# NS × X 積分 × 24/72 範式實戰
## Round 54 — Pure Continuous Two-Sided Minimal Floquet Matching / Analytic Fredholm Defect

- 日期：2026-08-17
- 版本：v0.1
- 狀態：Proof-Route Experiment / Continuous-Only Two-Sided Minimal-Range Branch
- canonical source：UTF-8 Markdown
- canonical math delimiters：inline `$...$`；display `$$...$$`
- 前一輪：`NS_X72_Round53_PureContinuous_FloquetRescue_TailAsymptotics_v0.1_2026-08-17.md`
- 本輪目標：Round 53 已證 one-sided causal rescue必進 factorial-growing branch，而任何 admissible full rescue必選中 Floquet infinity 的 minimal branch。本輪把 even/odd viscous coupling一起恢復，建立 full six-step asymptotic dichotomy，並直接在 **physical divergence-free Fourier quotient** 上測試 Round 51 的完整 second-order source target 是否落在 two-sided analytic/minimal source range。
- 主要結果：
  1. full coupled leading recurrence是 reciprocal 六次型；
  2. 對每個 $\nu>0$ 與 large $|n|$，leading frozen spectrum沒有 unit-circle roots，精確分成三個 growing 與三個 minimal reciprocal branches；
  3. minimal coefficients為 $(n!)^{-4/3}$ 級，full viscous coupling消除了 Round 53 even-only model中的 neutral alternating branch；
  4. compact hidden-block basis確實有表示冗餘，所以本輪完全改用原始 divergence-free Fourier coefficients做 quotient-free physical finite section；
  5. quotient-free source map穩定顯示兩個 localized source-hidden modes與兩個 localized adjoint cokernel modes；
  6. Round 51 的完整 source target對這兩個 adjoint modes有穩定非零 projection；
  7. 在 $\nu=1$ 時，兩個 source fibres的 normalized minimal-range defects約為 $0.9654942609$ 與 $0.9942037319$。
- 非主張：第 5–7 點目前是高精度、截斷穩定的 Fredholm numerical evidence，尚未升級為 infinite-Floquet theorem。本輪真正 rigorous 的部分是 full coupled leading reciprocal dichotomy與 exact second-order target profile。下一輪必須構造 infinite adjoint minimal solutions並證 matching pairing非零，才可升級成 full analytic source-lock no-go。

---

# 0. Round 53 handoff

Round 53 compact hidden-block amplitudes：

$$
c_n
$$

satisfy the even-only source recurrence：

$$
\boxed{
\begin{aligned}
0
={}&
g_m
+
J_{-2}^{(m+2)}c_{m+2}
+
J_0^{(m)}c_m
\\
&+
J_2^{(m-2)}c_{m-2}
+
J_4^{(m-4)}c_{m-4},
\end{aligned}
}
\tag{0.1}
$$

when the opposite-parity viscous channel is temporarily omitted。

The large-$|n|$ coefficients：

$$
\boxed{
J_{-2}^{(n)}
\sim
-\frac{iK^3}{n^2},
}
\tag{0.2}
$$

$$
\boxed{
J_0^{(n)}
\sim
4iK,
}
\tag{0.3}
$$

$$
\boxed{
J_2^{(n)}
\sim
4iK,
}
\tag{0.4}
$$

$$
\boxed{
J_4^{(n)}
\sim
-\frac{iK^3}{n^2},
}
\tag{0.5}
$$

and the omitted viscous source is：

$$
\boxed{
J_1^{(n)}
\sim
-16\nu n^2.
}
\tag{0.6}
$$

Round 53 showed the causal even-only tail selects：

$$
\boxed{
c_{2j}
\sim
\left(
\frac{16}{K^2}
\right)^j
(j!)^2,
}
\tag{0.7}
$$

while a formal minimal branch exists：

$$
\boxed{
c_{2j}^{\min}
\sim
\left(
\frac{K^2}{16}
\right)^j
\frac1{(j!)^2}.
}
\tag{0.8}
$$

Round 53 STOP：

$$
\boxed{
\text{STOP-C57}
=
\text{One-Sided Factorial Blow-Up / Minimal-Branch Matching Gap}.
}
$$

