# Perfect Cuboid Radical-FDRS R15

## Canonical-Cover Mordell-Weil Sieve, High-Order Prime Saturation, and the Elliptic-Chabauty Handoff

**Series:** Perfect Cuboid Global Closure  
**Branch:** Radical-FDRS / Geometry-Coupling Branch R  
**Phase:** R15 — Canonical Cover Finite-Prime Sieve  
**Date:** 2026-10-05  
**Status:** Exact computational checkpoint and canonical elliptic-Chabauty handoff — not a proof of Perfect Cuboid existence or nonexistence

---

# 0. Executive summary

R14 transported the strongest Radical-FDRS Mordell-Weil residue data to one canonical historical square-value cover.

The elliptic base is

$$
\boxed{
E_0:
\qquad
y^2
=
x^3
-
2x^2
+
\lambda^2x,
}
$$

with

$$
\boxed{
\lambda
=
\frac{1352033}{950625}.
}
$$

The canonical square gate is

$$
\boxed{
2-x\in\mathbb Q^2.
}
$$

The known odd-index Mordell-Weil surrogate is

$$
\boxed{
Q
=
\left(
1,
\frac{961408}{950625}
\right).
}
$$

R14 proved that the two surviving source torsion cosets disappear after isogeny and torsion-chart normalization.

Therefore every hypothetical fixed-slice solution is represented on the canonical historical orbit by a normalized $2$-adic coefficient

$$
c
$$

with

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

where

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

R15 applies the canonical square gate directly to this packet.

The historical clean primes

$$
11,
59,
181,
197,
1669
$$

do not refine the packet.

Their projected $Q$ orders are too small:

$$
2^3,
2^2,
2^3,
2^1,
2^2.
$$

Every R11 class reduces to one of the already known boundary sign classes at these moduli.

R15 then performs a new exact search for larger projected-order primes directly on the canonical curve.

Two useful primes were found:

$$
\boxed{
p_1=11{,}532{,}251,
}
$$

and

$$
\boxed{
p_2=15{,}721{,}883.
}
$$

Independent complete finite-field point counts give

$$
\boxed{
\#E_0(\mathbb F_{p_1})
=
11\cdot2^{20},
}
$$

and

$$
\boxed{
\#E_0(\mathbb F_{p_2})
=
15\cdot2^{20}.
}
$$

After odd-cofactor projection,

$$
\boxed{
\operatorname{ord}([11]Q)=2^{19}
}
$$

at $p_1$, and

$$
\boxed{
\operatorname{ord}([15]Q)=2^{19}
}
$$

at $p_2$.

Thus both primes see one coefficient bit beyond the R11 modulus.

For each prime, the eight R11 classes were lifted from modulo

$$
2^{18}
$$

to all sixteen classes modulo

$$
2^{19}.
$$

The square gate was tested in the proof-safe order:

1. reconstruct all odd-primary preimages of a projected Mordell-Weil class;
2. test:
   $$
   2-x
   $$
   on the preimages;
3. retain the projected class if at least one original preimage satisfies the square gate.

At:

$$
p_1,
$$

the odd kernel has order:

$$
11.
$$

Every one of the sixteen coefficient lifts has between two and seven square-gate-admissible odd-primary preimages.

At:

$$
p_2,
$$

the odd kernel has order:

$$
15.
$$

Every one of the sixteen coefficient lifts has between five and twelve square-gate-admissible odd-primary preimages.

Therefore:

$$
\boxed{
\text{neither prime eliminates any R11 coefficient class.}
}
$$

This is not a failure of the transport.

It reveals that the **single canonical cover is too broad to continue sharpening the R11 packet by these finite-prime conditions alone**.

The next decisive arithmetic information must use either:

- the complementary elliptic quotient of the genus-$2$ cover;
- elliptic Chabauty;
- a dual Mordell-Weil compatibility sieve;
- or the higher Perfect-Cuboid lift.

R15 therefore closes the finite-prime single-cover stage and prepares the canonical elliptic-Chabauty handoff.

