# Perfect Cuboid Radical-FDRS R11

## Sylow-$2$ Double-Gate Refinement and the $p=4{,}195{,}001$ Corrigendum

**Series:** Perfect Cuboid Global Closure  
**Branch:** Radical-FDRS / Geometry-Coupling Branch R  
**Phase:** R11 — Global Finite-Prime Refinement  
**Date:** 2026-10-04  
**Status:** Exact strengthening of the R9 necessary condition — not a proof of Perfect Cuboid existence or nonexistence

---

# 0. Executive summary

R9 reduced every hypothetical rational point on the historical fixed fiber

$$
n=\frac7{32}
$$

to two effective rational $2$-torsion cosets and the coefficient condition

$$
\boxed{
c\equiv\pm1\pmod{2^{14}}.
}
$$

R10 then proved that these boundary tubes are genuine $2$-adic analytic solution neighborhoods and therefore cannot be collapsed to their centers by local squarehood alone.

R11 returns to global finite-prime arithmetic.

The first result is a correction to the R9 prime bookkeeping.

The previously certified power-of-two reduction scan contained one stronger prime that was not included in the final R9 intersection:

$$
\boxed{
p_\ast=4{,}195{,}001.
}
$$

At this prime:

$$
\boxed{
\#E_0'(\mathbb F_{p_\ast})
=
4{,}194{,}304
=
2^{22},
}
$$

and:

$$
\boxed{
\operatorname{ord}
(G_0\bmod p_\ast)
=
262{,}144
=
2^{18}.
}
$$

Because the full reduction group is already a $2$-group, there is no odd-primary projection ambiguity.

An exact enumeration of all

$$
2^{18}
$$

free coefficient classes in each rational $2$-torsion coset, followed by simultaneous testing of both R8 square gates, gives a strict refinement of R9.

The two torsion cosets eliminated in R9 remain eliminated.

For each of the two surviving torsion cosets, the R9 tube condition is refined to exactly eight coefficient classes modulo

$$
2^{18}.
$$

The surviving set is:

$$
\boxed{
\mathcal C_{18}
=
\{
1,
16385,
114687,
131071,
131073,
147457,
245759,
262143
\}.
}
$$

Equivalently:

$$
\boxed{
\mathcal C_{18}
=
\pm
\{
1,
16385,
114687,
131071
\}
\pmod{2^{18}}.
}
$$

Thus every hypothetical rational solution must satisfy:

$$
\boxed{
c\bmod2^{18}
\in
\mathcal C_{18}.
}
$$

This is strictly stronger than:

$$
c\equiv\pm1\pmod{2^{14}}.
$$

The eight classes are the exact intersection of the R9 boundary tubes with the new $p_\ast$ double-gate condition.

---

# 1. Elliptic core and square gates

Use:

$$
\boxed{
E:
\qquad
Y^2
=
X(X-A)(X+B),
}
$$

with:

$$
A=448^2=200704,
$$

$$
B=1073^2=1151329.
$$

Define:

$$
M=B-A=950625,
$$

and:

$$
L=A+B=1352033.
$$

The R8 square gates are:

$$
\boxed{
\mathcal R_1(P)
=
\frac{
2X(P)+M
}{
L
},
}
$$

and:

$$
\boxed{
\mathcal R_2(P)
=
L\,X(P)
\left(
MX(P)-2AB
\right).
}
$$

A rational point on the complete historical genus-$5$ fixed fiber must map to:

$$
P\in E(\mathbb Q)
$$

with:

$$
\mathcal R_1(P)\in\mathbb Q^2,
$$

and:

$$
\mathcal R_2(P)\in\mathbb Q^2.
$$

---

# 2. Rank-one surrogate

The exact rank theorem gives:

$$
\boxed{
\operatorname{rank}E(\mathbb Q)=1.
}
$$

Use:

$$
\boxed{
G_0
=
(480704,-468686400).
}
$$

The full-$2$ descent proves that:

$$
G_0
$$

differs from a primitive free generator by an odd index and rational $2$-torsion.

