# DCRP47 — Dual-Current Critical Scaling NO-GO and the Flat-Pancake Characteristic Window

**Series:** Independent Navier–Stokes Research Series  
**Date:** 2026-08-17  
**Status:** Proof-development checkpoint / same-parent scaling audit + equality-mode classification  
**Immediate predecessor:** `NS_DCRP46_GaugeFlat_AnnularScalar_TransportGap_2026-08-17.md`

**Primary internal dependencies**
- DCRP-30 — exact same-parent reroot identity and strict Type-II exponent window
- DCRP-31 — finite-radius inward PFET matching layer and raw PFET scaling
- DCRP-41 — fixed-plane zero-shape canonical pancake tensor
- DCRP-42 — planar shear scalar reduction
- DCRP-44 — gauge-flat scalar-connection theorem
- DCRP-45 — canonical-ray compression
- DCRP-46 — strict finite-annulus scalar-transport gap

**External calibration checked before this round**
- D. S. Agafontsev, E. A. Kuznetsov, A. A. Mailybaev, *Asymptotic solution for high vorticity regions in incompressible 3D Euler equations*, arXiv:1609.07782.
- G. Seregin, *On potential Type II blowups for the Navier-Stokes equations*, arXiv:2606.29468.

The Agafontsev–Kuznetsov–Mailybaev exact Euler pancake solution combines shear and asymmetric strain and is explicitly infinite-energy on $\mathbb R^3$; this remains a useful external calibration that local pancake geometry need not be contradictory by itself. Seregin's 2026 Type-II work continues to use Euler-scale local limits and Liouville-type exclusions, consistent with the local-limit setting used here.

No full Navier–Stokes regularity theorem is claimed.

---

# 0. Executive result

DCRP46 left a final rank-two branch carrying two compulsory finite-annulus observables:

1. a normalized inward PFET current;
2. a normalized canonical scalar-transport current.

The first task in DCRP47 was to test whether their simultaneous positivity creates a new same-parent return-depletion contradiction.

It does not.

The exact same-parent scaling gives a clean **critical two-current orbit**.

Let

$$
\ell_n
$$

be the physical root scale and

$$
a_n
$$

the Type-II amplitude-normalization parameter of DCRP-30.

Define

$$
\lambda_n
=
\frac{\ell_{n+1}}{\ell_n},
\qquad
\mu_n
=
\frac{a_{n+1}}{a_n}.
$$

On an exact DSS return,

$$
c_\ast
=
\frac{\lambda_\ast}{\mu_\ast}
=
\lambda_\ast^\alpha,
$$

hence

$$
\boxed{
\mu_\ast
=
\lambda_\ast^{1-\alpha}.
}
$$

The raw physical PFET payment scales as

$$
\boxed{
\mathcal P_{n+1}^{\rm phys}
=
\lambda_\ast^{3-2\alpha}
\mathcal P_n^{\rm phys}.
}
$$

The natural physical strain-unit representative of the DCRP46 scalar-transport current scales as

$$
\boxed{
\mathcal S_{n+1}^{\rm phys}
=
\lambda_\ast^{-(1+\alpha)}
\mathcal S_n^{\rm phys}.
}
$$

Writing

$$
\gamma
=
\frac1{\alpha+1},
$$

these become

$$
\boxed{
\mathcal P_{n+1}^{\rm phys}
=
e^{-(5\gamma-2)S_0}
\mathcal P_n^{\rm phys},
}
$$

$$
\boxed{
\mathcal S_{n+1}^{\rm phys}
=
e^{S_0}
\mathcal S_n^{\rm phys}.
}
$$

Thus one current decays while the strain-scale current grows, but the pair lies on an exact scale orbit.

With

$$
\boxed{
\theta
=
\frac{3-2\alpha}{1+\alpha}
=
5\gamma-2
\in
(0,1/2),
}
$$

the dimensionless return ratios satisfy

$$
\boxed{
\frac{\mathcal P_n^{\rm phys}}{\mathcal P_0^{\rm phys}}
\left(
\frac{\mathcal S_n^{\rm phys}}{\mathcal S_0^{\rm phys}}
\right)^\theta
=
1.
}
$$

Therefore:

$$
\boxed{
\textbf{
positive PFET + positive scalar transport does not produce a scale-only depletion theorem.
}
}
$$

This closes the naive two-current summability route.

The round then shifts from **payment size** to **equality-mode structure**.

