# NS × X 積分 × 24/72 範式實戰
## Round 58 — Pure Continuous Small-Viscosity Singular Adjoint Limit / Bounded-Neutral Corrector

- 日期：2026-08-17
- 版本：v0.1
- 狀態：Proof-Route Experiment / Continuous-Only Singular-Endpoint Branch
- canonical source：UTF-8 Markdown
- canonical math delimiters：inline `$...$`；display `$$...$$`
- 前一輪：`NS_X72_Round57_PureContinuous_ViscosityContinuation_AdjointPositivityMap_v0.1_2026-08-17.md`
- 本輪目標：Round 57 的全 viscosity scan顯示
  $$
  a_{3,\pm}(\nu)>0
  $$
  且
  $$
  a_3(\nu)=O(\nu)
  $$
  as
  $$
  \nu\to0^+.
  $$
  本輪不再以小正 viscosity 的 finite-section SVD直接外插，而是把 adjoint recurrence做 singular parity rescaling，導出正確的 $\nu=0$ endpoint problem。
- 主要結果：
  1. exact rescaling
     $$
     u_{2j}=e_j,
     \qquad
     u_{2j+1}=\nu o_j
     $$
     將 canonical adjoint equation變成只依賴
     $$
     \mu=\nu^2
     $$
     的系統；
  2. $\mu=0$ 時 even sector是 superfactorial minimal Euler mode；
  3. odd first corrector不是 analytic/minimal tail，而是 bounded-neutral mode；
  4. direct recurrence asymptotics gives
     $$
     o_j
     =
     L
     -
     \frac{3K^2L}{4j^2}
     +
     O(j^{-3});
     $$
  5. endpoint central slope constant由一個「minimal-even forced bounded-odd」BVP決定；
  6. direct rescaled endpoint solve stabilizes to
     $$
     c_{0,-}
     =
     5.79052557842265\ldots,
     $$
     $$
     c_{0,+}
     =
     5.33175254587449\ldots;
     $$
  7. hence the predicted Fredholm-pairing slopes are
     $$
     \Pi'_-(0^+)
     \approx
     -1.76280262464,
     $$
     $$
     \Pi'_+(0^+)
     \approx
     532.651652172;
     $$
  8. the small-fibre value
     $$
     5.7905357\ldots
     $$
     quoted from Round 57's extremely-small-$\nu$ raw SVD is identified as singular-cutoff contamination，not the endpoint constant；
  9. the correct rigorous endpoint space is
     $$
     \boxed{
     \text{minimal even}
     \times
     \text{bounded-neutral odd}
     }
     $$
     rather than the fixed-$\nu$ analytic tail space。
- 非主張：本輪尚未證明
  $$
  \lim_{\nu\to0^+}
  a_3(\nu)/\nu
  =
  c_0
  $$
  in the full infinite operator topology。The singular endpoint system and asymptotics are exact；the numerical values of $c_0$ are stable direct BVP computations。The remaining proof obligation is a rigorous Green/Jost matching theorem connecting the fixed-$\nu$ minimal branches to the bounded-neutral endpoint corrector。

---

# 0. Round 57 handoff

Round 57：

$$
\boxed{
q_n(K,\nu)
=
q_n(K,1)/\nu.
}
\tag{0.1}
$$

So every fixed：

$$
\nu>0
$$

still has a contractive tail at sufficiently large Floquet depth。

But：

$$
N_{\rm tail}
\sim
\nu^{-1/2}
$$

diverges as：

$$
\nu\to0^+.
$$

Numerically：

$$
a_3(\nu)>0
$$

through：

$$
10^{-5}\le\nu\le10^3.
$$

The suggested small-viscosity law was：

$$
\boxed{
a_3(\nu)
\sim
c_0\nu.
}
\tag{0.2}
$$

Round 57 STOP：

$$
\boxed{
\text{STOP-C61}
=
\text{Viscosity-Uniform Positivity / Endpoint-Continuation Gap}.
}
$$

