# Perfect Cuboid Radical-FDRS R21

## Coupled Bad-Prime Local Images and the Exact Finite Quadratic-Chabauty Target Modulo $11^5$

**Series:** Perfect Cuboid Global Closure  
**Branch:** Radical-FDRS / Geometry-Coupling Branch R  
**Phase:** R21 — Exact Coupled Local Arithmetic  
**Date:** 2026-10-05  
**Status:** Exact coupled local-image theorem and finite target compression — final height audit and $11$-adic root isolation remain open

---

# 0. Executive summary

R20 computed the independent theorem-level local-height upper bounds

$$
\Omega_q
$$

for every bad prime. Their global Minkowski sum was maximally coarse:

$$
\boxed{
\Omega_{\rm coarse}
\bmod11^5
=
11\mathbb Z_{11}/11^5\mathbb Z_{11}.
}
$$

Hence:

$$
\boxed{
|\Omega_{\rm coarse}\bmod11^5|
=
11^4
=
14641.
}
$$

The reason was identified precisely: the theorem-level construction allows the local height on the first elliptic quotient, the local height on the second quotient, and the valuation of the genus-$2$ coordinate to vary independently.

For a true local point on the canonical genus-$2$ curve, those quantities arise from the same local coordinate.

R21 keeps this coupling.

For every bad prime

$$
q\in
\{2,3,5,7,13,29,37,73,18521\},
$$

define the actual coupled local image

$$
\boxed{
\Omega_q^{\rm exact}
=
\left\{
-\lambda_q^{E_1}(\varphi_1(P))
+
\lambda_q^{E_2}(\varphi_2(P))
+
2\log|r(P)|_q
:
P\in\mathcal H(\mathbb Q_q)
\right\}.
}
$$

The canonical even sextic is

$$
\boxed{
\mathcal H:
\qquad
\mathcal Y^2
=
(2-r^2)
\left(
M^2r^4
-
2M^2r^2
+
L^2
\right),
}
$$

where

$$
M=950625,
\qquad
L=1352033.
$$

The explicit theorem quotients are

$$
\boxed{
\varphi_1(r,\mathcal Y)
=
(a_6r^2,a_6\mathcal Y),
}
$$

and

$$
\boxed{
\varphi_2(r,\mathcal Y)
=
(a_0r^{-2},a_0\mathcal Yr^{-3}),
}
$$

with

$$
a_6=-M^2,
\qquad
a_0=2L^2.
$$

R21 computes all nine coupled local images exactly:

$$
\boxed{
\Omega_2^{\rm exact}
=
\left\{
-\frac{29}{40},
-\frac35,
-\frac{21}{40},
-\frac12
\right\}\log2,
}
$$

$$
\boxed{
\Omega_3^{\rm exact}
=
\{2,3\}\log3,
}
$$

$$
\boxed{
\Omega_5^{\rm exact}
=
\left\{
4,
\frac92,
6
\right\}\log5,
}
$$

$$
\boxed{
\Omega_7^{\rm exact}
=
\{0\},
}
$$

$$
\boxed{
\Omega_{13}^{\rm exact}
=
\{2,3\}\log13,
}
$$

$$
\boxed{
\Omega_{29}^{\rm exact}
=
\Omega_{37}^{\rm exact}
=
\{0\},
}
$$

and

$$
\boxed{
\Omega_{73}^{\rm exact}
=
\Omega_{18521}^{\rm exact}
=
\left\{
-2,
-\frac32,
-\frac98,
-1
\right\}\log q.
}
$$

Thus the local cardinalities are:

$$
\boxed{
4,\ 2,\ 3,\ 1,\ 2,\ 1,\ 1,\ 4,\ 4.
}
$$

Compare these with the R20 independent upper-bound cardinalities:

$$
41,\ 8,\ 29,\ 3,\ 12,\ 3,\ 3,\ 24,\ 24.
$$

The coupling removes the vast majority of formally possible but unrealizable local combinations.

