# NS × X 積分 × 24/72 範式實戰
## Round 62 — Pure Continuous Neutral-Residual Cancellation / Restored Quarter-Power Matching Law

- 日期：2026-08-18
- 版本：v0.1
- 狀態：Proof-Route Experiment / Continuous-Only Schur-Renormalized Matching Branch
- canonical source：UTF-8 Markdown
- canonical math delimiters：inline `$...$`；display `$$...$$`
- 前一輪：`NS_X72_Round61_PureContinuous_FastSlowStableBundle_OptimalMatchingExponent_v0.1_2026-08-18.md`
- 本輪目標：Round 61 若把所有 same-parity far-neighbor terms當成 generic $O(j^{-2})$ perturbation，得到安全 overlap
  $$
  j_m=\nu^{-2/7}
  $$
  與 coarse
  $$
  O(\nu^{1/7})
  $$
  matching target。本輪檢查 fast-Schur reduction真正作用在 **neutral direction** 上的 remainder，並發現一階 $j^{-2}$ contribution具有 exact cancellation，因此 slow self-drift降為 $O(j^{-3})$。
- 主要結果：
  1. 定義 neutral residual
     $$
     S_n
     =
     -A_{-2}^{(n)}
     +
     A_0^{(n)}
     -
     A_2^{(n)}
     +
     A_4^{(n)};
     $$
     then
     $$
     \boxed{
     \lim_{n\to\infty}
     n^3S_n
     =
     48K^3;
     }
     $$
  2. 若 frozen same-parity characteristic polynomial為
     $$
     F_n(r)
     =
     -A_{-2}^{(n)}
     +
     A_0^{(n)}r
     -
     A_2^{(n)}r^2
     +
     A_4^{(n)}r^3,
     $$
     then
     $$
     F_n'(1)
     =
     -4K+O(n^{-1});
     $$
  3. 因此 neutral root具有
     $$
     \boxed{
     r_{\rm neu}(n)
     =
     1+
     \frac{12K^2}{n^3}
     +
     O(n^{-4});
     }
     $$
  4. in parity index $n\sim2j$，
     $$
     \boxed{
     r_{\rm neu}
     =
     1+
     \frac{3K^2}{2j^3}
     +
     O(j^{-4});
     }
     $$
  5. exact rescaled viscous coupling coefficients
     $$
     a_e(j,\nu)
     =
     -\nu
     \frac{
     b_{2j}
     }{
     A_2^{(2j)}
     },
     $$
     $$
     a_o(j,\nu)
     =
     -\nu
     \frac{
     b_{2j+1}
     }{
     A_2^{(2j+1)}
     }
     $$
     satisfy
     $$
     \boxed{
     a_e
     =
     a_j
     \left(
     1+\frac1j+O(j^{-2})
     \right),
     }
     $$
     $$
     \boxed{
     a_o
     =
     a_j
     \left(
     1+\frac2j+O(j^{-2})
     \right),
     }
     $$
     where
     $$
     a_j
     =
     \frac{16\nu j^2}{K};
     $$
  6. after fast-Schur renormalization，the conservative slow matching budget becomes
     $$
     \nu j_m^3,
     \qquad
     j_m^{-1},
     \qquad
     \frac{j_m^{-3}}{\nu j_m^2};
     $$
  7. for
     $$
     j_m=\nu^{-\alpha},
     $$
     these are
     $$
     \nu^{1-3\alpha},
     \qquad
     \nu^\alpha,
     \qquad
     \nu^{5\alpha-1};
     $$
  8. all three balance exactly at
     $$
     \boxed{
     \alpha_\ast=\frac14;
     }
     $$
  9. hence the Schur-renormalized first theorem target returns to
     $$
     \boxed{
     j_m=\nu^{-1/4},
     \qquad
     \operatorname{dist}
     \left(
     E_\nu^{\min},
     E_0^{\rm Jost}
     \right)
     =
     O(\nu^{1/4});
     }
     $$
  10. Round 61's $2/7$ law remains valid as a **pre-Schur generic-error safety budget**；Round 62 explains why it is not expected to be sharp after the exact neutral cancellation is exploited；
  11. full six-dimensional physical stable-plane diagnostics at
      $$
      j_m=\nu^{-1/4}
      $$
      are substantially stronger than the quarter-power target；
  12. at
      $$
      \nu=10^{-8},
      $$
      the largest principal angles are
      $$
      \boxed{
      \theta_{\max,-}
      \approx
      7.27\times10^{-4},
      }
      $$
      and
      $$
      \boxed{
      \theta_{\max,+}
      \approx
      1.82\times10^{-3};
      }
      $$
  13. the normalized quantities
      $$
      \theta_{\max}/\nu^{1/4}
      $$
      decrease strongly over the tested small-viscosity sequence；
  14. empirical log-slopes over the final four points are approximately
      $$
      0.52
      $$
      and
      $$
      0.56\text{--}0.58,
      $$
      again faster than the conservative exponent $1/4$；
  15. Remaining proof obligation：build an actual fast Schur graph and prove that its induced slow projective/Riccati remainder obeys the neutral-cancelled bounds uniformly，then propagate the resulting $O(\nu^{1/4})$ interval to Round 59's positive center functional。
- 非主張：本輪 does not yet prove the full
  $$
  O(\nu^{1/4})
  $$
  stable-bundle theorem，nor a positive-viscosity interval。The exact parts are the neutral residual cancellation，neutral-root asymptotic coefficient and viscous-coupling asymptotics；the restored quarter-power theorem is the quantitatively corrected proof target supported by full 6D diagnostics。

