# Perfect Cuboid Radical-FDRS R14 ## Canonical Elliptic-Chabauty Cover, Exact Residue Transport, and Elimination of the Torsion Ambiguity **Series:** Perfect Cuboid Global Closure **Branch:** Radical-FDRS / Geometry-Coupling Branch R **Phase:** R14 — Canonical Cover Selection and Mordell-Weil Transport **Date:** 2026-10-05 **Status:** Exact structural and arithmetic transport theorem — not a proof of Perfect Cuboid existence or nonexistence --- # 0. Executive summary R13 proved that the R12 quotient gate and the historical G2E.5 square-value cover are two torsion/Kummer charts of the same obstruction after pullback through the historical $2$-isogeny. R14 chooses one canonical chart and transports the strongest R9–R12 Mordell-Weil information into it. The canonical elliptic curve is the historical normalized rank-one curve $$ \boxed{ E_0: \qquad y^2 = x^3 - 2x^2 + \lambda^2x, } $$ with $$ \boxed{ \lambda = \frac{1352033}{950625}. } $$ The canonical square-value cover is $$ \boxed{ r^2=2-x. } $$ The full-$2$-torsion isogenous source is $$ \boxed{ E_0': \qquad Y^2 = X(X-448^2)(X+1073^2). } $$ Let $$ \boxed{ G_0=(480704,-468686400)\in E_0'(\mathbb Q). } $$ Under the normalized degree-$2$ isogeny $$ \phi:E_0'\to E_0, $$ one obtains the exact point $$ \boxed{ Q = \phi(G_0) = \left( 1, \frac{961408}{950625} \right). } $$ Thus the historical boundary anchor $$ x=1 $$ is exactly the image of the known odd-index Mordell-Weil surrogate. The two R9–R11 surviving rational $2$-torsion cosets on $E_0'$ were $$ T_A=(448^2,0), $$ and $$ T_{-B}=(-1073^2,0). $$ Both map to the same nontrivial rational $2$-torsion point $$ \boxed{ T=(0,0)\in E_0(\mathbb Q). } $$ Therefore every R11 surviving point $$ [c]G_0+T_A $$ or $$ [c]G_0+T_{-B} $$ has the same image $$ [c]Q+T. $$ Applying the R13 torsion translation $$ \tau_T $$ removes this final torsion term and gives $$ \boxed{ [c]Q. } $$ Hence the two surviving $E_0'$ torsion cosets collapse to **one torsion-free Mordell-Weil orbit** on the canonical historical cover. Moreover, $Q$ itself is an odd-index surrogate for the primitive free generator of $E_0(\mathbb Q)$. This is certified without assuming saturation. Modulo $17$, $$ E_0: \qquad y^2=x^3-2x^2+9x. $$ The reduction has $$ \boxed{ \#E_0(\mathbb F_{17})=24. } $$ The image of doubling is $$ \boxed{ 2E_0(\mathbb F_{17}) = \{ O, (0,0), (2,\pm1), (13,\pm2) \}. } $$ But $$ \boxed{ Q\bmod17=(1,12), } $$ and $$ \boxed{ (Q+T)\bmod17=(9,11) } $$ are both outside this set. Therefore the free coefficient of $Q$ is odd. Consequently the same $2$-adic normalized Mordell-Weil coefficient used in R9–R12 transports directly to the canonical historical orbit $$ [c]Q. $$ The R11 necessary condition becomes $$ \boxed{ c\bmod2^{18} \in \mathcal C_{18}, } $$ where $$ \boxed{ \mathcal C_{18} = \pm \{ 1, 16385, 114687, 131071 \}. } $$ No torsion ambiguity remains. Since the old square-value function depends only on $x$, elliptic negation identifies the sign-paired classes at the first cover level. Thus the canonical