On the DCRP44 fully flat canonical scalar branch, write

$$
\boxed{
w
=
B(q,s)-2A(s)z,
}
$$

where $A(s)$ is the canonical pancake strain amplitude and $B$ is the remaining scalar response.

Using only

$$
q=w-\phi_z
$$

and incompressibility, one obtains a new exact compatibility PDE:

$$
\boxed{
\nabla_h\cdot
\left[
(B_q-1)\nabla_hq
\right]
+
\partial_z
\left[
B_q q_z
\right]
=
0.
}
$$

Equivalently,

$$
\boxed{
B_q q_{zz}
+
(B_q-1)\Delta_hq
+
B_{qq}|\nabla q|^2
=
0.
}
$$

This equation has a sharp type change at

$$
\boxed{
B_q=0,\qquad B_q=1.
}
$$

The coefficient matrix is

$$
\boxed{
M_B
=
\operatorname{diag}
(
B_q-1,
B_q-1,
B_q
).
}
$$

Hence:

- $B_q>1$: elliptic sign;
- $B_q<0$: elliptic sign after multiplication by $-1$;
- $0<B_q<1$: Lorentzian/hyperbolic signature;
- $B_q=0$ or $1$: degenerate endpoints.

A new coercive rigidity follows.

If a connected flat patch has zero compatibility flux through its boundary and

$$
B_q\ge1+\delta
$$

everywhere, or

$$
B_q\le-\delta
$$

everywhere, then

$$
\nabla q=0.
$$

Therefore the planar vorticity vanishes.

Thus any nonzero closed flat rank-two pancake must satisfy at least one of:

$$
\boxed{
\text{response slope enters }[0,1]
}
$$

or

$$
\boxed{
\text{nonzero finite boundary compatibility flux}
}
$$

or

$$
\boxed{
\text{the flat/regular scalar chart breaks}.
}
$$

This is the new frontier.

The interval

$$
\boxed{
0\le B_q\le1
}
$$

is the **flat-pancake characteristic response window**.

For a constant response slope

$$
B_q=c,
$$

the equality mode obeys

$$
\boxed{
c q_{zz}
+
(c-1)\Delta_hq
=
0.
}
$$

For

$$
0<c<1,
$$

the real Fourier characteristic cone is

$$
\boxed{
\xi_z^2
=
\frac{1-c}{c}
|\xi_h|^2.
}
$$

Thus the final flat survivor has been converted into a characteristic-cone / boundary-feed problem rather than a scale-payment problem.

---

# 1. Exact same-parent root variables

DCRP-30 defines

$$
v_n(y,\tau)
=
\frac{\ell_n}{a_n}
U
\left(
x_n+\ell_ny,
t_n+\frac{\ell_n^2}{a_n}\tau
\right).
$$

Therefore:

### physical velocity scale

$$
\boxed{
U_n
=
\frac{a_n}{\ell_n};
}
\tag{1.1}
$$

### physical Euler-time scale

$$
\boxed{
T_n
=
\frac{\ell_n^2}{a_n};
}
\tag{1.2}
$$

### physical strain scale

$$
\boxed{
\Sigma_n
=
\frac{U_n}{\ell_n}
=
\frac{a_n}{\ell_n^2}
=
T_n^{-1};
}
\tag{1.3}
$$

### raw physical core-energy scale

$$
\boxed{
\beta_n
=
\ell_n a_n^2.
}
\tag{1.4}
$$

For consecutive same-parent roots,

$$
\boxed{
\lambda_n
=
\frac{\ell_{n+1}}{\ell_n},
}
\tag{1.5}
$$

$$
\boxed{
\mu_n
=
\frac{a_{n+1}}{a_n}.
}
\tag{1.6}
$$

Hence

$$
\boxed{
\frac{\beta_{n+1}}{\beta_n}
=
\lambda_n\mu_n^2,
}
\tag{1.7}
$$

and

$$
\boxed{
\frac{\Sigma_{n+1}}{\Sigma_n}
=
\frac{\mu_n}{\lambda_n^2}.
}
\tag{1.8}
$$

These are pure same-parent scaling identities.

