# NS × X Integration × 24/72 Paradigm In Action
## Round 10 — Pure Continuous Triad Phase Dynamics / Phase-Locking Route

- Date: 2026-08-16
- Version: v0.1
- Status: Proof-Route Experiment / Continuous-Only Phase-Dynamics Branch
- canonical source: UTF-8 Markdown
- canonical math delimiters: inline `$...$`; display `$$...$$`
- Previous round: `NS_X72_Round09_PureContinuous_FourierTriad_PhaseCoherence_v0.1_2026-08-16.md`
- This round's objective: Exact time differentiation of the translation-invariant triad interaction phase from Round 09, to determine the respective roles of viscosity, the nonlinear network, quartic lifting, and phase-locking; and to use the nonstationary-phase identity to verify the exact conditions under which "sustained signed transfer must be accompanied by phase locking or strong modulation."
- Non-claims: This round does not prove that 3D Navier–Stokes triad phases necessarily dephase, nor does it prove that phase locking is necessarily insufficient to support a finite-time singularity. Instead, this round proves that viscosity itself does not directly rotate the triad phase, pushing the remaining problem to the nonlinear phase-locking network.

# 0. Round 09 handoff

Round 09 defined the ordered interaction product for the continuous Fourier triad:

$$
k=p+q
$$

Writing:

$$
Z(k;p,q)
=
\left(
k\cdot\widehat u(p)
\right)
\left(
\widehat u(q)\cdot
\overline{\widehat u(k)}
\right).
$$

Writing:

$$
\boxed{
Z
=
\mathcal A e^{i\Phi},
}
\tag{0.1}
$$

where:

$$
\mathcal A=|Z|,
$$

and the signed triad transfer:

$$
\boxed{
\mathcal T
=
\operatorname{Im}Z
=
\mathcal A\sin\Phi.
}
\tag{0.2}
$$

yielding the analytic weighted covariance:

$$
G
\operatorname{Cov}(r,\vartheta)
=
\iint
w_k(r_k-m)
\mathcal A
\sin\Phi
\,dp\,dk.
$$

Thus, the Round 09 STOP was:

$$
\boxed{
\text{STOP-C13}
=
\text{Triad Phase-Coherence / Commutator-Sign Gap}.
}
$$

This round directly investigates:

$$
\boxed{
\partial_t\Phi.
}
$$

---

# 1. Gauge-safe triad phase

Directly choosing a scalar phase for each complex vector Fourier mode is unnatural, because:

$$
\widehat u(k)
$$

lies in the two-dimensional complex polarization plane orthogonal to:

$$
k
$$

Therefore, this round does not define any arbitrary modal scalar phase.

Instead, we use the scalar interaction product from Round 09:

$$
Z(k;p,q).
$$

Its phase:

$$
\boxed{
\Phi
=
\arg Z
}
\tag{1.1}
$$

is an interaction-level phase.

---

# 2. Translation invariance of the interaction phase

Performing a physical translation:

$$
u(x)
\mapsto
u(x+x_0).
$$

Under the current Fourier convention:

$$
\widehat u(r)
\mapsto
e^{ir\cdot x_0}
\widehat u(r).
$$

Therefore:

$$
k\cdot\widehat u(p)
\mapsto
e^{ip\cdot x_0}
k\cdot\widehat u(p),
$$

and:

$$
\widehat u(q)\cdot
\overline{\widehat u(k)}
\mapsto
e^{i(q-k)\cdot x_0}
\widehat u(q)\cdot
\overline{\widehat u(k)}.
$$

From:

$$
k=p+q,
$$

we have:

$$
p+q-k=0.
$$

Hence:

$$
\boxed{
Z
\mapsto
Z.
}
\tag{2.1}
$$

Thus:

$$
\boxed{
\Phi
}
$$

is not an artificial phase gauge arising from the physical origin.

It is a translation-invariant triad interaction phase.