---

# 0. Round 61 handoff

Round 61 factorized the endpoint leading parity cubic：

$$
\boxed{
\epsilon+r-r^2-\epsilon r^3
=
-(r-1)
[
\epsilon r^2+(1+\epsilon)r+\epsilon
].
}
\tag{0.1}
$$

Thus each parity has：

$$
\boxed{
1\text{ neutral}
+
1\text{ fast minimal}
+
1\text{ fast growing}.
}
\tag{0.2}
$$

For：

$$
\nu>0,
$$

the two neutral directions couple into a slow stable/unstable pair，while the two fast minimal lines persist。

Round 61 therefore proposed：

$$
E_\nu^{\min}
=
E_{\rm fast,e}^{\min}
\oplus
E_{\rm fast,o}^{\min}
\oplus
E_{\rm slow}^{-}.
$$

Without yet exploiting neutral cancellation，its generic roughness budget gave：

$$
\boxed{
j_m=\nu^{-2/7},
\qquad
O(\nu^{1/7}).
}
\tag{0.3}
$$

Round 61 STOP：

$$
\boxed{
\text{STOP-C65}
=
\text{Fast-Schur / Slow-Riccati Quantitative Gap}.
}
$$

---

# 1. The neutral residual is not $O(j^{-2})$

The same-parity frozen recurrence is：

$$
\boxed{
-
A_{-2}^{(n)}
x_{m-1}
+
A_0^{(n)}
x_m
-
A_2^{(n)}
x_{m+1}
+
A_4^{(n)}
x_{m+2}
=
0.
}
\tag{1.1}
$$

A pure neutral constant profile：

$$
x_m\equiv1
$$

has residual：

$$
\boxed{
S_n
=
-
A_{-2}^{(n)}
+
A_0^{(n)}
-
A_2^{(n)}
+
A_4^{(n)}.
}
\tag{1.2}
$$

Individually：

$$
A_{-2}^{(n)},
A_4^{(n)}
=
O(n^{-2}),
$$

and：

$$
A_0^{(n)}-A_2^{(n)}
$$

also contains $O(n^{-2})$ structure。

But these contributions cancel on the neutral vector。

Exact symbolic evaluation gives：

$$
\boxed{
\lim_{n\to\infty}
n^3S_n
=
48K^3.
}
\tag{1.3}
$$

Therefore：

$$
\boxed{
S_n
=
\frac{
48K^3
}{
n^3
}
+
O(n^{-4}).
}
\tag{1.4}
$$

命名：

$$
\boxed{
\textbf{Neutral Residual Cancellation}.
}
$$

---

# 2. Neutral frozen root

Define：

$$
\boxed{
F_n(r)
=
-
A_{-2}^{(n)}
+
A_0^{(n)}r
-
A_2^{(n)}r^2
+
A_4^{(n)}r^3.
}
\tag{2.1}
$$