---

# 2. Exact DSS return parameters

On the compact DSS branch,

$$
\boxed{
c_\ast
=
\frac{\lambda_\ast}{\mu_\ast}.
}
\tag{2.1}
$$

DCRP-30 writes

$$
\boxed{
c_\ast
=
\lambda_\ast^\alpha.
}
\tag{2.2}
$$

Thus

$$
\boxed{
\mu_\ast
=
\lambda_\ast^{1-\alpha}.
}
\tag{2.3}
$$

The strict Type-II atom-free range is

$$
\boxed{
1<\alpha<\frac32.
}
\tag{2.4}
$$

Equivalently,

$$
\boxed{
\frac25<\gamma<\frac12,
\qquad
\gamma=\frac1{\alpha+1}.
}
\tag{2.5}
$$

If

$$
\lambda_\ast=e^{-\gamma S_0},
$$

then

$$
\boxed{
\mu_\ast
=
e^{(1-2\gamma)S_0}.
}
\tag{2.6}
$$

---

# 3. PFET raw scaling

DCRP-31 proves that a fixed normalized profile-level PFET event corresponds to raw physical kinetic-energy transfer of order

$$
\beta_n.
$$

Let

$$
\boxed{
\mathsf P
>0
}
$$

denote the fixed normalized inward PFET observable on the exact DSS profile.

Define its raw physical representative

$$
\boxed{
\mathcal P_n^{\rm phys}
=
\beta_n\mathsf P.
}
\tag{3.1}
$$

Then

$$
\frac{
\mathcal P_{n+1}^{\rm phys}
}{
\mathcal P_n^{\rm phys}
}
=
\lambda_\ast\mu_\ast^2.
$$

Using (2.3),

$$
\boxed{
\frac{
\mathcal P_{n+1}^{\rm phys}
}{
\mathcal P_n^{\rm phys}
}
=
\lambda_\ast^{3-2\alpha}.
}
\tag{3.2}
$$

Define

$$
\boxed{
\kappa
=
3-2\alpha
>0.
}
\tag{3.3}
$$

Then

$$
\boxed{
\mathcal P_n^{\rm phys}
=
\lambda_\ast^{\kappa n}
\mathcal P_0^{\rm phys}.
}
\tag{3.4}
$$

This is the DCRP31 critical summability mechanism.

---

# 4. Scalar-transport strain-unit scaling

DCRP46 defines a dimensionless normalized period current

$$
\boxed{
\mathsf T
=
-\int_0^{S_0}
\mathcal T_\psi(s)\,ds
>0
}
\tag{4.1}
$$

on the exact aligned flat branch.

Its natural physical reconstruction is in the same units as the affine strain moment whose normalized transport it maintains.

Therefore define the physical strain-unit representative

$$
\boxed{
\mathcal S_n^{\rm phys}
=
\Sigma_n\mathsf T
=
\frac{a_n}{\ell_n^2}\mathsf T.
}
\tag{4.2}
$$

This is **not** being declared an energy payment.

It is the physical-scale representative of the canonical scalar/strain transport observable.

Its same-parent ratio is exact:

$$
\boxed{
\frac{
\mathcal S_{n+1}^{\rm phys}
}{
\mathcal S_n^{\rm phys}
}
=
\frac{\mu_\ast}{\lambda_\ast^2}.
}
\tag{4.3}
$$

Using (2.3),

$$
\boxed{
\frac{
\mathcal S_{n+1}^{\rm phys}
}{
\mathcal S_n^{\rm phys}
}
=
\lambda_\ast^{-(1+\alpha)}.
}
\tag{4.4}
$$

Since

$$
\gamma(1+\alpha)=1,
$$

$$
\boxed{
\frac{
\mathcal S_{n+1}^{\rm phys}
}{
\mathcal S_n^{\rm phys}
}
=
e^{S_0}.
}
\tag{4.5}
$$

The physical strain scale grows by the universal one-period factor $e^{S_0}$.

This is the expected singular-time strain scaling and is not by itself contradictory.

---

# 5. Dual-current critical orbit

Rewrite the PFET exponent in $\gamma$.

Because

$$
\alpha
=
\frac1\gamma-1,
$$

$$
\kappa
=
3-2\alpha
=
\frac{5\gamma-2}{\gamma}.
$$

Therefore

$$
\lambda_\ast^\kappa
=
e^{-(5\gamma-2)S_0}.
$$

Hence

$$
\boxed{
\mathcal P_{n+1}^{\rm phys}
=
e^{-(5\gamma-2)S_0}
\mathcal P_n^{\rm phys},
}
\tag{5.1}
$$

while

$$
\boxed{
\mathcal S_{n+1}^{\rm phys}
=
e^{S_0}
\mathcal S_n^{\rm phys}.
}
\tag{5.2}
$$

Define

$$
\boxed{
\theta
=
5\gamma-2.
}
\tag{5.3}
$$

Since

$$
\frac25<\gamma<\frac12,
$$

$$
\boxed{
0<\theta<\frac12.
}
\tag{5.4}
$$

Then

$$
\boxed{
\frac{\mathcal P_n^{\rm phys}}
{\mathcal P_0^{\rm phys}}
\left(
\frac{\mathcal S_n^{\rm phys}}
{\mathcal S_0^{\rm phys}}
\right)^\theta
=
1.
}
\tag{5.5}
$$

This formulation uses dimensionless ratios, so no dimensional fractional-power issue occurs.

