---
title: "Navier–Stokes C6-J：Log-Scale Renormalized Defect Flow、Telescoping Potentials 與 Critical-Cycle Closure Tests"
subtitle: "Backward Leray Dynamics Turns UV Zeno into Infinite Scale-Time; Physical Energy Telescopes Only with a Subcritical Weight; Critical Field-Compact Recurrence Is Excluded, So Any Surviving Defect Cycle Must Escape Along a Noncompact Critical Fiber"
version: "v0.1"
date: "2026-08-16"
author: "Neo.K / EveMissLab"
language: "zh-TW"
status: "C6 log-scale renormalization / Lyapunov audit / field-compact recurrence no-go"
epistemic_status: "Exact backward-Leray rescaling and weighted-energy identities + external critical-norm blow-up necessities and self-similar/DSS Liouville barriers. Does NOT prove existence or nonexistence of all recurrent defect orbits and does NOT prove Navier–Stokes global regularity."
---

# Navier–Stokes C6-J
# Log-Scale Renormalized Defect Flow、Telescoping Potentials 與 Critical-Cycle Closure Tests

## 0. 本輪定位

C6-I 建立：

$$
\boxed{
Q^{crit}=r^{d_Q}Q
}
$$

的 Criticalization Operator，

並把：

- middle load；
- operator load；
- pressure；
- Duhamel capacity；
- derivative-chain roots；
- shell vorticity toll；
- CKN local quantities；

全部放進同一套 N–S critical scaling ledger。

但 C6-I 也證：

$$
\boxed{
\textbf{fixed nonzero critical toll per shrinking scale}
\not\Rightarrow
\textbf{finite-time contradiction}.
}
$$

因為 geometric scale ladder：

$$
r_n=r_0a^{-n}
$$

可同時滿足：

$$
\sum_nr_n^2<\infty,
$$

以及：

$$
\sum_nr_n<\infty.
$$

所以：

- infinitely many parabolic events；
- fixed $O(1)$ critical toll；
- finite physical time；
- finite raw energy cost；

在 scaling architecture上可共存。

因此 C6-I 的結論是：

> **criticalization修正了 scaling type，但沒有提供跨 scale 的方向性。**

C6-J 正式將：

$$
\boxed{
s=-\log r
}
$$

升為 scale-time，

並問：

1. N–S 在 log-scale中是否形成 autonomous renormalized flow？
2. fixed point / periodic orbit對應什麼 physical scenario？
3. 是否存在 natural telescoping potential？
4. 它是否同時 critical + monotone？
5. 已知 self-similar / discretely self-similar no-go可以排除哪些 cycle？
6. 若 compact defect orbit仍可能 recurrence，底層 field必如何逃逸？

本輪主要結果：

1. 標準 backward Leray variables將 hypothetical finite-time blow-up horizon映成：
   $$
   s\to\infty;
   $$
2. renormalized N–S變成 autonomous equation；
3. fixed point = backward self-similar profile；
4. periodic orbit = backward discretely self-similar profile；
5. renormalized $L^2$ balance exact：
   $$
   \boxed{
   \frac12E'
   +
   \nu D
   -
   \frac14E
   =
   0;
   }
   $$
6. 因此 criticalized renormalized $L^2$ energy本身不是 monotone；
7. weighted family：
   $$
   V_\alpha=e^{-\alpha s}E
   $$
   obeys exact identity；
8. universal monotonicity in this family starts at：
   $$
   \alpha\ge1/2;
   $$
9. $\alpha=1/2$ exactly recovers physical energy；
10. hence：
    $$
    \boxed{
    \textbf{Criticality–Monotonicity Tradeoff}
    }
    $$
    for the natural $L^2$ family；
11. physical-energy telescoping exists but carries weight：
    $$
    e^{-s/2}=r;
    $$
12. this exactly reproduces the C6-I Zeno summability；
13. periodic renormalized $L^2$ states are not contradicted by the $L^2$ balance alone；
14. critical $L^3$ and $\dot H^{1/2}$ norms are invariant under the backward rescaling；
15. potential blow-up requires both to diverge；
16. therefore：
    $$
    \boxed{
    \textbf{no blow-up renormalized orbit can remain precompact in }
    L^3
    \textbf{ or }
    \dot H^{1/2};
    }
    $$
17. fixed / periodic / compact recurrent field orbits in those critical topologies are excluded；
18. known self-similar / asymptotically DSS Liouville results independently exclude important profile classes；
19. however compact **defect metadata** can still recur while the field critical norm diverges；
20. define：
    $$
    \boxed{
    \textbf{Critical Fiber Escape};
    }
    $$
21. any surviving C6 defect recurrence must occur over noncompact fibers of the projection：
    $$
    \pi:
    \mathcal X_{crit}
    \to
    \mathcal K_{defect};
    $$
22. if a compact defect set has uniformly bounded critical fibers，it cannot support hypothetical blow-up recurrence；
23. thus the remaining C6 problem becomes a skew-product/noncompact-fiber problem：
    $$
    \boxed{
    \text{compact recurrent base}
    +
    \text{critical field escape in the fiber}.
    }
    $$

---

# 1. Backward parabolic variables

Assume for contradiction/research analysis that：

$$
T^\ast<\infty
$$

is a potential blow-up time，

and fix a candidate singular center：

$$
x^\ast\in\mathbb R^3.
$$

Set：

$$
\boxed{
\tau
=
T^\ast-t.
}
$$

Define standard backward logarithmic time：

$$
\boxed{
s
=
-\log\tau.
}
$$

Then：

$$
\tau=e^{-s}.
$$

Define parabolic scale：

$$
\boxed{
r(s)
=
\sqrt{\tau}
=
e^{-s/2}.
}
$$

### Relation to C6-I scale-time

C6-I used：

$$
s_r=-\log r.
$$

Therefore：

$$
\boxed{
s=2s_r.
}
$$

The two variables differ only by a factor：

$$
2.
$$

---

# 2. Backward Leray coordinates

Define：

$$
\boxed{
y
=
\frac{
x-x^\ast
}{
\sqrt{T^\ast-t}
}
=
\frac{
x-x^\ast
}{
r(s)
}.
}
$$