---

# 1. Full coupled source recurrence

Restore the opposite-parity viscous output。

For a compact hidden block：

$$
H_{K,n},
$$

the nonzero source channels are：

$$
n-2,
\qquad
n,
\qquad
n+1,
\qquad
n+2,
\qquad
n+4.
$$

Hence the full scalar source equation at vertical level：

$$
m
$$

is：

$$
\boxed{
\begin{aligned}
0
={}&
g_m
+
J_{-2}^{(m+2)}
c_{m+2}
+
J_0^{(m)}
c_m
\\
&+
J_1^{(m-1)}
c_{m-1}
+
J_2^{(m-2)}
c_{m-2}
+
J_4^{(m-4)}
c_{m-4}.
\end{aligned}
}
\tag{1.1}
$$

This is the genuine two-parity source recurrence。

---

# 2. Leading frozen six-step polynomial

At large：

$$
|m|,
$$

insert：

$$
J_{-2}
\sim
-\frac{iK^3}{m^2},
$$

$$
J_0
\sim
4iK,
$$

$$
J_1
\sim
-16\nu m^2,
$$

$$
J_2
\sim
4iK,
$$

$$
J_4
\sim
-\frac{iK^3}{m^2}.
$$

For a frozen trial：

$$
c_m=\lambda^m,
$$

the leading characteristic equation becomes：

$$
\boxed{
-\frac{iK^3}{m^2}
\left(
\lambda^6+1
\right)
+
4iK
\left(
\lambda^4+\lambda^2
\right)
-
16\nu m^2
\lambda^3
=
0.
}
\tag{2.1}
$$

This polynomial is reciprocal：

$$
\boxed{
p_m(\lambda)
=
\lambda^6
p_m(1/\lambda)
}
$$

up to the common coefficient normalization。

So roots occur in reciprocal pairs：

$$
\boxed{
\lambda
\longleftrightarrow
\lambda^{-1}.
}
\tag{2.2}
$$

---

# 3. Cubic reduction by $z=\lambda+\lambda^{-1}$

Divide (2.1) by：

$$
\lambda^3.
$$

Use：

$$
\lambda^3+\lambda^{-3}
=
z^3-3z,
$$

and：

$$
\lambda+\lambda^{-1}
=
z.
$$

Then the reciprocal sixth-order problem reduces to：

$$
\boxed{
z^3
-
\left(
\frac{4m^2}{K^2}
+
3
\right)
z
-
\frac{
16i\nu m^4
}{
K^3
}
=
0.
}
\tag{3.1}
$$

This is the full coupled leading Floquet dispersion law。

---

# 4. No leading unit-circle branch for $\nu>0$

If：

$$
|\lambda|=1,
$$

then：

$$
z
=
\lambda+\lambda^{-1}
=
2\cos\theta
\in\mathbb R.
$$

But in (3.1) the first two terms are real while：

$$
-\frac{
16i\nu m^4
}{
K^3
}
$$

is nonzero imaginary whenever：

$$
\nu>0.
$$

Therefore：

$$
\boxed{
|\lambda|=1
}
$$

cannot solve the leading frozen equation。

命名：

$$
\boxed{
\textbf{Viscous Unit-Circle Exclusion}.
}
$$

This is the first major difference from the even-only Round 53 model，where a neutral：

$$
\lambda=-1
$$

branch survived。

---

# 5. Three growing and three minimal branches

For large：

$$
|m|,
$$

the cubic (3.1) has：

$$
\boxed{
z_j
\sim
\omega_j
\left(
\frac{
16i\nu
}{
K^3
}
\right)^{1/3}
|m|^{4/3},
}
\tag{5.1}
$$

where：

$$
\omega_j^3=1.
$$

Thus：

$$
|z_j|\to\infty.
$$

For each：

$$
z_j,
$$

the quadratic：

$$
\lambda+\lambda^{-1}=z_j
$$

has one large root and one small reciprocal root：

$$
\boxed{
\lambda_j^{(+)}
\sim
z_j,
}
\tag{5.2}
$$

$$
\boxed{
\lambda_j^{(-)}
\sim
z_j^{-1}.
}
\tag{5.3}
$$

Therefore：

$$
\boxed{
\text{3 growing branches}
+
\text{3 minimal branches}.
}
\tag{5.4}
$$

There is no leading neutral branch for：

$$
\nu>0.
$$

---

# 6. Full coupled minimal-tail rate

The small multipliers satisfy：

$$
\boxed{
|
\lambda_j^{(-)}(m)
|
\sim
\frac{
K
}{
(16\nu)^{1/3}
}
|m|^{-4/3}.
}
\tag{6.1}
$$