---

# 1. Real canonical adjoint recurrence

Round 56 writes the $\mathcal C$-even canonical adjoint as：

$$
\boxed{
\psi_n=i^nu_n,
\qquad
u_n\in\mathbb R.
}
\tag{1.1}
$$

Reflection：

$$
\boxed{
u_{-n}=(-1)^nu_n.
}
\tag{1.2}
$$

Normalization：

$$
\boxed{
u_0=1,
\qquad
u_1=0.
}
\tag{1.3}
$$

The recurrence：

$$
\boxed{
-
A_{-2}^{(n)}
u_{n-2}
+
A_0^{(n)}
u_n
-
B_1^{(n)}
u_{n+1}
-
A_2^{(n)}
u_{n+2}
+
A_4^{(n)}
u_{n+4}
=
0.
}
\tag{1.4}
$$

Here：

$$
A_{-2},
A_0,
A_2,
A_4
$$

are independent of viscosity，while：

$$
\boxed{
B_1^{(n)}(K,\nu)
=
\nu b_n(K).
}
\tag{1.5}
$$

---

# 2. Exact parity rescaling

Set：

$$
\boxed{
u_{2j}=e_j,
}
\tag{2.1}
$$

and：

$$
\boxed{
u_{2j+1}
=
\nu o_j.
}
\tag{2.2}
$$

Then the even equations，at：

$$
n=2j,
$$

become：

$$
\boxed{
-
A_{-2}^{(2j)}
e_{j-1}
+
A_0^{(2j)}
e_j
-
A_2^{(2j)}
e_{j+1}
+
A_4^{(2j)}
e_{j+2}
-
\nu^2
b_{2j}
o_j
=
0.
}
\tag{2.3}
$$

The odd equations，at：

$$
n=2j+1,
$$

after dividing by：

$$
\nu,
$$

become：

$$
\boxed{
\begin{aligned}
0
={}&
-
A_{-2}^{(2j+1)}
o_{j-1}
+
A_0^{(2j+1)}
o_j
\\
&-
b_{2j+1}
e_{j+1}
-
A_2^{(2j+1)}
o_{j+1}
+
A_4^{(2j+1)}
o_{j+2}.
\end{aligned}
}
\tag{2.4}
$$

Therefore the rescaled system depends on viscosity only through：

$$
\boxed{
\mu=\nu^2.
}
\tag{2.5}
$$

This explains the observed structural expansion：

$$
\boxed{
u_{2j}
=
e_j^{(0)}
+
O(\nu^2),
}
\tag{2.6}
$$

$$
\boxed{
u_{2j+1}
=
\nu
\left[
o_j^{(0)}
+
O(\nu^2)
\right]
}
\tag{2.7}
$$

provided the singular endpoint matching is justified。

---

# 3. The $\mu=0$ even problem

At：

$$
\mu=0,
$$

the even mode solves：

$$
\boxed{
-
A_{-2}^{(2j)}
e_{j-1}
+
A_0^{(2j)}
e_j
-
A_2^{(2j)}
e_{j+1}
+
A_4^{(2j)}
e_{j+2}
=
0.
}
\tag{3.1}
$$

with：

$$
\boxed{
e_0=1.
}
\tag{3.2}
$$

The admissible branch is the minimal Euler branch。

Let：

$$
\boxed{
R_j
=
e_j/e_{j-1}.
}
\tag{3.3}
$$

Then：

$$
\boxed{
R_j
=
\frac{
A_{-2}^{(2j)}
}{
A_0^{(2j)}
-
A_2^{(2j)}
R_{j+1}
+
A_4^{(2j)}
R_{j+2}R_{j+1}
}.
}
\tag{3.4}
$$