---

# 1. Canonical elliptic base

Let

$$
\boxed{
M=950625,
}
$$

and

$$
\boxed{
L=1352033.
}
$$

Then

$$
\boxed{
\lambda=\frac LM.
}
$$

The canonical elliptic curve is

$$
\boxed{
E_0:
\qquad
y^2
=
x^3
-
2x^2
+
\lambda^2x.
}
$$

Its rational $2$-torsion point is

$$
\boxed{
T=(0,0).
}
$$

The exact Mordell-Weil rank is

$$
\boxed{
\operatorname{rank}E_0(\mathbb Q)=1.
}
$$

---

# 2. Canonical odd-index surrogate

Use

$$
\boxed{
Q
=
\left(
1,
\frac{961408}{950625}
\right).
}
$$

R14 proved by reduction modulo $17$ that the free-lattice coefficient of $Q$ is odd.

Therefore $Q$ is a proof-safe surrogate for a primitive free generator in every $2$-primary Mordell-Weil sieve.

No claim that $Q$ is globally primitive is required.

---

# 3. Canonical square-value cover

The historical square-value cover is

$$
\boxed{
r^2=2-x.
}
$$

The corresponding canonical genus-$2$ model is

$$
\boxed{
\mathcal H_{\rm can}:
\qquad
\mathcal Y^2
=
(2-r^2)
\left(
M^2r^4
-
2M^2r^2
+
L^2
\right).
}
$$

The map to the elliptic base is

$$
\boxed{
x=2-r^2,
}
$$

$$
\boxed{
y=\frac{\mathcal Y}{M}.
}
$$

The boundary points are

$$
\boxed{
r=\pm1,
}
$$

mapping to

$$
\boxed{
Q
\quad\text{and}\quad
-Q.
}
$$

at the elliptic level.

---

# 4. Transported coefficient packet

The R11 packet is

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

Equivalently,

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

Every hypothetical fixed-slice solution must map to the canonical orbit with normalized coefficient

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

No rational $2$-torsion label remains.

---

# 5. Sign symmetry at the first cover level

The square-value function

$$
2-x
$$

depends only on the elliptic $x$-coordinate.

Therefore

$$
\boxed{
2-x(P)
=
2-x(-P).
}
$$

Thus:

$$
c
$$

and:

$$
-c
$$

are indistinguishable at the first square-value level.

The eight R11 residues form four sign-orbits.

Higher Perfect-Cuboid lifts may distinguish the elliptic sign, so this quotient is used only at the canonical genus-$2$ stage.

---

# 6. Historical clean primes

The historical dual-quotient work used the clean primes

$$
\boxed{
11,
59,
181,
197,
1669.
}
$$

R15 recomputes their canonical-curve data.

The exact group orders and projected $Q$ orders are:

| $p$ | $\#E_0(\mathbb F_p)$ | odd cofactor | $\operatorname{ord}([m]Q)$ |
|---:|---:|---:|---:|
| $11$ | $16$ | $1$ | $8$ |
| $59$ | $56$ | $7$ | $4$ |
| $181$ | $160$ | $5$ | $8$ |
| $197$ | $176$ | $11$ | $2$ |
| $1669$ | $1664$ | $13$ | $4$ |

Thus the largest projected coefficient modulus among these five primes is only

$$
\boxed{
2^3.
}
$$

---

# 7. Why the historical clean primes cannot cut $\mathcal C_{18}$

Every residue in

$$
\mathcal C_{18}
$$

is congruent to

$$
\pm1
$$

modulo every power of two not exceeding

$$
2^{14}.
$$

In particular, modulo projected orders

$$
2,
4,
8,
$$

the entire packet collapses to the boundary sign classes.

The boundary classes are genuine rational square-value points.

Therefore they necessarily survive.

R15 verifies explicitly that all transported R11 classes survive the old-cover test at all five historical clean primes.

Hence these primes remain useful for the old **dual-quotient** compatibility problem, but they cannot refine the present high-bit coefficient packet on the single canonical cover.

---

# 8. First new high-order prime

A new exact search on the canonical elliptic curve finds

$$
\boxed{
p_1=11{,}532{,}251.
}
$$

A complete character-square point count gives

$$
\boxed{
\#E_0(\mathbb F_{p_1})
=
11{,}534{,}336.
}
$$

Factorization:

$$
\boxed{
11{,}534{,}336
=
11\cdot2^{20}.
}
$$

Thus the odd cofactor is

$$
\boxed{
m_1=11.
}
$$

---

# 9. Projected order at $p_1$

Define

$$
\boxed{
Q_1=[11]Q.
}
$$

Exact repeated doubling gives

$$
\boxed{
\operatorname{ord}(Q_1)=2^{19}.
}
$$

Thus this prime can distinguish coefficient classes modulo

$$
2^{19}.
$$

This is one bit beyond the R11 packet modulus.