Define renormalized velocity：

$$
\boxed{
U(y,s)
=
\sqrt{T^\ast-t}
\,u(x,t)
=
r(s)u(x,t).
}
$$

Pressure：

$$
\boxed{
P(y,s)
=
(T^\ast-t)
p(x,t)
=
r(s)^2p(x,t).
}
$$

---

# 3. C6-J.1：Backward Leray Flow Equation

A direct change of variables gives：

$$
\boxed{
\partial_sU
+
\frac12U
+
\frac12
(y\cdot\nabla)U
+
(U\cdot\nabla)U
+
\nabla P
=
\nu\Delta U,
}
$$

with：

$$
\boxed{
\nabla\cdot U=0.
}
$$

This is autonomous in：

$$
s.
$$

### Interpretation

The finite physical-time endpoint：

$$
t\uparrow T^\ast
$$

becomes：

$$
\boxed{
s\to+\infty.
}
$$

Thus finite-time Zeno in physical coordinates becomes an infinite-time dynamical problem in renormalized scale-time。

---

# 4. Fixed point

If：

$$
\boxed{
U(y,s)=U_\ast(y)
}
$$

independent of：

$$
s,
$$

then：

$$
U_\ast
$$

solves the stationary backward Leray equation：

$$
\boxed{
\frac12U_\ast
+
\frac12(y\cdot\nabla)U_\ast
+
(U_\ast\cdot\nabla)U_\ast
+
\nabla P_\ast
=
\nu\Delta U_\ast.
}
$$

In physical variables：

$$
\boxed{
u(x,t)
=
\frac1{
\sqrt{T^\ast-t}
}
U_\ast
\left(
\frac{
x-x^\ast
}{
\sqrt{T^\ast-t}
}
\right).
}
$$

This is a backward self-similar blow-up profile。

---

# 5. Periodic orbit

Suppose：

$$
\boxed{
U(y,s+L)
=
U(y,s).
}
$$

Then：

$$
\tau(s+L)
=
e^{-L}\tau(s),
$$

and：

$$
r(s+L)
=
e^{-L/2}r(s).
$$

Define：

$$
\boxed{
\lambda
=
e^{L/2}>1.
}
$$

Then the physical solution obeys a backward discrete self-similarity relation between scales separated by：

$$
\lambda.
$$

Thus：

$$
\boxed{
\textbf{periodic orbit in }s
=
\textbf{backward DSS scenario}.
}
$$

---

# 6. External fixed/periodic-profile barriers

Known Liouville-type results exclude broad classes of nontrivial backward self-similar profiles。

Known asymptotically discrete-self-similar results also exclude locally asymptotically DSS blow-up under suitable critical-profile integrability/regularity assumptions。

Therefore：

$$
\boxed{
\textbf{some field-level fixed points and periodic orbits of the backward Leray flow are externally impossible}.
}
$$

### Guard

These theorems require field-level profile assumptions。

They do not automatically apply to a periodic orbit of only the C6 defect metadata。

---

# 7. Renormalized $L^p$ scaling

For：

$$
1\le p\le\infty,
$$

using：

$$
U(y,s)
=
r
u(x^\ast+ry,t),
$$

$$
dy
=
r^{-3}dx,
$$

we get：

$$
\boxed{
\|U(s)\|_{L^p_y}
=
r^{1-\frac3p}
\|u(t)\|_{L^p_x}.
}
$$

In terms of：

$$
\tau=r^2,
$$

$$
\boxed{
\|U(s)\|_p
=
\tau^{\frac12-\frac3{2p}}
\|u(t)\|_p.
}
$$

---

# 8. Critical $L^3$ invariance

At：

$$
p=3,
$$

$$
1-\frac33=0.
$$

Hence：

$$
\boxed{
\|U(s)\|_{L^3}
=
\|u(t)\|_{L^3}.
}
$$

Thus：

$$
L^3
$$

is exactly critical under backward Leray rescaling。

---

# 9. Critical $\dot H^{1/2}$ invariance

For：

$$
U(y)=r\,u(x^\ast+ry),
$$

the homogeneous Sobolev scaling is：

$$
\boxed{
\|U\|_{\dot H^\alpha}
=
r^{\alpha-\frac12}
\|u\|_{\dot H^\alpha}.
}
$$

At：

$$
\alpha=\frac12,
$$

$$
\boxed{
\|U(s)\|_{\dot H^{1/2}}
=
\|u(t)\|_{\dot H^{1/2}}.
}
$$

So：

$$
\dot H^{1/2}
$$

is another critical field topology。

---

# 10. External critical-norm blow-up necessities

For a potential blow-up time：

$$
T^\ast,
$$

known necessary conditions imply：

$$
\boxed{
\|u(t)\|_{L^3}
\to\infty
\qquad
(t\uparrow T^\ast),
}
$$

and：

$$
\boxed{
\|u(t)\|_{\dot H^{1/2}}
\to\infty.
}
$$

By §§8–9：

$$
\boxed{
\|U(s)\|_{L^3}
\to\infty,
}
$$

$$
\boxed{
\|U(s)\|_{\dot H^{1/2}}
\to\infty
\qquad
(s\to\infty).
}
$$

---

# 11. C6-J.2：Critical Field-Compact Recurrence No-Go

## Theorem

Let：

$$
U(s)
$$

be the backward Leray rescaling of a hypothetical finite-time blow-up solution。

Then no tail：

$$
\{U(s):s\ge s_0\}
$$

can be precompact in：

$$
L^3(\mathbb R^3)
$$

or：

$$
\dot H^{1/2}(\mathbb R^3).
$$

### Proof

A precompact subset of a normed space is bounded。

But hypothetical blow-up requires：

$$
\|U(s)\|_{L^3}\to\infty
$$

and：

$$
\|U(s)\|_{\dot H^{1/2}}\to\infty.
$$

Contradiction。$\square$

---

# 12. Consequences for fixed and periodic field orbits

A fixed point：

$$
U(s)=U_\ast
$$

with finite：

$$
L^3
$$

or：

$$
\dot H^{1/2}
$$

norm is bounded，

hence cannot represent the hypothetical blow-up。

Likewise a periodic orbit：

$$
U(s+L)=U(s)
$$

is bounded in any topology in which the periodic map is continuous and one period has finite norm。

Therefore：

$$
\boxed{
\textbf{finite-critical-norm fixed and periodic field orbits are impossible blow-up orbits}.
}
$$

This is consistent with the specialized backward self-similar / DSS Liouville literature。

---

# 13. Asymptotically periodic field orbit

Suppose：

$$
U(s)
-
U_{per}(s)
\to0
$$

in：

$$
L^3
$$

as：

$$
s\to\infty,
$$

with：

$$
U_{per}
$$

periodic and bounded in：

$$
L^3.
$$

Then：

$$
\|U(s)\|_3
$$

would remain bounded。

Therefore：

$$
\boxed{
\textbf{asymptotically periodic recurrence in critical }L^3
\textbf{ is also incompatible with hypothetical blow-up}.
}
$$