Named:

$$
\boxed{
\textbf{Triad-Phase Gauge Invariance}.
}
$$

---

# 3. Fourier equation with nonlinear source

Write:

$$
\boxed{
\partial_t\widehat u(r)
=
-\nu|r|^2\widehat u(r)
+
N(r),
}
\tag{3.1}
$$

where:

$$
\boxed{
N(r)
=
-iP_r
\int_{\mathbb R^3}
\left(
r\cdot\widehat u(a)
\right)
\widehat u(r-a)
\,da.
}
\tag{3.2}
$$

All pressure effects have been handled by the Leray projector:

$$
P_r
$$

Therefore, no additional pressure phase is added to the triad phase dynamics.

---

# 4. Exact triad-product evolution

Fix:

$$
k=p+q.
$$

Define:

$$
A
=
k\cdot\widehat u(p),
$$

$$
B
=
\widehat u(q)\cdot
\overline{\widehat u(k)}.
$$

Then:

$$
Z=AB.
$$

From (3.1):

$$
A'
=
-\nu|p|^2A
+
k\cdot N(p).
$$

and:

$$
\boxed{
\begin{aligned}
B'
={}&
-\nu
\left(
|q|^2+|k|^2
\right)B
\\
&+
N(q)\cdot\overline{\widehat u(k)}
+
\widehat u(q)\cdot\overline{N(k)}.
\end{aligned}
}
\tag{4.1}
$$

Therefore:

$$
\boxed{
Z'
+
\nu\Sigma_{kpq}Z
=
Q,
}
\tag{4.2}
$$

where:

$$
\boxed{
\Sigma_{kpq}
=
|k|^2+|p|^2+|q|^2,
}
\tag{4.3}
$$

and:

$$
\boxed{
\begin{aligned}
Q
={}&
\left(
k\cdot N(p)
\right)
B
\\
&+
A
\left[
N(q)\cdot
\overline{\widehat u(k)}
+
\widehat u(q)\cdot
\overline{N(k)}
\right].
\end{aligned}
}
\tag{4.4}
$$

This equation is exact.

---

# 5. Viscosity-Neutral Phase Rotation Theorem

At:

$$
Z\neq0
$$

From:

$$
Z
=
\mathcal A e^{i\Phi}
$$

and (4.2):

$$
\frac{Z'}{Z}
=
-\nu\Sigma_{kpq}
+
\frac QZ.
$$

Taking the real and imaginary parts:

$$
\boxed{
\frac{\mathcal A'}{\mathcal A}
=
-\nu\Sigma_{kpq}
+
\operatorname{Re}
\frac QZ,
}
\tag{5.1}
$$

and:

$$
\boxed{
\Phi'
=
\operatorname{Im}
\frac QZ.
}
\tag{5.2}
$$

Define the nonlinear phase angular velocity:

$$
\boxed{
\Omega_\Phi
=
\operatorname{Im}
\frac QZ.
}
\tag{5.3}
$$

Therefore:

$$
\boxed{
\Phi'
=
\Omega_\Phi.
}
$$

Most importantly:

$$
\boxed{
-\nu\Sigma_{kpq}
}
$$

is purely real.

Thus:

$$
\boxed{
\textbf{
viscosity directly damps triad amplitude but does not directly rotate triad phase.
}
}
\tag{5.4}
$$

If:

$$
N\equiv0,
$$

then:

$$
Q=0
$$

and:

$$
\boxed{
\Phi'=0.
}
\tag{5.5}
$$

That is, pure heat evolution preserves the phase of every nonzero interaction product.

---

# 6. Consequence — no universal viscous dephasing mechanism

Round 09 proposed a possible candidate for:

$$
\text{viscous phase dispersion / dephasing}
$$

The exact equation (5.2) in this round shows:

$$
\boxed{
\text{viscosity alone cannot be that mechanism}.
}
$$

Any:

- phase drift;
- phase locking;
- phase synchronization;
- phase decoherence;

at the exact modal interaction phase level must be determined by:

$$
\boxed{
Q
}
$$

which is the nonlinear network coupling.

Therefore:

$$
\boxed{
\textbf{
dissipation and dephasing are distinct mechanisms.
}
}
\tag{6.1}
$$

---

# 7. Exact transfer-kernel evolution without dividing by $Z$

The phase equation is not suitable for direct use at:

$$
Z=0
$$

However, the signed transfer:

$$
\mathcal T
=
\operatorname{Im}Z
$$

can always be used.