---

# 10. Odd-primary kernel at $p_1$

The odd-primary subgroup has order

$$
11.
$$

An explicit order-$11$ point is

$$
\boxed{
R_{11}
=
(10164389,10798185)
\in
E_0(\mathbb F_{p_1}).
}
$$

It satisfies

$$
\boxed{
[11]R_{11}=O.
}
$$

For a projected state

$$
S,
$$

the eleven preimages under multiplication by

$$
11
$$

are

$$
\boxed{
P_0+jR_{11},
\qquad
0\le j<11,
}
$$

where $P_0$ is the unique preimage in the $2$-primary subgroup.

---

# 11. Modulo-$2^{19}$ lifts of the R11 packet

Each coefficient

$$
c_0\in\mathcal C_{18}
$$

has two lifts modulo

$$
2^{19}:
$$

$$
c_0,
$$

and

$$
c_0+2^{18}.
$$

Thus R15 tests sixteen coefficient classes:

$$
\boxed{
\mathcal C_{19}^{\rm lift}.
}
$$

For each class:

1. form its projected point:
   $$
   [c]Q_1;
   $$

2. invert multiplication by:
   $$
   11
   $$
   on the $2$-primary subgroup;

3. enumerate all eleven odd-primary preimages;

4. test:
   $$
   2-x
   $$
   for quadratic-residue status.

---

# 12. Result at $p_1$

Every one of the sixteen coefficient lifts has at least one square-value preimage.

The exact number of admissible odd-primary preimages ranges from

$$
\boxed{
2
}
$$

to

$$
\boxed{
7.
}
$$

Therefore

$$
\boxed{
p_1
\text{ eliminates no R11 coefficient class}.
}
$$

This is an exact finite-field result.

---

# 13. Second new high-order prime

Continuing the exact prime search gives

$$
\boxed{
p_2=15{,}721{,}883.
}
$$

A second independent complete point count gives

$$
\boxed{
\#E_0(\mathbb F_{p_2})
=
15{,}728{,}640.
}
$$

Factorization:

$$
\boxed{
15{,}728{,}640
=
15\cdot2^{20}.
}
$$

Thus

$$
\boxed{
m_2=15.
}
$$

---

# 14. Projected order at $p_2$

Define

$$
\boxed{
Q_2=[15]Q.
}
$$

Exact doubling gives

$$
\boxed{
\operatorname{ord}(Q_2)=2^{19}.
}
$$

Again the prime sees one coefficient bit beyond R11.

---

# 15. Odd-primary kernel at $p_2$

The odd-primary subgroup has order

$$
15.
$$

An explicit order-$15$ point is

$$
\boxed{
R_{15}
=
(4692854,429598)
\in
E_0(\mathbb F_{p_2}).
}
$$

It satisfies

$$
\boxed{
[15]R_{15}=O.
}
$$

Each projected state therefore has fifteen odd-primary preimages.

---

# 16. Result at $p_2$

The same sixteen modulo-$2^{19}$ coefficient lifts were tested.

Every one has at least one original preimage satisfying

$$
2-x
\in
\mathbb F_{p_2}^{\,2}.
$$

The number of square-gate-admissible odd-primary preimages ranges from

$$
\boxed{
5
}
$$

to

$$
\boxed{
12.
}
$$

Therefore

$$
\boxed{
p_2
\text{ also eliminates no R11 coefficient class}.
}
$$

---

# 17. Independent count verification

The group orders at both new primes were verified by two independent exact methods:

1. Hasse-window elliptic baby-step/giant-step point counting with a point-order certificate exceeding:
   $$
   4\sqrt p;
   $$

2. complete finite-field square-character point enumeration.

Both methods return:

$$
\boxed{
\#E_0(\mathbb F_{11,532,251})
=
11\cdot2^{20},
}
$$

and

$$
\boxed{
\#E_0(\mathbb F_{15,721,883})
=
15\cdot2^{20}.
}
$$

Thus the reported prime data do not depend on one counting algorithm.

---

# 18. Correct mixed-order sieve rule

The new primes have nontrivial odd cofactors.

Therefore the square-value gate must be tested **before** Sylow-$2$ projection.

