# NS × X Integration × 24/72 Paradigm Practice
## Round 59 — Pure Continuous Endpoint Jost Graph / Rigorous Positive Green Functional

- Date:  2026-08-17
- Version:  v0.1
- Status:  Proof-Route Experiment / Continuous-Only Endpoint-Jost Branch
- canonical source: UTF-8 Markdown
- canonical math delimiters: inline `$...$`; display `$$...$$`
- Previous round:  `NS_X72_Round58_PureContinuous_SmallViscosity_BoundedNeutralAdjointLimit_v0.1_2026-08-17.md`
- This round's objective:  Round 58 has derived the small-viscosity singular endpoint
  $$
  u_{2j}=e_j,
  \qquad
  u_{2j+1}=\nu o_j,
  $$
  and identified
  $$
  e_j
  $$
  as the minimal Euler mode,
  $$
  o_j
  $$
  as the forced bounded-neutral corrector, but the endpoint Green/Jost positivity still only has numerical BVP evidence. This round will rewrite the odd endpoint's bounded two-dimensional Jost family into an **affine graph pullback**, and use exact algebraic coefficient bounds + outward interval arithmetic to directly prove that the endpoint Green functional is strictly positive.
- Main result: 
  1. the even minimal ratio tail can be written as a nonautonomous pullback contraction;
  2. the endpoint odd bounded-Jost family can be written as an affine graph
     $$
     o_{j+2}
     =
     P_j o_{j+1}
     +
     Q_j o_j
     +
     R_j;
     $$
  3. the graph parameters satisfy an exact rational pullback recurrence;
  4. for the two source fibres, on all
     $$
     j\ge10
     $$
     there exist fixed rational invariant boxes, and the pullback Lipschitz constants are respectively less than
     $$
     0.01
     $$
     and
     $$
     0.04;
     $$
  5. therefore the endpoint Jost affine plane is the unique pullback attractor, independent of far-tail initialization;
  6. pushing the entire invariant tail box back to the center using outward interval arithmetic yields:
     $$
     c_{0,-}
     \in
     [
     5.7905255784226477,\,
     5.7905255784226478
     ],
     $$
     $$
     c_{0,+}
     \in
     [
     5.331752543272241,\,
     5.331752548672376
     ];
     $$
  7. hence
     $$
     c_{0,\pm}>0
     $$
     is now a rigorous endpoint theorem;
  8. Round 58's numerical endpoint BVP constants are recovered inside these intervals;
  9. Remaining gap is **not endpoint positivity anymore**, but the singular matching theorem:
     $$
     a_3(\nu)/\nu
     \longrightarrow
     c_0
     \qquad
     (\nu\to0^+).
     $$
- Non-claims:  This round does not prove the aforementioned singular matching limit. Therefore it does not yet prove a positive-viscosity interval
  $$
  a_3(\nu)>0
  $$
  near zero. It proves the $\nu=0$ Jost/Green endpoint functional itself is rigorously positive.

---

# 0. Round 58 handoff

The two horizontal source fibres are:

$$
\boxed{
K_-
=
\sqrt{17}-3,
}
\tag{0.1}
$$

and:

$$
\boxed{
K_+
=
\sqrt{17}+3.
}
\tag{0.2}
$$

Round 58 parity rescaling:

$$
\boxed{
u_{2j}=e_j,
}
\tag{0.3}
$$

$$
\boxed{
u_{2j+1}
=
\nu o_j.
}
\tag{0.4}
$$

At:

$$
\nu=0,
$$

the even mode solves a homogeneous minimal recurrence, while the odd first corrector solves:

$$
\boxed{
\begin{aligned}
0
={}&
-
A_{-2}^{(2j+1)}
o_{j-1}
+
A_0^{(2j+1)}
o_j
\\
&-
b_{2j+1}
e_{j+1}
-
A_2^{(2j+1)}
o_{j+1}
+
A_4^{(2j+1)}
o_{j+2}.
\end{aligned}
}
\tag{0.5}
$$