Specialized literature provides stronger versions under profile regularity assumptions。

---

# 14. Important forward-DSS guard

Forward discretely self-similar Navier–Stokes solutions are known to exist for large data in suitable classes。

Therefore：

$$
\boxed{
\textbf{log-periodicity / discrete scale invariance is not intrinsically forbidden by the Navier--Stokes equation as an abstract phenomenon}.
}
$$

The backward blow-up setting has different dynamical and regularity constraints。

This prevents an invalid argument of the form：

> N–S can never have a periodic renormalized structure。

---

# 15. Renormalized $L^2$ identity

Assume sufficient decay/integrability so all integrations by parts are legitimate。

Set：

$$
\boxed{
E(s)
=
\|U(s)\|_2^2.
}
$$

Take the：

$$
L^2
$$

inner product of the backward Leray equation with：

$$
U.
$$

Nonlinearity：

$$
\int
U\cdot
(U\cdot\nabla)U
=
0.
$$

Pressure：

$$
\int
U\cdot\nabla P
=
0.
$$

Viscosity：

$$
\nu
\int
U\cdot\Delta U
=
-\nu
\|\nabla U\|_2^2.
$$

---

# 16. Dilation term

$$
\frac12
\int
U\cdot
(y\cdot\nabla)U
=
\frac14
\int
y\cdot\nabla
|U|^2.
$$

In：

$$
\mathbb R^3,
$$

$$
\boxed{
\int
y\cdot\nabla f
=
-3
\int f.
}
$$

Therefore：

$$
\boxed{
\frac12
\int
U\cdot
(y\cdot\nabla)U
=
-\frac34
\|U\|_2^2.
}
$$

Combined with：

$$
\frac12\|U\|_2^2,
$$

the drift contribution is：

$$
-\frac14E.
$$

---

# 17. C6-J.3：Renormalized $L^2$ Balance

Thus：

$$
\boxed{
\frac12
E'(s)
+
\nu
\|\nabla U(s)\|_2^2
-
\frac14
E(s)
=
0.
}
$$

Equivalently：

$$
\boxed{
E'
=
\frac12E
-
2\nu
\|\nabla U\|_2^2.
}
$$

### Main point

$$
\boxed{
E(s)
}
$$

is not a universal monotone quantity。

The scale-dilation term creates an anti-dissipative contribution：

$$
+\frac12E.
$$

---

# 18. Periodic $L^2$ balance

If：

$$
E(s+L)=E(s)
$$

for a period：

$$
L,
$$

integrate C6-J.3 over one period：

$$
0
+
\nu
\int_0^L
\|\nabla U\|_2^2ds
-
\frac14
\int_0^L
E(s)ds
=
0.
$$

Therefore：

$$
\boxed{
\nu
\int_0^L
\|\nabla U\|_2^2ds
=
\frac14
\int_0^L
\|U\|_2^2ds.
}
$$

### Consequence

The renormalized $L^2$ balance **alone** does not contradict a periodic renormalized orbit。

It only requires an average balance between：

- dilation；
- viscosity。

---

# 19. Weighted $L^2$ potential family

For：

$$
\alpha\in\mathbb R,
$$

define：

$$
\boxed{
V_\alpha(s)
=
e^{-\alpha s}
E(s).
}
$$

Differentiate：

$$
\begin{aligned}
V_\alpha'
&=
e^{-\alpha s}
\left(
E'
-
\alpha E
\right)
\\
&=
e^{-\alpha s}
\left[
\left(
\frac12-\alpha
\right)
E
-
2\nu
\|\nabla U\|_2^2
\right].
\end{aligned}
$$

Thus：

# 20. C6-J.4：Weighted Renormalized Energy Identity

$$
\boxed{
V_\alpha'
=
e^{-\alpha s}
\left[
\left(
\frac12-\alpha
\right)
\|U\|_2^2
-
2\nu
\|\nabla U\|_2^2
\right].
}
$$

---

# 21. Universal monotonicity threshold

If：

$$
\boxed{
\alpha\ge\frac12,
}
$$

then：

$$
\boxed{
V_\alpha'(s)\le0.
}
$$

At：

$$
\alpha=\frac12,
$$

$$
\boxed{
V_{1/2}'
=
-2\nu
e^{-s/2}
\|\nabla U\|_2^2.
}
$$

For：

$$
\alpha<\frac12,
$$

the sign is not controlled solely by the identity。

---

# 22. Physical energy identification

Recall：

$$
U(y,s)=r\,u(x,t),
$$

with：

$$
r=e^{-s/2}.
$$

Compute：

$$
\|U\|_2^2
=
r^{-1}
\|u\|_2^2.
$$

Thus：

$$
\boxed{
e^{-s/2}
\|U(s)\|_2^2
=
r
\|U\|_2^2
=
\|u(t)\|_2^2.
}
$$

Therefore：

$$
\boxed{
V_{1/2}
}
$$

is exactly the physical kinetic energy。

---

# 23. C6-J.5：Criticality–Monotonicity Tradeoff for Natural $L^2$ Potentials

Within the family：

$$
V_\alpha
=
e^{-\alpha s}\|U\|_2^2,
$$

the unweighted renormalized energy：

$$
V_0=E
$$

retains full log-scale sensitivity but is not universally monotone。

Universal monotonicity begins at：

$$
\boxed{
\alpha\ge1/2.
}
$$

But every such potential carries an explicit decaying factor：

$$
e^{-\alpha s}.
$$

The weakest monotone weight：

$$
\alpha=1/2
$$

is exactly：

$$
r=e^{-s/2}.
$$

Therefore：

$$
\boxed{
\textbf{within the natural exponential }L^2\textbf{ family，
critical scale sensitivity and universal monotonicity do not coexist}.
}
$$

---

# 24. Why this exactly reproduces C6-I Zeno

Define renormalized critical dissipation density：

$$
\boxed{
D_{crit}(s)
=
\nu
\|\nabla U(s)\|_2^2.
}
$$

Physical-energy telescoping gives：

$$
\boxed{
V_{1/2}(s_1)
-
V_{1/2}(s_2)
=
2
\int_{s_1}^{s_2}
e^{-s/2}
D_{crit}(s)ds.
}
$$