Form the exact global target:

$$
\boxed{
\Omega^{\rm exact}
=
\sum_{q\ {\rm bad}}
\Omega_q^{\rm exact}.
}
$$

Using the Iwasawa $11$-adic logarithm and reducing modulo

$$
11^5=161051,
$$

R21 obtains:

$$
\boxed{
|\Omega^{\rm exact}\bmod11^5|
=
721.
}
$$

Every element remains divisible by

$$
11.
$$

Equivalently:

$$
\boxed{
\left|
\frac{\Omega^{\rm exact}}{11}
\bmod11^4
\right|
=
721.
}
$$

This is a reduction from

$$
14641
$$

coarse classes to

$$
721.
$$

The retained fraction is approximately

$$
\boxed{
4.92\%.
}
$$

Thus more than

$$
\boxed{
95\%
}
$$

of the R20 theorem-level local-height combinations are proved impossible.

The normalized residue packet modulo

$$
11^4=14641
$$

is recorded in the appendix for direct use in the next root-isolation phase.

Its SHA-256 checksum is:

$$
\boxed{
\texttt{94bae376fa7d8876fa2b01b584c3c405590cef56255cf3c42c316ed741d5c65f}.
}
$$

---

# 1. Local-height algorithm used

R21 uses the non-archimedean Néron-height normalization of the explicit bielliptic quadratic-Chabauty implementation.

For a minimal integral generalized Weierstrass model

$$
y^2+a_1xy+a_3y
=
x^3+a_2x^2+a_4x+a_6,
$$

define

$$
A(P)
=
v_q
\left(
3x^2+2a_2x+a_4-a_1y
\right),
$$

$$
B(P)
=
v_q
\left(
2y+a_1x+a_3
\right),
$$

and let

$$
N=v_q(\Delta).
$$

At multiplicative reduction, if

$$
A(P)>0
$$

and

$$
B(P)>0,
$$

then

$$
n(P)
=
\min
\left(
B(P),
\frac N2
\right),
$$

and

$$
\boxed{
\lambda_q(P)
=
-\frac{
n(P)(N-n(P))
}{
N
}
\log q.
}
$$

If the point has nonsingular integral reduction, the finite local value is

$$
0.
$$

For a point in the formal neighborhood of infinity, the nonintegral contribution is controlled by

$$
\log\max\{1,|x(P)|_q\}.
$$

For a nonminimal theorem model, the exact discriminant-ratio correction is applied before forming the coupled difference.

---

# 2. Why coupling is controlled by one genus-$2$ coordinate

The theorem quotient $x$-coordinates are:

$$
\boxed{
x_1
=
a_6r^2
=
-M^2r^2,
}
$$

and

$$
\boxed{
x_2
=
a_0r^{-2}
=
\frac{
2L^2
}{
r^2
}.
}
$$

Therefore:

$$
\boxed{
v_q(x_1)
=
v_q(a_6)+2v_q(r),
}
$$

and:

$$
\boxed{
v_q(x_2)
=
v_q(a_0)-2v_q(r).
}
$$

The two quotient $y$-coordinates arise from the same

$$
\mathcal Y.
$$

Hence the component-group indices of $E_1$ and $E_2$ and the valuation of $r$ are not independent.

This is the exact information discarded by the R20 Minkowski upper bound.

---

# 3. Exact $q=2$ geometry

At

$$
q=2,
$$

both

$$
M
$$

and

$$
L
$$

are odd.

If

$$
v_2(r)>0,
$$

then

$$
v_2(2-r^2)=1,
$$

while the second factor is a unit.

Hence:

$$
v_2(F(r))=1,
$$

which is odd.

Therefore no $\mathbb Q_2$ point has:

$$
v_2(r)>0.
$$

If

$$
v_2(r)<0,
$$

the dominant term is:

$$
-M^2r^6.
$$

Its unit squareclass is:

$$
[-1].
$$

Since:

$$
-1
$$

is not a square in:

$$
\mathbb Q_2,
$$

there are no such local points.

Thus every local point satisfies:

$$
\boxed{
v_2(r)=0.
}
$$

So:

$$
\boxed{
r\in\mathbb Z_2^\times.
}
$$

---

# 4. The two $2$-minimal quotient models

The theorem quotient models each lose discriminant valuation $12$ under one $u=2$ Weierstrass transformation.