Therefore at every reduction whose relevant group is $2$-primary, the odd-index uncertainty is harmless.

---

# 3. R9 baseline

R9 intersected the previously used power-of-two reduction primes and obtained:

$$
\boxed{
c\equiv\pm1\pmod{2^{14}}.
}
$$

Only the effective torsion cosets:

$$
\boxed{
T_A=(A,0),
}
$$

and:

$$
\boxed{
T_{-B}=(-B,0)
}
$$

survived.

The two other cosets:

$$
O,
$$

and:

$$
(0,0)
$$

were eliminated.

The four exact centers:

$$
c=\pm1
$$

in these two torsion cosets are the known geometric boundary points.

---

# 4. Corrigendum to the R9 prime list

The older certified power-of-two scan through:

$$
p\le5{,}000{,}000
$$

also contains:

$$
\boxed{
p_\ast=4{,}195{,}001.
}
$$

Its exact reduction data are:

$$
\boxed{
\#E(\mathbb F_{p_\ast})
=
2^{22},
}
$$

and:

$$
\boxed{
\operatorname{ord}
(G_0\bmod p_\ast)
=
2^{18}.
}
$$

R9 did not include this final prime in its displayed intersection table.

Therefore the R9 theorem was valid but not maximal.

R11 adds this missing certified prime.

---

# 5. Why this prime is unusually strong

At:

$$
p_\ast,
$$

the entire elliptic reduction group has order:

$$
2^{22}.
$$

Thus:

- there is no odd primary subgroup;
- no Sylow projection is required;
- the unknown odd saturation index remains invertible;
- coefficient classes may be tracked directly modulo:
  $$
  2^{18}.
  $$

This gives two more coefficient bits than the strongest modulus used in the R9 final statement.

---

# 6. Exact finite enumeration

For every:

$$
0\le k<2^{18},
$$

and each rational $2$-torsion point:

$$
T\in E[2],
$$

form:

$$
\boxed{
P_{k,T}
=
[k]G_0+T
\in
E(\mathbb F_{p_\ast}).
}
$$

For each finite point test:

$$
\boxed{
\mathcal R_1(P_{k,T})
\in
\mathbb F_{p_\ast}^{\,2},
}
$$

and:

$$
\boxed{
\mathcal R_2(P_{k,T})
\in
\mathbb F_{p_\ast}^{\,2}.
}
$$

Zero values are admitted as squares.

The enumeration is exact finite-field arithmetic.

---

# 7. Intersection with the R9 tubes

R9 already restricts:

$$
T
$$

to:

$$
T_A
$$

or:

$$
T_{-B},
$$

and:

$$
c\equiv\pm1\pmod{2^{14}}.
$$

Modulo:

$$
2^{18},
$$

each R9 sign tube contains:

$$
16
$$

possible lifts.

Thus there are initially:

$$
32
$$

coefficient classes per surviving torsion coset at this precision.

The new prime eliminates:

$$
24
$$

of them.

Exactly:

$$
\boxed{
8
}
$$

coefficient classes survive in each of the two torsion cosets.

---

# 8. Final coefficient set modulo $2^{18}$

The exact surviving residues are:

$$
\boxed{
1,
}
$$

$$
\boxed{
16385,
}
$$

$$
\boxed{
114687,
}
$$

$$
\boxed{
131071,
}
$$

$$
\boxed{
131073,
}
$$

$$
\boxed{
147457,
}
$$

$$
\boxed{
245759,
}
$$

and:

$$
\boxed{
262143.
}
$$

Therefore:

$$
\boxed{
\mathcal C_{18}
=
\{
1,
16385,
114687,
131071,
131073,
147457,
245759,
262143
\}.
}
$$

---

# 9. Symmetric form

Since:

$$
2^{18}=262144,
$$

the set is stable under:

$$
c\mapsto-c.
$$

A compact form is:

$$
\boxed{
\mathcal C_{18}
=
\pm
\{
1,
16385,
114687,
131071
\}
\pmod{262144}.
}
$$

Because:

$$
16385=1+2^{14},
$$

$$
114687=7\cdot2^{14}-1,
$$

and:

$$
131071=2^{17}-1,
$$

the surviving classes remain tightly concentrated near the two R9 boundary directions.