Thus the monotone potential integrates critical dissipation with weight：

$$
\boxed{
e^{-s/2}=r.
}
$$

If：

$$
D_{crit}(s)
$$

is：

$$
O(1)
$$

on one unit interval of：

$$
s
$$

per generation，

the energy cost is：

$$
O(r).
$$

For：

$$
r_n\sim a^{-n},
$$

these costs are summable。

This is the continuous log-scale version of C6-I's geometric Zeno lemma。

---

# 25. Telescoping exists but is subcritical-weighted

Therefore the problem is not：

$$
\boxed{
\text{no telescoping quantity exists}.
}
$$

Physical energy already telescopes。

The problem is：

$$
\boxed{
\textbf{the available universal telescoping weight decays with scale}.
}
$$

It cannot assign a fixed positive price to a scale-invariant recurrent event。

---

# 26. Simple critical Lyapunov test

At：

$$
\alpha=0,
$$

the criticalized renormalized energy：

$$
E(s)
$$

obeys：

$$
E'
=
\frac12E
-
2D_{crit}.
$$

Thus any proof of：

$$
E'\le0
$$

would require an additional inequality：

$$
\boxed{
D_{crit}
\ge
\frac14E.
}
$$

There is no universal whole-space inequality of this form without an additional confinement/normalization condition。

### Guard

C6-J does not claim no other critical Lyapunov functional can exist。

It only closes the most natural $L^2$ family and identifies the missing coercivity。

---

# 27. Renormalized critical field capacity

Define：

$$
\boxed{
\mathfrak F_{crit}(s)
=
\|U(s)\|_{L^3}
+
\|U(s)\|_{\dot H^{1/2}}.
}
$$

Both coordinates have N–S scaling degree：

$$
0.
$$

Hypothetical blow-up requires：

$$
\boxed{
\mathfrak F_{crit}(s)\to\infty.
}
$$

Compactify：

$$
\boxed{
\widehat{\mathfrak F}_{crit}
=
\frac{
\mathfrak F_{crit}
}{
1+\mathfrak F_{crit}
}
\in[0,1).
}
$$

Then any hypothetical blow-up renormalized orbit satisfies：

$$
\boxed{
\widehat{\mathfrak F}_{crit}(s)
\to1.
}
$$

---

# 28. Field capacity at infinity is genuine critical infinity

Unlike raw：

$$
A_k\to\infty
$$

without criticalization，

$$
\|U\|_3
$$

and：

$$
\|U\|_{\dot H^{1/2}}
$$

are already scale invariant。

Therefore：

$$
\boxed{
\mathfrak F_{crit}\to\infty
}
$$

is a genuine：

$$
\boxed{
CAP^{crit,\infty}
}
$$

field boundary，

not a scaling artifact。

---

# 29. Defect projection

Let：

$$
\boxed{
\mathcal X_{crit}
}
$$

be a suitable renormalized field state space carrying at least：

- local smoothness before the singular time；
- critical field norms；
- the spatial/provenance data needed by C6。

Let：

$$
\boxed{
\mathcal K_{def}
}
$$

be the compactified C6 defect state space：

- $TS$；
- $GP$；
- $HF$；
- critical boundary faces；
- composition reserves。

Define a projection：

$$
\boxed{
\pi:
\mathcal X_{crit}
\to
\mathcal K_{def}.
}
$$

C6-A–I have largely studied：

$$
\boxed{
\pi(U(s)),
}
$$

not the full：

$$
U(s).
$$

---

# 30. Defect recurrence vs field recurrence

A defect recurrence means：

$$
\boxed{
\pi(U(s_n))
\to
\theta_\ast
}
$$

or returns near a compact subset：

$$
K\subset\mathcal K_{def}.
$$

This does **not** imply：

$$
\boxed{
U(s_n)
}
$$

is precompact in：

$$
\mathcal X_{crit}.
$$

The fiber：

$$
\boxed{
\pi^{-1}(\theta)
}
$$

may be noncompact。

This is the exact mathematical place where a compact defect cycle can coexist with critical field norm blow-up。

---

# 31. Critical fiber radius

For：

$$
K\subset
\mathcal K_{def},
$$

define：

$$
\boxed{
\mathfrak R_{fiber}(K)
=
\sup
\left\{
\mathfrak F_{crit}(U):
\pi(U)\in K
\right\}
\in[0,\infty].
}
$$

If：

$$
\mathfrak R_{fiber}(K)<\infty,
$$

the defect set controls the critical field norm uniformly。

---

# 32. C6-J.6：Bounded-Fiber Recurrence No-Go

Let：

$$
K\subset\mathcal K_{def}
$$

be compact。

Suppose a hypothetical blow-up orbit satisfies：

$$
\boxed{
\pi(U(s))
\in K
}
$$

for all sufficiently large：

$$
s,
$$

and：

$$
\boxed{
\mathfrak R_{fiber}(K)<\infty.
}
$$

Then：

$$
\boxed{
\mathfrak F_{crit}(s)
}
$$

remains bounded，

contradicting the necessary blow-up divergence。

Therefore：

$$
\boxed{
\textbf{no compact defect trap with uniformly bounded critical fibers can support hypothetical blow-up}.
}
$$

---

# 33. Recurrent-subsequence version

Suppose：

$$
s_n\to\infty
$$

and：

$$
\pi(U(s_n))
\in K
$$

for a compact defect set：

$$
K.
$$

If：

$$
\mathfrak R_{fiber}(K)<\infty,
$$

then：

$$
\mathfrak F_{crit}(s_n)
$$

is bounded，

contradicting：

$$
\mathfrak F_{crit}(s)\to\infty.
$$

Thus even recurrent visits to a bounded-fiber defect set are impossible arbitrarily late in a blow-up orbit。

---

# 34. C6-J.7：Critical Fiber Escape Theorem

Any compact defect set：

$$
K
$$

visited infinitely often by a hypothetical blow-up renormalized orbit must satisfy：

$$
\boxed{
\mathfrak R_{fiber}(K)
=
\infty.
}
$$

Equivalently：

$$
\boxed{
\textbf{every surviving compact defect recurrence requires critical noncompactness in the fiber}.
}
$$

This is：

$$
\boxed{
\textbf{Critical Fiber Escape}.
}
$$

---

# 35. What can escape inside the fiber?

A defect projection may forget：

- critical amplitude；
- profile multiplicity；
- spatial translation after recentering；
- secondary concentration scales；
- high-frequency oscillation；
- pressure/nonlinear fine structure；
- derivative-order escape；
- noncompact tails。