Normalization:

$$
\boxed{
o_0=0.
}
\tag{0.6}
$$

The correct endpoint condition is bounded-neutral rather than decaying:

$$
\boxed{
o_j
=
L
-
\frac{
3K^2L
}{
4j^2
}
+
O(j^{-3}).
}
\tag{0.7}
$$

Round 58 numerical candidates:

$$
c_{0,-}
=
-\,
o_1
\approx
5.79052557842265,
$$

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

Round 58 STOP:

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

---

# 1. Even minimal-ratio pullback

Define:

$$
\boxed{
R_j
=
\frac{
e_j
}{
e_{j-1}
}.
}
\tag{1.1}
$$

The even endpoint recurrence gives:

$$
\boxed{
R_j
=
\frac{
A_{-2}^{(2j)}
}{
A_0^{(2j)}
-
A_2^{(2j)}
R_{j+1}
+
A_4^{(2j)}
R_{j+2}R_{j+1}
}.
}
\tag{1.2}
$$

This is a two-step nonautonomous pullback map.

For both:

$$
K=K_-,
\qquad
K=K_+,
$$

and every real:

$$
j\ge10,
$$

the exact algebraic coefficient certificate proves the sign pattern:

$$
\boxed{
A_{-2}^{(2j)}<0,
}
\tag{1.3}
$$

$$
\boxed{
A_0^{(2j)}>0,
}
\tag{1.4}
$$

$$
\boxed{
A_2^{(2j)}>0,
}
\tag{1.5}
$$

$$
\boxed{
A_4^{(2j)}<0.
}
\tag{1.6}
$$

---

# 2. Even invariant ratio box

Let:

$$
\boxed{
\mathcal B_R
=
[-1/10,0]^2.
}
\tag{2.1}
$$

For the small fibre, the coefficient bounds:

$$
\boxed{
-0.01
<
A_{-2}^{(2j)}
<0,
}
\tag{2.2}
$$

$$
\boxed{
4
<
A_0^{(2j)},
A_2^{(2j)}
<
5,
}
\tag{2.3}
$$

$$
\boxed{
-0.01
<
A_4^{(2j)}
<0
}
\tag{2.4}
$$

hold for all:

$$
j\ge10.
$$

For the large fibre:

$$
\boxed{
-1.4
<
A_{-2}^{(2j)}
<0,
}
\tag{2.5}
$$

$$
\boxed{
25
<
A_0^{(2j)}
<
29,
}
\tag{2.6}
$$

$$
\boxed{
24
<
A_2^{(2j)}
<
29,
}
\tag{2.7}
$$

$$
\boxed{
-0.4
<
A_4^{(2j)}
<0.
}
\tag{2.8}
$$

For:

$$
R_{j+1},R_{j+2}
\in
[-0.1,0],
$$

these imply:

$$
\boxed{
R_j\in[-0.1,0].
}
\tag{2.9}
$$

Moreover the Jacobian sum is uniformly below:

$$
0.004
$$

for:

$$
K_-,
$$

and below:

$$
0.067
$$

for:

$$
K_+.
$$

Therefore the ratio pullback is a strict contraction on:

$$
\mathcal B_R.
$$

This constructs a unique minimal-ratio pullback tail.

---

# 3. Rigorous finite ratio enclosure

Because the true minimal tail satisfies:

$$
(R_{11},R_{12})
\in
\mathcal B_R,
$$

outward interval evaluation of (1.2) from:

$$
j=10
$$

down to:

$$
j=1
$$

gives rigorous enclosures for:

$$
R_1,\ldots,R_{10}.
$$

Multiplying interval ratios yields:

### small fibre

$$
\boxed{
|e_{11}|
<
10^{-19},
}
\tag{3.1}
$$

### large fibre

$$
\boxed{
|e_{11}|
<
10^{-8}.
}
\tag{3.2}
$$

These deliberately coarse bounds are sufficient for the odd Jost graph proof.

---

# 4. Tail forcing bound

Recall:

$$
\boxed{
f_j
=
b_{2j+1}
e_{j+1}.
}
\tag{4.1}
$$

The exact coefficient certificate proves:

$$
\boxed{
b_{2j+1}<0
}
\tag{4.2}
$$

and:

$$
\boxed{
|b_{2j+1}|
<
100
(
2j+1
)^2
}
\tag{4.3}
$$

for both fibres and all:

$$
j\ge10.
$$

Since:

$$
|R_j|\le0.1
$$

through the even tail,

$$
|e_{j+1}|
$$

decreases by at least one factor:

$$
0.1
$$

per further endpoint level.