## Theorem D47.1 — Exact Dual-Current Critical Orbit

On an exact same-parent strict DSS return chain, the raw physical representatives of the mandatory PFET current and mandatory canonical scalar-transport current lie on the one-dimensional scaling orbit (5.5).

Therefore their simultaneous nonzero normalized gaps are fully compatible with exact DSS scale recurrence.

---

# 6. Homogeneous scale-only depletion NO-GO

Consider a monomial observable built only from the two raw magnitudes:

$$
\mathcal J_n
=
(\mathcal P_n^{\rm phys})^x
(\mathcal S_n^{\rm phys})^y.
$$

Then

$$
\boxed{
\frac{\mathcal J_{n+1}}{\mathcal J_n}
=
\lambda_\ast^{x(3-2\alpha)-y(1+\alpha)}.
}
\tag{6.1}
$$

If

$$
x(3-2\alpha)
=
y(1+\alpha),
$$

then $\mathcal J_n$ is exactly scale invariant.

If the exponent is nonzero, its monotonicity is completely predetermined by the root contraction $\lambda_\ast$ and does not measure a new dynamical loss.

Thus:

## Theorem D47.2 — Scale-Only Joint-Depletion NO-GO

No homogeneous functional of the two current magnitudes alone can yield a new same-parent return-depletion theorem beyond their known DSS homogeneity.

Any genuine contradiction must use information not contained in scale magnitude alone, such as:

- support geometry;
- phase;
- pressure coupling;
- material ancestry;
- boundary transport;
- failure of the flat scalar equality mode.

This closes the naive DCRP47 “two positive currents imply cumulative depletion” route.

---

# 7. Shift from payment to equality-mode structure

Because scale-only depletion fails, the correct next question is:

> What exact spatial structure is required for the fully flat canonical pancake mode itself?

DCRP44 proves that on the nondegenerate flat branch there is a unique periodic gauge in which

$$
\boxed{
F_z=-2A(s),
}
\tag{7.1}
$$

and

$$
\boxed{
D_sq+k(s)q=0.
}
\tag{7.2}
$$

Here $A(s)$ is the canonical pancake strain amplitude.

Integrating (7.1) at fixed $q$ gives

$$
\boxed{
F(q,z,s)
=
B(q,s)-2A(s)z
}
\tag{7.3}
$$

for some scalar response function $B$.

Hence the normal velocity is

$$
\boxed{
w
=
B(q,s)-2A(s)z.
}
\tag{7.4}
$$

This is the starting point of the new compatibility calculation.

---

# 8. Potential relation

Recall

$$
\boxed{
q=w-\phi_z.
}
\tag{8.1}
$$

Therefore

$$
\phi_z
=
w-q.
$$

Using (7.4),

$$
\boxed{
\phi_z
=
B(q,s)-q-2A(s)z.
}
\tag{8.2}
$$

This is exact on the fully flat canonical patch.

---

# 9. Incompressibility

The velocity is

$$
V_h=\nabla_h\phi,
\qquad
V_3=w.
$$

Incompressibility gives

$$
\boxed{
\Delta_h\phi+w_z=0.
}
\tag{9.1}
$$

Because

$$
w=B(q,s)-2Az,
$$

the total $z$ derivative is

$$
\boxed{
w_z
=
B_q q_z-2A.
}
\tag{9.2}
$$

Therefore

$$
\boxed{
\Delta_h\phi
=
2A-B_q q_z.
}
\tag{9.3}
$$

---

# 10. Differentiate the incompressibility constraint

Differentiate (9.3) in $z$.