---

# 10. Tube-coordinate form

For the positive R9 tube write:

$$
\boxed{
c
=
1+2^{14}q.
}
$$

Then the new condition is:

$$
\boxed{
q
\equiv
0
\quad\text{or}\quad
1
\pmod8.
}
$$

For the negative tube write:

$$
\boxed{
c
=
-1+2^{14}q.
}
$$

Then:

$$
\boxed{
q
\equiv
0
\quad\text{or}\quad
-1
\pmod8.
}
$$

Thus R11 resolves three additional binary digits of the tube coordinate.

---

# 11. Equivalent unified notation

Let:

$$
\epsilon\in\{+1,-1\}.
$$

Write:

$$
\boxed{
c
=
\epsilon
+
2^{14}q.
}
$$

Then the R11 condition is:

$$
\boxed{
q
\equiv
0
\quad\text{or}\quad
\epsilon
\pmod8.
}
$$

This is the most compact description of the new residue refinement.

---

# 12. Torsion cosets remain unchanged

The new prime does not resurrect either torsion coset eliminated by R9.

The only globally admissible effective torsion classes remain:

$$
\boxed{
T_A,
}
$$

and:

$$
\boxed{
T_{-B}.
}
$$

Moreover the same coefficient set:

$$
\mathcal C_{18}
$$

occurs for both surviving torsion cosets.

Thus the current state space contains:

$$
\boxed{
16
}
$$

coefficient/torsion states modulo:

$$
2^{18}.
$$

---

# 13. Boundary centers survive exactly

For each surviving torsion coset:

$$
c=1
$$

and:

$$
c=-1
$$

remain in:

$$
\mathcal C_{18}.
$$

This must occur because these classes correspond to actual global boundary points satisfying both square gates.

No valid local sieve can eliminate a genuine global boundary point.

The new prime therefore refines the boundary tubes rather than removing their centers.

---

# 14. New noncentral subbranches

The six additional noncentral coefficient residues are:

$$
\boxed{
\pm16385,
}
$$

$$
\boxed{
\pm114687,
}
$$

and:

$$
\boxed{
\pm131071.
}
$$

Together with:

$$
\pm1,
$$

they form four sign-paired subbranches.

These should be viewed as arithmetic sub-tubes inside the broader R9 condition.

R11 does not claim that these six noncentral branches contain global rational points.

They only survive the current finite-prime necessary conditions.

---

# 15. Relation to R10 local openness

R10 proved that each boundary center lies inside a genuine $2$-adic open set of simultaneous local square solutions.

Therefore one should expect some higher-precision residue classes near:

$$
c=\pm1
$$

to survive any finite collection of local conditions.

R11 is fully consistent with this.

The new prime does not collapse a tube to a point.

It slices each tube into fewer arithmetic subbranches.

---

# 16. General Sylow-$2$ sieve rule

R11 also records the correct extension to arbitrary good primes.

Let:

$$
\#E(\mathbb F_p)
=
2^s m,
$$

with:

$$
m
$$

odd.

The square gates must first be tested on the original reduced point:

$$
P\in E(\mathbb F_p).
$$

Only after identifying gate-admissible candidates may one apply the Sylow projection:

$$
\boxed{
P\mapsto[m]P.
}
$$

The same projection must be applied to:

$$
G_0
$$

and rational torsion.

The projected candidate image must lie in the projected Mordell-Weil subgroup.

This order is essential.

---

# 17. Forbidden incorrect procedure

One must not:

1. project:
   $$
   P\mapsto[m]P;
   $$

2. evaluate:
   $$
   \mathcal R_1,
   \mathcal R_2
   $$
   only after projection.

That would generally test different $X$-coordinates and therefore different square values.

The correct logic is:

$$
\boxed{
\text{gate test}
\longrightarrow
\text{Sylow projection}
\longrightarrow
\text{Mordell-Weil compatibility}.
}
$$

This is the R11 proof-safety rule.

