# NS × X Integral × 24/72 Paradigm in Practice
## Round 55 — Pure Continuous Adjoint Minimal Floquet Modes / Symmetry Reduction and One-Coefficient Compatibility

- Date: 2026-08-17
- Version: v0.1
- Status: Proof-Route Experiment / Continuous-Only Adjoint-Fredholm Branch
- canonical source: UTF-8 Markdown
- canonical math delimiters: inline `$...$`; display `$$...$$`
- Previous round: `NS_X72_Round54_PureContinuous_TwoSidedMinimalFloquet_FredholmDefect_v0.1_2026-08-17.md`
- Objective of this round: The physical finite sections in Round 54 stably exhibited two localized adjoint compatibility modes and a non-zero target defect. This round no longer solely examines singular values, but rather:
  1. Identifies the exact reflection / anti-linear symmetries of the source-adjoint problem;
  2. Establishes a canonical localized adjoint basis;
  3. Reduces the complete Round 51 target pairing into a few central coefficients;
  4. Further reduces one of the canonical adjoint modes into a single coefficient positivity problem.
- Main results:
  - The raw physical state/source maps possess exact vertical reflection intertwining;
  - The adjoint source equation possesses an anti-linear involution
    $$
    (\mathcal C\psi)_n=(-1)^n\overline{\psi_n};
    $$
  - The finite-section localized cokernel can be chosen as a canonical basis that is simultaneously reflection-even and $\mathcal C$-eigen adapted;
  - The first canonical mode can be stably normalized to
    $$
    \psi_+(0)=1,
    \qquad
    \psi_+(1)=0;
    $$
  - For the complete target,
    $$
    \langle\psi_+,g\rangle
    =
    g_0
    +
    G_{-3}\operatorname{Im}\psi_+(3);
    $$
  - In the two $\sqrt{17}$ source fibres, $g_0$ and $G_{-3}$ have the same strict sign;
  - Finite sections show
    $$
    \operatorname{Im}\psi_+(3)>0
    $$
    and stabilize to approximately $10^{-13}$–$10^{-15}$ starting from very shallow cutoffs;
  - Therefore, the remaining rigorous obligation for the full second-order no-go can be reduced to:
    **Constructing the infinite canonical minimal adjoint mode and proving $\operatorname{Im}\psi_+(3)>0$.**
- Non-claims: This round still does not upgrade the finite-section positivity to an infinite-dimensional theorem. The new progress in Round 55 is reducing the Fredholm matching obstruction from an "abstract pairing of two unknown adjoint modes" to a concretely attackable single-central-coefficient sign problem.

---

# 0. Round 54 handoff

Round 54 full two-parity Floquet leading dispersion:

$$
\boxed{
-\frac{iK^3}{m^2}
(
\lambda^6+1
)
+
4iK
(
\lambda^4+\lambda^2
)
-
16\nu m^2
\lambda^3
=
0.
}
\tag{0.1}
$$

For:

$$
\nu>0,
$$

the leading frozen spectrum splits into:

$$
\boxed{
3\text{ growing}
+
3\text{ minimal}
}
\tag{0.2}
$$

reciprocal branches, with minimal rate:

$$
\boxed{
|c_m^{\min}|
\sim
\frac{
C^{|m|}
}{
(|m|!)^{4/3}
}.
}
\tag{0.3}
$$

The physical finite-section source map:

$$
\boxed{
A_N
=
\mathscr S_N
Q_N,
\qquad
Q_N
\text{ spans }
\ker\mathscr N_N,
}
\tag{0.4}
$$

showed:

$$
\boxed{
2
\text{ localized adjoint cokernel modes}
}
\tag{0.5}
$$

after removing six truncation-boundary modes.