The coefficient asymptotics imply：

$$
\boxed{
R_j
\sim
-\frac{
K^2
}{
16j^2
}.
}
\tag{3.5}
$$

Hence：

$$
\boxed{
|e_j|
\asymp
C
\frac{
(K^2/16)^j
}{
(j!)^2
}
}
\tag{3.6}
$$

up to subfactorial factors。

So the even endpoint remains superfactorially minimal。

---

# 4. The $\mu=0$ odd corrector

The odd first corrector solves：

$$
\boxed{
\begin{aligned}
0
={}&
-
A_{-2}^{(2j+1)}
o_{j-1}
+
A_0^{(2j+1)}
o_j
\\
&-
b_{2j+1}
e_{j+1}
-
A_2^{(2j+1)}
o_{j+1}
+
A_4^{(2j+1)}
o_{j+2}.
\end{aligned}
}
\tag{4.1}
$$

Normalization：

$$
\boxed{
o_0=0.
}
\tag{4.2}
$$

Reflection is：

$$
\boxed{
o_{-j-1}
=
-o_j.
}
\tag{4.3}
$$

The forcing：

$$
b_{2j+1}e_{j+1}
$$

decays superfactorially。

However，the homogeneous odd operator possesses the neutral Euler branch inherited from Round 53。

Therefore the correct endpoint condition is：

$$
\boxed{
o_j
\text{ bounded as }
j\to+\infty,
}
\tag{4.4}
$$

not：

$$
o_j\to0.
$$

---

# 5. Exact constant-residual asymptotic

Define：

$$
\boxed{
S_j
=
-
A_{-2}^{(2j+1)}
+
A_0^{(2j+1)}
-
A_2^{(2j+1)}
+
A_4^{(2j+1)}.
}
\tag{5.1}
$$

Direct exact symbolic expansion gives：

$$
\boxed{
\lim_{j\to\infty}
j^3S_j
=
6K^3.
}
\tag{5.2}
$$

So a strictly constant odd mode is not exact；its residual is：

$$
O(j^{-3}).
$$

This is precisely weak enough to generate a bounded neutral correction rather than an exponentially separated tail。

---

# 6. Exact plateau law

Assume：

$$
\boxed{
o_j
=
L
+
\frac{C}{j^2}
+
o(j^{-2}).
}
\tag{6.1}
$$

The forcing is superfactorially smaller and does not contribute to the algebraic tail balance。

Substitute (6.1) into the homogeneous part of (4.1)。

The exact symbolic limit is：

$$
\boxed{
\lim_{j\to\infty}
j^3
\mathcal O
\left[
L+C/j^2
\right]
=
8CK
+
6K^3L.
}
\tag{6.2}
$$

Thus：

$$
\boxed{
C
=
-\frac34
K^2L.
}
\tag{6.3}
$$

Hence：

$$
\boxed{
o_j
=
L
-
\frac{
3K^2L
}{
4j^2
}
+
O(j^{-3}).
}
\tag{6.4}
$$

命名：

$$
\boxed{
\textbf{Bounded-Neutral Plateau Law}.
}
$$

This is an exact asymptotic balance of the endpoint recurrence。

---

# 7. Why Round 57 direct small-$\nu$ extrapolation is delicate

For every：

$$
\nu>0,
$$

the full canonical adjoint is analytic/minimal at infinity。

But the scale at which viscosity dominates is：

$$
j
\sim
\nu^{-1/2}.
$$

At fixed：

$$
j,
$$

the rescaled odd variable：

$$
o_j=u_{2j+1}/\nu
$$

approaches the bounded-neutral endpoint profile。

At：

$$
j
\gg
\nu^{-1/2},
$$

the fixed-$\nu$ minimal tail bends back toward superfactorial decay。

Therefore the limits：

$$
\nu\to0^+
$$

and：

$$
j\to\infty
$$

do not commute。

This is the source of singular-cutoff contamination in raw finite-section extrapolation。