Then：

$$
F_n(1)=S_n.
$$

Also：

$$
\boxed{
F_n'(1)
=
A_0^{(n)}
-
2A_2^{(n)}
+
3A_4^{(n)}.
}
\tag{2.2}
$$

Exact asymptotics give：

$$
\boxed{
F_n'(1)
=
-4K
+
O(n^{-1}).
}
\tag{2.3}
$$

Since the limiting derivative is nonzero，the nearby neutral root is simple。

Put：

$$
r
=
1+
\frac c{n^3}.
$$

Then：

$$
\boxed{
\lim_{n\to\infty}
n^3
F_n
\left(
1+\frac c{n^3}
\right)
=
48K^3
-
4Kc.
}
\tag{2.4}
$$

Hence：

$$
\boxed{
c=12K^2.
}
\tag{2.5}
$$

Therefore：

$$
\boxed{
r_{\rm neu}(n)
=
1+
\frac{
12K^2
}{
n^3
}
+
O(n^{-4}).
}
\tag{2.6}
$$

For：

$$
n=2j+O(1),
$$

$$
\boxed{
r_{\rm neu}
=
1+
\frac{
3K^2
}{
2j^3
}
+
O(j^{-4}).
}
\tag{2.7}
$$

---

# 3. Numerical neutral-root audit

For each parity，let：

$$
r_{\rm neu}^{e/o}(j)
$$

be the local frozen eigenvalue closest to：

$$
1.
$$

Then：

$$
\boxed{
\frac{
j^3
(
r_{\rm neu}^{e/o}(j)-1
)
}{
(3/2)K^2
}
\to1.
}
\tag{3.1}
$$

The included verification checks this directly for both source fibres and both parity sectors。

This confirms that the $j^{-3}$ residual controls the actual local neutral eigenvalue drift，not merely the constant test profile。

---

# 4. Exact viscous coupling normalization

In the desingularized variables：

$$
E_j=e_j/\nu,
\qquad
o_j,
$$

define the one-step parity couplings：

$$
\boxed{
a_e(j,\nu)
=
-\nu
\frac{
b_{2j}
}{
A_2^{(2j)}
},
}
\tag{4.1}
$$

$$
\boxed{
a_o(j,\nu)
=
-\nu
\frac{
b_{2j+1}
}{
A_2^{(2j+1)}
}.
}
\tag{4.2}
$$

Let：

$$
\boxed{
a_j
=
\frac{
16\nu j^2
}{
K
}.
}
\tag{4.3}
$$

Exact symbolic limits give：

$$
\boxed{
\lim_{j\to\infty}
j
\left[
\frac{
a_e(j,\nu)
}{
a_j
}
-
1
\right]
=
1,
}
\tag{4.4}
$$

and：

$$
\boxed{
\lim_{j\to\infty}
j
\left[
\frac{
a_o(j,\nu)
}{
a_j
}
-
1
\right]
=
2.
}
\tag{4.5}
$$

Thus：

$$
\boxed{
a_e
=
a_j
[
1+j^{-1}+O(j^{-2})
],
}
\tag{4.6}
$$

$$
\boxed{
a_o
=
a_j
[
1+2j^{-1}+O(j^{-2})
].
}
\tag{4.7}
$$

The dominant relative coupling correction is therefore：

$$
\boxed{
O(j^{-1}).
}
\tag{4.8}
$$

---

# 5. Schur-renormalized slow error scales

The two fast modes are generated by the $O(j^{-2})$ same-parity coefficients。

But on the neutral direction，their direct self-action cancels to：

$$
O(j^{-3}).
$$

After the fast pair is absorbed into the reference Schur graph，the conservative slow matching budget becomes：

## E1 — cumulative viscous attenuation before matching

$$
\boxed{
\mathcal E_{\rm WKB}
\sim
\nu j_m^3.
}
\tag{5.1}
$$

## E2 — relative coupling / basis drift

$$
\boxed{
\mathcal E_{\rm coup}
\sim
j_m^{-1}.
}
\tag{5.2}
$$

## E3 — neutral self-drift relative to slow gap

The slow gap is：

$$
\asymp
\nu j_m^2.
$$

The neutral drift is：

$$
\asymp
j_m^{-3}.
$$

Hence：

$$
\boxed{
\mathcal E_{\rm neu}
\sim
\frac{
j_m^{-3}
}{
\nu j_m^2
}
=
\frac1{
\nu j_m^5
}.
}
\tag{5.3}
$$