Therefore the worst forcing is at:

$$
j=10.
$$

Using Sections 3–4:

$$
\boxed{
|f_j|
<
10^{-3}
\qquad
(j\ge10)
}
\tag{4.4}
$$

for both fibres.

---

# 5. Affine Jost graph

Instead of separately constructing neutral and minimal homogeneous solutions, represent the entire two-dimensional bounded endpoint family by an affine graph:

$$
\boxed{
o_{j+2}
=
P_j
o_{j+1}
+
Q_j
o_j
+
G_j.
}
\tag{5.1}
$$

The notation:

$$
G_j
$$

is used here for the affine source offset, to avoid confusion with the even ratio:

$$
R_j.
$$

Substitute (5.1) into the odd endpoint recurrence.

The exact pullback is:

$$
\boxed{
P_{j-1}
=
\frac{
A_0^{(2j+1)}
+
A_4^{(2j+1)}
Q_j
}{
A_2^{(2j+1)}
-
A_4^{(2j+1)}
P_j
},
}
\tag{5.2}
$$

$$
\boxed{
Q_{j-1}
=
-
\frac{
A_{-2}^{(2j+1)}
}{
A_2^{(2j+1)}
-
A_4^{(2j+1)}
P_j
},
}
\tag{5.3}
$$

$$
\boxed{
G_{j-1}
=
\frac{
A_4^{(2j+1)}
G_j
-
f_j
}{
A_2^{(2j+1)}
-
A_4^{(2j+1)}
P_j
}.
}
\tag{5.4}
$$

Designation:

$$
\boxed{
\textbf{Endpoint Affine Jost Pullback}.
}
$$

---

# 6. Small-fibre invariant graph box

For:

$$
K=K_-,
$$

define:

$$
\boxed{
\mathcal B_-
=
[
0.7,1.3
]
\times
[
0,0.01
]
\times
[
-1,1
].
}
\tag{6.1}
$$

Using the exact coefficient bounds from Sections 1–4, for:

$$
j\ge10
$$

the denominator satisfies:

$$
\boxed{
A_2-A_4P
>
4.
}
\tag{6.2}
$$

The pullback obeys:

$$
\boxed{
0.7
<
P_{j-1}
<
1.3,
}
\tag{6.3}
$$

$$
\boxed{
0
<
Q_{j-1}
<
0.01,
}
\tag{6.4}
$$

$$
\boxed{
|G_{j-1}|
<
1.
}
\tag{6.5}
$$

The infinity-norm Jacobian satisfies the coarse uniform estimate:

$$
\boxed{
\|D\Phi_j\|_\infty
<
0.01.
}
\tag{6.6}
$$

Therefore:

$$
\boxed{
\Phi_j(
\mathcal B_-
)
\subset
\mathcal B_-,
}
\tag{6.7}
$$

and the pullback is a strict contraction.

---

# 7. Large-fibre invariant graph box

For:

$$
K=K_+,
$$

take:

$$
\boxed{
\mathcal B_+
=
[
0.8,1.25
]
\times
[
0,0.06
]
\times
[
-1,1
].
}
\tag{7.1}
$$

The coefficient bounds yield:

$$
\boxed{
A_2-A_4P
>
24.
}
\tag{7.2}
$$

Then:

$$
\boxed{
0.8
<
P_{j-1}
<
1.25,
}
\tag{7.3}
$$

$$
\boxed{
0
<
Q_{j-1}
<
0.06,
}
\tag{7.4}
$$

$$
\boxed{
|G_{j-1}|
<
1.
}
\tag{7.5}
$$

and:

$$
\boxed{
\|D\Phi_j\|_\infty
<
0.04.
}
\tag{7.6}
$$

Thus:

$$
\boxed{
\Phi_j(
\mathcal B_+
)
\subset
\mathcal B_+.
}
\tag{7.7}
$$

---

# 8. Pullback-attractor existence and uniqueness

Fix:

$$
J>10
$$

and choose any terminal graph:

$$
(P_J,Q_J,G_J)
\in
\mathcal B_\pm.
$$

Pull it back to:

$$
j=10
$$

using:

$$
\Phi_J,
\Phi_{J-1},
\ldots,
\Phi_{11}.
$$

If two terminal graphs are chosen, their images at level:

$$
10
$$

differ by at most:

$$
\boxed{
C
q^{J-10}
}
\tag{8.1}
$$

where:

$$
q=0.01
$$

or:

$$
0.04.
$$

Therefore as:

$$
J\to\infty,
$$

the level-$10$ graph converges to a unique limit independent of terminal data.

Finite pullback then gives a unique affine Jost graph at every finite level.

Designation:

$$
\boxed{
\textbf{Endpoint Jost Pullback Theorem}.
}
$$

This theorem selects precisely the endpoint Green/Jost family that Round 58 was approximating with far-cutoff BVPs.

---

# 9. Central Green functional

At:

$$
j=0,
$$

the Jost graph is:

$$
\boxed{
o_2
=
P_0o_1
+
Q_0o_0
+
G_0.
}
\tag{9.1}
$$

Canonical endpoint normalization:

$$
o_0=0.
$$

The reflected central recurrence is:

$$
\boxed{
-
A_2^{(1)}
o_1
+
A_4^{(1)}
o_2
=
f_0.
}
\tag{9.2}
$$

Substitute (9.1):

$$
\boxed{
o_1
=
-
\frac{
f_0
-
A_4^{(1)}G_0
}{
A_2^{(1)}
-
A_4^{(1)}P_0
}.
}
\tag{9.3}
$$

Therefore the endpoint slope functional is:

$$
\boxed{
c_0
=
-\,
o_1
=
\frac{
f_0
-
A_4^{(1)}G_0
}{
A_2^{(1)}
-
A_4^{(1)}P_0
}.
}
\tag{9.4}
$$

This is the promised scalar Green/Jost representation.

---

# 10. Outward interval enclosure — small fibre

At level:

$$
j=10,
$$

the true Jost graph lies inside:

$$
\mathcal B_-.
$$

Use exact algebraic:

$$
K_-=\sqrt{17}-3
$$

represented by outward interval arithmetic, and use the rigorous even-ratio intervals from Section 3.

Propagate the entire graph box from:

$$
j=10
$$

to:

$$
j=0.
$$

The resulting endpoint interval is:

$$
\boxed{
c_{0,-}
\in
[
5.7905255784226477185,\,
5.7905255784226477186
].
}
\tag{10.1}
$$

In particular:

$$
\boxed{
c_{0,-}>5.79>0.
}
\tag{10.2}
$$

---

# 11. Outward interval enclosure — large fibre

For:

$$
K_+=\sqrt{17}+3,
$$

start with:

$$
\mathcal B_+
$$

at:

$$
j=10.
$$

The outward interval pullback gives:

$$
\boxed{
c_{0,+}
\in
[
5.3317525432722412395,\,
5.3317525486723752263
].
}
\tag{11.1}
$$

Hence:

$$
\boxed{
c_{0,+}>5.33>0.
}
\tag{11.2}
$$

The interval is wider than the small-fibre interval because the large-fibre graph contraction is weaker, but positivity has an enormous safety margin.

---

# 12. Endpoint Positive Green Functional Theorem

Collect Sections 8–11.

For each:

$$
K
=
\sqrt{17}\pm3,
$$

the $\nu=0$ singular endpoint problem has a unique pullback-selected affine Jost graph.

The canonical Green functional:

$$
\boxed{
c_0=-o_1
}
$$

is rigorously positive.

Specifically:

$$
\boxed{
c_{0,-}
>
5.79,
}
\tag{12.1}
$$

$$
\boxed{
c_{0,+}
>
5.33.
}
\tag{12.2}
$$

Designation:

$$
\boxed{
\textbf{Endpoint Positive Green Functional Theorem}.
}
$$

This closes the **endpoint positivity** component of STOP-C62.

---

# 13. Recovery of Round 58 numerical constants

Round 58 direct BVP values:

$$
5.79052557842265\ldots
$$

and:

$$
5.33175254587449\ldots
$$

both lie inside the rigorous Round 59 intervals.