---

# 8. Direct endpoint BVP computation

The verification script solves：

1. the even minimal mode by backward continued-ratio iteration；
2. the forced odd endpoint equation with：
   $$
   o_0=0;
   $$
3. a far cutoff whose boundary influence on the central value is empirically superfactorially small。

The endpoint central coefficient is：

$$
\boxed{
c_0
=
-\,
o_1.
}
\tag{8.1}
$$

For the small source fibre：

$$
\boxed{
K_-
=
\sqrt{17}-3,
}
\tag{8.2}
$$

the stable value is：

$$
\boxed{
c_{0,-}
=
5.79052557842265\ldots.
}
\tag{8.3}
$$

For the large source fibre：

$$
\boxed{
K_+
=
\sqrt{17}+3,
}
\tag{8.4}
$$

$$
\boxed{
c_{0,+}
=
5.33175254587449\ldots.
}
\tag{8.5}
$$

---

# 9. Endpoint cutoff stability

For the small fibre，the direct endpoint BVP gives：

$$
\boxed{
\begin{array}{c|c}
J
&
-c_1=-o_1
\\
\hline
3
&
5.79052558969675
\\
4
&
5.79052557841899
\\
5
&
5.79052557842265
\\
10
&
5.79052557842265
\\
20
&
5.79052557842265
\end{array}
}
\tag{9.1}
$$

For the large fibre：

$$
\boxed{
\begin{array}{c|c}
J
&
-o_1
\\
\hline
3
&
5.33374367637498
\\
4
&
5.33157043152890
\\
5
&
5.33175948323330
\\
6
&
5.33175232358105
\\
8
&
5.33175254575307
\\
10
&
5.33175254587446
\\
20
&
5.33175254587449
\end{array}
}
\tag{9.2}
$$

These are direct endpoint equations，not extrapolation from small positive viscosity。

They are still numerical until a rigorous endpoint Green/Jost tail estimate is supplied。

---

# 10. Small positive viscosity in the rescaled variables

Solve the finite rescaled system directly at small positive：

$$
\nu.
$$

Representative values：

### small fibre

$$
\boxed{
\begin{array}{c|c}
\nu
&
a_3(\nu)/\nu
\\
\hline
10^{-2}
&
5.82043633064
\\
10^{-3}
&
5.80013506899
\\
10^{-4}
&
5.79153420592
\\
10^{-5}
&
5.79061554981
\\
10^{-6}
&
5.79052700090
\end{array}
}
\tag{10.1}
$$

### large fibre

$$
\boxed{
\begin{array}{c|c}
\nu
&
a_3(\nu)/\nu
\\
\hline
10^{-2}
&
5.26561015994
\\
10^{-3}
&
5.32799430246
\\
10^{-4}
&
5.33141366867
\\
10^{-5}
&
5.33174497726
\\
10^{-6}
&
5.33175246896
\end{array}
}
\tag{10.2}
$$

The rescaled solve converges toward (8.3)–(8.5)。

---

# 11. Correction to the Round 57 small-fibre extrapolation

Round 57's raw physical finite section at：

$$
\nu=10^{-6}
$$

reported：

$$
a_{3,-}/\nu
\approx
5.790535712.
$$

The rescaled endpoint solve shows the true endpoint candidate is：

$$
\boxed{
5.790525578\ldots.
}
$$

The difference：

$$
\sim10^{-5}
$$

is entirely consistent with the diverging minimal-tail cutoff：

$$
N_{\rm tail}
\sim
\nu^{-1/2}.
$$

This does not change the positivity conclusion。

It is a useful audit correction：near a singular endpoint，raw finite sections should not be used to estimate derivatives without parity rescaling。

---

# 12. Fredholm-pairing slope candidate

Round 55：

$$
\boxed{
\langle
\psi_+,
g
\rangle
=
g_0(\nu)
+
a_3(\nu)G_{-3}.
}
\tag{12.1}
$$