Any of these can make：

$$
U(s_n)
$$

noncompact while：

$$
\pi(U(s_n))
$$

recurs。

Therefore the next phase must classify the **mechanism of fiber noncompactness**。

---

# 36. Fiber escape and C6-I boundary classes

Critical fiber escape is naturally related to：

$$
\boxed{
CAP^{crit,\infty}.
}
$$

But it is more specific：

$$
\boxed{
\text{field-critical norm infinity}
}
$$

rather than any required source capacity inflation。

C6-J distinguishes：

## FIELD-CAP∞

$$
\|U\|_3
+
\|U\|_{\dot H^{1/2}}
\to\infty.
$$

## EDGE-CAP∞

e.g.：

$$
\Gamma^{-1}\to\infty.
$$

Both are critical，

but have different meanings。

---

# 37. Fixed/periodic defect cycles

Suppose：

$$
\theta(s)
=
\pi(U(s))
$$

is periodic：

$$
\theta(s+L)=\theta(s).
$$

This is not excluded if：

$$
U(s)
$$

moves to infinity in the fiber every cycle。

Thus：

$$
\boxed{
\textbf{periodic defect metadata}
}
$$

can correspond to：

$$
\boxed{
\textbf{nonperiodic field dynamics with critical fiber drift}.
}
$$

This is the main reason field-level DSS Liouville theorems do not automatically kill C6 defect cycles。

---

# 38. Skew-product model

The correct schematic dynamics is：

$$
\boxed{
(\theta(s),\kappa(s)),
}
$$

where：

$$
\theta
\in
\mathcal K_{def}
$$

is compact defect metadata，

and：

$$
\kappa
$$

is an unbounded critical fiber coordinate。

A simple representative：

$$
\boxed{
\kappa(s)
=
\log
\left(
1+
\mathfrak F_{crit}(s)
\right).
}
$$

Hypothetical blow-up requires：

$$
\boxed{
\kappa(s)\to\infty.
}
$$

The base：

$$
\theta(s)
$$

may remain recurrent。

---

# 39. C6-J.8：Projected-Cycle Reframing

A C6 recurrent defect cycle compatible with hypothetical blow-up is not a closed orbit in the full critical field state space。

It must instead be a skew-product orbit：

$$
\boxed{
\theta(s+L)
\approx
\theta(s),
}
$$

while：

$$
\boxed{
\kappa(s+L)
>
\kappa(s)
}
$$

on average or along a subsequence。

Thus the remaining cycle question becomes：

> **can critical field capacity drift to infinity while all compact defect reserves recur indefinitely？**

---

# 40. Candidate telescoping potential in the fiber

A genuine cycle-killing potential：

$$
\Phi(\theta,\kappa)
$$

would need：

1. scale-critical sensitivity；
2. bounded below；
3. a fixed sign drift per defect cycle；
4. enough coercivity in：

$$
\kappa.
$$

Physical energy fails item 1。

Critical norms：

$$
L^3,\dot H^{1/2}
$$

have the correct scaling but no known universal monotonicity in the present framework。

Therefore the missing object is precisely：

$$
\boxed{
\textbf{a critical fiber Lyapunov / telescoping potential}.
}
$$

---

# 41. Field-level periodic orbit barriers

Known backward self-similar / discretely self-similar Liouville results can be reinterpreted：

they exclude certain stationary/periodic or asymptotically periodic subsets of：

$$
\mathcal X_{crit}
$$

under profile integrability/regularity assumptions。

C6-J uses them as：

$$
\boxed{
\textbf{field-level recurrence kill gates}.
}
$$

They do not yet control a projected defect orbit with fiber escape。

---

# 42. Chae locally asymptotically DSS barrier

A known result excludes locally asymptotically discretely self-similar blow-up when the periodic backward profile lies in：

$$
C^1(
\mathbb R;
L^3(\mathbb R^3)
\cap
C^2(\mathbb R^3)
).
$$

Thus any C6 defect periodic orbit that could be lifted to such a periodic field profile would be killed externally。

The unresolved case requires failure of this lift，

precisely consistent with Critical Fiber Escape。

---

# 43. Chae–Wolf Liouville barrier

Liouville-type results exclude broad classes of nontrivial backward self-similar profiles，

including profile spaces extending earlier：

- Nečas–Růžička–Šverák；
- Tsai；

settings。

Again：

$$
\boxed{
\textbf{field compactness/integrability is powerful enough to kill self-similar recurrence}.
}
$$

The C6 challenge is obtaining such field control from defect metadata。

---

# 44. Forward DSS caution

Forward DSS solutions are known to exist。

Thus the autonomous/periodic scale-time language itself does not guarantee contradiction。

One must exploit the backward blow-up boundary conditions and critical regularity constraints。

This is a useful no-go against purely dynamical-systems intuition：

$$
\boxed{
\text{periodic in renormalized time}
\not\Rightarrow
\text{impossible for N–S in every setting}.
}
$$

---

# 45. Critical compactness bridge criterion

Suppose a C6 interior/boundary state family：

$$
K\subset\mathcal K_{def}
$$

implies:

1. local compactness after recentering/rescaling；
2. tightness of tails；
3. uniform critical field norm bound；
4. pressure/source provenance compactness。

Then：

$$
\boxed{
\pi^{-1}(K)
}
$$

would be precompact in a critical field topology。

C6-J.2 would eliminate recurrent visits to：

$$
K
$$

near hypothetical blow-up。

Therefore：

$$
\boxed{
\textbf{critical compactness lifting is a complete cycle-kill strategy}.
}
$$

---

# 46. Current obstacle to compactness lifting

C6 states preserve many dimensionless quantities：

- overlap；
- sign geometry；
- pressure signature；
- source coherence；
- mean/axis reserves；
- clock/order geometry。

But they do not yet uniformly bound：

$$
\boxed{
\|U\|_3
}
$$

or：

$$
\boxed{
\|U\|_{\dot H^{1/2}}.
}
$$

Thus：

$$
\boxed{
\mathfrak R_{fiber}(K)
}
$$

is not known finite for：

- $GP^\circ$；
- $HF^\circ$；
- the six critical boundary faces。

This is the main remaining noncompactness channel。

---

# 47. Relation to uniform GP recurrence

Suppose：

$$
GP^\circ
$$

recurs with all geometry/provenance reserves uniformly positive。

If one could prove：

$$
\boxed{
GP^\circ
\Rightarrow
\mathfrak F_{crit}\le C_{GP},
}
$$

then GP recurrence would be impossible by C6-J.7。

Currently no such bound is known。

Thus：

$$
\boxed{
GP_{\rm uniform}
}
$$

survival is equivalent to the possibility of unbounded critical field norms inside a compact geometry-pressure metadata fiber。