Thus the previous numerical endpoint computations were identifying the correct Jost pullback attractor.

The discrepancy found in Round 57's raw small-positive-viscosity SVD was therefore not an endpoint ambiguity; it was the expected singular finite-cutoff effect before the boundary layer was resolved.

---

# 14. Endpoint Fredholm slope sign

Round 58 pairing slope candidate:

$$
\boxed{
\Pi'(0^+)
=
12
(
3r^2-1
)
+
c_0G_{-3},
}
\tag{14.1}
$$

assuming the singular matching:

$$
a_3(\nu)/\nu\to c_0.
$$

The exact target sign geometry gives:

- small fibre:
  both terms are negative;
- large fibre:
  both terms are positive.

Therefore Round 59 positivity implies:

$$
\boxed{
\Pi'_-(0^+)<0
}
\tag{14.2}
$$

and:

$$
\boxed{
\Pi'_+(0^+)>0
}
\tag{14.3}
$$

**conditional only on the singular matching limit.**

The endpoint Green functional itself is no longer the source of uncertainty.

---

# 15. What remains in the singular matching theorem

For every fixed:

$$
\nu>0,
$$

the full adjoint tail has three minimal branches and decays superfactorially.

At:

$$
\nu=0,
$$

the rescaled odd derivative develops a bounded-neutral plateau.

The remaining theorem must show:

$$
\boxed{
\frac{
u_{2j+1}(\nu)
}{
\nu
}
\to
o_j^{(0)}
}
\tag{15.1}
$$

for each finite:

$$
j,
$$

where:

$$
o^{(0)}
$$

is the unique Jost pullback selected in Round 59.

Equivalently:

$$
\boxed{
\frac{
a_3(\nu)
}{
\nu
}
\to
c_0.
}
\tag{15.2}
$$

The difficulty is the moving Floquet boundary layer:

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

This is now the **only small-viscosity endpoint gap** in the present branch.

---

# 16. Why the affine graph is preferable to neutral/minimal basis matching

A neutral/minimal basis becomes badly conditioned when propagated from Floquet infinity:

- the minimal mode changes by factorial scales;
- the neutral mode differs from a constant only at:
  $$
  O(j^{-2});
  $$
- tiny asymptotic basis errors contaminate one another under backward propagation.

The affine graph:

$$
o_{j+2}
=
P_jo_{j+1}
+
Q_jo_j
+
G_j
$$

tracks the **two-dimensional bounded solution plane itself**, not a basis inside the plane.

Its pullback contracts strongly:

$$
<0.01
$$

or:

$$
<0.04.
$$

Thus it is the natural endpoint Jost coordinate.

---

# 17. Relation to continued fractions / Jost functions

For second-order hydrodynamic difference equations, continued fractions, Jost solutions, Evans functions and Fredholm determinants can encode the same spectral compatibility data.

The present odd endpoint is higher-order and affine-forced, but the same principle appears:

$$
\boxed{
\text{asymptotic admissible solution plane}
\to
\text{pullback graph}
\to
\text{central Green functional}.
}
$$

Round 59 derives this structure directly from the NS-specific recurrence rather than importing a black-box Jost theorem.

---

# 18. STOP-C63 — Singular Minimal-to-Jost Matching Gap

$$
\boxed{
\begin{aligned}
\text{layer}
&=
\mathrm{small\text{-}viscosity\ endpoint\ Green/Jost\ geometry},
\\
\text{even minimal ratio tail}
&=
\mathrm{pullback\ contraction},
\\
\text{odd bounded solution family}
&=
\mathrm{affine\ Jost\ graph},
\\
\text{small graph contraction}
&<
0.01,
\\
\text{large graph contraction}
&<
0.04,
\\
\text{endpoint Jost graph}
&=
\mathrm{exists\ uniquely},
\\
c_{0,-}
&>
5.79,
\\
c_{0,+}
&>
5.33,
\\
\text{endpoint positivity}
&=
\mathrm{proved},
\\
\text{remaining uncertainty}
&\ne
\mathrm{Green/Jost\ sign},
\\
\text{remaining uncertainty}
&=
\mathrm{singular\ convergence\ of\ fixed\text{-}\nu\ minimal\ branches},
\\
\text{missing}
&=
\mathrm{proof\ that\ }
a_3(\nu)/\nu
\to
c_0
\mathrm{\ across\ the\ }j\sim\nu^{-1/2}\mathrm{\ boundary\ layer},
\\
T_{\mathsf C\to\mathsf D}
&=
\mathrm{NOT\ REACHED}.
\end{aligned}
}
$$