If：

$$
a_3(\nu)
=
c_0\nu
+
o(\nu),
$$

then：

$$
\boxed{
\frac{
\langle
\psi_+,
g
\rangle
}{
\nu
}
\to
12
(
3r^2-1
)
+
c_0G_{-3}.
}
\tag{12.2}
$$

Using the endpoint BVP constants：

### small fibre

$$
\boxed{
\Pi'_-(0^+)
\approx
-1.76280262464.
}
\tag{12.3}
$$

### large fibre

$$
\boxed{
\Pi'_+(0^+)
\approx
532.651652172.
}
\tag{12.4}
$$

Both are far from zero。

Thus the vanishing-viscosity endpoint numerically reinforces，rather than weakens，the Round 56 Fredholm obstruction。

---

# 13. Endpoint space must change

The fixed positive-viscosity canonical mode belongs to an analytic/minimal Floquet space。

Its first viscosity derivative does not。

The correct singular tangent space is：

$$
\boxed{
\mathcal X_0
=
\mathcal X_{\rm even}^{\rm minimal}
\times
\mathcal X_{\rm odd}^{\rm bounded\ neutral}.
}
\tag{13.1}
$$

This resolves an apparent paradox：

- for every：
  $$
  \nu>0,
  $$
  all components decay；
- at：
  $$
  \nu=0,
  $$
  the odd first corrector approaches a nonzero plateau。

The boundary layer sits at Floquet depth：

$$
j\sim\nu^{-1/2}
$$

and escapes to infinity as viscosity vanishes。

---

# 14. Formal endpoint analyticity in $\mu=\nu^2$

The rescaled equations (2.3)–(2.4) contain：

$$
\nu
$$

only through：

$$
\mu=\nu^2.
$$

This suggests the canonical singular branch，when formulated in：

$$
\mathcal X_0,
$$

should have：

$$
\boxed{
e_j(\nu)
=
e_j^{(0)}
+
\mu e_j^{(1)}
+\cdots,
}
\tag{14.1}
$$

$$
\boxed{
o_j(\nu)
=
o_j^{(0)}
+
\mu o_j^{(1)}
+\cdots.
}
\tag{14.2}
$$

Therefore：

$$
\boxed{
a_3(\nu)
=
-\nu o_1^{(0)}
+
O(\nu^3).
}
\tag{14.3}
$$

So：

$$
\boxed{
c_0
=
-\,
o_1^{(0)}.
}
\tag{14.4}
$$

The missing theorem is the invertibility / matching result that justifies this expansion uniformly across the singular tail。

---

# 15. Why a Jost / continued-fraction formulation is natural

The even minimal mode already admits the backward continued-ratio equation：

$$
R_j
=
\frac{
A_{-2}^{(2j)}
}{
A_0^{(2j)}
-
A_2^{(2j)}R_{j+1}
+
A_4^{(2j)}R_{j+2}R_{j+1}
}.
$$

The odd endpoint problem is a forced bounded-neutral recurrence。

Thus a natural rigorous endpoint construction is：

1. construct the even Jost/minimal solution by continued fractions；
2. construct the neutral homogeneous odd solution with：
   $$
   h_j\to1;
   $$
3. construct the forced odd particular solution：
   $$
   p_j\to0;
   $$
4. choose：
   $$
   o_j=Lh_j+p_j
   $$
   so：
   $$
   o_0=0;
   $$
5. prove the central quantity：
   $$
   -o_1>0.
   $$