These are the three leading conservative post-Schur errors。

---

# 6. General post-Schur overlap exponent

Set：

$$
\boxed{
j_m=\nu^{-\alpha}.
}
\tag{6.1}
$$

Then：

$$
\boxed{
\mathcal E_{\rm WKB}
=
\nu^{1-3\alpha},
}
\tag{6.2}
$$

$$
\boxed{
\mathcal E_{\rm coup}
=
\nu^\alpha,
}
\tag{6.3}
$$

and：

$$
\boxed{
\mathcal E_{\rm neu}
=
\nu^{5\alpha-1}.
}
\tag{6.4}
$$

A direct perturbative overlap requires：

$$
\boxed{
\alpha<1/3
}
$$

and：

$$
\boxed{
\alpha>1/5.
}
$$

Define：

$$
\boxed{
\beta_{\rm Schur}(\alpha)
=
\min
\{
1-3\alpha,\,
\alpha,\,
5\alpha-1
\}.
}
\tag{6.5}
$$

---

# 7. Restored optimal quarter-power law

Set the three exponents equal：

$$
1-3\alpha
=
\alpha,
$$

and：

$$
5\alpha-1
=
\alpha.
$$

Both give：

$$
\boxed{
4\alpha=1.
}
$$

Therefore：

$$
\boxed{
\alpha_\ast
=
\frac14.
}
\tag{7.1}
$$

At：

$$
\alpha=1/4,
$$

$$
\boxed{
1-3\alpha
=
\alpha
=
5\alpha-1
=
\frac14.
}
\tag{7.2}
$$

Hence the Schur-renormalized matching target is：

$$
\boxed{
j_m
=
\nu^{-1/4},
}
\tag{7.3}
$$

and：

$$
\boxed{
\operatorname{dist}
\left(
E_\nu^{\min}(j_m),
E_0^{\rm Jost}(j_m)
\right)
=
O(
\nu^{1/4}
).
}
\tag{7.4}
$$

命名：

$$
\boxed{
\textbf{Restored Quarter-Power Matching Law}.
}
$$

The statement (7.4) remains a theorem target；the exponent balance is now structurally self-consistent after neutral cancellation。

---

# 8. Relation to the Round 61 $2/7$ law

Round 61 used：

$$
\boxed{
\mathcal E_{\rm generic}
\sim
\frac{
j_m^{-2}
}{
\nu j_m^2
}
=
\frac1{
\nu j_m^4
}.
}
\tag{8.1}
$$

This corresponds to treating all $O(j^{-2})$ coefficients as generic slow perturbations。

Balancing：

$$
\nu j_m^3
$$

with：

$$
1/(\nu j_m^4)
$$

gave：

$$
\alpha=2/7.
$$

Round 62 shows：

$$
\boxed{
j^{-2}
\text{ is not the correct slow self-drift after neutral projection}.
}
$$

Instead：

$$
\boxed{
j^{-3}
}
$$

appears because of exact cancellation。

Therefore：

- $2/7$ is a valid conservative **pre-Schur** proof budget；
- $1/4$ is the refined **post-Schur** target。

There is no contradiction。

---

# 9. Full six-dimensional quarter-power diagnostics

The verification script computes the physical six-dimensional minimal three-plane and the $\nu=0$ endpoint selected Jost three-plane directly，with no compact hidden-block basis quotient。

Use：

$$
\boxed{
j_m
=
\operatorname{round}
(
\nu^{-1/4}
).
}
\tag{9.1}
$$

The largest principal angle is：

$$
\theta_{\max}
=
\theta_{\max}
(
E_\nu^{\min},
E_0^{\rm Jost}
).
$$

---

# 10. Small fibre

For：

$$
K_-=\sqrt{17}-3,
$$

$$
\boxed{
\begin{array}{c|c|c|c}
\nu
&
j_m
&
\theta_{\max}
&
\theta_{\max}/\nu^{1/4}
\\
\hline
10^{-4}
&
10
&
8.5906\times10^{-2}
&
8.5906\times10^{-1}
\\
10^{-5}
&
18
&
2.5709\times10^{-2}
&
4.5717\times10^{-1}
\\
10^{-6}
&
32
&
7.7570\times10^{-3}
&
2.4532\times10^{-1}
\\
10^{-7}
&
56
&
2.3143\times10^{-3}
&
1.3015\times10^{-1}
\\
10^{-8}
&
100
&
7.2665\times10^{-4}
&
7.2665\times10^{-2}
\end{array}
}
\tag{10.1}
$$