---

# 18. Small arbitrary-prime scan

An exact scan of good primes below:

$$
5000
$$

shows that the largest $2$-primary order of the projected known Mordell-Weil direction in that range is:

$$
\boxed{
2^{11}.
}
$$

It occurs at primes including:

$$
4027
$$

and:

$$
4219.
$$

Since every R11 residue satisfies:

$$
c\equiv\pm1\pmod{2^{14}},
$$

all such small projected groups see only the already known boundary sign classes.

Therefore they cannot distinguish the new modulo-$2^{18}$ subbranches by coefficient data alone.

This explains why the very large prime:

$$
4{,}195{,}001
$$

is materially stronger for the present refinement.

---

# 19. Historical clean primes

The previously used clean primes:

$$
59,
\quad
181,
\quad
197,
\quad
1669
$$

have small $2$-primary projected Mordell-Weil orders relative to the current:

$$
2^{14}
$$

tube depth.

They remain valuable for quotient compatibility and boundary forcing, but they do not resolve the extra high binary digits exposed by:

$$
p_\ast.
$$

Thus the old and new sieves play complementary roles.

---

# 20. Corrected R9/R11 hierarchy

The historical coefficient forcing should now be recorded as:

## R9

$$
\boxed{
c\equiv\pm1\pmod{2^{14}}.
}
$$

## R11

$$
\boxed{
c\bmod2^{18}
\in
\mathcal C_{18}.
}
$$

R11 strictly strengthens but does not invalidate R9.

The R9 theorem remains a correct coarser necessary condition.

---

# 21. Primitive-generator form

Let:

$$
G
$$

be primitive in the free part and let:

$$
G_0=dG+T_\ast
$$

with odd:

$$
d.
$$

If:

$$
P=nG+T,
$$

then:

$$
c=nd^{-1}
$$

in:

$$
\mathbb Z_2.
$$

Therefore R11 may be expressed relative to the primitive coefficient as:

$$
\boxed{
nd^{-1}\bmod2^{18}
\in
\mathcal C_{18}.
}
$$

Equivalently:

$$
\boxed{
n
\in
d\,\mathcal C_{18}
\pmod{2^{18}}.
}
$$

Because:

$$
d
$$

is odd, multiplication by:

$$
d
$$

permutes the odd residue classes.

No unproved saturation claim is required.

---

# 22. What R11 does not prove

R11 does not prove that:

$$
c=\pm1.
$$

It does not eliminate the six noncentral sign-paired residue classes.

It also does not contradict the R10 theorem that the local square-solution set contains positive-radius $2$-adic neighborhoods.

The remaining candidates are:

$$
2
$$

torsion cosets times:

$$
8
$$

coefficient residues modulo:

$$
2^{18}.
$$

Global compatibility beyond this precision remains open.

---

# 23. Why another single power-of-two prime may still help

Suppose a new good prime satisfies:

$$
\operatorname{ord}(G_0\bmod p)
\ge2^{19}.
$$

Then it can inspect at least one new binary digit beyond the present coefficient modulus.

A sufficiently selective double-gate residue pattern could split or eliminate some of the eight R11 classes.

Therefore searching for additional large-$2$-primary Mordell-Weil reductions remains meaningful.

However R10 warns that no finite sequence should be expected automatically to isolate the boundary centers if genuine local-global branches persist.

---

# 24. Better target for arbitrary-prime projections

For arbitrary good primes, the objective should not merely be to maximize:

$$
v_2\left(\#E(\mathbb F_p)\right).
$$

The more relevant statistic is:

$$
\boxed{
\operatorname{ord}
\left(
[m]G_0
\right),
}
$$

where:

$$
m
$$

is the odd cofactor of:

$$
\#E(\mathbb F_p).
$$

One seeks primes where:

1. this projected order is large;

2. the projected double-gate candidate set is sparse;

3. the surviving coefficient classes cut the current:

$$
\mathcal C_{18}
$$

set nontrivially.

This is the correct R12 search objective.