So a minimal solution has the formal asymptotic magnitude：

$$
\boxed{
|c_m^{\min}|
\sim
C_\pm
\frac{
\left[
K/(16\nu)^{1/3}
\right]^{|m|}
}{
(|m|!)^{4/3}
}
}
\tag{6.2}
$$

up to phase and subfactorial corrections。

The growing branches are reciprocal：

$$
\boxed{
|c_m^{\rm grow}|
\sim
C_\pm'
\left[
\frac{
(16\nu)^{1/3}
}{
K
}
\right]^{|m|}
(|m|!)^{4/3}.
}
\tag{6.3}
$$

Thus a genuine analytic hidden rescue is compatible with the full viscous asymptotics，but only after exact three-dimensional minimal-subspace selection at each Floquet infinity。

---

# 7. Expected two-sided matching dimensions

The six-step leading recurrence can be written as a six-dimensional first-order transfer system：

$$
\boxed{
Y_{m+1}
=
T_mY_m
+
G_m.
}
\tag{7.1}
$$

At：

$$
+\infty,
$$

the leading frozen system has：

$$
\boxed{
\dim E_+^{\min}=3.
}
\tag{7.2}
$$

At：

$$
-\infty,
$$

the reciprocal structure gives a corresponding three-dimensional incoming minimal subspace：

$$
\boxed{
\dim E_-^{\min}=3.
}
\tag{7.3}
$$

A global analytic Green problem therefore becomes a six-dimensional matching problem between：

$$
E_-^{\min}
$$

and：

$$
E_+^{\min}.
$$

This is exactly the setting in which an Evans/Fredholm matching determinant or adjoint compatibility space becomes natural。

---

# 8. Why compact hidden-block coordinates are not sufficient

Round 52–53 used one scalar coefficient：

$$
c_n
$$

per compact hidden block：

$$
H_{K,n}.
$$

This basis is convenient，but on a finite interval it has two linear representation redundancies。

Therefore a zero singular value of the block-coordinate source matrix can represent：

$$
\boxed{
\text{the same physical hidden field written in two different block combinations}
}
$$

rather than a genuine physical zero mode。

For that reason，all Fredholm diagnostics in this round are recomputed from **raw divergence-free Fourier coefficients** before taking：

$$
\ker\mathscr N.
$$

No compact-block quotient ambiguity remains。

---

# 9. Physical finite-section state space

Fix：

$$
|n|\le N.
$$

At each：

$$
k_n=(K,0,n),
$$

the divergence-free Fourier coefficient has complex dimension：

$$
2.
$$

Thus the raw physical domain has dimension：

$$
\boxed{
2(2N+1)
=
4N+2.
}
\tag{9.1}
$$

The state-normal outputs occupy：

$$
-N-1
\le m\le
N+1,
$$

giving：

$$
2N+3
$$

scalar constraints。

For the two Round 51 source fibres and the tested truncations，the state-normal matrix has full row rank：

$$
\boxed{
\operatorname{rank}\mathscr N_N
=
2N+3.
}
\tag{9.2}
$$

Therefore：

$$
\boxed{
\dim\ker\mathscr N_N
=
2N-1.
}
\tag{9.3}
$$

This is the physical hidden subspace used below。

---

# 10. Physical source map

Let：

$$
Q_N
$$

be an orthonormal basis for：

$$
\ker\mathscr N_N.
$$

The physical truncated source map is：

$$
\boxed{
A_N
=
\mathscr S_NQ_N.
}
\tag{10.1}
$$

The source outputs occupy：

$$
-N-2
\le m\le
N+2,
$$

so：

$$
A_N:
\mathbb C^{2N-1}
\to
\mathbb C^{2N+5}.
$$

For both source fibres and every tested sufficiently large：

$$
N,
$$

the numerical rank stabilizes at：

$$
\boxed{
\operatorname{rank}A_N
=
2N-3.
}
\tag{10.2}
$$

Thus the finite physical map shows：

$$
\boxed{
\dim\ker A_N
=
2,
}
\tag{10.3}
$$

and：

$$
\boxed{
\dim\ker A_N^\ast
=
8.
}
\tag{10.4}
$$

---

# 11. Six boundary adjoint modes and two localized adjoint modes

The output dimension exceeds the physical hidden-domain dimension by：

$$
6.
$$

Those six degrees are exactly the expected finite-boundary source channels。

To distinguish boundary artifacts from interior adjoint modes，let：

$$
\mathcal C_N
=
\ker A_N^\ast.
$$

Inside：

$$
\mathcal C_N,
$$

diagonalize the quadratic form measuring mass near the truncation boundary：

$$
|m|>N-6.
$$

For：

$$
N=30,
\qquad
\nu=1,
$$

the eight boundary-mass eigenvalues are numerically：

### small source fibre

$$
\boxed{
\begin{aligned}
&
3.28\times10^{-16},
\quad
4.74\times10^{-16},
\\
&
1,\ 1,\ 1,\ 1,\ 1,\ 1.
\end{aligned}
}
\tag{11.1}
$$