---

# 48. Relation to uniform HF recurrence

Similarly，

if uniform coherent：

$$
HF^\circ
$$

implied：

$$
\mathfrak F_{crit}\le C_{HF},
$$

the cycle would be killed。

But HF currently controls：

- re-entry coherence；
- sign geometry；
- theorem-window metadata；
- forcing/source ratios；

not the full global critical norm。

Therefore HF recurrence can survive only through critical fiber escape。

---

# 49. Relation to boundary-saturated recurrence

For：

$$
LOAD^{crit},
SEG,
GEOM^{res},
MEAN,
PROV,
CAP^{crit,\infty},
$$

the same dichotomy applies。

If a boundary face plus its critical ledger controls the field norm：

$$
\Rightarrow
$$

no late recurrent visits。

Otherwise its fiber must remain noncompact。

Thus C6-J upgrades every boundary SCC question into：

$$
\boxed{
\textbf{boundary transition}
+
\textbf{critical fiber escape}.
}
$$

---

# 50. Cycle closure hierarchy

C6-J identifies four levels。

## Level 1 — Defect recurrence

$$
\pi(U(s_n))
\to K.
$$

## Level 2 — Typed dynamic recurrence

edge metadata compose across generations。

## Level 3 — Critical field compact recurrence

$$
U(s_n)
$$

precompact in：

$$
L^3/\dot H^{1/2}.
$$

This is impossible for hypothetical blow-up。

## Level 4 — Projected recurrence with fiber escape

defect metadata recur，

but：

$$
\mathfrak F_{crit}(s_n)\to\infty.
$$

Only Level 4 remains compatible with the current blow-up necessities。

---

# 51. C6-J.9：Critical-Cycle Closure Test

For any proposed recurrent C6 cycle：

$$
C,
$$

ask in this order：

### Test 1 — Dynamic composition

Is：

$$
C
$$

a certified typed cycle？

If no：

cycle remains a proof obligation。

### Test 2 — Field compactness lift

Does uniform recurrence in：

$$
C
$$

imply bounded/precompact：

$$
L^3
$$

or：

$$
\dot H^{1/2}
$$

renormalized field state？

If yes：

$$
\boxed{
C
\text{ is incompatible with blow-up}.
}
$$

### Test 3 — Fiber escape

If not，

identify exactly which critical fiber coordinate escapes。

### Test 4 — Fiber debt

Does that escape trigger：

- an external regularity barrier；
- capacity incompatibility；
- a telescoping critical potential；
- profile decomposition contradiction？

Only after Test 4 can a surviving projected cycle be eliminated。

---

# 52. No critical field recurrence without escape

This gives a clean statement：

$$
\boxed{
\textbf{a genuine blow-up cycle cannot be both recurrent and compact in the full critical field state}.
}
$$

Thus if the C6 research program ever derives a compact invariant set in：

$$
L^3
$$

or：

$$
\dot H^{1/2},
$$

the phase closes immediately through an external critical-norm contradiction。

---

# 53. Why defect compactness was still useful

C5 compactified motifs without controlling full critical field norm。

C6-J shows this was not wasted：

compact defect base isolates the only remaining freedom into the fibers。

Instead of an unstructured infinite-dimensional flow，

the problem becomes：

$$
\boxed{
\text{compact finite defect base}
+
\text{classified noncompact critical fiber}.
}
$$

This is a much sharper target for concentration-compactness/profile decomposition。

---

# 54. Critical fiber mechanisms to classify

Candidate fiber escape mechanisms：

## J-F1 — amplitude escape

critical：

$$
L^3/\dot H^{1/2}
$$

mass grows in the same normalized core。

## J-F2 — multiplicity escape

many separated critical packets。

## J-F3 — secondary-scale escape

within the primary rescaling，a smaller unresolved scale appears。

## J-F4 — translation/tail escape

critical mass escapes spatially after the chosen recentering。

## J-F5 — frequency escape

mass moves to higher renormalized frequencies。

## J-F6 — profile splitting

critical norm divides into multiple asymptotically orthogonal profiles。

These are not yet proved exhaustive。

They are the natural next compactness audit。

---

# 55. Why concentration-compactness is now natural

Critical norm divergence/noncompactness is precisely the setting where profile decomposition and concentration-compactness methods become relevant。

C6's defect metadata can act as extra labels on each profile：

- GP geometry；
- HF coherence；
- TS shared source；
- pressure provenance；
- boundary face。

This suggests a hybrid：

$$
\boxed{
\textbf{profile decomposition + typed defect labels}.
}
$$

---

# 56. Revised role of $CAP^{crit,\infty}$

C6-I treated：

$$
CAP^{crit,\infty}
$$

as a critical boundary。

C6-J now distinguishes：

## Edge capacity infinity

$$
\Gamma^{-1}\to\infty.
$$

## Field capacity infinity

$$
\mathfrak F_{crit}\to\infty.
$$

For actual hypothetical blow-up，

field capacity infinity is mandatory。

Thus：

$$
\boxed{
CAP_{field}^{crit,\infty}
}
$$

is not merely one optional boundary：

it is a required fiber direction。

---

# 57. Consequence for boundary graph

The boundary graph should not ask merely：

$$
B_i\to B_j.
$$

It should ask：

$$
\boxed{
(B_i,\kappa)
\to
(B_j,\kappa')
}
$$

with：

$$
\kappa
=
\log(1+\mathfrak F_{crit}).
$$

Any recurrent base cycle compatible with blow-up must have：

$$
\boxed{
\kappa_n\to\infty.
}
$$

This adds a directional coordinate missing from C6-G/H。

---

# 58. Candidate drift theorem

A future C6 proof could succeed by showing：

for every recurrent base transition：

$$
\theta_n\to\theta_{n+1},
$$

critical fiber coordinate satisfies either：

$$
\boxed{
\kappa_{n+1}
\le
\kappa_n+C
}
$$

plus a global upper bound，

or：

$$
\boxed{
\kappa_{n+1}-\kappa_n
}
$$

forces some kill barrier。

Neither is currently proved。

---

# 59. C6 phase strategic correction

C6-I proposed：

$$
\text{log-scale telescoping potential}.
$$

C6-J finds：

- physical energy telescopes；
- but with subcritical weight；
- field-critical norms have correct scaling；
- but must diverge at blow-up and are not monotone。

