---
title: "Navier–Stokes C6-J: Log-Scale Renormalized Defect Flow, Telescoping Potentials, and 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: "en"
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, and Critical-Cycle Closure Tests

## 0. Positioning of the Current Phase

C6-I established the:

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

Criticalization Operator,

and placed:

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

all into the same N–S critical scaling ledger.

However, C6-I also proved:

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

Because the geometric scale ladder:

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

can simultaneously satisfy:

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

and:

$$
\sum_nr_n<\infty.
$$

Therefore:

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

can coexist within the scaling architecture.

Thus, the conclusion of C6-I is:

> **Criticalization corrects the scaling type, but does not provide directionality across scales.**

C6-J formally elevates:

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

to scale-time,

and asks:

1. Does N–S form an autonomous renormalized flow in log-scale?
2. What physical scenarios do fixed points / periodic orbits correspond to?
3. Does a natural telescoping potential exist?
4. Is it simultaneously critical and monotone?
5. Which cycles can be excluded by known self-similar / discretely self-similar no-go theorems?
6. If a compact defect orbit can still recur, how must the underlying field escape?

Main results of this phase:

1. Standard backward Leray variables map the hypothetical finite-time blow-up horizon to:
   $$
   s\to\infty;
   $$
2. renormalized N–S becomes an autonomous equation;
3. fixed point = backward self-similar profile;
4. periodic orbit = backward discretely self-similar profile;
5. renormalized $L^2$ balance is exact:
   $$
   \boxed{
   \frac12E'
   +
   \nu D
   -
   \frac14E
   =
   0;
   }
   $$
6. therefore, the criticalized renormalized $L^2$ energy itself is not monotone;
7. the weighted family:
   $$
   V_\alpha=e^{-\alpha s}E
   $$
   obeys an 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 Update

## 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. Conclusion

C6-I proposed:

$$
s=-\log r
$$

as the natural cycle time.

C6-J now truly connects this back to the N–S PDE.

Using the standard backward Leray time:

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

the renormalized velocity:

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

satisfies the autonomous equation:

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

Therefore:

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

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

Next, the natural $L^2$ balance:

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

shows that:

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

is not monotone.

The weighted family:

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

satisfies:

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

Therefore, only:

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

has universal monotonicity.

The weakest monotone case:

$$
\alpha=1/2
$$

is exactly the physical energy:

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

This:

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

is exactly the weight of the C6-I Zeno summability.

Thus:

$$
\boxed{
\textbf{a telescoping potential exists,
but the universal one is subcritical-weighted.}
}
$$

The genuine critical field norms are completely different:

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

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

And hypothetical blow-up requires:

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

Therefore:

$$
\boxed{
\textbf{any critical-field precompact recurrent orbit is excluded.}
}
$$

This includes finite-critical-norm fixed / periodic field cycles.

It is also compatible with known backward self-similar / asymptotically DSS Liouville no-go theorems.

But this does not kill C6 defect cycles.

Because:

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

The true survivor can only be:

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

Formally defined as:

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

If:

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

is the C6 defect projection,

any compact defect set visited infinitely often by a hypothetical blow-up:

$$
K
$$

must satisfy:

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

This means the C6 cycle problem is ultimately no longer an ordinary finite graph.

Instead, it is:

$$
\boxed{
\textbf{a skew-product recurrence problem of a compact base + noncompact critical fiber.}
}
$$

The next paper therefore formally turns to:

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

---

# 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.
8. Z. Grujić, L. Xu, *Asymptotic Criticality of the Navier-Stokes Regularity Problem*, arXiv:1911.00974; J. Math. Fluid Mech. 26, 53 (2024).
9. E. Miller, *On the interaction of strain and vorticity for solutions of the Navier-Stokes equation*, arXiv:2407.02691; Pure and Applied Analysis 8 (2026).

# 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:

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