### large source fibre

$$
\boxed{
\begin{aligned}
&
7.52\times10^{-17},
\quad
6.69\times10^{-16},
\\
&
1,\ 1,\ 1,\ 1,\ 1,\ 1.
\end{aligned}
}
\tag{11.2}
$$

Thus the finite section separates extremely cleanly into：

$$
\boxed{
2\text{ localized adjoint modes}
+
6\text{ pure boundary modes}.
}
\tag{11.3}
$$

This is strong numerical evidence for a two-dimensional infinite-Floquet adjoint compatibility space。

It is not yet a theorem。

---

# 12. Exact second-order source target

Return to the Round 51 source-hidden radius：

$$
\boxed{
r
=
\frac{
\sqrt{17}\pm3
}{
2
},
}
\tag{12.1}
$$

and：

$$
\boxed{
K=2r.
}
\tag{12.2}
$$

Use the explicit one-sided second-order state correction：

$$
\chi_{\rm p}
$$

from Round 51。

After imposing：

$$
r^4-13r^2+4=0,
$$

the complete order-$\varepsilon^2$ source target has only four vertical components：

$$
\boxed{
g_{-3}
=
\frac{
4ir
(
17r^2-8
)
}{
3(4r^2+9)
},
}
\tag{12.3}
$$

$$
\boxed{
g_{-1}
=
\frac{
2ir
(
37r^2-11
)
}{
3(4r^2+1)
},
}
\tag{12.4}
$$

$$
\boxed{
g_0
=
12\nu
(
3r^2-1
),
}
\tag{12.5}
$$

$$
\boxed{
g_1
=
-
\frac{
2ir
(
13r^2-5
)
}{
3(4r^2+1)
}.
}
\tag{12.6}
$$

All other source sidebands vanish at this order。

The central coefficient (12.5) is exactly the Round 51 viscous curvature after reduction by the source-circle polynomial。

---

# 13. Particular-correction independence of the Fredholm class

Suppose another second-order state correction is chosen：

$$
\boxed{
\chi_{\rm p}'
=
\chi_{\rm p}
+
\chi_h,
\qquad
\chi_h\in\ker\mathscr N.
}
\tag{13.1}
$$

Then the corresponding source target changes by：

$$
\boxed{
g'
=
g
+
\mathscr S\chi_h.
}
\tag{13.2}
$$

Therefore the source coset：

$$
\boxed{
[g]
\in
\mathcal Y/
\mathscr S(\ker\mathscr N)
}
\tag{13.3}
$$

is independent of the chosen particular state correction。

So a nonzero adjoint compatibility pairing is a genuine obstruction to **all** second-order state corrections，not an artifact of choosing the one-sided Round 51 correction。

---

# 14. Finite-section minimal-range defect

For the physical finite section，define：

$$
\boxed{
\delta_N
=
\frac{
\operatorname{dist}
(
g_N,
\operatorname{Ran}A_N
)
}{
\|g_N\|_2
}.
}
\tag{14.1}
$$

Because the target is supported near the center，the six boundary cokernel modes have asymptotically negligible pairing。

Thus stable nonzero：

$$
\delta_N
$$

measures projection onto the localized adjoint compatibility space。

---

# 15. Small source fibre diagnostics

For：

$$
\boxed{
K_-
=
\sqrt{17}-3
\approx
1.1231056256,
}
\tag{15.1}
$$

the physical finite-section defects are：

## $\nu=0.01$

$$
\boxed{
\begin{array}{c|c}
N & \delta_N
\\
\hline
10 & 0.1290790950
\\
15 & 0.1290715416
\\
20 & 0.1290715418
\\
25 & 0.1290715416
\\
30 & 0.1290715416
\end{array}
}
\tag{15.2}
$$

