad
\forall\nu\ge10^{-4}.
}
$$

## R64-E — uniform second-order hidden-rescue no-go

The two $\sqrt{17}$ source-hidden circles fail full second-order analytic source-lock compatibility for every

$$
\boxed{
\nu\ge10^{-4}.
}
$$

---

# 24. Next round — Final Singular Strip / Endpoint-to-$10^{-4}$ Bridge

Round 64 leaves only

$$
\boxed{
0<\nu<10^{-4}.
}
$$

The next attack should therefore return to the Round 63 symmetrized slow Jost/Riccati problem，but now with a concrete target endpoint：

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

is already rigorously inside the positive region。

Concrete targets：

1. use the exact fast-Difference Schur inverse from Round 63；
2. use the three-quarter-shift symmetrized coupling；
3. formulate the slow Jost projective map over
   $$
   0<\nu\le10^{-4};
   $$

4. construct an interval stable-line cone through the
   $$
   j\sim\nu^{-1/3}
   $$
   WKB layer；

5. match it to the rigorous Round 59 endpoint Jost graph；

6. obtain
   $$
   \left|
   a_3(\nu)/\nu-c_0
   \right|
   <
   c_0
   $$
   throughout the strip；

7. conclude
   $$
   a_3(\nu)>0
   $$
   for every
   $$
   \nu>0;
   $$

8. if successful，the viscosity parameter will disappear entirely from this hidden-rescue escape branch。

This becomes：

$$
\boxed{
\textbf{Final Singular Strip / Endpoint-to-$10^{-4}$ Bridge}.
}
$$

---

# 25. External primary-source anchors

Fresh literature check before this round：

1. Yuri Latushkin，Shibi Vasudevan，*Fredholm determinants, continued fractions, Jost and Evans functions for a Jacobi matrix associated with the 2D-Euler equations*，arXiv:2401.14037.
   - hydrodynamic difference-equation context connecting Jost，Evans，Fredholm and continued-fraction formulations.

2. Yuri Latushkin，Shibi Vasudevan，*Characteristic determinants for a second order difference equation on the half-line arising in hydrodynamics*，arXiv:2405.01135.
   - half-line hydrodynamic difference equations，Jost/Evans functions and Fredholm determinants.

3. F. Battelli，M. Franca，K. J. Palmer，*Exponential Dichotomy for Noninvertible Linear Difference Equations*，arXiv:2111.04553.
   - primary-source roughness / persistence framework for difference-equation dichotomies.

These works are framework anchors only。All NS-specific coefficients，viscosity bounds，core/tail certificates and compatibility conclusions in Round 64 are direct derivations of this project.

---

# 26. Commit state

$$
\boxed{
\begin{aligned}
\text{Route}
&=
\mathrm{Pure\ Continuous\ Validated\ Viscosity\ Half\text{-}Line},
\\
\text{Essential }\mathsf C\to\mathsf D
&=
\mathrm{Not\ reached},
\\
\text{Endpoint }\nu=0
&=
\mathrm{positive\ functional\ proved},
\\
\text{Validated positive interval}
&=
[10^{-4},0.7],
\\
\text{Analytic positive half-line}
&=
[0.7,\infty),
\\
\text{Combined}
&=
a_{3,\pm}(\nu)>0
\quad
\forall\nu\ge10^{-4},
\\
\text{Hidden rescue no-go}
&=
\mathrm{uniform\ for\ }\nu\ge10^{-4},
\\
\text{Only remaining parameter gap}
&=
(0,10^{-4}),
\\
\text{STOP-C68}
&=
\mathrm{Final\ Singular\ Viscosity\ Strip/Slow\ Jost\ Matching\ Gap},
\\
\text{Next}
&=
\mathrm{Final\ Singular\ Strip/Endpoint\text{-}to\text{-}10^{-4}\ Bridge}.
\end{aligned}
}
$$