Therefore the missing object is not simply：

$$
\boxed{
V(s)\text{ monotone}.
}
$$

It is more specifically：

$$
\boxed{
\textbf{a critical cross-fiber potential coupling compact defect recurrence to field-norm escape}.
}
$$

---

# 60. Proposed C6-K

The natural next paper：

$$
\boxed{
\textbf{C6-K — Critical Fiber Escape,
Defect-Fiber Compactness,
and Profile-Splitting Closure}.
}
$$

---

# 61. C6-K proof obligations

## K1 — choose critical field topology

Use：

$$
L^3,
\quad
\dot H^{1/2},
$$

and possibly local critical spaces consistent with pressure/profile decomposition。

## K2 — define fiber projection rigorously

$$
\pi:
\mathcal X_{crit}
\to
\mathcal K_{def}.
$$

## K3 — tightness

Determine when uniform GP/HF/TS reserves prevent spatial tail escape after recentering。

## K4 — multiplicity

Relate C5-H spectral-cell multiplicity / bad-core packing to critical profile splitting。

## K5 — secondary scale

Detect unresolved nested concentration inside the primary renormalized event。

## K6 — frequency escape

Relate Cheskidov–Dai shell toll / derivative chain to renormalized frequency noncompactness。

## K7 — labeled profile decomposition

Attach：

- GP；
- HF；
- TS；
- boundary metadata；

to concentration profiles。

## K8 — compactness lift or fiber classification

Either prove：

$$
\mathfrak R_{fiber}(K)<\infty
$$

for some candidate recurrence，

or classify the exact fiber escape route。

## K9 — external Liouville barrier

If a profile limit becomes fixed/periodic/asymptotically DSS with sufficient integrability，apply known no-blowup results。

## K10 — cycle update

Recompute minimal survivor cycles in the skew-product：

$$
(\text{defect base},\text{critical fiber}).
$$

---

# 62. Major no-go audit

### NG-J1

$$
\text{log-scale autonomous flow}
\Rightarrow
\text{a Lyapunov function exists}.
$$

FALSE。

### NG-J2

$$
\|U\|_2^2
\text{ is monotone}.
$$

FALSE。

### NG-J3

$$
\text{physical energy telescoping}
\Rightarrow
\text{critical event count finite}.
$$

FALSE；the telescoping weight is：

$$
e^{-s/2}.
$$

### NG-J4

$$
\text{periodic renormalized }L^2\text{ orbit}
\Rightarrow
\text{contradiction from }L^2\text{ balance}.
$$

FALSE。

### NG-J5

$$
\text{field-level periodic orbit in finite }L^3
\text{ can represent blow-up}.
$$

FALSE under the critical-norm blow-up necessity。

### NG-J6

$$
\text{periodic defect metadata}
\Rightarrow
\text{periodic field}.
$$

FALSE。

### NG-J7

$$
\text{compact defect recurrence}
\Rightarrow
\text{critical field compactness}.
$$

FALSE unless the fibers are uniformly bounded/precompact。

### NG-J8

$$
\text{Critical Fiber Escape}
\Rightarrow
\text{contradiction}.
$$

FALSE；it is in fact required by hypothetical blow-up。

### NG-J9

$$
\text{known self-similar Liouville results kill all defect cycles}.
$$

FALSE；they kill only liftable field-level profile scenarios satisfying their hypotheses。

---

# 63. X-Integration guards 更新

## G-LERAYTIME

Distinguish：

$$
s=-\log(T^\ast-t)
$$

from：

$$
-\log r=s/2.
$$

## G-FIELDDEF

Keep full renormalized field state distinct from compact defect projection。

## G-L2TRADE

Physical-energy monotonicity is subcritical-weighted。

## G-CRITFIELD

Preserve：

$$
L^3,
\dot H^{1/2}
$$

critical field norms。

## G-FIBER

Every recurrent defect set stores its critical fiber radius。

## G-PERDEF

Periodic defect metadata does not imply DSS field profile。

## G-LIOUVILLE

Apply self-similar/DSS no-go only after a legitimate field-level lift with required integrability。

## G-CAPFIELD

Distinguish field-critical infinity from edge-capacity inflation。

---

# 64. True ETN update

Backward renormalized field state：

$$
\boxed{
\Theta_U^{C6J}(s)
=
\left\langle
U(s),
P(s),
\|U\|_3,
\|U\|_{\dot H^{1/2}},
E(s),
D_{crit}(s),
\pi(U(s))
\right\rangle.
}
$$

Skew-product state：

$$
\boxed{
\Theta_{\rm skew}^{C6J}
=
\left(
\theta_{def},
\kappa_{fiber}
\right),
}
$$

where：

$$
\boxed{
\kappa_{fiber}
=
\log
\left[
1+
\|U\|_3
+
\|U\|_{\dot H^{1/2}}
\right].
}
$$

Hypothetical blow-up requires：

$$
\boxed{
\kappa_{fiber}\to\infty.
}
$$

---

# 65. Formal status

$$
\boxed{
\begin{aligned}
\text{backward Leray autonomous flow}
&:\ \mathrm{PROVED},\\
\text{fixed point}\leftrightarrow\text{backward self-similar}
&:\ \mathrm{PROVED},\\
\text{periodic orbit}\leftrightarrow\text{backward DSS}
&:\ \mathrm{PROVED},\\
\text{renormalized }L^2\text{ identity}
&:\ \mathrm{PROVED},\\
V_\alpha\text{ weighted identity}
&:\ \mathrm{PROVED},\\
\alpha\ge1/2\Rightarrow V_\alpha\text{ monotone}
&:\ \mathrm{PROVED},\\
V_{1/2}=\text{physical energy}
&:\ \mathrm{PROVED},\\
\text{critical/monotone tradeoff in }V_\alpha\text{ family}
&:\ \mathrm{PROVED},\\
L^3\text{ backward-rescaling invariance}
&:\ \mathrm{PROVED},\\
\dot H^{1/2}\text{ invariance}
&:\ \mathrm{PROVED},\\
\text{potential blow-up}\Rightarrow
L^3,\dot H^{1/2}\text{ divergence}
&:\ \mathrm{EXTERNAL/VERIFIED},\\
\text{critical field-precompact recurrence}
&:\ \mathrm{NO\mbox{-}GO/PROVED},\\
\text{bounded-fiber compact defect recurrence}
&:\ \mathrm{NO\mbox{-}GO/PROVED},\\
\text{Critical Fiber Escape}
&:\ \mathrm{PROVED\ AS\ NECESSARY\ STATE\ PROPERTY},\\
\text{all fiber-escape mechanisms classified}
&:\ \mathrm{NOT\ PROVED},\\
\text{universal critical fiber Lyapunov}
&:\ \mathrm{NOT\ FOUND},\\
\text{global regularity}
&:\ \mathrm{OPEN}.
\end{aligned}
}
$$