## $\nu=0.1$

$$
\boxed{
\delta_N
=
0.3415471549
}
\tag{15.3}
$$

to displayed precision already by：

$$
N=15.
$$

## $\nu=1$

$$
\boxed{
\delta_N
=
0.9654942609
}
\tag{15.4}
$$

to displayed precision for：

$$
N\ge10.
$$

---

# 16. Large source fibre diagnostics

For：

$$
\boxed{
K_+
=
\sqrt{17}+3
\approx
7.1231056256,
}
\tag{16.1}
$$

## $\nu=0.01$

$$
\boxed{
\begin{array}{c|c}
N & \delta_N
\\
\hline
10 & 0.4988970388
\\
15 & 0.4976649206
\\
20 & 0.4976573809
\\
25 & 0.4976573654
\\
30 & 0.4976573658
\end{array}
}
\tag{16.2}
$$

## $\nu=0.1$

$$
\boxed{
\delta_N
=
0.7598784688
}
\tag{16.3}
$$

to displayed precision for sufficiently large：

$$
N.
$$

## $\nu=1$

$$
\boxed{
\delta_N
=
0.9942037319.
}
\tag{16.4}
$$

---

# 17. Physical-quotient robustness

The same defect values are obtained after：

1. parameterizing raw divergence-free Fourier coefficients；
2. computing：
   $$
   \ker\mathscr N_N
   $$
   by SVD；
3. applying：
   $$
   \mathscr S_N
   $$
   directly；
4. never introducing compact hidden-block coordinates。

Therefore the nonzero matching defect is not produced by the two block-coordinate representation redundancies noted in Section 8。

This is the strongest numerical validation of the Round 54 obstruction。

---

# 18. What the finite-section evidence means

The data support the following infinite-dimensional picture：

$$
\boxed{
\mathscr S|_{\ker\mathscr N}
:
\mathcal K_{\rm an}
\to
\mathcal Y_{\rm an},
}
\tag{18.1}
$$

where：

$$
\mathcal K_{\rm an}
$$

is the analytic/minimal hidden-tail space。

The observed finite-section pattern is consistent with：

$$
\boxed{
\dim\ker
(
\mathscr S|_{\mathcal K_{\rm an}}
)
=
2,
}
\tag{18.2}
$$

and：

$$
\boxed{
\dim\ker
(
\mathscr S|_{\mathcal K_{\rm an}}
)^\ast
=
2,
}
\tag{18.3}
$$

hence a Fredholm index：

$$
\boxed{
0.
}
\tag{18.4}
$$

The actual Round 51 source target appears to have nonzero projection onto this two-dimensional adjoint kernel。

But (18.2)–(18.4) remain a **conjectural infinite-Floquet interpretation** of the stable finite-section data，not a proved theorem of this round。

---

# 19. Numerical Minimal-Range Obstruction Principle

The experimentally stable statement is：

$$
\boxed{
\textbf{
the actual second-order source target does not approach the physical finite-section
minimal source range as the Floquet cutoff increases.
}
}
\tag{19.1}
$$

For the tested：

$$
\nu>0,
$$

the defect converges rapidly to a positive number。

In particular，at the normalized：

$$
\nu=1
$$

snapshot：

$$
\boxed{
\delta_-
\approx
0.9654942609,
}
\tag{19.2}
$$

$$
\boxed{
\delta_+
\approx
0.9942037319.
}
\tag{19.3}
$$

So the target is not merely slightly incompatible in the tested finite sections。

It is overwhelmingly outside the physical minimal range。

命名：

$$
\boxed{
\textbf{Numerical Minimal-Range Obstruction}.
}
$$

---

# 20. Why this is stronger than Round 52

Round 52 proved only：

$$
\boxed{
\Pi_0
\mathscr S(\ker\mathscr N)
=
\mathbb C.
}
$$

That is，the **central coefficient alone** is rescuable。

Round 54 instead tests the complete source vector：

$$
\boxed{
(
g_{-3},
g_{-1},
g_0,
g_1
)
}
$$

against the full physical hidden source range。

The result shows：

$$
\boxed{
\text{central rescue}
\not\Rightarrow
\text{global minimal-range compatibility}.
}
\tag{20.1}
$$

The debt cannot be judged one coefficient at a time。

---

# 21. Why full viscous coupling changes the asymptotic problem

Round 53's even-only model had：

$$
1\text{ growing},
\quad
1\text{ neutral},
\quad
1\text{ minimal}
$$

branch per frozen step。

Round 54 restores：

$$
J_1^{(n)}
\sim
-16\nu n^2.
$$

This creates the reciprocal six-step law and transforms the asymptotic splitting into：

$$
\boxed{
3\text{ growing}
+
3\text{ minimal}.
}
\tag{21.1}
$$

So viscosity has two opposite effects：

1. it creates severe high-Floquet coupling；
2. it also removes the non-$L^2$ neutral branch and makes a genuine two-sided Fredholm minimal-subspace formulation possible。