Since $A=A(s)$,

$$
\boxed{
\partial_z\Delta_h\phi
=
-B_{qq}q_z^2
-
B_q q_{zz}.
}
\tag{10.1}
$$

On the other hand, from (8.2),

$$
\begin{aligned}
\Delta_h\phi_z
&=
\Delta_h
\left[
B(q,s)-q-2Az
\right]
\\
&=
(B_q-1)\Delta_hq
+
B_{qq}|\nabla_hq|^2.
\end{aligned}
$$

Because derivatives commute,

$$
\partial_z\Delta_h\phi
=
\Delta_h\phi_z.
$$

Thus

$$
\boxed{
B_q q_{zz}
+
(B_q-1)\Delta_hq
+
B_{qq}
\left(
q_z^2+|\nabla_hq|^2
\right)
=
0.
}
\tag{10.2}
$$

Since

$$
q_z^2+|\nabla_hq|^2
=
|\nabla q|^2,
$$

we obtain:

## Theorem D47.3 — Flat-Pancake Compatibility Equation

Every smooth fully flat canonical scalar patch satisfies

$$
\boxed{
B_q q_{zz}
+
(B_q-1)\Delta_hq
+
B_{qq}|\nabla q|^2
=
0.
}
\tag{10.3}
$$

No pressure estimate is used.

This follows only from:

1. planar potential representation;
2. scalar definition $q=w-\phi_z$;
3. DCRP44 flatness;
4. incompressibility.

---

# 11. Divergence form

Equation (10.3) has the exact divergence form

$$
\boxed{
\nabla_h\cdot
\left[
(B_q-1)\nabla_hq
\right]
+
\partial_z
\left[
B_q q_z
\right]
=
0.
}
\tag{11.1}
$$

Define the compatibility current

$$
\boxed{
J_B(q)
=
\left(
(B_q-1)\nabla_hq,
B_q q_z
\right).
}
\tag{11.2}
$$

Then

$$
\boxed{
\nabla\cdot J_B=0.
}
\tag{11.3}
$$

This is a new finite/local flux structure attached specifically to the fully flat pancake equality mode.

---

# 12. Type of the compatibility operator

Define the coefficient matrix

$$
\boxed{
M_B(q,s)
=
\begin{pmatrix}
B_q-1&0&0\\
0&B_q-1&0\\
0&0&B_q
\end{pmatrix}.
}
\tag{12.1}
$$

Then

$$
\boxed{
\nabla\cdot
\left(
M_B(q,s)\nabla q
\right)
=
0.
}
\tag{12.2}
$$

Its signature is determined completely by

$$
c=B_q.
$$

### Region E+

If

$$
\boxed{
c>1,
}
$$

then all three eigenvalues of $M_B$ are positive.

The operator is elliptic.

### Region E-

If

$$
\boxed{
c<0,
}
$$

then all three eigenvalues are negative.

After multiplication by $-1$, the operator is elliptic.

### Region H

If

$$
\boxed{
0<c<1,
}
$$

then

$$
c-1<0,
\qquad
c>0,
$$

so the signature is

$$
\boxed{
(-,-,+).
}
$$

This is a hyperbolic/Lorentzian-type spatial compatibility equation with the normal coordinate as the distinguished direction.

### Degenerate endpoints

$$
\boxed{
c=0
}
$$

and

$$
\boxed{
c=1
}
$$

make the principal matrix singular.

---

# 13. Characteristic response window

This motivates the definition

$$
\boxed{
\mathcal W_{\rm char}
=
[0,1].
}
\tag{13.1}
$$

Outside this interval the compatibility operator is sign-definite.

Inside it the operator is indefinite or degenerate and can carry characteristic sheet/shear propagation.

This interval is not imposed by hand.

It emerges from the exact incompressibility compatibility of the gauge-flat pancake branch.

---

# 14. Boundary energy identity

Let

$$
D
\subset\mathbb R^3
$$

be a smooth bounded domain contained in one flat canonical patch.

Multiply (11.1) by $q$ and integrate:

$$
0
=
\int_D
q\,\nabla\cdot J_B\,dy.
$$

Integration by parts gives

$$
\boxed{
\int_D
\left[
(B_q-1)|\nabla_hq|^2
+
B_q q_z^2
\right]dy
=
\int_{\partial D}
q\,J_B\cdot\nu\,dS.
}
\tag{14.1}
$$

Define the compatibility boundary flux

$$
\boxed{
\mathcal B_D[q]
=
\int_{\partial D}
q\,J_B\cdot\nu\,dS.
}
\tag{14.2}
$$

Then

$$
\boxed{
\mathcal B_D[q]
=
\int_D
\left[
(B_q-1)|\nabla_hq|^2
+
B_q q_z^2
\right]dy.
}
\tag{14.3}
$$

This is an exact local identity.