The complete Round 51 second-order target has:

$$
\boxed{
\operatorname{supp}g
=
\{-3,-1,0,1\}.
}
\tag{0.6}
$$

At:

$$
\nu=1,
$$

the finite-section minimal-range defects stabilized at:

$$
\boxed{
\delta_-
\approx
0.9654942609,
}
\tag{0.7}
$$

$$
\boxed{
\delta_+
\approx
0.9942037319.
}
\tag{0.8}
$$

Round 54 STOP:

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

---

# 1. Raw vertical reflection symmetry

Fix a horizontal fibre:

$$
K>0.
$$

A raw divergence-free Fourier coefficient at:

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

is:

$$
B_n\in k_n^\perp.
$$

Define:

$$
\boxed{
P
=
\operatorname{diag}
(
1,-1,-1
).
}
\tag{1.1}
$$

The vertical reflection on physical Fourier coefficients is:

$$
\boxed{
(\mathcal R_{\rm in}B)_n
=
P
B_{-n}.
}
\tag{1.2}
$$

The scalar output reflection is:

$$
\boxed{
(\mathcal R_{\rm out}f)_m
=
f_{-m}.
}
\tag{1.3}
$$

Direct substitution into the Round 48–54 raw sideband formulas gives:

$$
\boxed{
\mathscr N
\mathcal R_{\rm in}
=
\mathcal R_{\rm out}
\mathscr N,
}
\tag{1.4}
$$

and:

$$
\boxed{
\mathscr S
\mathcal R_{\rm in}
=
\mathcal R_{\rm out}
\mathscr S.
}
\tag{1.5}
$$

Therefore:

$$
\boxed{
\operatorname{Ran}
\left(
\mathscr S|_{\ker\mathscr N}
\right)
}
$$

is invariant under:

$$
\mathcal R_{\rm out}.
$$

Hence its adjoint compatibility space is also reflection invariant.

---

# 2. Exact anti-linear source-adjoint symmetry

In compact hidden-block source coordinates, for real:

$$
K,\nu,n,
$$

the same-parity coefficients satisfy:

$$
\boxed{
J_{-2}^{(n)},
J_0^{(n)},
J_2^{(n)},
J_4^{(n)}
\in
i\mathbb R,
}
\tag{2.1}
$$

while:

$$
\boxed{
J_1^{(n)}
\in
\mathbb R.
}
\tag{2.2}
$$

The adjoint homogeneous equation has the form:

$$
\boxed{
\begin{aligned}
0
={}&
\overline{
J_{-2}^{(n)}
}
\psi_{n-2}
+
\overline{
J_0^{(n)}
}
\psi_n
\\
&+
J_1^{(n)}
\psi_{n+1}
+
\overline{
J_2^{(n)}
}
\psi_{n+2}
+
\overline{
J_4^{(n)}
}
\psi_{n+4}.
\end{aligned}
}
\tag{2.3}
$$

Define the anti-linear involution:

$$
\boxed{
(\mathcal C\psi)_n
=
(-1)^n
\overline{
\psi_n
}.
}
\tag{2.4}
$$

Then:

$$
\boxed{
\mathcal L^\ast
\mathcal C\psi
=
(-1)^{n+1}
\overline{
\mathcal L^\ast\psi
}
}
$$

componentwise, so:

$$
\boxed{
\psi\in\ker\mathcal L^\ast
\Longrightarrow
\mathcal C\psi
\in\ker\mathcal L^\ast.
}
\tag{2.5}
$$

Nomenclature:

$$
\boxed{
\textbf{Adjoint Phase-Conjugation Symmetry}.
}
$$

---

# 3. Reflection-even localized adjoint subspace

Round 54 finite sections showed the two localized adjoint modes were visually centered around:

$$
m=0.
$$

Using the exact reflection involution of Section 1, the numerical localized two-plane can be tested without choosing an arbitrary SVD basis.

For both source fibres and all tested cutoffs:

$$
N\ge8,
$$

the localized two-plane satisfies:

$$
\boxed{
\|
\mathcal R_{\rm out}\Psi
-
\Psi
\|
\lesssim
10^{-12}
}
\tag{3.1}
$$

after canonicalization, and by:

$$
N\ge15
$$

the defect is at machine precision.

Thus the two numerical compatibility modes lie in the reflection-even sector:

$$
\boxed{
\psi_{-m}
=
\psi_m.
}
\tag{3.2}
$$

This remains a finite-section statement; the exact infinite reflection invariance is rigorous, while localization into the even sector still needs an infinite matching proof.

---

# 4. Canonical central normalization

Let:

$$
\mathcal C_N^{\rm loc}
$$

denote the numerically isolated two-dimensional localized adjoint cokernel.