For $E_1$ use:

$$
\boxed{
x_1=4X_1+3,
}
$$

$$
\boxed{
y_1=8Y_1+4X_1+4.
}
$$

The resulting minimal generalized model has split reduction:

$$
\boxed{
I_8.
}
$$

Its generalized coefficients are:

$$
\boxed{
a_1=1,
}
$$

$$
\boxed{
a_2=903687890627,
}
$$

$$
\boxed{
a_3=1,
}
$$

$$
\boxed{
a_4=307408910222356795239259,
}
$$

$$
\boxed{
a_6=46651061590372478437769413454425049.
}
$$

For $E_2$ use:

$$
\boxed{
x_2=4X_2+2,
}
$$

$$
\boxed{
y_2=8Y_2+4X_2.
}
$$

The resulting minimal model has split reduction:

$$
\boxed{
I_{10}.
}
$$

Its coefficients are:

$$
\boxed{
a_1=1,
}
$$

$$
\boxed{
a_2=-1360686198896,
}
$$

$$
\boxed{
a_3=0,
}
$$

$$
\boxed{
a_4=825967674442125495246416,
}
$$

$$
\boxed{
a_6=-188732914953705879117342557045695232.
}
$$

Because both theorem-model discriminant offsets are equal, the nonminimal translation terms cancel in the coupled difference:

$$
-\lambda_2^{E_1}+\lambda_2^{E_2}.
$$

---

# 5. Complete $2$-adic residue stratification

The equation

$$
\mathcal Y^2=F(r)
$$

for odd

$$
r
$$

has four square-solvable strata.

## Stratum A

$$
\boxed{
r\equiv\pm3\pmod8.
}
$$

Then:

$$
v_2(F(r))=6.
$$

The minimal component-depth parameters are:

$$
B_1=1,
\qquad
B_2=2.
$$

Therefore:

$$
\lambda_2^{E_1}
=
-\frac78\log2,
$$

$$
\lambda_2^{E_2}
=
-\frac85\log2.
$$

Thus:

$$
\boxed{
w_2
=
-\frac{29}{40}\log2.
}
$$

## Stratum B

$$
\boxed{
r\equiv\pm7\pmod{16}.
}
$$

Then:

$$
v_2(F(r))=8,
$$

and:

$$
B_1=2,
\qquad
B_2=3.
$$

Hence:

$$
\boxed{
w_2
=
-\frac35\log2.
}
$$

## Stratum C

$$
\boxed{
r\equiv\pm15\pmod{32}.
}
$$

Then:

$$
v_2(F(r))=10,
$$

and:

$$
B_1=3,
\qquad
B_2=4.
$$

Therefore:

$$
\boxed{
w_2
=
-\frac{21}{40}\log2.
}
$$

## Stratum D

The remaining square branch is:

$$
\boxed{
r\equiv\pm1\pmod{256}.
}
$$

The component depths have saturated:

$$
n_1=4,
\qquad
n_2=5.
$$

Thus:

$$
\boxed{
w_2
=
-\frac12\log2.
}
$$

The intermediate shells:

$$
r\equiv\pm31\pmod{64},
$$

$$
r\equiv\pm63\pmod{128},
$$

and

$$
r\equiv\pm127\pmod{256}
$$

fail the $\mathbb Q_2$ square criterion.

Thus the four values are complete.

---

# 6. Exact $\Omega_2^{\rm exact}$

Therefore:

$$
\boxed{
\Omega_2^{\rm exact}
=
\left\{
-\frac{29}{40},
-\frac35,
-\frac{21}{40},
-\frac12
\right\}\log2.
}
$$

The theorem-level upper bound had $41$ values.

The exact coupled image has only:

$$
\boxed{
4.
}
$$

---

# 7. Prime $q=3$

We have:

$$
v_3(M)=2,
\qquad
v_3(L)=0.
$$

A local point can occur only with:

$$
\boxed{
v_3(r)=0
}
$$

or:

$$
\boxed{
v_3(r)=-1.
}
$$

For:

$$
v_3(r)=0,
$$

the coupled contribution is:

$$
\boxed{
2\log3.
}
$$

For:

$$
v_3(r)=-1,
$$

the second quotient lies in the nontrivial rational component of the nonsplit $I_4$ fiber and the coupled contribution is:

$$
\boxed{
3\log3.
}
$$

For:

$$
v_3(r)\le-2,
$$

the dominant unit squareclass is $[-1]$, nonsquare in $\mathbb Q_3$.

Positive valuation is impossible because $2$ is nonsquare modulo $3$.