---

# 66. 結論

C6-I 將：

$$
s=-\log r
$$

提出為 natural cycle time。

C6-J現在把這件事真正接回 N–S PDE。

使用標準 backward Leray time：

$$
s=-\log(T^\ast-t),
$$

renormalized velocity：

$$
U
=
\sqrt{T^\ast-t}
\,u
$$

滿足 autonomous equation：

$$
\boxed{
\partial_sU
+
\frac12U
+
\frac12(y\cdot\nabla)U
+
(U\cdot\nabla)U
+
\nabla P
=
\nu\Delta U.
}
$$

所以：

$$
\boxed{
\text{fixed point}
=
\text{backward self-similar},
}
$$

$$
\boxed{
\text{periodic orbit}
=
\text{backward DSS}.
}
$$

接著 natural $L^2$ balance：

$$
\boxed{
\frac12E'
+
\nu D
-
\frac14E
=
0.
}
$$

說明：

$$
\boxed{
E=\|U\|_2^2
}
$$

不是 monotone。

Weighted family：

$$
V_\alpha
=
e^{-\alpha s}E
$$

滿足：

$$
\boxed{
V_\alpha'
=
e^{-\alpha s}
\left[
\left(
\frac12-\alpha
\right)E
-
2\nu D
\right].
}
$$

所以：

$$
\boxed{
\alpha\ge1/2
}
$$

才有 universal monotonicity。

最弱 monotone case：

$$
\alpha=1/2
$$

就是 physical energy：

$$
\boxed{
V_{1/2}
=
e^{-s/2}E.
}
$$

這個：

$$
e^{-s/2}=r
$$

正是 C6-I Zeno summability的 weight。

因此：

$$
\boxed{
\textbf{telescoping potential存在，
但 universal one是 subcritical-weighted。}
}
$$

真正 critical field norms則完全不同：

$$
\boxed{
\|U\|_3
=
\|u\|_3,
}
$$

$$
\boxed{
\|U\|_{\dot H^{1/2}}
=
\|u\|_{\dot H^{1/2}}.
}
$$

而 hypothetical blow-up需要：

$$
\boxed{
\|U(s)\|_3,
\|U(s)\|_{\dot H^{1/2}}
\to\infty.
}
$$

所以：

$$
\boxed{
\textbf{任何 critical-field precompact recurrent orbit都被排除。}
}
$$

這包括 finite-critical-norm fixed / periodic field cycles。

也和已知 backward self-similar / asymptotically DSS Liouville no-go相容。

但這沒有殺掉 C6 defect cycles。

因為：

$$
\boxed{
\textbf{compact defect recurrence}
\neq
\textbf{compact field recurrence}.
}
$$

真正 survivor只能是：

$$
\boxed{
\text{compact recurrent defect base}
+
\text{critical noncompact field fiber}.
}
$$

正式定義：

$$
\boxed{
\textbf{Critical Fiber Escape}.
}
$$

若：

$$
\pi:
\mathcal X_{crit}
\to
\mathcal K_{def}
$$

是 C6 defect projection，

任何被 hypothetical blow-up infinitely often訪問的 compact defect set：

$$
K
$$

都必：

$$
\boxed{
\sup_{\pi(U)\in K}
\left(
\|U\|_3
+
\|U\|_{\dot H^{1/2}}
\right)
=
\infty.
}
$$

這表示 C6 的 cycle問題最後已經不是普通 finite graph。

而是：

$$
\boxed{
\textbf{compact base + noncompact critical fiber 的 skew-product recurrence問題。}
}
$$

下一篇因此正式轉向：

$$
\boxed{
\textbf{C6-K — Critical Fiber Escape,
Defect-Fiber Compactness,
and Profile-Splitting Closure}.
}
$$

---

# References

1. G. Seregin, *A certain necessary condition of potential blow up for Navier-Stokes equations*, arXiv:1104.3615.
2. G. Seregin, *Necessary conditions of potential blow up for Navier-Stokes equations*, arXiv:1101.1869.
3. D. Chae, *Remarks on the asymptotically discretely self-similar solutions of the Navier-Stokes and the Euler equations*, arXiv:1306.0305.
4. D. Chae, J. Wolf, *On the Liouville type theorems for self-similar solutions to the Navier-Stokes equations*, arXiv:1609.06962.
5. D. Chae, J. Wolf, *Removing discretely self-similar singularities for the 3D Navier-Stokes equations*, arXiv:1610.09464.
6. T.-P. Tsai, *Forward Discretely Self-Similar Solutions of the Navier-Stokes Equations*, arXiv:1210.2783.
7. A. Cheskidov, M. Dai, *Regularity criteria for the 3D Navier-Stokes and MHD equations*, arXiv:1507.06611.
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# Internal dependencies

- `NS_C6I_CriticalDebt_CapacityInfinity_BarrierCycles_v0.1.md`
- `NS_C6H_BoundaryFaces_DebtCoercivity_CycleElimination_v0.1.md`
- `NS_C6G_TypedCrossDomainGraph_SCC_BoundarySurvivors_v0.1.md`
- `NS_C6F_SharedSource_CoreExtraction_CrossDomainRouting_v0.1.md`
- `NS_C6E_TemporalSpatial_SharedSource_TTrap_v0.1.md`
- `NS_C6D_GeometryPressure_Provenance_SignatureReturn_v0.1.md`
- `NS_C6C_DuhamelCoherence_ReentryCriticalSaturation_v0.1.md`
- `NS_C6B_ForcingReentry_HF_CycleTest_v0.1.md`
- `NS_C6A_CertifiedDefectGraph_TypedCycles_MinimalSurvivors_v0.1.md`
- `NS_C5M_UnifiedDefectGraph_C5PhaseClosure_v0.1.md`

Next:

$$
\boxed{
\textbf{C6-K — Critical Fiber Escape,
Defect-Fiber Compactness,
and Profile-Splitting Closure}
}
$$