The normalized quarter-power ratio decreases rapidly。

---

# 11. Large fibre

For：

$$
K_+=\sqrt{17}+3,
$$

$$
\boxed{
\begin{array}{c|c|c|c}
\nu
&
j_m
&
\theta_{\max}
&
\theta_{\max}/\nu^{1/4}
\\
\hline
10^{-4}
&
10
&
3.8287\times10^{-1}
&
3.8287
\\
10^{-5}
&
18
&
9.1156\times10^{-2}
&
1.6210
\\
10^{-6}
&
32
&
2.2736\times10^{-2}
&
7.1899\times10^{-1}
\\
10^{-7}
&
56
&
6.3743\times10^{-3}
&
3.5842\times10^{-1}
\\
10^{-8}
&
100
&
1.8174\times10^{-3}
&
1.8174\times10^{-1}
\end{array}
}
\tag{11.1}
$$

Again the normalized ratio decreases strongly。

---

# 12. Empirical principal-angle exponents

A log-log fit over the smallest four viscosities gives approximately：

### small fibre

$$
\boxed{
\theta_{\max}
\sim
\nu^{0.517\ldots}
}
\tag{12.1}
$$

### large fibre

$$
\boxed{
\theta_{\max}
\sim
\nu^{0.566\ldots}
}
\tag{12.2}
$$

These numerical exponents are not claimed as true asymptotic invariants。

They only show：

$$
\boxed{
\nu^{1/4}
}
$$

is a very conservative target in the tested range。

---

# 13. Direct local neutral-drift diagnostic

Let：

$$
r_{\rm neu}^{e/o}(j)
$$

denote the local frozen root nearest：

$$
1.
$$

The verification computes：

$$
\boxed{
D_{e/o}(j)
=
\frac{
j^3
[
r_{\rm neu}^{e/o}(j)-1
]
}{
(3/2)K^2
}.
}
\tag{13.1}
$$

For both fibres and both parities：

$$
\boxed{
D_{e/o}(j)\to1.
}
\tag{13.2}
$$

This independently confirms the exact neutral-residual derivation。

---

# 14. Why full stable-plane convergence can be faster

The quarter-power budget still treats：

$$
j^{-1}
$$

coupling corrections generically。

But：

1. the Round 59 Jost graph already incorporates nontrivial algebraic tail geometry；
2. the two parity coupling corrections are structured rather than arbitrary；
3. the fast Schur graph can absorb part of the local basis variation；
4. reflection and $\mathcal C$ symmetry eliminate several possible mixing channels。

Thus the actual remainder may begin beyond the first conservative：

$$
O(j^{-1})
$$

term in the projective coordinate relevant to the Fredholm functional。

This is consistent with empirical exponents around：

$$
1/2.
$$

No improved exponent is claimed without an explicit Schur/Riccati calculation。

---

# 15. Slow projective coordinate

After eliminating the two fast lines，let the effective slow vector be：

$$
\boxed{
Y_j^{\rm slow}
=
\begin{pmatrix}
E_j\\
o_j
\end{pmatrix}.
}
\tag{15.1}
$$

Define：

$$
\boxed{
z_j
=
\frac{
E_j
}{
o_j
}.
}
\tag{15.2}
$$

For the Round 60 reduced matrix：

$$
M(a)
=
\begin{pmatrix}
1&a\\
a&1+a^2
\end{pmatrix},
$$

the exact projective map is：

$$
\boxed{
z_{j+1}
=
\frac{
z_j+a_j
}{
a_jz_j+1+a_j^2
}.
}
\tag{15.3}
$$

Its two frozen fixed points solve：

$$
\boxed{
z^2+a z-1=0.
}
\tag{15.4}
$$

The stable-amplitude eigenline corresponds to the negative projective root：

$$
\boxed{
z_-(a)
=
-\frac{
a+\sqrt{a^2+4}
}{
2
}.
}
\tag{15.5}
$$

The next rigorous proof should perturb this scalar Riccati map by the exact neutral-cancelled Schur remainder。