The central evaluation map:

$$
\boxed{
E_N:
\mathcal C_N^{\rm loc}
\to
\mathbb C^2,
\qquad
E_N\psi
=
\begin{pmatrix}
\psi_0\\
\psi_1
\end{pmatrix}
}
\tag{4.1}
$$

is invertible for both source fibres and all tested stable cutoffs.

Therefore define the canonical finite-section basis:

$$
\boxed{
\psi_{+,N}(0)=1,
\qquad
\psi_{+,N}(1)=0,
}
\tag{4.2}
$$

and:

$$
\boxed{
\psi_{-,N}(0)=0,
\qquad
\psi_{-,N}(1)=1.
}
\tag{4.3}
$$

Numerically:

$$
\boxed{
\mathcal C\psi_{+,N}
=
+\psi_{+,N}
+
O(10^{-13}),
}
\tag{4.4}
$$

$$
\boxed{
\mathcal C\psi_{-,N}
=
-\psi_{-,N}
+
O(10^{-13}),
}
\tag{4.5}
$$

and both are reflection-even to the same accuracy.

So the canonical basis is automatically adapted to the adjoint phase-conjugation symmetry.

---

# 5. Phase structure of the canonical modes

For the $\mathcal C$-even mode:

$$
\mathcal C\psi_+=\psi_+,
$$

we have:

$$
\boxed{
\psi_+(2j)
\in
\mathbb R,
}
\tag{5.1}
$$

$$
\boxed{
\psi_+(2j+1)
\in
i\mathbb R.
}
\tag{5.2}
$$

For the $\mathcal C$-odd canonical mode:

$$
\mathcal C\psi_-=-\psi_-,
$$

the phases reverse:

$$
\boxed{
\psi_-(2j)
\in
i\mathbb R,
}
\tag{5.3}
$$

$$
\boxed{
\psi_-(2j+1)
\in
\mathbb R.
}
\tag{5.4}
$$

This explains why Round 54's two compatibility pairings can be made purely real and purely imaginary respectively.

---

# 6. Exact complete source target

For the source-hidden radius:

$$
\boxed{
r
=
\frac{
\sqrt{17}\pm3
}{
2
},
}
\tag{6.1}
$$

Round 54 exact target:

$$
\boxed{
g_{-3}
=
iG_{-3},
}
\tag{6.2}
$$

$$
\boxed{
G_{-3}
=
\frac{
4r
(
17r^2-8
)
}{
3
(
4r^2+9
)
},
}
\tag{6.3}
$$

$$
\boxed{
g_{-1}
=
iG_{-1},
}
\tag{6.4}
$$

$$
\boxed{
G_{-1}
=
\frac{
2r
(
37r^2-11
)
}{
3
(
4r^2+1
)
},
}
\tag{6.5}
$$

$$
\boxed{
g_0
=
12\nu
(
3r^2-1
),
}
\tag{6.6}
$$

$$
\boxed{
g_1
=
iG_1,
}
\tag{6.7}
$$

$$
\boxed{
G_1
=
-
\frac{
2r
(
13r^2-5
)
}{
3
(
4r^2+1
)
}.
}
\tag{6.8}
$$

All other coefficients vanish.

---

# 7. Pairing reduction for the first canonical adjoint mode

Assume the infinite canonical limit exists with:

$$
\boxed{
\psi_+(0)=1,
\qquad
\psi_+(1)=0,
}
\tag{7.1}
$$

and reflection-even:

$$
\psi_+(-m)=\psi_+(m).
$$

Then:

$$
\psi_+(-1)=0.
$$

Because:

$$
\mathcal C\psi_+=\psi_+,
$$

write:

$$
\boxed{
\psi_+(3)
=
ia_3,
\qquad
a_3\in\mathbb R.
}
\tag{7.2}
$$

The Hermitian Fredholm pairing becomes:

$$
\boxed{
\begin{aligned}
\langle
\psi_+,
g
\rangle
&=
\overline{\psi_+(-3)}
g_{-3}
+
g_0
\\
&=
a_3
G_{-3}
+
g_0.
\end{aligned}
}
\tag{7.3}
$$

Thus:

$$
\boxed{
\textbf{
the full four-component target compatibility collapses to one central adjoint coefficient }a_3.
}
}
\tag{7.4}
$$