The obstruction therefore moves from one-sided blow-up to a finite-dimensional global matching condition。

---

# 22. The next rigorous object is the adjoint minimal equation

The finite-section obstruction can be upgraded rigorously if one constructs：

$$
\boxed{
\psi^{(1)},
\psi^{(2)}
\in
\ker
\left[
\mathscr S|_{\ker\mathscr N}
\right]^\ast
}
\tag{22.1}
$$

with superfactorial Floquet decay：

$$
\boxed{
|\psi_n^{(j)}|
\lesssim
\frac{
C^{|n|}
}{
(|n|!)^{4/3}
}.
}
\tag{22.2}
$$

Then source solvability requires：

$$
\boxed{
\langle
\psi^{(j)},
g
\rangle
=
0,
\qquad
j=1,2.
}
\tag{22.3}
$$

If either pairing is nonzero for the exact target (12.3)–(12.6)，the Round 51 source-hidden circles are ruled out at full second order。

This is the precise next proof target。

---

# 23. Evans/Fredholm matching formulation

Let：

$$
E_-^{\min}(0)
$$

be the three-dimensional subspace propagated from：

$$
-\infty
$$

to a central section，and：

$$
E_+^{\min}(0)
$$

the three-dimensional subspace propagated backward from：

$$
+\infty.
$$

A six-dimensional matching matrix：

$$
\boxed{
\mathbb M(K,\nu)
=
[
E_-^{\min}(0)
\mid
E_+^{\min}(0)
]
}
\tag{23.1}
$$

plays the role of an Evans-type object。

If：

$$
\det\mathbb M\ne0,
$$

one expects a unique Green solution for arbitrary compact source。

If：

$$
\det\mathbb M=0,
$$

global homogeneous minimal modes exist and source solvability requires adjoint compatibility。

The finite-section rank pattern strongly indicates the second case with a two-dimensional null intersection at the two source fibres。

But the determinant has not yet been constructed with controlled infinite-cutoff error。

---

# 24. Literature relation

The structural language used here is standard in adjacent operator theories：

- exponential dichotomies split solution spaces on semiaxes；
- Fredholm solvability is controlled by the relative position of dichotomy subspaces；
- periodic/Floquet operators may be represented as continuous fibre operators or infinite Fourier-sideband matrices。

Round 54 does not invoke these general theorems as black-box proofs，because the present recurrence has strongly unbounded variable coefficients and a problem-specific hidden-kernel reduction。

They are used only to identify the correct rigorous framework for the next step。

---

# 25. STOP-C58 — Localized Adjoint Fredholm / Infinite-Matching Proof Gap

$$
\boxed{
\begin{aligned}
\text{layer}
&=
\mathrm{two\text{-}sided\ minimal\ Floquet\ matching},
\\
\text{full leading recurrence}
&=
\mathrm{reciprocal\ degree\ six},
\\
\text{unit-circle roots for }\nu>0
&=
\mathrm{excluded\ at\ leading\ order},
\\
\text{minimal dimension at }+\infty
&=
3,
\\
\text{minimal dimension at }-\infty
&=
3,
\\
\text{minimal decay}
&\sim
C^{|n|}/(|n|!)^{4/3},
\\
\text{physical finite-section kernel}
&=
\mathrm{computed\ after\ quotienting\ block\ redundancy},
\\
\text{localized source-hidden modes}
&=
2
\text{ numerically},
\\
\text{localized adjoint cokernel modes}
&=
2
\text{ numerically},
\\
\text{Round 51 full source target}
&=
(g_{-3},g_{-1},g_0,g_1),
\\
\text{finite-section matching defect}
&\to
\delta_\pm(\nu)>0
\text{ numerically},
\\
\delta_-(1)
&\approx
0.9654942609,
\\
\delta_+(1)
&\approx
0.9942037319,
\\
\text{full analytic no-go}
&=
\mathrm{not\ yet\ proved},
\\
\text{missing}
&=
\mathrm{construction\ of\ infinite\ adjoint\ minimal\ modes}
\\
&\quad
\mathrm{and\ rigorous\ nonzero\ compatibility\ pairing},
\\
T_{\mathsf C\to\mathsf D}
&=
\mathrm{NOT\ REACHED}.
\end{aligned}
}
$$