Hence:

$$
\boxed{
\Omega_3^{\rm exact}
=
\{2,3\}\log3.
}
$$

---

# 8. Prime $q=5$

Here:

$$
v_5(M)=4,
\qquad
v_5(L)=0.
$$

There are three coupled regimes.

If:

$$
v_5(r)=0,
$$

then:

$$
\boxed{
w_5=4\log5.
}
$$

If:

$$
v_5(r)=-1,
$$

then:

$$
\boxed{
w_5=\frac92\log5.
}
$$

For all deeper negative-valuation branches:

$$
v_5(r)\le-2,
$$

including the local branches approaching infinity:

$$
\boxed{
w_5=6\log5.
}
$$

Positive valuation is impossible because $2$ is nonsquare modulo $5$.

Thus:

$$
\boxed{
\Omega_5^{\rm exact}
=
\left\{
4,
\frac92,
6
\right\}\log5.
}
$$

---

# 9. Prime $q=13$

The valuation pattern is:

$$
v_{13}(M)=2,
\qquad
v_{13}(L)=0.
$$

For unit $r$:

$$
\boxed{
w_{13}=2\log13.
}
$$

For every negative-valuation local branch, including the local points at infinity:

$$
\boxed{
w_{13}=3\log13.
}
$$

There are no positive-valuation points because $2$ is nonsquare modulo $13$.

Therefore:

$$
\boxed{
\Omega_{13}^{\rm exact}
=
\{2,3\}\log13.
}
$$

---

# 10. Primes $q=7,29,37$

At each of these primes:

$$
a_0,
\qquad
a_6
$$

are units.

Both theorem quotient models are minimal split:

$$
I_2.
$$

For unit $r$ away from the common singular branch, both local heights vanish.

At a common singular branch, the two quotient component depths are identical:

$$
n_1=n_2=1.
$$

Hence:

$$
-\lambda_q^{E_1}
+
\lambda_q^{E_2}
=
0.
$$

For positive or negative valuation of $r$, the term:

$$
2\log|r|_q
$$

cancels the unique nonintegral quotient contribution.

Therefore:

$$
\boxed{
\Omega_7^{\rm exact}
=
\Omega_{29}^{\rm exact}
=
\Omega_{37}^{\rm exact}
=
\{0\}.
}
$$

These three bad primes contribute nothing to the exact global finite target.

---

# 11. Primes $q=73$ and $q=18521$

At both primes:

$$
v_q(M)=0,
\qquad
v_q(L)=1.
$$

The first quotient has split:

$$
I_4
$$

reduction, while the second has split:

$$
I_8.
$$

The two primes have identical local valuation geometry.

There are four coupled values.

For the positive-valuation / $r=0$ branch:

$$
\boxed{
w_q=-2\log q.
}
$$

For the negative-valuation / infinity branch:

$$
\boxed{
w_q=-\frac32\log q.
}
$$

For a generic unit branch, the same value occurs:

$$
\boxed{
w_q=-\frac32\log q.
}
$$

For the common singular unit branch, let:

$$
b=v_q(\mathcal Y).
$$

If:

$$
b=1,
$$

then:

$$
n_1=1,
\qquad
n_2=3,
$$

so:

$$
\boxed{
w_q=-\frac98\log q.
}
$$

If:

$$
b\ge2,
$$

then:

$$
n_1=2,
\qquad
n_2=4,
$$

so:

$$
\boxed{
w_q=-\log q.
}
$$

Therefore, for both primes:

$$
\boxed{
\Omega_q^{\rm exact}
=
\left\{
-2,
-\frac32,
-\frac98,
-1
\right\}\log q.
}
$$

---

# 12. Exact local-image table

| $q$ | R20 coarse size | R21 exact size | $\Omega_q^{\rm exact}/\log q$ |
|---:|---:|---:|---|
| $2$ | $41$ | $4$ | $\{-29/40,-3/5,-21/40,-1/2\}$ |
| $3$ | $8$ | $2$ | $\{2,3\}$ |
| $5$ | $29$ | $3$ | $\{4,9/2,6\}$ |
| $7$ | $3$ | $1$ | $\{0\}$ |
| $13$ | $12$ | $2$ | $\{2,3\}$ |
| $29$ | $3$ | $1$ | $\{0\}$ |
| $37$ | $3$ | $1$ | $\{0\}$ |
| $73$ | $24$ | $4$ | $\{-2,-3/2,-9/8,-1\}$ |
| $18521$ | $24$ | $4$ | $\{-2,-3/2,-9/8,-1\}$ |

The product of the local exact cardinalities is:

$$
4\cdot2\cdot3\cdot1\cdot2\cdot1\cdot1\cdot4\cdot4
=
\boxed{
768.
}
$$

Thus, before $11$-adic collisions, there are at most $768$ formal combinations of exact local values.