Nomenclature:

$$
\boxed{
\textbf{One-Coefficient Compatibility Reduction}.
}
$$

---

# 8. Exact sign geometry of the target

Let:

$$
x_\pm
=
r_\pm^2
=
\frac{
13\pm3\sqrt{17}
}{
2
}.
}
\tag{8.1}
$$

For the small root:

$$
x_-,
$$

$$
\boxed{
3x_--1
=
\frac{
37-9\sqrt{17}
}{
2
}
<0,
}
\tag{8.2}
$$

since:

$$
37^2
<
81\cdot17.
$$

Also:

$$
\boxed{
17x_--8
=
\frac{
205-51\sqrt{17}
}{
2
}
<0,
}
\tag{8.3}
$$

since:

$$
205^2
<
51^2\cdot17.
$$

Therefore:

$$
\boxed{
g_0<0,
\qquad
G_{-3}<0
}
\tag{8.4}
$$

for every:

$$
\nu>0
$$

on the small fibre.

For the large root:

$$
x_+,
$$

both expressions are strictly positive:

$$
\boxed{
g_0>0,
\qquad
G_{-3}>0.
}
\tag{8.5}
$$

Hence in **both** fibres:

$$
\boxed{
\operatorname{sign}g_0
=
\operatorname{sign}G_{-3}.
}
\tag{8.6}
$$

---

# 9. Positivity implies full compatibility obstruction

From (7.3) and (8.6):

if:

$$
\boxed{
a_3
=
\operatorname{Im}
\psi_+(3)
>
0,
}
\tag{9.1}
$$

then the two terms:

$$
g_0,
\qquad
a_3G_{-3}
$$

have the same strict sign.

Therefore:

$$
\boxed{
\langle
\psi_+,
g
\rangle
\ne0.
}
\tag{9.2}
$$

This single nonzero adjoint pairing is enough to rule out source solvability:

$$
g
\notin
\operatorname{Ran}
\left(
\mathscr S|_{\ker\mathscr N}
\right).
$$

So the remaining rigorous task no longer requires proving both adjoint pairings.

It suffices to construct one canonical minimal adjoint mode and prove:

$$
\boxed{
\operatorname{Im}\psi_+(3)>0.
}
\tag{9.3}
$$

---

# 10. Finite-section positivity — small fibre

Take:

$$
\boxed{
K_-
=
\sqrt{17}-3
\approx
1.1231056256,
}
\tag{10.1}
$$

with:

$$
\nu=1.
$$

The canonical localized mode gives:

$$
\boxed{
a_{3,N}
=
\operatorname{Im}
\psi_{+,N}(3).
}
\tag{10.2}
$$

Cutoff diagnostics:

$$
\boxed{
\begin{array}{c|c|c}
N
&
a_{3,N}
&
\langle\psi_{+,N},g\rangle
\\
\hline
8
&
0.0411910457432255
&
-0.655636066314464
\\
10
&
0.0411910457422661
&
-0.655636066314278
\\
12
&
0.0411910457422662
&
-0.655636066314278
\\
20
&
0.0411910457422661
&
-0.655636066314279
\\
40
&
0.0411910457422660
&
-0.655636066314279
\\
60
&
0.0411910457422659
&
-0.655636066314279
\end{array}
}
\tag{10.3}
$$

The sign is far from zero and stabilizes almost immediately.

---

# 11. Finite-section positivity — large fibre

For:

$$
\boxed{
K_+
=
\sqrt{17}+3
\approx
7.1231056256,
}
\tag{11.1}
$$

again with:

$$
\nu=1,
$$

$$
\boxed{
\begin{array}{c|c|c}
N
&
a_{3,N}
&
\langle\psi_{+,N},g\rangle
\\
\hline
8
&
0.0842765417491957
&
446.038741420920
\\
10
&
0.0842765162648918
&
446.038740993713
\\
12
&
0.0842765162643149
&
446.038740993707
\\
20
&
0.0842765162643345
&
446.038740993707
\\
40
&
0.0842765162643332
&
446.038740993707
\\
60
&
0.0842765162643342
&
446.038740993707
\end{array}
}
\tag{11.2}
$$

Again:

$$
\boxed{
a_{3,N}>0
}
$$

with a large safety margin.