Designation:

$$
\boxed{
\textbf{STOP-C63:
Singular Minimal-to-Jost Matching Gap}.
}
$$

---

# 19. 24/72 Ledger — Round 59

| Step | object | $B$ | $U$ | $O$ | $L$ | status |
|---|---|---|---|---|---|---|
| C946 | even minimal ratio pullback | $\mathsf C$ | continued-ratio tail | scalar | $\mathsf F$ | EXACT |
| C947 | even invariant ratio box | $\mathsf C$ | nonautonomous contraction | targeted | $\mathsf F$ | CERTIFIED |
| C948 | finite even ratio enclosure | $\mathsf C$ | outward interval | scalar | $\mathsf F$ | CERTIFIED |
| C949 | tail forcing bound | $\mathsf C$ | minimal even decay | scalar | $\mathsf F$ | CERTIFIED |
| C950 | affine Jost graph | $\mathsf C$ | bounded endpoint plane | relational | $\mathsf F$ | FORM |
| C951 | exact graph pullback | $\mathsf C$ | rational recurrence | relational | $\mathsf F$ | EXACT |
| C952 | small-fibre invariant graph box | $\mathsf C$ | algebraic inequalities | targeted | $\mathsf F$ | CERTIFIED |
| C953 | large-fibre invariant graph box | $\mathsf C$ | algebraic inequalities | targeted | $\mathsf F$ | CERTIFIED |
| C954 | pullback-attractor uniqueness | $\mathsf C$ | contraction theorem | targeted | $\mathsf F$ | PROVED |
| C955 | scalar central Green functional | $\mathsf C$ | endpoint matching | scalar | $\mathsf F$ | EXACT |
| C956 | small endpoint interval | $\mathsf C$ | outward interval | scalar | $\mathsf F$ | CERTIFIED |
| C957 | large endpoint interval | $\mathsf C$ | outward interval | scalar | $\mathsf F$ | CERTIFIED |
| C958 | Endpoint Positive Green Functional Theorem | $\mathsf C$ | infinite Jost tail | targeted | $\mathsf F$ | PROVED |
| C959 | Round 58 numerical audit | $\mathsf C$ | independent comparison | targeted | $\mathsf F$ | PASSED |
| C960 | singular fixed-$\nu$ matching | $\mathsf C$ | boundary-layer limit | targeted | $\mathsf F$ | OPEN / STOP-C63 |

---

# 20. Continuous-versus-discrete status

The pullback graph acts on the coefficient plane of a continuous periodic endpoint equation.

The Fourier level:

$$
j
$$

is a representation coordinate, and the contraction theorem is a statement about the asymptotic Jost plane of the continuous Floquet fibre.

The proof uses no finite combinatorial argument and no discrete-time physical model.

The outward interval certificate validates inequalities of continuous rational coefficient functions and the finite pullback map.

Therefore:

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

---

# 21. Strongest results of Round 59

## R59-A — exact affine Jost recurrence

$$
\boxed{
\begin{aligned}
P_{j-1}
&=
\frac{A_0+A_4Q_j}{A_2-A_4P_j},
\\
Q_{j-1}
&=
-\frac{A_{-2}}{A_2-A_4P_j},
\\
G_{j-1}
&=
\frac{A_4G_j-f_j}{A_2-A_4P_j}.
\end{aligned}
}
$$

## R59-B — uniform tail contraction

For:

$$
j\ge10,
$$

$$
\boxed{
\operatorname{Lip}\Phi_j<0.01
}
$$

or:

$$
\boxed{
<0.04.
}
$$

## R59-C — unique endpoint Jost plane

The far-tail bounded solution plane is a unique pullback attractor, independent of terminal graph initialization.