---

# 13. Exact Iwasawa logarithms modulo $11^5$

The bad-prime logarithms used in the global target assembly are:

$$
\boxed{
\log_{11}(2)
\equiv
75086
\pmod{11^5},
}
$$

$$
\boxed{
\log_{11}(3)
\equiv
155243
\pmod{11^5},
}
$$

$$
\boxed{
\log_{11}(5)
\equiv
14685
\pmod{11^5},
}
$$

$$
\boxed{
\log_{11}(7)
\equiv
50798
\pmod{11^5},
}
$$

$$
\boxed{
\log_{11}(13)
\equiv
136136
\pmod{11^5},
}
$$

$$
\boxed{
\log_{11}(29)
\equiv
28468
\pmod{11^5},
}
$$

$$
\boxed{
\log_{11}(37)
\equiv
23826
\pmod{11^5},
}
$$

$$
\boxed{
\log_{11}(73)
\equiv
23980
\pmod{11^5},
}
$$

and:

$$
\boxed{
\log_{11}(18521)
\equiv
50050
\pmod{11^5}.
}
$$

The logarithms were cross-checked by multiplicativity on independent integer products.

---

# 14. Global exact target

Define:

$$
\boxed{
\Omega^{\rm exact}
=
\left\{
\sum_q
\omega_q
:
\omega_q\in\Omega_q^{\rm exact}
\right\}.
}
$$

Modulo:

$$
11^5,
$$

the exact dynamic-programming sum has cardinality:

$$
\boxed{
721.
}
$$

Hence only $721$ of the $14641$ possible multiples of $11$ remain.

The compression factor is:

$$
\boxed{
\frac{14641}{721}
\approx20.31.
}
$$

The retained proportion is:

$$
\boxed{
\frac{721}{14641}
\approx4.92\%.
}
$$

---

# 15. Progressive global compression

The distinct residue count modulo $11^5$ evolves as follows.

| after adding $q$ | exact local size | global residue count |
|---:|---:|---:|
| $2$ | $4$ | $4$ |
| $3$ | $2$ | $8$ |
| $5$ | $3$ | $24$ |
| $7$ | $1$ | $24$ |
| $13$ | $2$ | $48$ |
| $29$ | $1$ | $48$ |
| $37$ | $1$ | $48$ |
| $73$ | $4$ | $192$ |
| $18521$ | $4$ | $\boxed{721}$ |

The final count is smaller than the raw product $768$ because different exact local combinations collide modulo $11^5$.

---

# 16. Normalized exact packet

Every exact global value is divisible by:

$$
11.
$$

Define:

$$
\boxed{
\overline\Omega
=
\frac{
\Omega^{\rm exact}
}{
11
}
\bmod11^4.
}
$$

Then:

$$
\boxed{
|\overline\Omega|=721.
}
$$

This normalized packet is the preferred finite target for the next prototype-disk root-isolation phase.

---

# 17. Machine checksum

Write the normalized residues as one comma-separated increasing list.

Its SHA-256 checksum is:

$$
\boxed{
\texttt{94bae376fa7d8876fa2b01b584c3c405590cef56255cf3c42c316ed741d5c65f}.
}
$$

This provides a compact machine-verification target for future local runs.

---

# 18. Global-minimal-model audit progress

R20 noted that the convenient historical integral $E_0$ model is not globally minimal at $2$.

R21 identifies an explicit $2$-minimal model.

Starting from:

$$
Y^2
=
X^3
-
2MX^2
+
L^2X,
$$

use:

$$
\boxed{
X=4X'+1,
}
$$

$$
\boxed{
Y=8Y'+4X'.
}
$$

The resulting generalized Weierstrass coefficients are:

$$
\boxed{
a_1=1,
}
$$

$$
\boxed{
a_2=-475312,
}
$$

$$
\boxed{
a_3=0,
}
$$

$$
\boxed{
a_4=114249339412,
}
$$

$$
\boxed{
a_6=28562364560.
}
$$

Its discriminant valuation at $2$ is:

$$
\boxed{
8,
}
$$

matching the split $I_8$ fiber.