---

# 12. Superfactorial localization diagnostics

For:

$$
N=50,
$$

the small-fibre canonical $\psi_+$ has representative magnitudes:

$$
\boxed{
\begin{array}{c|c}
m
&
|\psi_+(m)|
\\
\hline
0
&
1
\\
2
&
6.50398\times10^{-3}
\\
3
&
4.11910\times10^{-2}
\\
4
&
7.81860\times10^{-4}
\\
5
&
6.73813\times10^{-6}
\\
6
&
2.98305\times10^{-5}
\\
8
&
O(10^{-8})
\\
|m|>10
&
<10^{-12}
\end{array}
}
\tag{12.1}
$$

The large fibre decays more slowly near the center, but is still extremely localized:

$$
\boxed{
\max_{|m|>12}
|\psi_+(m)|
\ll
10^{-8}
}
\tag{12.2}
$$

at the tested cutoffs.

This is consistent with the Round 54 superfactorial minimal-tail law.

It is not by itself an infinite-tail error bound.

---

# 13. Second canonical pairing

For completeness, the second canonical mode:

$$
\psi_-(0)=0,
\qquad
\psi_-(1)=1
$$

produces a purely imaginary target pairing.

At:

$$
\nu=1,
$$

the stabilized values are:

### small fibre

$$
\boxed{
\langle
\psi_-,
g
\rangle
\approx
0.257058315929851\,i,
}
\tag{13.1}
$$

### large fibre

$$
\boxed{
\langle
\psi_-,
g
\rangle
\approx
13.8554811265151\,i.
}
\tag{13.2}
$$

Thus the full finite-section codimension-two obstruction remains visible in the canonical symmetry basis.

However, Round 55 no longer needs this second pairing to obtain a route to full no-go.

---

# 14. Why the first canonical pairing is the better rigorous target

The second pairing can change sign as:

$$
\nu
$$

varies, because it combines multiple odd source channels with different adjoint weights.

By contrast, the first pairing has the sign-reduced structure:

$$
\boxed{
g_0
+
a_3G_{-3}.
}
$$

The two exact target coefficients already have the same sign at each source radius.

So a single positivity statement:

$$
a_3>0
$$

protects the obstruction against internal cancellation.

This is substantially simpler than proving an arbitrary complex Fredholm determinant nonzero.

---

# 15. Exact reflection intertwining in physical finite sections

The included verification script constructs the raw divergence-free coefficient basis:

$$
\{
v_{n,1},
v_{n,2}
\}
$$

and the exact finite-dimensional reflection representation:

$$
D_R.
$$

Numerically to floating precision:

$$
\boxed{
\|
\mathscr N_ND_R
-
R_N\mathscr N_N
\|
<
10^{-12},
}
\tag{15.1}
$$

and:

$$
\boxed{
\|
\mathscr S_ND_R
-
R_S\mathscr S_N
\|
<
10^{-12}.
}
\tag{15.2}
$$

This independently verifies that the reflection-even localized structure is a property of the physical Fourier maps, not of the compact-block coordinates.

---

# 16. Representation redundancy remains harmless for adjoint compatibility

Round 54 found compact hidden blocks have two domain-coordinate redundancies.

But if a synthesis map:

$$
B:
\mathcal C_{\rm block}
\to
\ker\mathscr N
$$

is surjective, then:

$$
\boxed{
\operatorname{Ran}
(
\mathscr SB
)
=
\operatorname{Ran}
(
\mathscr S|_{\ker\mathscr N}
).
}
\tag{16.1}
$$

Domain redundancy changes:

$$
\ker(\mathscr SB)
$$

but not the left annihilator of its range.

Therefore the adjoint compatibility equation may still be analyzed in block source coordinates for tail asymptotics, provided the physical quotient is used to verify localization and dimension.

This clarifies the division of labor:

- raw physical coefficients for rank / quotient;
- compact block coordinates for recurrence asymptotics.

---

# 17. The rigorous proof target is now local at the center

Round 54 appeared to require:

1. construction of two global adjoint minimal modes;
2. a two-dimensional matching determinant;
3. two nonzero pairings.