Taking the imaginary part of (4.2):

$$
\boxed{
\mathcal T'
+
\nu\Sigma_{kpq}\mathcal T
=
\operatorname{Im}Q.
}
\tag{7.1}
$$

Therefore:

- viscosity applies linear damping to the existing signed transfer amplitude;
- the nonlinear quartet forcing:

$$
\operatorname{Im}Q
$$

can generate, sustain, or reverse the signed transfer.

This equation does not produce a division singularity at:

$$
Z=0
$$

---

# 8. Unit-circle phase-coherence dynamics

At:

$$
Z\neq0
$$

define:

$$
c_\Phi
=
\cos\Phi
=
\frac{\operatorname{Re}Z}{|Z|},
$$

$$
s_\Phi
=
\sin\Phi
=
\frac{\operatorname{Im}Z}{|Z|}.
$$

From:

$$
\Phi'=\Omega_\Phi
$$

we obtain:

$$
\boxed{
c_\Phi'
=
-\Omega_\Phi s_\Phi,
}
\tag{8.1}
$$

$$
\boxed{
s_\Phi'
=
\Omega_\Phi c_\Phi.
}
\tag{8.2}
$$

and:

$$
\boxed{
c_\Phi^2+s_\Phi^2=1.
}
$$

Thus, the normalized phase coherence is rotated on the unit circle by the nonlinear angular velocity:

$$
\Omega_\Phi
$$

Viscosity does not appear in the normalized phase ODE.

---

# 9. Quartet lifting

From (3.2):

$$
N(p)
$$

is already a quadratic convolution over:

$$
a\in\mathbb R^3
$$

of:

$$
\widehat u(a)
\widehat u(p-a).
$$

Thus:

$$
Q
$$

in:

$$
(k\cdot N(p))B
$$

contains:

$$
\boxed{
\widehat u(a)
\widehat u(p-a)
\widehat u(q)
\overline{\widehat u(k)}.
}
$$

Similarly:

$$
N(q)\cdot\overline{\widehat u(k)}
$$

and:

$$
\widehat u(q)\cdot\overline{N(k)}
$$

also generate quartic modal products.

Therefore:

$$
\boxed{
\textbf{
exact triad-phase dynamics lifts cubic triad products to quartic convolution forcing.
}
}
\tag{9.1}
$$

This is not an approximation.

It is a direct algebraic consequence of the quadratic PDE nonlinearity under phase differentiation.

---

# 10. Continuous neighboring-triad network

The quartic forcing does not need to be represented by a discrete graph.

For example:

$$
N(p)
=
\int_{\mathbb R^3}
\mathcal K_p(a,p-a)
\,da
$$

indicates that:

the phase velocity of the triad:

$$
(k,p,q)
$$

is influenced by all:

$$
(a,p-a,p)
$$

neighboring interactions.

Thus, we can define the continuous triad manifold:

$$
\boxed{
\mathfrak T
=
\left\{
(k,p,q)\in(\mathbb R^3)^3:
k=p+q
\right\}.
}
\tag{10.1}
$$

Its phase field:

$$
\boxed{
\Phi:
\mathfrak T\times[0,T)
\to
\mathbb S^1
}
\tag{10.2}
$$

satisfies:

$$
\boxed{
\partial_t\Phi
=
\Omega_\Phi[\widehat u].
}
\tag{10.3}
$$

where:

$$
\Omega_\Phi
$$

is a continuous integral operator depending on the full Fourier field sharing the triad vertices.

Therefore, quartet lifting:

$$
\not\Rightarrow
$$

essential discreteness.

---

# 11. Phase-only closure fails exactly

Although:

$$
\Phi'
=
\Omega_\Phi,
$$

yet:

$$
\Omega_\Phi
=
\operatorname{Im}(Q/Z)
$$

depends on:

- modal amplitudes;
- vector polarizations;
- neighboring-mode phases;
- neighboring triad amplitudes;
- Leray-projected convolution geometry.

Therefore, there is no scalar autonomous law automatically derived from this formulation:

$$
\boxed{
\Phi'
=
F(\Phi)
}
$$

or:

$$
\Phi'
=
F(k,p,q,\Phi)
$$

that relies solely on the current single triad phase closure.

