# NS × X Integration × 24/72 Paradigm in Practice
## Round 54 — Pure Continuous Two-Sided Minimal Floquet Matching / Analytic Fredholm Defect

- Date: 2026-08-17
- Version: v0.1
- Status: Proof-Route Experiment / Continuous-Only Two-Sided Minimal-Range Branch
- canonical source: UTF-8 Markdown
- canonical math delimiters: inline `$...$`; display `$$...$$`
- Previous round: `NS_X72_Round53_PureContinuous_FloquetRescue_TailAsymptotics_v0.1_2026-08-17.md`
- Goal of this round: Round 53 proved that the one-sided causal rescue must enter the factorial-growing branch, and any admissible full rescue must select the minimal branch at Floquet infinity. This round restores the even/odd viscous coupling together, establishes a full six-step asymptotic dichotomy, and directly tests whether the complete second-order source target from Round 51 falls within the two-sided analytic/minimal source range on the **physical divergence-free Fourier quotient**.
- Main results:
  1. The full coupled leading recurrence is of reciprocal degree six;
  2. For each $\nu>0$ and large $|n|$, the leading frozen spectrum has no unit-circle roots, and is exactly split into three growing and three minimal reciprocal branches;
  3. The minimal coefficients are of order $(n!)^{-4/3}$, and the full viscous coupling eliminates the neutral alternating branch present in the Round 53 even-only model;
  4. The compact hidden-block basis indeed possesses representation redundancy, so this round completely switches to using the raw divergence-free Fourier coefficients for the quotient-free physical finite section;
  5. The quotient-free source map stably exhibits two localized source-hidden modes and two localized adjoint cokernel modes;
  6. The complete source target from Round 51 has a stable nonzero projection onto these two adjoint modes;
  7. At $\nu=1$, the normalized minimal-range defects of the two source fibres are approximately $0.9654942609$ and $0.9942037319$.
- Non-claims: Points 5–7 are currently high-precision, truncation-stable Fredholm numerical evidence, and have not yet been upgraded to an infinite-Floquet theorem. The truly rigorous parts of this round are the full coupled leading reciprocal dichotomy and the exact second-order target profile. The next round must construct infinite adjoint minimal solutions and prove that the matching pairing is nonzero before it can be upgraded to a 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.

Named:

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

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.

Named:

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

---

# 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

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

Named:

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

---

# 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

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