Round 55 reduces the strongest sufficient route to:

## Step A

Construct one infinite reflection-even, $\mathcal C$-even minimal solution:

$$
\boxed{
\psi_+
}
$$

with:

$$
\psi_+(0)=1,
\qquad
\psi_+(1)=0.
$$

## Step B

Prove:

$$
\boxed{
\operatorname{Im}\psi_+(3)>0.
}
$$

Then automatically:

$$
\boxed{
\langle\psi_+,g\rangle\ne0
}
$$

at both source radii.

So the infinite-dimensional problem can be attacked by a tail-to-center sign enclosure rather than a full determinant evaluation.

---

# 18. A posteriori route toward a rigorous sign enclosure

The numerical margin is large:

$$
a_3
\approx
0.04119
$$

or:

$$
0.08428.
$$

Hence an a posteriori proof does not need extremely sharp tail control.

A viable route is:

1. choose a cutoff:
   $$
   N_\ast;
   $$
2. construct the three-dimensional minimal adjoint subspace at:
   $$
   +N_\ast
   $$
   and the reflected one at:
   $$
   -N_\ast;
   $$
3. use the full viscous large-$n$ dichotomy to bound the tail graph transform;
4. propagate interval/subspace enclosures to the center;
5. impose:
   $$
   \psi_0=1,\quad\psi_1=0;
   $$
6. obtain an interval:
   $$
   \operatorname{Im}\psi_3
   \in
   [a_-,a_+]
   $$
   with:
   $$
   a_->0.
   $$

Because the observed coefficient is orders of magnitude larger than the numerical truncation drift, this is a promising computer-assisted / analytic enclosure route.

---

# 19. Current strongest finite-section statement

At:

$$
\nu=1,
$$

both source fibres satisfy simultaneously:

$$
\boxed{
a_{3,N}>0
}
$$

for every tested:

$$
8\le N\le80,
$$

with convergence well beyond the displayed digits.

The corresponding first canonical pairings never approach zero:

$$
\boxed{
\langle\psi_{+,N},g\rangle
\to
-0.655636066314278\ldots
}
\tag{19.1}
$$

for the small fibre, and:

$$
\boxed{
\langle\psi_{+,N},g\rangle
\to
446.038740993707\ldots
}
\tag{19.2}
$$

for the large fibre.

This is still numerical evidence, but now attached to an exact sign-reduction identity.

---

# 20. STOP-C59 — Adjoint Central Positivity / Rigorous Tail-Enclosure Gap

$$
\boxed{
\begin{aligned}
\text{layer}
&=
\mathrm{adjoint\ minimal\ Floquet\ compatibility},
\\
\text{physical reflection}
&=
\mathrm{exact},
\\
\text{adjoint anti\text{-}linear symmetry}
&=
\mathcal C\psi_n=(-1)^n\bar\psi_n,
\\
\text{localized adjoint dimension}
&=
2
\text{ numerically},
\\
\text{canonical first mode}
&=
\psi_+(0)=1,\ \psi_+(1)=0,
\\
\text{phase}
&=
\psi_+(3)=ia_3,
\\
\text{pairing}
&=
g_0+a_3G_{-3},
\\
\text{target signs}
&=
\operatorname{sign}g_0
=
\operatorname{sign}G_{-3},
\\
\text{sufficient full obstruction}
&=
a_3>0,
\\
a_{3,-}^{\rm num}
&\approx
0.041191045742266,
\\
a_{3,+}^{\rm num}
&\approx
0.084276516264333,
\\
\text{finite pairing small}
&\approx
-0.655636066314278,
\\
\text{finite pairing large}
&\approx
446.038740993707,
\\
\text{missing}
&=
\mathrm{rigorous\ construction\ of\ }\psi_+
\mathrm{\ and\ a\ positive\ tail\text{-}to\text{-}center\ enclosure\ for\ }a_3,
\\
T_{\mathsf C\to\mathsf D}
&=
\mathrm{NOT\ REACHED}.
\end{aligned}
}
$$

Nomenclature:

$$
\boxed{
\textbf{STOP-C59:
Adjoint Central Positivity / Rigorous Tail-Enclosure Gap}.
}
$$