Therefore:

$$
\boxed{
\textbf{
phase-only observation is not an exact closed state for 3D NS triad dynamics.
}
}
\tag{11.1}
$$

This does not negate the use of phase-only reduced models as approximate/statistical models.

It only negates their qualification as an exact deterministic closure.

---

# 12. A phase-speed singularity at vanishing interaction amplitude

From:

$$
\Omega_\Phi
=
\operatorname{Im}(Q/Z),
$$

When:

$$
|Z|
$$

is very small, the phase velocity representation may become large or lose its meaning.

This is not a physical PDE singularity.

It indicates that:

$$
\boxed{
\text{phase of an almost-zero interaction product is a bad coordinate}.
}
$$

Therefore, an exact proof should not solely track:

$$
\Phi
$$

and forget:

$$
\mathcal A.
$$

A more stable primary carrier is the pair:

$$
\boxed{
(\mathcal A,\mathcal T)
}
$$

or the complex:

$$
\boxed{
Z.
}
$$

The phase is a derived coordinate in the region where:

$$
Z\neq0
$$

---

# 13. Nonstationary-Phase Cancellation Lemma

Now we investigate sustained signed transfer.

Let a fixed triad satisfy:

$$
Z(t)\neq0.
$$

on the interval:

$$
I=[t_0,t_1]
$$

Let:

$$
b(t)
$$

be any $C^1$ real amplitude weight.

Consider:

$$
\boxed{
\mathcal J_I
=
\int_{t_0}^{t_1}
b(t)\sin\Phi(t)\,dt.
}
\tag{13.1}
$$

If:

$$
\Omega_\Phi(t)=\Phi'(t)
$$

is non-zero on $I$,

from:

$$
\frac d{dt}
\cos\Phi
=
-\Omega_\Phi\sin\Phi
$$

we have:

$$
\sin\Phi
=
-
\frac1{\Omega_\Phi}
\frac d{dt}\cos\Phi.
$$

Thus, by integration by parts:

$$
\boxed{
\begin{aligned}
\mathcal J_I
={}&
-
\left[
\frac{
b\cos\Phi
}{
\Omega_\Phi
}
\right]_{t_0}^{t_1}
\\
&+
\int_{t_0}^{t_1}
\cos\Phi
\frac d{dt}
\left(
\frac b{\Omega_\Phi}
\right)
dt.
\end{aligned}
}
\tag{13.2}
$$

If:

$$
|\Omega_\Phi|\ge\omega_0>0,
$$

then:

$$
\boxed{
\begin{aligned}
|\mathcal J_I|
\le{}&
\frac{
|b(t_0)|+|b(t_1)|
}{
\omega_0
}
\\
&+
\frac1{\omega_0}
\int_I|b'|dt
\\
&+
\frac1{\omega_0^2}
\int_I
|b|
|\Omega_\Phi'|
dt.
\end{aligned}
}
\tag{13.3}
$$

Named:

$$
\boxed{
\textbf{Nonstationary-Phase Cancellation Lemma}.
}
$$

---

# 14. Meaning of the cancellation lemma

If the triad phase continuously rotates rapidly:

$$
|\Phi'|
\ge
\omega_0,
$$

and:

$$
b/\Phi'
$$

does not have drastic total variation,

then:

$$
\int
b\sin\Phi
$$

can only produce a finite residual from:

- boundary terms;
- amplitude modulation;
- phase-speed modulation.

Therefore, sustained large signed transfer cannot rely solely on "the phase constantly rotating."

It requires at least one of the following:

$$
\boxed{
\begin{aligned}
&\text{A. phase locking / slow phase: }|\Phi'|\approx0,
\\
&\text{B. strong amplitude modulation},
\\
&\text{C. strong phase-acceleration modulation}.
\end{aligned}
}
\tag{14.1}
$$

This is the first time-accumulation rigidity statement of the continuous phase route.

---

# 15. Phase-Locking Necessity for persistent coherent transfer

For the weighted triad contribution from Round 09, take:

$$
b(t)
=
\mathcal W_m(k,t)
\mathcal A(k;p,q,t).
$$

If a fixed triad provides a sustained, significant contribution of the same sign to:

$$
\int
\mathcal W_m
\mathcal A
\sin\Phi
\,dt
$$

over a long time, and:

$$
b/\Phi'
$$

does not vary exceptionally drastically,

then by Section 13, there must be periods where it enters:

$$
\boxed{
|\Phi'|
=
|\Omega_\Phi|
\ll1.
}
\tag{15.1}
$$

Therefore:

$$
\boxed{
\textbf{
persistent phase-coherent transfer requires phase locking,
near-locking, or compensating singular modulation.
}
}
\tag{15.2}
$$

This does not mean that every instantaneous forward-transfer triad must be phase-locked.