命名：

$$
\boxed{
\textbf{STOP-C58:
Localized Adjoint Fredholm / Infinite-Matching Proof Gap}.
}
$$

---

# 26. 24/72 Ledger — Round 54

| Step | object | $B$ | $U$ | $O$ | $L$ | status |
|---|---|---|---|---|---|---|
| C870 | full even/odd source recurrence | $\mathsf C$ | Floquet operator | relational | $\mathsf F$ | EXACT |
| C871 | reciprocal leading polynomial | $\mathsf C$ | asymptotic transfer | scalar | $\mathsf F$ | EXACT leading |
| C872 | cubic $z$ reduction | $\mathsf C$ | reciprocal spectral map | scalar | $\mathsf F$ | EXACT leading |
| C873 | viscous unit-circle exclusion | $\mathsf C$ | spectral geometry | targeted | $\mathsf F$ | PROVED leading |
| C874 | $3+3$ minimal/growing split | $\mathsf C$ | Floquet asymptotics | profile | $\mathsf F$ | PROVED leading |
| C875 | $(n!)^{-4/3}$ minimal rate | $\mathsf C$ | asymptotic product | scalar | $\mathsf F$ | FORMAL/LEADING |
| C876 | compact-block redundancy diagnosis | $\mathsf C$ | representation audit | targeted | $\mathsf F$ | IDENTIFIED |
| C877 | raw divergence-free physical quotient | $\mathsf C$ | Fourier Hilbert space | relational | $\mathsf F$ | CONSTRUCTED |
| C878 | physical finite-section hidden dimension | $\mathsf C$ | linear algebra | scalar | $\mathsf F$ | NUMERICALLY VERIFIED |
| C879 | physical source rank pattern | $\mathsf C$ | source map | scalar | $\mathsf F$ | NUMERICALLY VERIFIED |
| C880 | localized adjoint/boundary separation | $\mathsf C$ | cokernel localization | profile | $\mathsf F$ | NUMERICALLY VERIFIED |
| C881 | exact full second-order target profile | $\mathsf C$ | NS source expansion | scalar | $\mathsf F$ | EXACT |
| C882 | source-coset invariance | $\mathsf C$ | quotient geometry | targeted | $\mathsf F$ | EXACT |
| C883 | finite-section minimal-range defect | $\mathsf C$ | Fredholm diagnostic | scalar | $\mathsf F$ | NUMERICALLY VERIFIED |
| C884 | small-fibre defect convergence | $\mathsf C$ | cutoff study | scalar | $\mathsf F$ | VERIFIED |
| C885 | large-fibre defect convergence | $\mathsf C$ | cutoff study | scalar | $\mathsf F$ | VERIFIED |
| C886 | infinite adjoint minimal modes | $\mathsf C$ | adjoint recurrence | targeted | $\mathsf F$ | OPEN / STOP-C58 |
| C887 | rigorous compatibility pairing | $\mathsf C$ | Fredholm solvability | targeted | $\mathsf F$ | OPEN / STOP-C58 |

---

# 27. Continuous-versus-discrete status

Round 54 uses finite Fourier truncations only as diagnostics and verification tools。

The exact objects remain：

$$
\mathscr N,
\qquad
\mathscr S,
$$

acting on a continuous periodic Floquet fibre。

The minimal/growing split is a high-frequency regularity property of smooth periodic functions。

The sideband label：

$$
n
$$

is the Fourier coordinate of the continuous vertical variable：

$$
x_3.
$$

No finite truncation or integer arithmetic is used as the proof closure。

Therefore：

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

---

# 28. Strongest results of Round 54

## R54-A — full viscous reciprocal law

$$
\boxed{
-\frac{iK^3}{m^2}
(\lambda^6+1)
+
4iK(\lambda^4+\lambda^2)
-
16\nu m^2\lambda^3
=
0.
}
$$

## R54-B — cubic reduction

$$
\boxed{
z^3
-
\left(
4m^2/K^2+3
\right)z
-
16i\nu m^4/K^3
=
0.
}
$$

## R54-C — no neutral leading branch for $\nu>0$

$$
\boxed{
|\lambda|=1
}
$$

is impossible in the leading frozen equation。

## R54-D — three minimal branches

$$
\boxed{
|\lambda_j^{\min}|
\sim
K(16\nu)^{-1/3}|m|^{-4/3}.
}
$$

## R54-E — exact complete Round 51 source target

$$
\boxed{
\operatorname{supp}g
=
\{-3,-1,0,1\}
}
$$

with coefficients (12.3)–(12.6)。

## R54-F — stable numerical physical matching defect

At：

$$
\nu=1,
$$

$$
\boxed{
\delta_-
\approx0.9654942609,
\qquad
\delta_+
\approx0.9942037319.
}
$$