This avoids sending a fixed positive-viscosity contraction cutoff to infinity。

---

# 16. Neutral homogeneous asymptotic

Let：

$$
h_j\to1.
$$

Section 6 gives：

$$
\boxed{
h_j
=
1
-
\frac{
3K^2
}{
4j^2
}
+
O(j^{-3}).
}
\tag{16.1}
$$

The forced particular solution inherits the superfactorial scale of：

$$
e_j
$$

and may be chosen：

$$
\boxed{
p_j\to0.
}
\tag{16.2}
$$

Then：

$$
\boxed{
o_j
=
Lh_j+p_j.
}
\tag{16.3}
$$

The scalar：

$$
L
$$

is fixed by：

$$
o_0=0.
$$

This decomposition cleanly separates：

- the neutral Euler tail；
- the genuinely forced minimal correction。

---

# 17. Numerical plateau diagnostics

A large-cutoff endpoint solve gives approximate plateaux：

$$
\boxed{
L_-
\approx
2.97825,
}
\tag{17.1}
$$

and：

$$
\boxed{
L_+
\approx
-11.4795.
}
\tag{17.2}
$$

The observed approach is consistent with：

$$
j^2
(
o_j-L
)
\to
-\frac34K^2L.
$$

For the small fibre：

$$
-\frac34K_-^2L_-
\approx
-2.82.
$$

For the large fibre：

$$
-\frac34K_+^2L_+
\approx
4.37\times10^2.
$$

These large differences explain why the large-$K$ endpoint needs much deeper Floquet resolution before its neutral plateau is visually apparent。

---

# 18. What is now rigorously clear versus still open

## Exact / analytic

- parity-rescaled endpoint equations；
- dependence on：
  $$
  \mu=\nu^2;
  $$
- even minimal ratio asymptotic；
- neutral residual：
  $$
  j^3S_j\to6K^3;
  $$
- plateau coefficient：
  $$
  C=-3K^2L/4;
  $$
- exact target sign geometry of Rounds 55–57。

## Numerically stable but not yet infinite-endpoint theorem

- $c_{0,\pm}$ values；
- plateau values：
  $$
  L_\pm;
  $$
- actual convergence：
  $$
  a_3(\nu)/\nu\to c_0.
  $$

This distinction is maintained deliberately。

---

# 19. STOP-C62 — Bounded-Neutral Green/Jost Endpoint Gap

$$
\boxed{
\begin{aligned}
\text{layer}
&=
\mathrm{small\text{-}viscosity\ singular\ adjoint\ limit},
\\
u_{2j}
&=
e_j,
\\
u_{2j+1}
&=
\nu o_j,
\\
\mu
&=
\nu^2,
\\
\text{even endpoint}
&=
\mathrm{superfactorial\ minimal},
\\
\text{odd endpoint}
&=
\mathrm{bounded\ neutral},
\\
e_j/e_{j-1}
&\sim
-K^2/(16j^2),
\\
o_j
&=
L
-
3K^2L/(4j^2)
+
O(j^{-3}),
\\
c_{0,-}^{\rm num}
&=
5.79052557842265\ldots,
\\
c_{0,+}^{\rm num}
&=
5.33175254587449\ldots,
\\
\Pi'_-(0^+)^{\rm num}
&\approx
-1.76280262464,
\\
\Pi'_+(0^+)^{\rm num}
&\approx
532.651652172,
\\
\text{endpoint sign}
&=
\mathrm{strongly\ positive\ for\ }a_3/\nu
\mathrm{\ numerically},
\\
\text{missing}
&=
\mathrm{rigorous\ Green/Jost\ construction\ in\ }
\mathcal X_{\rm even}^{\rm minimal}
\times
\mathcal X_{\rm odd}^{\rm bounded}
\\
&\quad
\mathrm{and\ proof\ that\ the\ fixed\text{-}\nu\ minimal\ branch\ converges\ to\ it},
\\
T_{\mathsf C\to\mathsf D}
&=
\mathrm{NOT\ REACHED}.
\end{aligned}
}
$$