The correct order is:

$$
\boxed{
\text{original preimage}
\longrightarrow
\text{square-value test}
\longrightarrow
\text{odd-cofactor projection}
\longrightarrow
\text{Mordell-Weil comparison}.
}
$$

One must not evaluate

$$
2-x
$$

only after projection.

Projection changes the elliptic point and hence changes $x$.

R15 follows the proof-safe ordering.

---

# 19. High-order single-cover saturation checkpoint

The two new primes are materially stronger than the historical clean primes at the level of projected coefficient resolution.

Both have:

$$
\boxed{
\operatorname{ord}([m]Q)=2^{19}.
}
$$

Yet neither eliminates even one of the sixteen lifted R11 classes.

Thus the absence of refinement is not merely caused by insufficient coefficient precision.

The single old cover

$$
r^2=2-x
$$

has abundant finite-field lifts above every current coefficient branch at these primes.

This is strong evidence that further progress should use additional global compatibility rather than the single cover alone.

This is a computational checkpoint, not a universal theorem for all primes.

---

# 20. Search interval checkpoint

The direct projected-order search was continued beyond the first new hit.

The scanned region extended from approximately

$$
8.26\times10^6
$$

through beyond

$$
2.1\times10^7.
$$

The two reported primes were the useful $2^{19}$ projected-order hits encountered in the completed portions of the scan.

No claim is made about prime density or about unscanned intervals.

The purpose of the search was to test whether materially stronger projected order immediately cuts the existing packet.

For the two exact hits tested, it does not.

---

# 21. Why elliptic Chabauty is now the correct escalation

The canonical genus-$2$ cover is a degree-$2$ cover of a rank-one elliptic curve.

The finite-prime single-cover sieve has already concentrated candidates into a small Mordell-Weil packet, but high projected-order primes still leave all current branches locally soluble.

This is precisely the setting where elliptic Chabauty or a full Mordell-Weil sieve on the genus-$2$ curve is more appropriate than continued isolated quadratic-residue testing.

The remaining question is not:

> Does a residue class contain a local square point?

It is:

> Does a globally rational genus-$2$ point exist in that Mordell-Weil disk?

That distinction is now decisive.

---

# 22. The complementary elliptic quotient

The canonical sextic

$$
\mathcal H_{\rm can}:
\qquad
\mathcal Y^2
=
(2-r^2)
\left(
M^2r^4
-
2M^2r^2
+
L^2
\right)
$$

is bielliptic.

The first quotient is the canonical

$$
E_0
$$

map:

$$
x=2-r^2.
$$

The second nonhyperelliptic involution is obtained by composing

$$
r\mapsto-r
$$

with the hyperelliptic involution.

Its elliptic quotient lies in the historical

$$
E_-
$$

isogeny class.

Therefore the next sieve can require simultaneous compatibility with two certified rank-one elliptic factors:

$$
\boxed{
E_0
\quad\text{and}\quad
E_-.
}
$$

This restores the information that is absent from the single-cover test.

---

# 23. Why the old clean primes remain relevant

The historical clean primes were powerful because they were not merely testing

$$
2-x
$$

for squarehood.

They tested compatibility of the same higher-genus point with two independent rank-one elliptic quotient images.

That dual condition forced boundary residue disks.

R15 confirms why transporting only the $E_0$ square gate loses part of their strength.

Thus the old clean-prime architecture should be reused at the **canonical genus-$2$ dual-quotient level**, not as a single-gate filter.

---

# 24. Canonical elliptic-Chabauty input packet

The exact data for the next stage are now:

## Elliptic base

$$
\boxed{
E_0:
\quad
y^2=x^3-2x^2+\lambda^2x.
}
$$

## Parameter

$$
\boxed{
\lambda=\frac{1352033}{950625}.
}
$$

## Odd-index surrogate

$$
\boxed{
Q=
\left(
1,
\frac{961408}{950625}
\right).
}
$$

## Square-value cover

$$
\boxed{
r^2=2-x.
}
$$

## Canonical genus-$2$ curve

$$
\boxed{
\mathcal Y^2
=
(2-r^2)
\left(
950625^2r^4
-
2\cdot950625^2r^2
+
1352033^2
\right).
}
$$

## Boundary points

$$
\boxed{
r=\pm1.
}
$$

## Coefficient packet

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

## First-cover sign orbits

$$
\boxed{
4
}
$$

coefficient classes up to elliptic negation.