---

# 15. Elliptic rigidity above the characteristic window

Assume

$$
\boxed{
B_q\ge1+\delta
}
\tag{15.1}
$$

throughout $D$, with $\delta>0$.

Then

$$
B_q-1\ge\delta,
$$

and

$$
B_q\ge1+\delta.
$$

If

$$
\boxed{
\mathcal B_D[q]=0,
}
\tag{15.2}
$$

equation (14.3) gives

$$
0
\ge
\delta
\int_D|\nabla_hq|^2
+
(1+\delta)
\int_Dq_z^2.
$$

Hence

$$
\boxed{
\nabla q=0.
}
\tag{15.3}
$$

Therefore

$$
\boxed{
\Omega_h=J\nabla_hq=0.
}
\tag{15.4}
$$

---

# 16. Elliptic rigidity below the characteristic window

Assume

$$
\boxed{
B_q\le-\delta
}
\tag{16.1}
$$

throughout $D$.

Then

$$
B_q-1\le-(1+\delta).
$$

If again

$$
\mathcal B_D[q]=0,
$$

the right side of (14.3) is zero while the integrand is nonpositive:

$$
0
\le
-(1+\delta)
\int_D|\nabla_hq|^2
-
\delta
\int_Dq_z^2.
$$

Thus again

$$
\boxed{
\nabla q=0,
}
$$

and

$$
\boxed{
\Omega_h=0.
}
$$

---

# Theorem D47.4 — Sign-Definite Flat-Pancake Rigidity

Let $D$ be a connected fully flat canonical patch with zero compatibility boundary flux.

If either

$$
\boxed{
B_q\ge1+\delta
}
$$

everywhere or

$$
\boxed{
B_q\le-\delta
}
$$

everywhere, then

$$
\boxed{
q=\text{constant on }D,
}
$$

and hence

$$
\boxed{
\Omega_h=0.
}
$$

Therefore a nonzero rank-two pancake cannot be a closed sign-definite elliptic flat mode.

---

# 17. New trichotomy

Assume $B_q$ is continuous on a connected nonzero flat rank-two patch.

If its range does not intersect

$$
[0,1],
$$

then by connectedness it lies entirely in one component

$$
(-\infty,0)
$$

or

$$
(1,\infty).
$$

On a compact subpatch separated from the endpoints, Theorem D47.4 applies whenever the compatibility boundary flux vanishes.

Thus a nonzero flat pancake requires at least one of:

## Branch C — characteristic response

$$
\boxed{
B_q(q(y,s),s)
\in[0,1]
}
$$

somewhere on the active patch.

## Branch B — compatibility boundary feed

$$
\boxed{
\mathcal B_D[q]\neq0.
}
$$

## Branch T — chart/flatness transition

The regular flat scalar chart fails before the domain closes.

Hence:

## Theorem D47.5 — Characteristic / Boundary / Transition Dichotomy

A nonzero fully flat rank-two pancake cannot remain entirely in a closed sign-definite response region with zero compatibility boundary flux.

It must enter the characteristic window, export/import compatibility flux through a finite boundary, or leave the flat regular branch.

This is a new finite-domain closure statement.

---

# 18. Constant-slope equality modes

Suppose

$$
\boxed{
B(q,s)
=
c\,q+b(s)
}
\tag{18.1}
$$

with constant response slope $c$.

Then

$$
B_q=c,
\qquad
B_{qq}=0.
$$

Equation (10.3) becomes

$$
\boxed{
c q_{zz}
+
(c-1)\Delta_hq
=
0.
}
\tag{18.2}
$$

---

# 19. Constant-slope Fourier cone

For a plane wave

$$
q(y)
=
e^{i\xi\cdot y},
$$

equation (18.2) gives

$$
\boxed{
c\xi_z^2
+
(c-1)|\xi_h|^2
=
0.
}
\tag{19.1}
$$

If

$$
0<c<1,
$$

this is

$$
\boxed{
\xi_z^2
=
\frac{1-c}{c}
|\xi_h|^2.
}
\tag{19.2}
$$

Thus the flat constant-response mode has a genuine real characteristic cone.

If

$$
c>1
$$

or

$$
c<0,
$$

there is no nonzero real characteristic wavevector.

At

$$
c=0,
$$

the equation reduces to

$$
\boxed{
\Delta_hq=0.
}
\tag{19.3}
$$

At

$$
c=1,
$$

it reduces to

$$
\boxed{
q_{zz}=0.
}
\tag{19.4}
$$

These are the two degenerate characteristic endpoints.