It is a time-integrated statement.

---

# 16. Exact phase-locking condition

From:

$$
\Phi'
=
\operatorname{Im}
\frac QZ,
$$

exact phase lock:

$$
\Phi'=0
$$

is equivalent to:

$$
\boxed{
\operatorname{Im}
\left(
Q\overline Z
\right)
=
0
}
\tag{16.1}
$$

at:

$$
Z\neq0.
$$

Since $Q,Z$ are both complex scalars,

(16.1) is equivalent to:

$$
\boxed{
Q
=
\lambda Z
}
\tag{16.2}
$$

for some real:

$$
\lambda\in\mathbb R.
$$

Named:

$$
\boxed{
\textbf{Phase-Locked Ray Condition}.
}
$$

---

# 17. Dynamics on the phase-locked ray

If on some interval:

$$
Q=\lambda Z,
\qquad
\lambda\in\mathbb R,
$$

then from (4.2):

$$
\boxed{
Z'
=
\left(
\lambda
-
\nu\Sigma_{kpq}
\right)Z.
}
\tag{17.1}
$$

Therefore:

$$
\boxed{
\Phi'=0,
}
$$

and:

$$
\boxed{
\frac{\mathcal A'}{\mathcal A}
=
\lambda
-
\nu\Sigma_{kpq}.
}
\tag{17.2}
$$

Thus, on the exact phase-locking manifold:

- the nonlinear network only changes the interaction amplitude;
- viscosity also only changes the amplitude;
- the interaction complex ray remains unchanged.

If:

$$
\sin\Phi>0,
$$

then the phase sign of the signed forward transfer remains unchanged during the lock interval.

---

# 18. Maximal-transfer lock

If:

$$
\Phi
=
\frac\pi2
\quad
(\operatorname{mod}2\pi),
$$

then:

$$
\boxed{
\sin\Phi=1.
}
$$

If simultaneously:

$$
Q=\lambda Z
$$

holds,

then the triad interaction, at a fixed amplitude, lies in the maximal positive phase-coherence direction, and the phase does not rotate.

Therefore, the most dangerous coherent state can be compressed into:

$$
\boxed{
\Phi\approx\frac\pi2
\quad
\text{and}
\quad
\operatorname{Im}(Q\overline Z)\approx0.
}
\tag{18.1}
$$

This further compresses the:

$$
\text{positive phase coherence}
$$

from Round 09 into:

$$
\boxed{
\text{positive phase coherence + nonlinear phase locking}.
}
$$

---

# 19. Why viscosity cannot break an exact phase lock

Under the exact lock:

$$
Q=\lambda Z
$$

the viscosity contribution:

$$
-\nu\Sigma Z
$$

is parallel to the same complex ray as:

$$
Z
$$

Therefore, no matter how large:

$$
\nu>0
$$

is,

viscosity only changes:

$$
|Z|
$$

and does not change:

$$
\Phi.
$$

Thus, any attempt to use:

> viscosity will automatically disperse the coherent triad phase

as a deterministic proof mechanism is invalid.

Viscosity can:

- reduce the amplitude;
- reduce high-frequency mode energy;
- weaken the transfer kernel;

but:

$$
\boxed{
\text{not directly rotate the locked phase}.
}
$$

---

# 20. Network lock, not isolated-triad lock

In the full Navier–Stokes equations:

$$
Q
$$

is determined by a continuum of many neighboring interactions.

Therefore:

$$
Q=\lambda Z
$$

is not an isolated-triad algebraic trick.

It indicates that:

$$
\boxed{
\text{the entire surrounding nonlinear network produces a forcing
collinear with the current complex interaction ray}.
}
$$

Thus, the truly dangerous phase-locking object is the:

$$
\boxed{
\textbf{network-supported phase lock}.
}
$$

This is different from a single-triad truncation.