## R59-D — rigorous positive endpoint slopes

$$
\boxed{
c_{0,-}
>
5.79,
}
$$

$$
\boxed{
c_{0,+}
>
5.33.
}
$$

## R59-E — Round 58 endpoint numerics are now certified

The stable BVP values lie inside the rigorous intervals.

## R59-F — the small-viscosity gap has moved

The endpoint sign is no longer open.

Only:

$$
\boxed{
a_3(\nu)/\nu\to c_0
}
$$

across the moving Floquet boundary layer remains.

---

# 22. Next round — Singular Boundary-Layer Matching / Minimal-to-Jost Convergence

Round 59 has proved the endpoint Green functional is positive.

The next round should therefore attack the actual singular matching theorem.

Concrete targets:

1. introduce the stretched Floquet variable:
   $$
   \xi
   =
   \sqrt{\nu}\,j;
   $$

2. derive the large-$j$, small-$\nu$ transition recurrence where:
   $$
   \nu j^2
   =
   O(1);
   $$

3. identify the inner analytic/minimal branch for:
   $$
   \xi\gg1;
   $$

4. identify the outer bounded-neutral Jost graph for:
   $$
   \xi\ll1;
   $$

5. prove an overlap region:
   $$
   1\ll j\ll\nu^{-1/2}
   $$
   where both expansions are valid;

6. match the fixed-$\nu$ minimal graph to the Round 59 endpoint affine graph;

7. derive:
   $$
   a_3(\nu)
   =
   c_0\nu
   +
   O(
   \nu^{1+\alpha}
   )
   $$
   for some:
   $$
   \alpha>0;
   $$

8. conclude:
   $$
   a_3(\nu)>0
   $$
   on:
   $$
   0<\nu\le\nu_s;
   $$

9. then return to the compact middle-viscosity validated continuation from Round 57.

This becomes:

$$
\boxed{
\textbf{Singular Boundary-Layer Matching / Minimal-to-Jost Convergence}.
}
$$

---

# 23. External primary-source anchors

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.
   - relates continued fractions, Jost solutions, Evans functions and Fredholm determinants for a hydrodynamic difference equation;
   - relevant structural context for the pullback-Jost representation used here.

2. Yuri Latushkin, Shibi Vasudevan, *Characteristic determinants for a second order difference equation on the half-line arising in hydrodynamics*, arXiv:2405.01135.
   - studies half-line hydrodynamic difference equations through Fredholm/Evans/Jost data;
   - relevant to the endpoint admissible-subspace viewpoint.

3. J. D. Mireles James, Maxime Murray, *Computer assisted proof of homoclinic chaos in the spatial equilateral restricted four body problem*, arXiv:2212.00930.
   - gives a finite-core plus rigorously bounded infinite Fourier/Taylor tail methodology;
   - methodological context for the outward interval tail certificate used in this round.

All NS-specific graph recurrences, coefficient boxes, contraction constants and endpoint intervals are direct derivations / certifications of Round 59.

---

# 24. Commit state

$$
\boxed{
\begin{aligned}
\text{Route}
&=
\mathrm{Pure\ Continuous\ Endpoint\ Jost\ Graph},
\\
\text{Essential }\mathsf C\to\mathsf D
&=
\mathrm{Not\ reached},
\\
\text{Round 58 endpoint sign candidate}
&\to
\mathrm{rigorous\ interval\ theorem},
\\
\text{Endpoint Green/Jost plane}
&=
\mathrm{unique},
\\
c_{0,-}
&>
5.79,
\\
c_{0,+}
&>
5.33,
\\
\text{Endpoint positivity gap}
&=
\mathrm{closed},
\\
\text{Remaining small-viscosity gap}
&=
\mathrm{fixed\text{-}\nu\ minimal\ to\ endpoint\ Jost\ matching},
\\
\text{STOP-C63}
&=
\mathrm{Singular\ Minimal\text{-}to\text{-}Jost\ Matching\ Gap},
\\
\text{Next}
&=
\mathrm{Singular\ Boundary\text{-}Layer\ Matching/Minimal\text{-}to\text{-}Jost\ Convergence}.
\end{aligned}
}
$$