---

# 16. Corrected Riccati remainder target

The desired effective equation is：

$$
\boxed{
z_{j+1}
=
\frac{
z_j+a_j
}{
a_jz_j+1+a_j^2
}
+
\mathcal R_j(z_j;\nu).
}
\tag{16.1}
$$

Round 62 identifies the structural scales that should enter：

$$
\boxed{
|\mathcal R_j|
\lesssim
\frac1j
\cdot
a_j
+
\frac1{j^3}
+
\text{higher fast-Schur terms}.
}
\tag{16.2}
$$

Relative to the slow gap：

$$
a_j,
$$

this becomes：

$$
\boxed{
\frac{
|\mathcal R_j|
}{
a_j
}
\lesssim
\frac1j
+
\frac1{
\nu j^5
}
+
\cdots.
}
\tag{16.3}
$$

At：

$$
j=\nu^{-1/4},
$$

both displayed terms are：

$$
\boxed{
O(\nu^{1/4}).
}
\tag{16.4}
$$

This is the quantitative inequality the next round should prove with explicit constants。

---

# 17. Consequence if the quarter-power enclosure is completed

Round 59 rigorously proved：

$$
\boxed{
c_{0,-}>5.79,
\qquad
c_{0,+}>5.33.
}
\tag{17.1}
$$

Suppose Round 63 obtains：

$$
\boxed{
\left|
\frac{
a_3(\nu)
}{
\nu
}
-
c_0
\right|
\le
C_\ast
\nu^{1/4}.
}
\tag{17.2}
$$

Then any explicit：

$$
\nu_s
$$

satisfying：

$$
\boxed{
C_\ast
\nu_s^{1/4}
<
5
}
\tag{17.3}
$$

already yields：

$$
\boxed{
a_3(\nu)>0
\qquad
0<\nu\le\nu_s.
}
\tag{17.4}
$$

The exact Fredholm same-sign identity would then extend the Round 56 hidden-rescue no-go from：

$$
\nu=1
$$

to an actual open viscosity interval adjacent to：

$$
0.
$$

---

# 18. STOP-C66 — Neutral-Cancelled Schur/Riccati Enclosure Gap

$$
\boxed{
\begin{aligned}
\text{layer}
&=
\mathrm{small\text{-}viscosity\ fast\text{-}Schur/slow\text{-}Riccati\ matching},
\\
S_n
&=
-A_{-2}+A_0-A_2+A_4,
\\
n^3S_n
&\to
48K^3,
\\
r_{\rm neu}(n)
&=
1+12K^2n^{-3}+O(n^{-4}),
\\
a_e/a_j
&=
1+j^{-1}+O(j^{-2}),
\\
a_o/a_j
&=
1+2j^{-1}+O(j^{-2}),
\\
\text{post-Schur error budget}
&=
\nu j_m^3
+
j_m^{-1}
+
(\nu j_m^5)^{-1},
\\
\text{optimal overlap}
&=
j_m=\nu^{-1/4},
\\
\text{first theorem target}
&=
O(\nu^{1/4}),
\\
\text{6D physical diagnostics}
&=
\mathrm{faster\ than\ quarter\text{-}power},
\\
\text{remaining task}
&=
\mathrm{explicit\ invariant\ fast\ Schur\ graph}
+
\mathrm{slow\ Riccati\ interval\ remainder},
\\
\text{small-viscosity positivity interval}
&=
\mathrm{not\ yet\ rigorous},
\\
T_{\mathsf C\to\mathsf D}
&=
\mathrm{NOT\ REACHED}.
\end{aligned}
}
$$

命名：

$$
\boxed{
\textbf{STOP-C66:
Neutral-Cancelled Schur/Riccati Enclosure Gap}.
}
$$