---

# 21. External evidence does not justify a universal dephasing assumption

Existing 3D Navier–Stokes numerical/diagnostic works have studied Fourier triad phases and found:

- phase alignments are correlated with the direction of energy flux;
- in extreme 3D NS flows, the transfer to small scales can be carried by a small fraction of phase-preferred triads;
- the triad network, rather than an isolated triad, is the relevant object.

Therefore, one cannot take:

$$
\boxed{
\text{random phase / automatic dephasing}
}
$$

as an unconditional deterministic axiom.

These external results serve only as phenomenological and methodological support, not as proofs for the theorems in this round.

---

# 22. Interaction-order proliferation

If we now differentiate again:

$$
Q,
$$

each:

$$
N(r)
$$

will again use a quadratic convolution.

Thus, the raw polynomial degree continues to rise:

$$
\boxed{
3
\to
4
\to
5
\to
\cdots
}
\tag{22.1}
$$

A natural integer order appears in the interaction-product expansion.

This is the first time in the current Pure-C route that a seemingly "naturally discrete" index appears:

$$
n
=
3,4,5,\ldots
$$

But we cannot yet declare:

$$
T_{\mathsf C\to\mathsf D}.
$$

Reasons:

1. the exact full Fourier field:

$$
\widehat u(k,t)
$$

is already closed in itself;
2. $Q$ can be directly written as a continuous convolution operator;
3. the interaction-order expansion might be globally resummed using a continuous generating functional, without needing to expand order by order in $n$.

Therefore:

$$
\boxed{
\text{discrete interaction order appears},
}
$$

but:

$$
\boxed{
\text{essential discrete proof dependence has not yet been proved}.
}
$$

---

# 23. Candidate continuous resummation

The next Pure-C repair candidate is not:

$$
n=3,4,5,\ldots
$$

writing the interaction hierarchy order by order.

Instead, it is to establish a continuous functional source:

$$
\boxed{
\mathcal Z[\varphi,t]
=
\exp
\left(
\int_{\mathbb R^3}
\varphi(k)\cdot
\widehat u(k,t)
\,dk
\right).
}
\tag{23.1}
$$

Formally:

$$
\frac{
\delta\mathcal Z
}{
\delta\varphi(k)
}
=
\widehat u(k)
\mathcal Z,
$$

and:

$$
\frac{
\delta^2\mathcal Z
}{
\delta\varphi(p)\delta\varphi(q)
}
=
\widehat u(p)
\widehat u(q)
\mathcal Z.
$$

Therefore, the quadratic NS convolution could potentially be written as a:

$$
\boxed{
\text{second functional derivative}
}
$$

rather than explicitly listing:

$$
3\to4\to5\to\cdots.
$$

This is what will be formally verified in the next round:

$$
\boxed{
\textbf{Deterministic Hopf-Type Functional Resummation}.
}
$$

Currently, it serves only as a candidate; we do not prematurely claim closure in this round.

---

# 24. STOP-C14 — Nonlinear Phase-Locking / Quartet-Network Gap

Define:

$$
\boxed{
\bot_X^{\mathrm{C14}}
=
\left\langle
\begin{array}{l}
\text{layer}
=
\mathrm{continuous\ triad\ phase\ dynamics},
\\
\text{exact\ phase\ law}
=
\Phi'
=
\operatorname{Im}(Q/Z),
\\
\text{viscous\ phase\ rotation}
=
0,
\\
\text{raw\ nonlinear\ forcing}
=
\mathrm{quartic\ convolution},
\\
\text{persistent\ transfer}
=
\mathrm{phase\ lock}
\vee
\mathrm{strong\ modulation},
\\
\text{lock\ condition}
=
\operatorname{Im}(Q\overline Z)=0,
\\
\text{dangerous\ lock}
=
\Phi\approx\pi/2
\text{ with network-supported lock},
\\
\text{missing}
=
\mathrm{unconditional\ exclusion\ or\ integrable\ control\ of\ such\ locks},
\\
\text{essential\ discrete\ intrusion}
=
\mathrm{not\ yet\ established}.
\end{array}
\right\rangle.
}
$$

Named:

$$
\boxed{
\textbf{STOP-C14:
Nonlinear Phase-Locking / Quartet-Network Gap}.
}
$$

---

# 25. 24/72 interpretation

The substrate for this round:

$$
\boxed{
B=\mathsf C.
}
$$

Since all wavevectors:

$$
k,p,q,a\in\mathbb R^3
$$

remain continuous.