命名：

$$
\boxed{
\textbf{STOP-C62:
Bounded-Neutral Green/Jost Endpoint Gap}.
}
$$

---

# 20. 24/72 Ledger — Round 58

| Step | object | $B$ | $U$ | $O$ | $L$ | status |
|---|---|---|---|---|---|---|
| C931 | parity viscosity rescaling | $\mathsf C$ | singular operator family | relational | $\mathsf F$ | EXACT |
| C932 | $\mu=\nu^2$ endpoint system | $\mathsf C$ | parameter desingularization | scalar | $\mathsf F$ | EXACT |
| C933 | even Euler minimal recurrence | $\mathsf C$ | continued-ratio geometry | profile | $\mathsf F$ | EXACT |
| C934 | even minimal asymptotic | $\mathsf C$ | Floquet infinity | scalar | $\mathsf F$ | DERIVED |
| C935 | forced odd endpoint recurrence | $\mathsf C$ | singular corrector | relational | $\mathsf F$ | EXACT |
| C936 | bounded-neutral endpoint condition | $\mathsf C$ | function-space change | targeted | $\mathsf F$ | IDENTIFIED |
| C937 | constant residual limit | $\mathsf C$ | exact asymptotic algebra | scalar | $\mathsf F$ | PROVED |
| C938 | Bounded-Neutral Plateau Law | $\mathsf C$ | asymptotic matching | scalar | $\mathsf F$ | PROVED formally from recurrence |
| C939 | endpoint BVP constants | $\mathsf C$ | direct rescaled solve | scalar | $\mathsf F$ | NUMERICALLY VERIFIED |
| C940 | Round 57 endpoint audit | $\mathsf C$ | cutoff comparison | targeted | $\mathsf F$ | VERIFIED |
| C941 | Fredholm pairing slope candidate | $\mathsf C$ | endpoint compatibility | scalar | $\mathsf F$ | NUMERICAL |
| C942 | mixed endpoint function space | $\mathsf C$ | singular topology | relational | $\mathsf F$ | IDENTIFIED |
| C943 | $\mu$-analytic branch ansatz | $\mathsf C$ | singular perturbation | profile | $\mathsf F$ | CONJECTURAL / STRUCTURAL |
| C944 | Green/Jost endpoint decomposition | $\mathsf C$ | continued-fraction route | targeted | $\mathsf F$ | ROUTE DESIGNED |
| C945 | rigorous positive endpoint slope | $\mathsf C$ | infinite matching | targeted | $\mathsf F$ | OPEN / STOP-C62 |

---

# 21. Continuous-versus-discrete status

This round is fundamentally a continuous viscosity singular-limit problem：

$$
\nu\to0^+.
$$

The sideband index is still a spectral chart for a continuous periodic vertical coordinate。

The decisive new object is a change of infinite-dimensional function space at the endpoint，not a discrete computational mechanism。

The continued-ratio / Jost language is a representation of the same periodic continuous operator。

Therefore：

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

---

# 22. Strongest results of Round 58

## R58-A — exact desingularization

$$
\boxed{
u_{2j}=e_j,
\qquad
u_{2j+1}=\nu o_j
}
$$

turns the parameter into：

$$
\boxed{
\mu=\nu^2.
}
$$

## R58-B — even minimal endpoint

$$
\boxed{
e_j/e_{j-1}
\sim
-K^2/(16j^2).
}
$$

## R58-C — odd bounded-neutral endpoint

$$
\boxed{
o_j
=
L
-
\frac{3K^2L}{4j^2}
+
O(j^{-3}).
}
$$

## R58-D — refined positive slope constants

$$
\boxed{
c_{0,-}
\approx
5.79052557842265,
}
$$

$$
\boxed{
c_{0,+}
\approx
5.33175254587449.
}
$$

## R58-E — strong nonzero endpoint pairing slopes

$$
\boxed{
\Pi'_-(0^+)
\approx
-1.76280262464,
}
$$

$$
\boxed{
\Pi'_+(0^+)
\approx
532.651652172.
}
$$

## R58-F — endpoint difficulty is now localized

The remaining issue is not whether the formal endpoint is positive。

It is to justify the singular matching：

$$
\boxed{
\text{fixed-}\nu\text{ analytic minimal tail}
\longrightarrow
\text{minimal-even/bounded-odd endpoint}.
}
$$