---

# 19. 24/72 Ledger — Round 62

| Step | object | $B$ | $U$ | $O$ | $L$ | status |
|---|---|---|---|---|---|---|
| C991 | neutral residual $S_n$ | $\mathsf C$ | same-parity transfer | scalar | $\mathsf F$ | EXACT |
| C992 | $n^3S_n\to48K^3$ | $\mathsf C$ | asymptotic cancellation | scalar | $\mathsf F$ | PROVED |
| C993 | frozen neutral root drift | $\mathsf C$ | local spectrum | scalar | $\mathsf F$ | DERIVED |
| C994 | even viscous coupling asymptotic | $\mathsf C$ | slow coupling | scalar | $\mathsf F$ | PROVED |
| C995 | odd viscous coupling asymptotic | $\mathsf C$ | slow coupling | scalar | $\mathsf F$ | PROVED |
| C996 | Schur-renormalized error budget | $\mathsf C$ | fast–slow geometry | relational | $\mathsf F$ | DERIVED |
| C997 | general post-Schur exponent | $\mathsf C$ | matched asymptotics | scalar | $\mathsf F$ | DERIVED |
| C998 | restored $\alpha=1/4$ optimum | $\mathsf C$ | exponent optimization | scalar | $\mathsf F$ | PROVED within budget model |
| C999 | relation to $2/7$ safety law | $\mathsf C$ | route audit | targeted | $\mathsf F$ | CLARIFIED |
| C1000 | 6D quarter-power principal angles | $\mathsf C$ | physical stable bundle | profile | $\mathsf F$ | NUMERICALLY VERIFIED |
| C1001 | neutral-root numerical audit | $\mathsf C$ | frozen spectrum | scalar | $\mathsf F$ | VERIFIED |
| C1002 | reduced slow Riccati map | $\mathsf C$ | projective dynamics | scalar | $\mathsf F$ | EXACT reduced model |
| C1003 | neutral-cancelled Riccati remainder scale | $\mathsf C$ | Schur perturbation | scalar | $\mathsf F$ | TARGET IDENTIFIED |
| C1004 | explicit small-$\nu$ positivity interval | $\mathsf C$ | validated continuation | targeted | $\mathsf F$ | OPEN / STOP-C66 |

---

# 20. Continuous-versus-discrete status

The neutral cancellation is an asymptotic spectral identity of the continuous periodic Floquet operator。

The coefficient label：

$$
n
$$

is its Fourier chart，not a discrete physical substrate。

The Schur/Riccati reduction is a coordinate description of the invariant spectral subspaces of that continuous operator family。

The remaining validation is an interval enclosure of continuous parameter：

$$
\nu.
$$

Therefore：

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

---

# 21. Strongest results of Round 62

## R62-A — exact neutral cancellation

$$
\boxed{
S_n
=
48K^3n^{-3}
+
O(n^{-4}).
}
$$

## R62-B — neutral root drifts only at cubic order

$$
\boxed{
r_{\rm neu}
=
1+
12K^2n^{-3}
+
O(n^{-4}).
}
$$

## R62-C — exact first coupling corrections

$$
\boxed{
a_e/a_j
=
1+j^{-1}+O(j^{-2}),
}
$$

$$
\boxed{
a_o/a_j
=
1+2j^{-1}+O(j^{-2}).
}
$$

## R62-D — Schur-renormalized optimal overlap

$$
\boxed{
j_m
=
\nu^{-1/4}.
}
$$

## R62-E — corrected theorem target

$$
\boxed{
\operatorname{dist}
(
E_\nu^{\min},
E_0^{\rm Jost}
)
=
O(\nu^{1/4}).
}
$$

## R62-F — physical data are comfortably stronger

At：

$$
\nu=10^{-8},
$$

$$
\boxed{
\theta_{\max,-}
\approx
7.27\times10^{-4},
}
$$

$$
\boxed{
\theta_{\max,+}
\approx
1.82\times10^{-3}.
}
$$

---

# 22. Next round — Explicit Fast-Schur Graph / Validated Slow-Riccati Bound

Round 62 has now identified the cancellation needed to make the quarter-power proof self-consistent。

The next round should stop changing scaling laws and actually build the invariant graph。

Concrete targets：

1. choose the local same-parity eigenbasis：
   $$
   \{
   v_{\rm fast}^{\min},
   v_{\rm neu},
   v_{\rm fast}^{\max}
   \};
   $$