The update organization is more clearly:

$$
\boxed{
\mathsf P_{\rm convolution}
+
\mathsf S_{\rm time}.
}
$$

The observation route:

$$
\boxed{
\mathsf X_{\rm amplitude/geometry}
\to
\mathsf C_{\rm targeted\ interaction\ phase},
}
$$

But if only the phase is retained:

$$
\boxed{
\mathsf C_{\Phi}
\to
\mathsf X_{\rm phase-only}
}
$$

because the exact phase derivative still requires amplitude / polarization / network information.

The transition law remains:

$$
\boxed{
L=\mathsf F.
}
$$

There is no need for a probability kernel:

$$
\mathsf K
$$

to define the exact deterministic phase dynamics.

---

# 26. 24/72 Ledger — Round 10

| Step | X object / operation | $B$ | $U$ | $O$ | $L$ | status |
|---|---|---|---|---|---|---|
| C97 | gauge-safe triad product $Z$ | $\mathsf C$ | triadic | targeted complex scalar | $\mathsf F$ | FORM |
| C98 | translation invariance of $Z$ | $\mathsf C$ | — | targeted | $\mathsf F$ | PROVED |
| C99 | exact $Z'$ equation | $\mathsf C$ | $\mathsf S/\mathsf P$ | complex | $\mathsf F$ | EXACT |
| C100 | $\Phi'=\operatorname{Im}(Q/Z)$ | $\mathsf C$ | network | phase | $\mathsf F$ | EXACT where $Z\neq0$ |
| C101 | direct viscous dephasing | $\mathsf C$ | — | phase | $\mathsf F$ | REFUTED |
| C102 | transfer evolution $\mathcal T'+\nu\Sigma\mathcal T=\operatorname{Im}Q$ | $\mathsf C$ | network | signed transfer | $\mathsf F$ | EXACT |
| C103 | quartic lifting | $\mathsf C$ | continuous convolution | $\mathsf X$ | $\mathsf F$ | PROVED |
| C104 | phase-only exact closure | $\mathsf C$ | — | phase only | $\mathsf F$ | REFUTED |
| C105 | nonstationary-phase cancellation | $\mathsf C$ | temporal | targeted | $\mathsf F$ | PROVED |
| C106 | phase-locking necessity for sustained transfer | $\mathsf C$ | temporal/network | relational | $\mathsf F$ | CONDITIONAL RIGIDITY |
| C107 | phase-locked ray $Q=\lambda Z$ | $\mathsf C$ | network | complex relation | $\mathsf F$ | EXACT equivalence |
| C108 | universal exclusion of network-supported positive lock | $\mathsf C$ | network | targeted | $\mathsf F$ | OPEN / STOP-C14 |
| C109 | discrete interaction order $n$ | mixed representation issue | — | hierarchy | $\mathsf F$ | APPEARS BUT NOT ESSENTIAL YET |
| C110 | functional resummation candidate | $\mathsf C$ | functional | $\mathsf X$ | $\mathsf F$ | NEXT |

---

# 27. Pure-C path after ten rounds

$$
\boxed{
\begin{aligned}
\mathsf C_{\rm energy}
&\to
\mathsf C_{\rm critical}
\\
&\to
\mathsf C_{\rm relational}
\\
&\to
\mathsf C_{\rm nonlocal}
\\
&\to
\mathsf C_{\rm projected}
\\
&\to
\mathsf C_{\rm gradient\ geometry}
\\
&\to
\mathsf C_{\rm hierarchy}
\\
&\to
\mathsf C_{\rm Gevrey}
\\
&\to
\mathsf C_{\rm covariance}
\\
&\to
\mathsf C_{\rm triad\ phase}
\\
&\to
\mathsf C_{\rm phase\ network}.
\end{aligned}
}
\tag{27.1}
$$

Currently:

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

But for the first time, we see a:

$$
\boxed{
\text{natural discrete interaction order}
}
$$

emerge.