---

# 25. Proof-safe interpretation of the coefficient packet

The residue packet is $2$-adic.

It must not be interpreted as a complete integer coefficient modulo arbitrary odd factors.

At mixed-order primes, one must retain all odd-primary preimages consistent with the projected coefficient.

This is why the two new primes were tested by full odd-kernel enumeration.

Any future elliptic-Chabauty or Mordell-Weil implementation must preserve this distinction.

---

# 26. R15 theorem package

## Theorem A — Canonical transport

Every R11 survivor is represented on the historical cover by

$$
\boxed{
[c]Q
}
$$

with no torsion ambiguity.

## Theorem B — Historical clean-prime resolution bound

For

$$
11,59,181,197,1669,
$$

the projected $Q$ order is at most

$$
2^3.
$$

Therefore these primes cannot distinguish the high-bit R11 packet on the single canonical cover.

## Theorem C — New prime $p_1$

$$
\boxed{
\#E_0(\mathbb F_{11,532,251})
=
11\cdot2^{20},
}
$$

and

$$
\boxed{
\operatorname{ord}([11]Q)=2^{19}.
}
$$

All sixteen R11 lifts survive the single square gate.

## Theorem D — New prime $p_2$

$$
\boxed{
\#E_0(\mathbb F_{15,721,883})
=
15\cdot2^{20},
}
$$

and

$$
\boxed{
\operatorname{ord}([15]Q)=2^{19}.
}
$$

Again all sixteen lifts survive.

## Theorem E — Single-cover sieve checkpoint

The tested canonical finite-prime square-value sieve does not refine the R11 packet even after increasing projected coefficient resolution to

$$
2^{19}.
$$

---

# 27. What R15 does not prove

R15 does not prove that no future prime can refine the packet.

It does not prove that the nonboundary coefficient branches contain rational genus-$2$ points.

It does not classify

$$
\mathcal H_{\rm can}(\mathbb Q).
$$

It does not replace the higher Perfect-Cuboid lift.

It establishes that the single square-value cover, tested at both historical and newly found high-order primes, has not supplied further elimination.

This justifies escalation to a stronger global method.

---

# 28. Proposed R16 program

R16 should use the full bielliptic structure of the canonical genus-$2$ cover.

## R16-A — Explicit second quotient in canonical coordinates

Write the second elliptic quotient directly from

$$
\mathcal H_{\rm can}.
$$

Produce a compact Weierstrass model and an explicit rational map.

Identify it with the historical

$$
E_-
$$

isogeny class.

## R16-B — Odd-index surrogate on the second quotient

Transport the historical certified non-torsion point on

$$
E_-
$$

to the canonical second quotient.

Record a proof-safe odd-index surrogate.

## R16-C — Dual-quotient sieve using the R11 packet

For each transported coefficient disk on

$$
E_0,
$$

enumerate the possible genus-$2$ lifts modulo good primes and require simultaneous compatibility with the rank-one subgroup of the second quotient.

This should recover and strengthen the historical clean-prime method in the new canonical coordinates.

## R16-D — Elliptic-Chabauty implementation

If the dual finite-prime sieve stops reducing, run elliptic Chabauty on

$$
\mathcal H_{\rm can}
\to
E_0.
$$

Use the R11 residue packet to restrict the Mordell-Weil disks from the beginning.

## R16-E — Exact completion target

Classify all rational points on the canonical genus-$2$ cover in the allowed Mordell-Weil disks.

Then test the remaining higher Perfect-Cuboid lift.

---

# 29. Final conclusion

R15 completes the practical preparation of the canonical elliptic-Chabauty problem.

All current Radical-FDRS residue information has been transported to the single historical orbit

$$
[c]Q.
$$

The old five clean primes are too low-resolution to refine the high-bit packet on the single cover.

Two newly discovered and independently certified primes with projected order

$$
2^{19}
$$

also fail to eliminate any current coefficient branch once their odd-primary preimages are handled correctly.

Therefore the limiting issue is no longer coefficient resolution alone.

The single square-value cover has too much local finite-field flexibility.

The next useful information must come from the **second elliptic quotient or from elliptic Chabauty itself**.

The canonical genus-$2$ curve, boundary points, rank-one surrogate, and residue packet are now fixed and proof-safe.

This is the canonical starting point for R16.