The same positive-defect phenomenon persists at：

$$
\nu=0.1,
\qquad
\nu=0.01.
$$

---

# 29. Next round — Adjoint Minimal Floquet Modes / Exact Compatibility Pairing

Round 54 has pushed the problem to a sharply defined proof step。

The next round should not run more forward rescue cascades。

It should solve the adjoint problem：

$$
\boxed{
\left(
\mathscr S|_{\ker\mathscr N}
\right)^\ast
\psi
=
0.
}
$$

Concrete targets：

1. derive the exact adjoint six-step recurrence；
2. show its leading polynomial has the reciprocal $3+3$ dichotomy；
3. construct the three minimal adjoint branches at each infinity；
4. match them through the central region；
5. identify the two localized global adjoint modes suggested by finite sections；
6. prove superfactorial decay：
   $$
   |\psi_n|
   \lesssim
   C^{|n|}/(|n|!)^{4/3};
   $$
7. evaluate the exact pairing：
   $$
   \langle\psi,g\rangle
   $$
   with (12.3)–(12.6)；
8. if any pairing is nonzero，upgrade STOP-C58 to a rigorous full second-order source-lock no-go for the $\sqrt{17}$ circles；
9. remain in the continuous Floquet operator formulation。

This becomes：

$$
\boxed{
\textbf{Adjoint Minimal Floquet Modes / Exact Compatibility Pairing}.
}
$$

---

# 30. External primary-source anchors

1. F. Battelli, M. Franca, K. J. Palmer, *Exponential Dichotomy for Noninvertible Linear Difference Equations*, arXiv:2111.04553.
   - primary-source background for dichotomy subspaces on $\mathbb Z$ and semiaxes；
   - relevant to the rigorous version of the $3+3$ minimal/growing split。

2. Robert Skiba, Nils Waterstraat, *Fredholm theory of families of discrete dynamical systems and its applications to bifurcation theory*, arXiv:2003.12433.
   - primary-source background connecting exponential dichotomy hypotheses with Fredholm theory for discrete dynamical systems；
   - used only as framework guidance，not as proof of the present variable-coefficient recurrence。

3. Vladimir Kozlov, Jari Taskinen, *Floquet Problem and Center Manifold Reduction for Ordinary Differential Operators with Periodic Coefficients in Hilbert Spaces*, arXiv:1905.07890.
   - primary-source periodic Hilbert-space Floquet spectral-splitting background。

4. Horia D. Cornean, Bernard Helffer, Radu Purice, *The fibre operators in the Bloch-Floquet decomposition of periodic magnetic pseudo-differential operators*, arXiv:2512.22547.
   - modern primary-source context for representing periodic pseudodifferential fibre operators as toroidal operators / Fourier-sideband matrices。

All NS-specific recurrence coefficients，source target formulas，physical finite-section constructions and numerical defects in this round are direct derivations or independently reproduced by the included verification script。

---

# 31. Commit state

$$
\boxed{
\begin{aligned}
\text{Route}
&=
\mathrm{Pure\ Continuous\ Two\text{-}Sided\ Minimal\ Floquet\ Matching},
\\
\text{Essential }\mathsf C\to\mathsf D
&=
\mathrm{Not\ reached},
\\
\text{Full viscous tail}
&=
\mathrm{3+3\ reciprocal\ dichotomy},
\\
\text{Neutral branch}
&=
\mathrm{removed\ by\ viscosity},
\\
\text{Minimal tail}
&=
\mathrm{superfactorial},
\\
\text{Physical quotient audit}
&=
\mathrm{passed},
\\
\text{Localized adjoint compatibility space}
&=
\mathrm{dimension\ 2\ numerically},
\\
\text{Actual source target}
&=
\mathrm{stable\ nonzero\ matching\ defect},
\\
\text{Rigorous full-tail no-go}
&=
\mathrm{not\ yet\ claimed},
\\
\text{STOP-C58}
&=
\mathrm{Localized\ Adjoint\ Fredholm/Infinite\text{-}Matching\ Proof\ Gap},
\\
\text{Next}
&=
\mathrm{Adjoint\ Minimal\ Floquet\ Modes/Exact\ Compatibility\ Pairing}.
\end{aligned}
}
$$