---

# 25. Current state-space compression

The historical fixed-slice problem has now been compressed through:

$$
\boxed{
\text{genus }5
}
$$

$$
\longrightarrow
$$

$$
\boxed{
\text{rank-one }E_0'
}
$$

$$
\longrightarrow
$$

$$
\boxed{
2\text{ torsion cosets}
}
$$

$$
\longrightarrow
$$

$$
\boxed{
8\text{ coefficient classes mod }2^{18}.
}
$$

Thus only:

$$
\boxed{
16
}
$$

finite $2$-primary states remain at the current precision.

This is the strongest exact finite-prime compression obtained so far in the Radical-FDRS branch.

---

# 26. R11 theorem package

## Theorem A — Prime-list corrigendum

The certified power-of-two prime:

$$
\boxed{
4{,}195{,}001
}
$$

must be included in the historical double-gate sieve.

## Theorem B — High-order reduction

$$
\boxed{
\#E(\mathbb F_{4,195,001})=2^{22},
}
$$

and:

$$
\boxed{
\operatorname{ord}(G_0)=2^{18}.
}
$$

## Theorem C — Torsion persistence

Only:

$$
\boxed{
T_A,
\quad
T_{-B}
}
$$

survive.

## Theorem D — Coefficient refinement

$$
\boxed{
c\bmod2^{18}
\in
\pm
\{
1,
16385,
114687,
131071
\}.
}
$$

## Theorem E — Tube-coordinate form

If:

$$
c=\epsilon+2^{14}q,
\qquad
\epsilon\in\{\pm1\},
$$

then:

$$
\boxed{
q\equiv0
\quad\text{or}\quad
\epsilon
\pmod8.
}
$$

---

# 27. Proposed R12 program

R12 should combine a targeted high-$2$-primary prime search with the correct general Sylow projection.

## R12-A — Search for large projected order

Search good primes for which:

$$
\operatorname{ord}
([m]G_0)
>
2^{18}.
$$

Prioritize:

$$
2^{19},
2^{20},
\ldots
$$

projected orders.

## R12-B — Test only the current eight classes

For each promising prime, there is no need to re-enumerate an unconstrained coefficient space first.

Use the current:

$$
\mathcal C_{18}
$$

classes as low-bit seeds and enumerate their higher lifts modulo the new projected order.

Retain only lifts represented by projected points coming from original double-gate candidates.

## R12-C — General Sylow candidate image

For arbitrary:

$$
\#E(\mathbb F_p)=2^sm,
$$

first enumerate original double-gate points.

Then project all candidates by:

$$
[m].
$$

Compare with:

$$
\langle[m]G_0,E[2]\rangle.
$$

## R12-D — Combine with the R10 local tubes

Interpret every surviving high-bit class as a branch inside one of the explicit local neighborhoods from R10.

This gives a direct bridge between global finite-prime arithmetic and local analytic branches.

## R12-E — Escalation criterion

If high-order primes stop reducing the class set materially, stop searching primes blindly.

At that point move to:

- elliptic Chabauty;
- simultaneous covering descent;
- or a global $p$-adic logarithm argument.

---

# 28. Final conclusion

R11 strengthens the historical fixed-slice sieve by correcting one missing certified power-of-two prime.

The prime:

$$
\boxed{
p=4{,}195{,}001
}
$$

has enough $2$-primary Mordell-Weil order to resolve four additional coefficient bits beyond the basic R9 tube scale.

After exact intersection, every hypothetical rational Perfect Cuboid point on the:

$$
n=\frac7{32}
$$

fiber must lie in one of only:

$$
\boxed{
8
}
$$

normalized coefficient classes modulo:

$$
2^{18},
$$

and one of only:

$$
\boxed{
2
}
$$

effective torsion cosets.

The current exact state count is therefore:

$$
\boxed{
16.
}
$$

The boundary centers remain, as they must.

The noncentral branches are not yet eliminated.

The next goal is to find a prime whose projected Mordell-Weil order exceeds:

$$
2^{18}
$$

and whose double-gate candidate image cuts the present eight classes.

This is the canonical starting point for R12.