This is the correct global-minimal direction for the R19 height-normalization audit.

---

# 19. Global minimal model for theorem $E_2$ at $2$

The theorem $E_2$ model is nonminimal only at $2$.

A minimizing transformation is:

$$
\boxed{
x=4X+2,
}
$$

$$
\boxed{
y=8Y+4X.
}
$$

The resulting generalized coefficients are:

$$
\boxed{
a_1=1,
}
$$

$$
\boxed{
a_2=-1360686198896,
}
$$

$$
\boxed{
a_3=0,
}
$$

$$
\boxed{
a_4=825967674442125495246416,
}
$$

$$
\boxed{
a_6=-188732914953705879117342557045695232.
}
$$

Its discriminant valuation at $2$ is:

$$
\boxed{
10,
}
$$

matching split $I_{10}$.

Thus the two required $2$-minimal models are now explicit.

---

# 20. Proof-safety interpretation

The exact coupled sets in R21 are independent of the provisional R19 global height calibration.

They use only:

- the canonical genus-$2$ equation;
- exact local solvability;
- exact quotient maps;
- exact minimal-model transformations;
- exact non-archimedean local-height valuation formulas.

Therefore the reduction:

$$
14641
\longrightarrow
721
$$

is not affected by the R20 height-normalization corrigendum.

The height calibration and the finite target packet are logically separate layers.

---

# 21. Relation to the general bielliptic theorem

The general explicit bielliptic quadratic-Chabauty theorem constructs the sets:

$$
\Omega_q
$$

as independent upper bounds.

R21 computes a curve-specific refinement:

$$
\boxed{
\Omega_q^{\rm exact}
\subseteq
\Omega_q
}
$$

by retaining the fact that both elliptic quotient points arise from the same local genus-$2$ point.

Thus R21 strengthens the general theorem for this particular Perfect-Cuboid curve without changing the theorem itself.

---

# 22. Next arithmetic target

Two tasks remain before a fixed-slice rational-point theorem.

## Task A — Height normalization audit

Recompute the R19 global height constants on the explicit global minimal models from Sections 18–19.

This will promote:

$$
\alpha_1,
\qquad
\alpha_2
$$

from provisional to certified constants.

## Task B — Prototype-disk root isolation

Compute the quadratic-Chabauty series:

$$
\widetilde\rho(1+11u)
$$

to sufficient precision.

For only the $721$ target classes in:

$$
\Omega^{\rm exact},
$$

solve:

$$
\boxed{
\widetilde\rho(1+11u)
=
\omega.
}
$$

Use Newton polygons, Hensel lifting, or Strassmann's theorem to certify the number of roots in:

$$
u\in\mathbb Z_{11}.
$$

---

# 23. Preferred R22 program

R22 should first close the R19 minimal-model height audit.

The exact minimal models are now known.

Therefore:

1. rebuild the canonical $11$-adic sigma functions on those models;
2. transport $Q$ and the theorem-$E_2$ rank-one point exactly;
3. recompute the global $11$-adic heights;
4. recompute $\alpha_1,\alpha_2$;
5. compare with the provisional R19 residues;
6. after certification, build $\widetilde\rho(u)$;
7. intersect its image with the exact $721$-class target.

---

# 24. R21 theorem package

## Theorem A — Exact $2$-adic coupled image

$$
\boxed{
\Omega_2^{\rm exact}
=
\left\{
-\frac{29}{40},
-\frac35,
-\frac{21}{40},
-\frac12
\right\}\log2.
}
$$

## Theorem B — Vanishing bad-prime contributions

$$
\boxed{
\Omega_7^{\rm exact}
=
\Omega_{29}^{\rm exact}
=
\Omega_{37}^{\rm exact}
=
\{0\}.
}
$$

## Theorem C — Complete exact local table

Every:

$$
\Omega_q^{\rm exact}
$$

for the nine bad primes is explicitly determined.

## Theorem D — Global exact finite target size

$$
\boxed{
|\Omega^{\rm exact}\bmod11^5|
=
721.
}
$$

## Theorem E — Coupling compression

More than $95\%$ of the R20 independent local combinations are eliminated by enforcing that the two elliptic images arise from the same local genus-$2$ point.