---

# Theorem D47.6 — Flat-Pancake Wave-Cone Classification

For constant scalar response slope $c=B_q$:

- outside $[0,1]$, the compatibility symbol is elliptic;
- for $0<c<1$, it has the real cone (19.2);
- $c=0,1$ are degenerate endpoint modes.

Thus the fully flat pancake survivor admits an exact local wave-cone classification.

---

# 20. Whole-space $L^2$ constant-slope NO-GO

Assume

$$
q\in L^2(\mathbb R^3)
$$

solves (18.2) distributionally for constant $c$.

Fourier transformation gives

$$
\left[
c\xi_z^2
+
(c-1)|\xi_h|^2
\right]
\widehat q(\xi)
=
0.
$$

For every real $c$, the zero set of the polynomial symbol has Lebesgue measure zero unless the polynomial vanishes identically, which it never does.

Because

$$
\widehat q\in L^2,
$$

an $L^2$ function supported on a measure-zero set is zero.

Therefore

$$
\boxed{
q=0.
}
$$

## Theorem D47.7 — No Nonzero Global $L^2$ Constant-Response Flat Mode

Every nontrivial whole-space constant-slope solution of the fully flat compatibility equation must evade $L^2$ localization.

It must be nonlocalized, distributional, boundary-fed, variable-slope, or leave the flat branch.

This is consistent with known exact Euler pancake models being infinite-energy rather than finite-energy whole-space states.

---

# 21. Why this does not yet contradict the DSS profile

DCRP-30/31 already proved that the nonzero strict DSS Euler profile has infinite global normalized kinetic energy and requires a critical tail.

Therefore Theorem D47.7 does **not** eliminate the DSS limit profile.

It instead classifies how it must evade the most rigid constant-response flat mode:

$$
\boxed{
\text{critical nonlocalized tail}
}
$$

or

$$
\boxed{
\text{variable characteristic response}
}
$$

or

$$
\boxed{
\text{finite boundary/transition feed}.
}
$$

This is useful because these are precisely the structures already visible to DCRP31 and DCRP46.

---

# 22. Connection to the dual-current package

The scale-only two-current route failed because exact DSS homogeneity absorbs the two current magnitudes.

The compatibility equation introduces new information not contained in scale magnitude:

$$
\boxed{
B_q(q,s)
}
$$

and

$$
\boxed{
\mathcal B_D[q].
}
$$

These are structural/transport quantities.

Thus the updated finite-annulus observer package becomes

$$
\boxed{
\mathsf O_{47}
=
\left(
\mathsf O_{\rm PFET},
\mathsf O_{\rm tr},
B_q,
\mathcal B_D
\right).
}
\tag{22.1}
$$

The new question is no longer whether two currents are large enough.

It is whether their recurrent finite-annulus support can maintain the required characteristic response geometry without producing:

- nonzero boundary compatibility flux;
- variable-slope transition;
- loss of scalar flatness;
- a noncritical tail.

---

# 23. NTLA-O interpretation

DCRP47 is another direct use of the rebuilt NTLA-O logic.

At the coarse observer level:

$$
\boxed{
\text{PFET}>0,
\qquad
\text{scalar transport}>0.
}
$$

Both observables recur.

A naive observer therefore sees “two positive costs.”

The same-parent quotient reveals that their scale evolution lies on an exact critical orbit, so this coarse distinction does **not** imply depletion.

A finer structural observer reads the flat scalar constitutive slope:

$$
\boxed{
B_q.
}
$$

This reveals a new topology of the equality space:

$$
(-\infty,0)
\quad\cup\quad
[0,1]
\quad\cup\quad
(1,\infty),
$$

with two elliptic sectors separated by a characteristic window.

Thus the first new obstruction does not come from current magnitude.

It comes from the **type of the realizability equation**.

This is exactly the NTLA-O principle:

$$
\boxed{
\text{coarse admissibility}
\not\Rightarrow
\text{fine realizability}.
}
$$

---

# 24. Updated final rank-two survivor

After DCRP47, the most rigid survivor must satisfy:

$$
\boxed{
\begin{aligned}
&
\text{strict same-parent DSS}
\\
&+
\text{rank two}
\\
&+
\text{fixed plane}
\\
&+
\text{zero shape action}
\\
&+
\text{gauge-flat scalar connection}
\\
&+
\text{finite inward PFET}
\\
&+
\text{strict finite-annulus scalar transport}
\\
&+
\text{critical two-current same-parent scaling}
\\
&+
\Big[
\text{characteristic response }B_q\in[0,1]
\\
&\hspace{2cm}
\vee\
\text{nonzero compatibility boundary flux}
\\
&\hspace{2cm}
\vee\
\text{finite scalar-chart transition}
\Big].
\end{aligned}
}
$$

The sign-definite elliptic zero-boundary flat mode is excluded.