---

# 21. 24/72 Ledger — Round 55

| Step | object | $B$ | $U$ | $O$ | $L$ | status |
|---|---|---|---|---|---|---|
| C888 | raw vertical reflection | $\mathsf C$ | physical Fourier operator | relational | $\mathsf F$ | EXACT |
| C889 | state/source reflection intertwining | $\mathsf C$ | operator symmetry | targeted | $\mathsf F$ | EXACT |
| C890 | adjoint phase-conjugation involution | $\mathsf C$ | anti-linear symmetry | relational | $\mathsf F$ | EXACT |
| C891 | localized reflection-even two-plane | $\mathsf C$ | finite-section cokernel | profile | $\mathsf F$ | NUMERICALLY VERIFIED |
| C892 | canonical central evaluation basis | $\mathsf C$ | adjoint normalization | relational | $\mathsf F$ | CONSTRUCTED numerically |
| C893 | canonical phase sectors | $\mathsf C$ | anti-linear symmetry | targeted | $\mathsf F$ | NUMERICALLY VERIFIED |
| C894 | exact target coefficient signs | $\mathsf C$ | algebraic source geometry | scalar | $\mathsf F$ | PROVED |
| C895 | one-coefficient pairing reduction | $\mathsf C$ | Fredholm pairing | scalar | $\mathsf F$ | EXACT conditional on canonical infinite mode |
| C896 | small-fibre $a_3$ stabilization | $\mathsf C$ | cutoff study | scalar | $\mathsf F$ | VERIFIED |
| C897 | large-fibre $a_3$ stabilization | $\mathsf C$ | cutoff study | scalar | $\mathsf F$ | VERIFIED |
| C898 | first canonical pairing stabilization | $\mathsf C$ | adjoint compatibility | scalar | $\mathsf F$ | VERIFIED |
| C899 | second canonical pairing | $\mathsf C$ | adjoint compatibility | scalar | $\mathsf F$ | VERIFIED |
| C900 | infinite canonical minimal mode | $\mathsf C$ | adjoint dichotomy | targeted | $\mathsf F$ | OPEN |
| C901 | rigorous $a_3>0$ enclosure | $\mathsf C$ | tail graph / interval sign | targeted | $\mathsf F$ | OPEN / STOP-C59 |

---

# 22. Continuous-versus-discrete status

Round 55 continues to use Fourier sideband coordinates.

But the exact symmetries are spatial symmetries of the continuous periodic operators:

- vertical reflection;
- complex phase-conjugation induced by the real periodic coefficients;
- adjoint Fredholm orthogonality.

The finite section is only a diagnostic of the continuous Floquet fibre.

The remaining positivity proof can equally be phrased as a continuous periodic adjoint boundary-value problem with a minimal regularity condition at Floquet infinity.

Therefore:

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

---

# 23. Strongest results of Round 55

## R55-A — exact physical reflection

$$
\boxed{
\mathscr N
\mathcal R_{\rm in}
=
\mathcal R_{\rm out}
\mathscr N,
}
$$

$$
\boxed{
\mathscr S
\mathcal R_{\rm in}
=
\mathcal R_{\rm out}
\mathscr S.
}
$$

## R55-B — exact adjoint anti-linear symmetry

$$
\boxed{
(\mathcal C\psi)_n
=
(-1)^n\overline{\psi_n}
}
$$

preserves the adjoint homogeneous equation.

## R55-C — one-coefficient target reduction

For the first canonical mode:

$$
\boxed{
\langle\psi_+,g\rangle
=
g_0
+
G_{-3}
\operatorname{Im}\psi_+(3).
}
$$

## R55-D — exact same-sign target geometry

At both source radii:

$$
\boxed{
\operatorname{sign}g_0
=
\operatorname{sign}G_{-3}.
}
$$

## R55-E — numerical central positivity

At:

$$
\nu=1,
$$

$$
\boxed{
\operatorname{Im}\psi_{+,N}(3)
\to
0.041191045742266\ldots
}
$$

or:

$$
\boxed{
0.084276516264333\ldots
}
$$

for the two fibres.

## R55-F — sufficient rigorous closure condition

$$
\boxed{
\operatorname{Im}\psi_+(3)>0
}
$$

for the infinite minimal adjoint mode would immediately imply:

$$
\boxed{
g
\notin
\operatorname{Ran}
\left(
\mathscr S|_{\ker\mathscr N}
\right)
}
$$

and therefore close the full second-order source-lock escape for the two $\sqrt{17}$ circles.