---

# 25. What R21 does not prove

R21 does not yet compute the final zero set of the quadratic-Chabauty function.

It does not certify the provisional R19 height constants.

It does not prove that the prototype residue disk contains only the boundary point.

It does not yet prove the nonexistence of a nondegenerate rational point on the:

$$
n=\frac7{32}
$$

fixed slice.

It supplies the exact finite local target required for those final steps.

---

# 26. Appendix — normalized exact target residues

The following are the $721$ values of:

$$
\overline\Omega
=
\Omega^{\rm exact}/11
\bmod11^4,
$$

listed in increasing order.

```text
4, 8, 9, 23, 61, 71, 75, 88, 97, 107, 130, 192, 202, 211, 215, 221
225, 359, 369, 382, 383, 387, 396, 397, 409, 418, 428, 451, 482, 513, 520, 532
536, 537, 546, 603, 616, 623, 633, 718, 728, 753, 754, 764, 777, 787, 887, 897
915, 924, 925, 927, 937, 944, 1010, 1074, 1075, 1098, 1099, 1151, 1161, 1165, 1246, 1248
1250, 1256, 1260, 1282, 1292, 1305, 1315, 1332, 1449, 1455, 1459, 1465, 1472, 1486, 1544, 1571
1572, 1603, 1626, 1627, 1653, 1693, 1776, 1778, 1788, 1808, 1818, 1844, 1854, 1860, 1867, 1877
1939, 1949, 1958, 1962, 1968, 1972, 1977, 1987, 2014, 2072, 2099, 2100, 2165, 2181, 2188, 2336
2340, 2346, 2350, 2353, 2363, 2372, 2382, 2395, 2405, 2467, 2477, 2486, 2490, 2496, 2500, 2634
2661, 2662, 2674, 2693, 2716, 2717, 2811, 2831, 2868, 2878, 2881, 2891, 2895, 2898, 2908, 2993
3018, 3028, 3029, 3038, 3039, 3052, 3062, 3152, 3162, 3189, 3190, 3202, 3212, 3216, 3245, 3339
3359, 3423, 3426, 3430, 3436, 3440, 3521, 3525, 3535, 3557, 3566, 3567, 3580, 3590, 3597, 3607
3680, 3724, 3730, 3740, 3744, 3751, 3807, 3918, 3921, 3957, 3958, 3968, 4053, 4063, 4075, 4083
4119, 4125, 4128, 4129, 4135, 4142, 4152, 4214, 4233, 4237, 4242, 4252, 4278, 4279, 4335, 4396
4446, 4449, 4485, 4603, 4611, 4615, 4625, 4628, 4638, 4647, 4656, 4657, 4670, 4680, 4742, 4761
4765, 4770, 4775, 4806, 4897, 4924, 4982, 4992, 5011, 5096, 5106, 5143, 5153, 5156, 5160, 5165
5166, 5170, 5173, 5218, 5293, 5303, 5304, 5327, 5332, 5406, 5417, 5425, 5477, 5481, 5486, 5501
5510, 5520, 5539, 5624, 5634, 5638, 5688, 5693, 5698, 5701, 5705, 5715, 5727, 5746, 5800, 5822
5831, 5832, 5855, 5860, 5872, 5882, 5934, 5945, 5959, 6005, 6009, 6014, 6029, 6072, 6082, 6166
6186, 6196, 6222, 6232, 6233, 6243, 6255, 6328, 6340, 6350, 6393, 6394, 6400, 6410, 6417, 6487
6507, 6543, 6576, 6591, 6600, 6610, 6627, 6650, 6661, 6714, 6724, 6728, 6745, 6750, 6760, 6783
6868, 6878, 6890, 6903, 6912, 6921, 6922, 6945, 6948, 6971, 7035, 7040, 7049, 7066, 7071, 7104
7119, 7155, 7162, 7172, 7178, 7189, 7256, 7257, 7267, 7273, 7276, 7286, 7371, 7381, 7418, 7430
7431, 7435, 7440, 7445, 7476, 7483, 7499, 7568, 7577, 7594, 7597, 7671, 7681, 7690, 7700, 7751
7766, 7776, 7785, 7795, 7804, 7814, 7818, 7835, 7899, 7903, 7909, 7913, 7958, 7963, 7968, 7973
7980, 7992, 8002, 8011, 8087, 8125, 8139, 8147, 8156, 8199, 8209, 8213, 8224, 8279, 8294, 8304