---

# 25. Status ledger

## PROVED this round

### D47-P1 — Exact raw PFET root scaling

$$
\mathcal P_{n+1}^{\rm phys}
=
\lambda_\ast^{3-2\alpha}
\mathcal P_n^{\rm phys}.
$$

### D47-P2 — Exact physical strain-unit scalar-transport scaling

$$
\mathcal S_{n+1}^{\rm phys}
=
\lambda_\ast^{-(1+\alpha)}
\mathcal S_n^{\rm phys}.
$$

### D47-P3 — Exact dual-current critical orbit

$$
\frac{\mathcal P_n}{\mathcal P_0}
\left(
\frac{\mathcal S_n}{\mathcal S_0}
\right)^{5\gamma-2}
=
1.
$$

### D47-P4 — Homogeneous scale-only joint-depletion NO-GO

Current magnitude and same-parent scaling alone cannot produce a new normalized return tax.

### D47-P5 — Exact flat-pancake compatibility PDE

$$
\nabla_h\cdot[(B_q-1)\nabla_hq]
+
\partial_z(B_q q_z)
=
0.
$$

### D47-P6 — Sign-definite elliptic rigidity

Zero compatibility boundary flux plus $B_q>1$ or $B_q<0$ uniformly forces $\Omega_h=0$.

### D47-P7 — Characteristic / boundary / transition dichotomy

A nonzero flat rank-two patch must enter $B_q\in[0,1]$, carry boundary compatibility flux, or leave the flat chart.

### D47-P8 — Constant-slope wave-cone classification

For $0<B_q<1$,

$$
\xi_z^2
=
\frac{1-B_q}{B_q}
|\xi_h|^2.
$$

### D47-P9 — Whole-space $L^2$ constant-response NO-GO

A nonzero whole-space constant-slope flat mode cannot belong to $L^2(\mathbb R^3)$.

---

# 26. Closed / downgraded routes

## Closed

$$
\boxed{
\text{PFET gap}
+
\text{scalar transport gap}
\Rightarrow
\text{scale-only same-parent depletion}
}
$$

is false.

The two raw representatives sit on an exact critical scaling orbit.

## Still alive

Structural coupling through:

- characteristic response slope;
- compatibility boundary flux;
- pressure current;
- finite transition;
- variable-slope wave geometry.

---

# 27. New STOP

$$
\boxed{
\textbf{
STOP-D47:
The two-current magnitude route is exactly critical; the next genuine obstruction is the flat-pancake characteristic compatibility equation, whose nonzero closed solutions must enter the response window }0\le B_q\le1\textbf{ or export a finite boundary/transition flux.}
}
$$

---

# 28. Next autonomous step

## DCRP48 — Characteristic-Slope Dynamics and Boundary Coupling

**Working title**

> **Flat-Pancake Characteristic Cone, Response-Slope Transport, and PFET/Compatibility Boundary Coupling**

Primary tasks:

1. derive the evolution equation for
   $$
   c=B_q
   $$
   from the scalar transport and vertical Euler momentum equations;
2. determine whether the interval
   $$
   0\le c\le1
   $$
   is invariant, repelling, or requires boundary replenishment;
3. express the compatibility boundary flux
   $$
   \mathcal B_D[q]
   $$
   on the common finite supplier annulus;
4. test whether it couples to the DCRP46 scalar current or DCRP31 PFET;
5. classify the endpoint modes
   $$
   c=0,\qquad c=1
   $$
   and compare them with exact pancake/shear models.

Desired endpoint:

$$
\boxed{
\text{characteristic-window escape}
\ \vee\
\text{finite boundary coupling}
\ \vee\
\text{exact endpoint pancake mode}.
}
$$

---

# 29. One-line checkpoint

The dual-current payments are exactly critical under same-parent scaling, but the fully flat pancake equality mode satisfies a new anisotropic divergence equation that rules out closed elliptic response sectors and forces every nonzero survivor into a characteristic $0\le B_q\le1$ window or a finite boundary/transition carrier.

---

**End checkpoint:** DCRP47  
**Next:** DCRP48 — Characteristic-Slope Dynamics / Boundary Coupling.