---

# 24. Next round — Rigorous Adjoint Tail Enclosure / Positive Central Coefficient

Round 55 has reduced the proof burden enough that the next round should stop doing broad exploratory numerics.

The direct target is:

$$
\boxed{
\operatorname{Im}\psi_+(3)>0.
}
$$

Concrete route:

1. derive a normalized adjoint transfer system with bounded coefficients after factorial rescaling;
2. construct the three-dimensional minimal graph at:
   $$
   +\infty;
   $$
3. use reflection to recover the negative tail;
4. impose the two central normalizations:
   $$
   \psi_0=1,
   \qquad
   \psi_1=0;
   $$
5. derive an a posteriori tail contraction bound;
6. propagate an interval/subspace enclosure to:
   $$
   m=3;
   $$
7. prove:
   $$
   \operatorname{Im}\psi_+(3)
   \in
   [a_-,a_+]
   $$
   with:
   $$
   a_->0;
   $$
8. then use the exact same-sign identity of Round 55 to conclude a nonzero Fredholm compatibility pairing;
9. if completed, upgrade STOP-C59 to a rigorous full second-order no-go for the two source-hidden circles.

This becomes:

$$
\boxed{
\textbf{Rigorous Adjoint Tail Enclosure / Positive Central Coefficient}.
}
$$

---

# 25. External primary-source anchors

1. F. Battelli, M. Franca, K. J. Palmer, *Exponential Dichotomy for Noninvertible Linear Difference Equations*, arXiv:2111.04553.
   - studies dichotomy projections and roughness for finite-dimensional difference equations;
   - relevant framework for turning the asymptotic minimal/growing split into controlled semiaxis subspaces.

2. Robert Skiba, Nils Waterstraat, *Fredholm theory of families of discrete dynamical systems and its applications to bifurcation theory*, arXiv:2003.12433.
   - relates exponential-dichotomy hypotheses to Fredholm properties of discrete dynamical systems;
   - relevant to the eventual infinite source-range / adjoint compatibility theorem.

3. Fritz Gesztesy, Yuri Latushkin, Konstantin A. Makarov, *Evans Functions, Jost Functions, and Fredholm Determinants*, arXiv:math/0511372.
   - primary-source background for expressing global matching through finite-dimensional Evans/Jost data and Fredholm determinants;
   - used only as structural guidance, not as a black-box proof for the present unbounded Floquet recurrence.

4. George Bayliss, Jared C. Bronski, *The Evans function as a lower bound on the spectral distance function*, arXiv:2604.19938.
   - recent primary-source evidence that properly normalized Evans data can quantitatively control distance to spectral obstruction in suitable boundary-value settings;
   - relevant motivation for converting Round 54–55 numerical matching margins into rigorous enclosures.

All NS-specific reflection identities, canonical pairing reduction, target sign formulas and finite-section values in this round are direct derivations or independently reproduced by the included verification script.

---

# 26. Commit state

$$
\boxed{
\begin{aligned}
\text{Route}
&=
\mathrm{Pure\ Continuous\ Adjoint\ Minimal\ Floquet\ Compatibility},
\\
\text{Essential }\mathsf C\to\mathsf D
&=
\mathrm{Not\ reached},
\\
\text{Adjoint symmetries}
&=
\mathrm{reflection}
+
\mathrm{phase\ conjugation},
\\
\text{Abstract codim-2 defect}
&=
\mathrm{reduced\ to\ canonical\ symmetry\ basis},
\\
\text{First canonical pairing}
&=
g_0+a_3G_{-3},
\\
\text{Target coefficient signs}
&=
\mathrm{same\ and\ strict},
\\
\text{Finite }a_3
&=
\mathrm{positive\ with\ large\ margin},
\\
\text{Sufficient full no-go step}
&=
\mathrm{prove\ infinite\ }a_3>0,
\\
\text{STOP-C59}
&=
\mathrm{Adjoint\ Central\ Positivity/Rigorous\ Tail\text{-}Enclosure\ Gap},
\\
\text{Next}
&=
\mathrm{Rigorous\ Adjoint\ Tail\ Enclosure/Positive\ Central\ Coefficient}.
\end{aligned}
}
$$