8308, 8346, 8347, 8357, 8363, 8431, 8441, 8461, 8471, 8497, 8507, 8508, 8520, 8530, 8534, 8615
8625, 8629, 8667, 8675, 8684, 8741, 8752, 8836, 8841, 8856, 8866, 8875, 8885, 8892, 8902, 8915
8925, 8989, 8993, 8999, 9003, 9010, 9020, 9025, 9035, 9048, 9058, 9062, 9143, 9153, 9157, 9177
9213, 9236, 9246, 9314, 9331, 9369, 9370, 9384, 9394, 9398, 9420, 9430, 9437, 9443, 9447, 9453
9521, 9531, 9532, 9538, 9548, 9551, 9561, 9646, 9691, 9705, 9710, 9715, 9719, 9741, 9764, 9774
9842, 9859, 9898, 9926, 9946, 9956, 9965, 9975, 10041, 10051, 10060, 10079, 10083, 10089, 10093, 10100
10110, 10174, 10178, 10188, 10219, 10233, 10238, 10243, 10247, 10267, 10404, 10421, 10460, 10474, 10478, 10484
10488, 10569, 10573, 10579, 10583, 10611, 10621, 10622, 10628, 10638, 10706, 10716, 10736, 10772, 10781, 10795
10799, 10805, 10809, 10819, 10890, 10894, 10932, 10949, 10988, 11006, 11016, 11101, 11111, 11131, 11140, 11141
11150, 11167, 11177, 11190, 11200, 11264, 11268, 11278, 11285, 11295, 11300, 11309, 11323, 11327, 11333, 11337
11347, 11418, 11422, 11511, 11550, 11635, 11645, 11659, 11663, 11668, 11669, 11673, 11695, 11705, 11712, 11718
11728, 11796, 11806, 11813, 11823, 11826, 11871, 11909, 11945, 11956, 11980, 11984, 12039, 12040, 12059, 12063
12078, 12163, 12173, 12191, 12201, 12221, 12230, 12231, 12240, 12266, 12316, 12354, 12358, 12361, 12368, 12375
12380, 12384, 12385, 12399, 12437, 12453, 12473, 12484, 12508, 12512, 12568, 12587, 12591, 12725, 12735, 12749
12753, 12758, 12759, 12763, 12794, 12844, 12848, 12858, 12886, 12889, 12896, 12903, 12908, 12912, 12913, 12981
12999, 13046, 13070, 13074, 13084, 13094, 13130, 13153, 13253, 13263, 13281, 13291, 13303, 13320, 13376, 13386
13405, 13406, 13442, 13451, 13465, 13474, 13475, 13527, 13543, 13560, 13574, 13598, 13602, 13612, 13622, 13624
13626, 13658, 13681, 13815, 13825, 13831, 13848, 13910, 13920, 13933, 13934, 13938, 13947, 13948, 13970, 13979
13993, 14002, 14003, 14071, 14088, 14136, 14152, 14154, 14174, 14184, 14210, 14220, 14243, 14305, 14315, 14324
14328, 14334, 14338, 14343, 14353, 14438, 14448, 14466, 14475, 14476, 14495, 14496, 14531, 14541, 14564, 14626
14633
```

---

# 27. Final conclusion

R20 showed that the generic theorem-level bad-prime upper bound was too coarse to support root isolation.

R21 resolves that problem.

Every bad prime has now been treated with the two quotient maps coupled through one local genus-$2$ point.

The exact local images collapse from:

$$
41,8,29,3,12,3,3,24,24
$$

possible values to:

$$
\boxed{
4,2,3,1,2,1,1,4,4.
}
$$

The global finite quadratic-Chabauty target shrinks from the entire:

$$
11\mathbb Z_{11}/11^5\mathbb Z_{11}
$$

to only:

$$
\boxed{
721
}
$$

classes.

The local-coupling bottleneck is therefore closed.

The next proof-safe step is sharply defined:

$$
\boxed{
\text{minimal-model height audit}
\longrightarrow
\widetilde\rho(u)
\longrightarrow
721\text{-class root isolation}.
}
$$

This is the canonical starting point for R22.