The next round will determine whether it can be resummed away by a continuous generating functional.

---

# 28. Strongest result of Round 10

The strongest exact reduction of this round:

$$
\boxed{
Z'
+
\nu
\left(
|k|^2+|p|^2+|q|^2
\right)Z
=
Q,
}
$$

Thus:

$$
\boxed{
\Phi'
=
\operatorname{Im}(Q/Z).
}
$$

From this, we obtain:

$$
\boxed{
\textbf{
viscosity damps triad-transfer amplitude,
but all exact triad-phase rotation is nonlinear.
}
}
$$

Then, by the nonstationary-phase identity:

$$
\boxed{
\textbf{
persistent signed transfer requires
phase locking / near-locking
or compensating strong modulation.
}
}
$$

Therefore, the Pure-C frontier is further compressed from:

$$
\text{phase coherence}
$$

into:

$$
\boxed{
\textbf{
network-supported nonlinear phase locking.
}
}
$$

---

# 29. Next round — deterministic functional resummation

The sole main objective of the next round:

$$
\boxed{
\textbf{
Can the interaction-order hierarchy be exactly resummed
into a continuous functional PDE?
}
}
$$

Specifically:

1. Establish:

$$
\mathcal Z[\varphi,t]
$$

or an equivalent generating functional;

2. Replace quadratic products with functional derivatives;

3. Derive the exact deterministic functional evolution;

4. Determine whether:

$$
n=3,4,5,\ldots
$$

is merely an expansion artifact rather than an essential discrete structure;

5. If the functional equation closes, Pure-C continues;

6. If exact resummation cannot avoid a countable interaction order, then for the first time, seriously consider:

$$
T_{\mathsf C\to\mathsf D}.
$$

---

# 30. External primary-source anchors

1. Di Kang, Bartosz Protas, Miguel D. Bustamante, *Alignments of Triad Phases in 1D Burgers and 3D Navier-Stokes Flows*, arXiv:2105.09425.
   - Fourier triad phases are correlated with energy flux;
   - in 3D NS extreme flows, small-scale energy flux can be carried by a small fraction of phase-preferred triads;
   - an isolated triad is insufficient to represent the full network dynamics.

2. Santiago J. Benavides, Miguel D. Bustamante, *Triad phase dynamics determine cascade direction in two-dimensional turbulence*, arXiv:2605.03049.
   - in 2D turbulence, triad-phase dynamics can be used to predict the cascade direction;
   - this document only uses it as a cross-dimensional methodological comparison for phase dynamics, and does not smuggle a 2D closure into a 3D NS theorem.

3. Brendan P. Murray, Miguel D. Bustamante, *Energy flux enhancement, intermittency and turbulence via Fourier triad phase dynamics in 1D Burgers equation*, arXiv:1705.08960.
   - theoretical and numerical evidence correlating triad-phase synchronization/alignment with forward flux enhancement;
   - used solely for phase-locking mechanism comparison.

The formulas for $Z'$, the viscosity-neutral phase, nonstationary-phase cancellation, and the phase-locked ray in this round are all directly derived in this document.

---

# 31. Commit state

$$
\boxed{
\begin{aligned}
\text{Route}
&:
\mathrm{Pure\ Continuous\ Triad\ Phase\ Dynamics},
\\
\text{Essential }\mathsf C\to\mathsf D
&:
\mathrm{Not\ reached},
\\
\text{Gauge-safe phase}
&:
\arg Z,
\\
\text{Viscous phase rotation}
&:
0,
\\
\text{Nonlinear phase speed}
&:
\Omega_\Phi=\operatorname{Im}(Q/Z),
\\
\text{Raw lifting}
&:
\mathrm{triad}\to\mathrm{quartic\ network},
\\
\text{Persistent transfer}
&:
\mathrm{lock}
\vee
\mathrm{strong\ modulation},
\\
\text{Exact lock}
&:
Q=\lambda Z,\ \lambda\in\mathbb R,
\\
\text{STOP-C14}
&:
\mathrm{Nonlinear\ Phase\text{-}Locking/Quartet\text{-}Network\ Gap},
\\
\text{Next}
&:
\mathrm{Deterministic\ Functional\ Resummation}.
\end{aligned}
}
$$