---

# 23. Next round — Endpoint Green Functional / Rigorous Positive Slope

The next round should now prove the endpoint slope rather than re-scan viscosity。

Concrete targets：

1. rigorously construct the even minimal Jost solution through the continued-ratio fixed point；
2. construct the neutral odd homogeneous solution：
   $$
   h_j\to1;
   $$
3. prove：
   $$
   h_j
   =
   1
   -
   3K^2/(4j^2)
   +
   O(j^{-3});
   $$
4. construct the forced minimal odd particular solution：
   $$
   p_j\to0;
   $$
5. set：
   $$
   o_j=Lh_j+p_j,
   \qquad
   o_0=0;
   $$
6. derive a Green/Jost formula for：
   $$
   c_0=-o_1;
   $$
7. use a finite-core exact enclosure plus superfactorial forcing tail to prove a coarse but rigorous：
   $$
   c_{0,\pm}>0;
   $$
8. then establish a singular matching estimate：
   $$
   a_3(\nu)
   =
   c_0\nu
   +
   O(\nu^3)
   $$
   in a mixed weighted norm。

This becomes：

$$
\boxed{
\textbf{Endpoint Green Functional / Rigorous Positive Slope}.
}
$$

---

# 24. External primary-source anchors

1. Quansen Jiu, Milton C. Lopes Filho, Dongjuan Niu, Helena J. Nussenzveig Lopes, *The limit of vanishing viscosity for the incompressible 3D Navier-Stokes equations with helical symmetry*, arXiv:1706.10012.
   - rigorous 3D incompressible Navier–Stokes vanishing-viscosity analysis in a helical setting；
   - used only as external context showing that helical geometry and viscosity-dependent decompositions naturally lead to singular-limit estimates，not as a source for the Round 58 Floquet formulas.

2. 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.
   - relates continued fractions，Jost solutions，Evans functions and Fredholm determinants for a fluid-derived difference equation；
   - directly relevant methodological context for the planned endpoint continued-fraction / Jost formulation.

3. Christian Pötzsche, Robert Skiba, *Evans function, parity and nonautonomous bifurcations*, arXiv:2503.07221.
   - parameter-dependent Evans/parity framework；
   - relevant to the later step of reconnecting the rigorously solved singular endpoint to the positive-viscosity continuation branch.

All NS-specific endpoint recurrences，plateau asymptotics and numerical constants in Round 58 are direct derivations / computations of this project。

---

# 25. Commit state

$$
\boxed{
\begin{aligned}
\text{Route}
&=
\mathrm{Pure\ Continuous\ Small\text{-}Viscosity\ Singular\ Adjoint\ Limit},
\\
\text{Essential }\mathsf C\to\mathsf D
&=
\mathrm{Not\ reached},
\\
\text{Fixed-}\nu\text{ analytic tail}
&=
\mathrm{singular\ as\ }\nu\to0^+,
\\
\text{Correct endpoint topology}
&=
\mathrm{minimal\ even}
\times
\mathrm{bounded\ neutral\ odd},
\\
\text{Endpoint plateau law}
&=
\mathrm{derived},
\\
\text{Positive slope constants}
&=
\mathrm{strongly\ supported},
\\
\text{Round 57 raw endpoint estimate}
&=
\mathrm{refined},
\\
\text{Rigorous endpoint positivity}
&=
\mathrm{not\ yet\ claimed},
\\
\text{STOP-C62}
&=
\mathrm{Bounded\text{-}Neutral\ Green/Jost\ Endpoint\ Gap},
\\
\text{Next}
&=
\mathrm{Endpoint\ Green\ Functional/Rigorous\ Positive\ Slope}.
\end{aligned}
}
$$