2. write the full six-dimensional rescaled transfer in this biorthogonal basis；

3. solve the two fast coordinates as invariant graphs over the slow neutral pair；

4. exploit：
   $$
   r_{\rm fast}^{\min}=O(j^{-2}),
   \qquad
   r_{\rm fast}^{\max}=O(j^2)
   $$
   to obtain uniform graph contraction；

5. prove the induced neutral self-drift is：
   $$
   O(j^{-3})
   $$
   with explicit constants，not merely asymptotically；

6. prove the coupling corrections：
   $$
   |a_e/a_j-1|
   \le
   C/j,
   $$
   $$
   |a_o/a_j-1|
   \le
   C/j;
   $$

7. derive an explicit interval Riccati map：
   $$
   z_{j+1}
   =
   \frac{
   z_j+a_j
   }{
   a_jz_j+1+a_j^2
   }
   +
   \mathcal R_j;
   $$

8. establish：
   $$
   |\mathcal R_j|
   \le
   C_1a_j/j
   +
   C_2/j^3;
   $$

9. evaluate the bound at：
   $$
   j_m=\nu^{-1/4};
   $$

10. propagate the resulting interval through the Round 59 endpoint pullback and obtain the first explicit：
    $$
    0<\nu\le\nu_s
    \Longrightarrow
    a_3(\nu)>0.
    $$

This becomes：

$$
\boxed{
\textbf{Explicit Fast-Schur Graph / Validated Slow-Riccati Bound}.
}
$$

---

# 23. External primary-source anchors

1. F. Battelli, M. Franca, K. J. Palmer, *Exponential Dichotomy for Noninvertible Linear Difference Equations*, arXiv:2111.04553.
   - primary-source roughness results for exponential dichotomies of finite-dimensional difference equations；
   - relevant to invariant fast/slow graph persistence under coefficient perturbations。

2. Evans M. Harrell II, Manwah Lilian Wong, *On the behavior at infinity of solutions to difference equations in Schroedinger form*, arXiv:1109.4691.
   - develops variation-of-constants comparison and a discrete Liouville–Green/WKB transformation；
   - relevant to comparing the exact slow projective dynamics with a WKB reference equation。

3. Pierre Del Moral, Emma Horton, *A note on Riccati matrix difference equations*, arXiv:2107.12918.
   - studies time-varying Riccati difference equations，duality formulae and uniform bounds；
   - relevant structural context for the planned nonautonomous interval Riccati enclosure。

4. Yuri Latushkin, Shibi Vasudevan, *Fredholm determinants, continued fractions, Jost and Evans functions for a Jacobi matrix associated with the 2D-Euler equations*, arXiv:2401.14037.
   - connects Jost / Evans / Fredholm / continued-fraction descriptions in a hydrodynamic difference equation；
   - relevant to connecting the slow Riccati line back to the Round 59 endpoint Jost functional。

All NS-specific cancellation identities，coupling asymptotics，matching exponent balances and Grassmannian diagnostics are direct results of this project。

---

# 24. Commit state

$$
\boxed{
\begin{aligned}
\text{Route}
&=
\mathrm{Pure\ Continuous\ Neutral\text{-}Cancelled\ Fast\text{-}Schur/Slow\text{-}Riccati},
\\
\text{Essential }\mathsf C\to\mathsf D
&=
\mathrm{Not\ reached},
\\
\text{Generic pre-Schur law}
&=
2/7,
\\
\text{Exact neutral cancellation}
&=
j^{-2}
\to
j^{-3},
\\
\text{Post-Schur optimal overlap}
&=
1/4,
\\
\text{First corrected error target}
&=
O(\nu^{1/4}),
\\
\text{6D physical convergence}
&=
\mathrm{stronger\ than\ corrected\ target},
\\
\text{Remaining proof object}
&=
\mathrm{explicit\ fast\ invariant\ graph}
+
\mathrm{validated\ slow\ Riccati\ remainder},
\\
\text{STOP-C66}
&=
\mathrm{Neutral\text{-}Cancelled\ Schur/Riccati\ Enclosure\ Gap},
\\
\text{Next}
&=
\mathrm{Explicit\ Fast\text{-}Schur\ Graph/Validated\ Slow\text{-}Riccati\ Bound}.
\end{aligned}
}
$$