square-value stage has only four coefficient sign-orbits modulo $2^{18}$. Finally, the historical genus-$2$ cover admits a much shorter canonical sextic model than the older large-coefficient form. With $$ M=950625, \qquad L=1352033, $$ set $$ \boxed{ \mathcal H_{\rm can}: \qquad \mathcal Y^2 = (2-r^2) \left( M^2r^4 - 2M^2r^2 + L^2 \right). } $$ The map to $E_0$ is simply $$ \boxed{ x=2-r^2, \qquad y=\frac{\mathcal Y}{M}. } $$ The historical G2E.5 even sextic is obtained from this model by $$ \boxed{ r=\frac{L}{M}s. } $$ Therefore R14 formally selects one canonical cover, one canonical rank-one orbit, and one transported residue packet for all future elliptic-Chabauty work. --- # 1. Source elliptic curve Use $$ \boxed{ E_0': \qquad Y^2 = X(X-A)(X+B), } $$ where $$ \boxed{ A=448^2=200704, } $$ and $$ \boxed{ B=1073^2=1151329. } $$ Define $$ \boxed{ M=B-A=950625=975^2, } $$ and $$ \boxed{ L=A+B=1352033. } $$ Then $$ E_0': \qquad Y^2 = X^3 + MX^2 - ABX. $$ --- # 2. Normalized target elliptic curve The standard $2$-isogenous target is $$ \widehat E: \qquad V^2 = U^3 - 2MU^2 + L^2U. $$ Normalize by $$ \boxed{ U=Mx, } $$ and $$ \boxed{ V=975^3y. } $$ Because $$ M=975^2, $$ the target becomes $$ \boxed{ E_0: \qquad y^2 = x^3 - 2x^2 + \lambda^2x, } $$ with $$ \boxed{ \lambda = \frac{L}{M} = \frac{1352033}{950625}. } $$ --- # 3. Explicit normalized isogeny For a finite point $$ P=(X,Y)\in E_0', $$ with $$ X\ne0, $$ define $$ \boxed{ U = \frac{Y^2}{X^2}. } $$ The corresponding unnormalized $y$-coordinate is $$ \boxed{ V = - \frac{ Y(X^2+AB) }{ X^2 }. } $$ Therefore the normalized map is $$ \boxed{ x = \frac{ Y^2 }{ MX^2 }, } $$ and $$ \boxed{ y = - \frac{ Y(X^2+AB) }{ 975^3X^2 }. } $$ The kernel is $$ \boxed{ \{O,(0,0)\}. } $$ --- # 4. Image of the known Mordell-Weil surrogate Use $$ \boxed{ G_0 = (480704,-468686400). } $$ Since $$ \frac{ 468686400^2 }{ 480704^2 } = 950625, $$ we obtain $$ \boxed{ x(\phi(G_0))=1. } $$ The normalized $y$-coordinate is $$ \boxed{ \frac{961408}{950625}. } $$ Therefore $$ \boxed{ Q = \phi(G_0) = \left( 1, \frac{961408}{950625} \right). } $$ This is precisely the historical boundary anchor. --- # 5. Boundary-anchor verification At $$ x=1, $$ the elliptic equation gives $$ y^2 = \lambda^2-1. $$ Now $$ \boxed{ L^2-M^2 = 961408^2. } $$ Hence $$ \lambda^2-1 = \left( \frac{961408}{950625} \right)^2. $$ So the point $$ Q $$ lies on $$ E_0(\mathbb Q) $$ exactly. The canonical old square function satisfies $$ \boxed{ 2-x(Q)=1. } $$ Thus $Q$ is a rational point on the canonical genus-$2$ cover. --- # 6. Rational $2$-torsion on the target The normalized target curve has one nontrivial rational $2$-torsion point: $$ \boxed{ T=(0,0). } $$ Translation by this point is $$ \boxed{ \tau_T(x,y) = \left( \frac{\lambda^2}{x}, - \frac{\lambda^2y}{x^2} \right). } $$ In particular, $$ \boxed{ x(Q+T)=\lambda^2. } $$ This is the R13 torsion-dual boundary chart. --- # 7. Images of the surviving source torsion points On $$ E_0', $$ the three nonzero rational $2$-torsion points are $$ T_0=(0,0), $$ $$ T_A=(A,0), $$ and $$ T_{-B}=(-B,0). $$ The isogeny kernel is $$ \boxed{ \{O,T_0\}. } $$ Therefore $$ T_A $$ and $$ T_{-B} $$ have the same image. Directly from the isogeny formulas, $$ \boxed{ \phi(T_A)=T, } $$ and $$ \boxed{ \phi(T_{-B})=T. } $$ This is also forced by $$ T_A+T_{-B}=T_0. $$ --- # 8. Transport of the R11 surviving cosets R11 proved that every hypothetical full fixed-fiber rational point must lie in one of the two effective source cosets $$ [c]G_0+T_A, $$ or $$ [c]G_0+T_{-B}, $$ with $$ c\bmod2^{18} \in \mathcal C_{18}. $$ Apply the isogeny. Both cases give $$ \boxed{ [c]Q+T. } $$ Thus the two source torsion cosets become one target torsion coset. Now apply the R13 torsion translation: $$ \boxed{ \tau_T([c]Q+T)=[c]Q. } $$ Therefore all surviving arithmetic data can be transported to the single orbit $$ \boxed{ \{[c]Q\}. } $$ --- # 9. Exact torsion-ambiguity elimination theorem ## Theorem The two R11 source torsion cosets $$ T_A, \qquad T_{-B} $$ produce the same canonical point on the historical square-value chart after: 1. the degree-$2$ isogeny; 2. translation by the nontrivial target $2$-torsion point. Explicitly: $$ \boxed{ \tau_T \phi \left( [c]G_0+T_A \right) = [c]Q, } $$ and $$ \boxed{ \tau_T \phi \left( [c]G_0+T_{-B} \right) = [c]Q. } $$ Therefore no torsion label survives in the canonical historical cover. --- # 10. The odd-index question on $E_0$ To use $$ Q $$ as a proof-safe $2$-primary Mordell-Weil surrogate, we must verify that its free-lattice coefficient is odd. This must be proved independently of any unverified saturation assumption. R14 uses reduction modulo $$ 17. $$ --- # 11. Reduction modulo $17$ Since $$ M\equiv2\pmod{17}, $$ and $$ L\equiv6\pmod{17}, $$ we have $$ \lambda \equiv3\pmod{17}. $$ Hence $$ \lambda^2 \equiv9\pmod{17}. $$ The reduced curve is $$ \boxed{ E_0/\mathbb F_{17}: \qquad y^2 = x^3 - 2x^2 + 9x. } $$ Direct exact point counting gives $$ \boxed{ \#E_0(\mathbb F_{17})=24. } $$ --- # 12. Image of doubling modulo $17$ Exact enumeration gives $$ \boxed{ 2E_0(\mathbb F_{17}) = \{ O, (0,0), (2,1), (2,16), (13,2), (13,15) \}. } $$ Meanwhile $$ \boxed{ Q\bmod17=(1,12). } $$ Also $$ \boxed{ (Q+T)\bmod17=(9,11). } $$ Neither point lies in $$ 2E_0(\mathbb F_{17}). $$ --- # 13. Odd-index theorem for $Q$ Write $$ E_0(\mathbb Q) \cong \mathbb Z H \oplus \langle T\rangle, $$ where $$ H $$ is primitive in the free part. Write $$ Q=kH+\epsilon T. $$ Suppose $$ k $$ were even. If $$ \epsilon=0, $$ then $$ Q\in2E_0(\mathbb Q). $$ If $$ \epsilon=1, $$ then $$ Q+T\in2E_0(\mathbb Q). $$ Either conclusion would remain true after good reduction modulo $$ 17. $$ But neither reduced point is in $$ 2E_0(\mathbb F_{17}). $$ Contradiction. Therefore $$ \boxed{ k\text{ is odd}. } $$ Thus $Q$ has odd index in the primitive free subgroup modulo torsion. --- # 14. Consequence for the normalized coefficient R9 defined a normalized $2$-adic coefficient $$ c $$ using the fact that $$ G_0 $$ has odd free index. R14 proves that $$ Q $$ also has odd free index. Therefore the same normalized coefficient transports through the isogeny without a hidden factor of two. The R11 congruence data may be used directly on the orbit $$ [c]Q. $$ This is a proof-safe transport. --- # 15. Transported R11 coefficient packet The R11 coefficient set is $$ \boxed{ \mathcal C_{18} = \{ 1, 16385, 114687, 131071, 131073, 147457, 245759, 262143 \}. } $$ Equivalently, $$ \boxed{ \mathcal C_{18} = \pm \{ 1, 16385, 114687, 131071 \} \pmod{2^{18}}. } $$ R14 transports this unchanged to the canonical historical orbit: $$ \boxed{ P_c=[c]Q. } $$ Thus every hypothetical nonboundary fixed-slice solution must arise above a coefficient satisfying $$ \boxed{ c\bmod2^{18} \in \mathcal C_{18}. } $$ --- # 16. Negation symmetry The canonical old square function is $$ f_{\rm old}=2-x. $$ Elliptic negation sends $$ (x,y)\mapsto(x,-y). $$ Therefore $$ \boxed{ f_{\rm old}(-P)=f_{\rm old}(P). } $$ At the first genus-$2$ square-value level, the coefficient classes $$ c $$ and $$ -c $$ therefore have the same $x$-coordinate and the same square condition. Hence the eight R11 coefficient residues form four sign-orbits: $$ \boxed{ \{1,-1\}, } $$ $$ \boxed{ \{16385,-16385\}, } $$ $$ \boxed{ \{114687,-114687\}, } $$ and $$ \boxed{ \{131071,-131071\} } $$ modulo $$ 2^{18}. $$ This reduces the first-cover bookkeeping to four coefficient classes up to elliptic sign. Higher lifts may still distinguish the sign of $y$, so the sign quotient must not be applied blindly beyond the square-value stage. --- # 17. Canonical historical square-value cover The historical cover is $$ \boxed{ r^2=2-x. } $$ Eliminate $$ x $$ using $$ \boxed{ x=2-r^2. } $$ Substitute into $$ E_0: \qquad y^2 = x \left( x^2-2x+\lambda^2 \right). $$ The second factor becomes $$ \boxed{ x^2-2x+\lambda^2 = r^4-2r^2+\lambda^2. } $$ Therefore $$ \boxed{ y^2 = (2-r^2) \left( r^4-2r^2+\lambda^2 \right). } $$ --- # 18. Integral-scaled canonical sextic Multiply the square root by $$ M. $$ Define $$ \boxed{ \mathcal Y=My. } $$ Then $$ \boxed{ \mathcal H_{\rm can}: \qquad \mathcal Y^2 = (2-r^2) \left( M^2r^4 - 2M^2r^2 + L^2 \right). } $$ This is an even sextic genus-$2$ model. It is the preferred canonical model for future elliptic-Chabauty work. --- # 19. Direct quotient map The map from the canonical sextic to $$ E_0 $$ is simply $$ \boxed{ x=2-r^2, } $$ and $$ \boxed{ y=\frac{\mathcal Y}{M}. } $$ The involution $$ r\mapsto-r $$ is the degree-$2$ quotient map to the elliptic base. The old square-value coordinate is therefore visible directly. --- # 20. Boundary anchor in canonical coordinates At $$ Q, $$ we have $$ x=1. $$ Therefore $$ r^2=1. $$ The two boundary lifts are $$ \boxed{ r=1, } $$ and $$ \boxed{ r=-1. } $$ Thus the historical boundary anchor becomes the simplest possible pair of rational points on the canonical sextic. This is another reason to prefer the old chart. --- # 21. Relation to the historical G2E.5 even model The historical even genus-$2$ model was $$ \boxed{ \bar Y^2 = - \left( L^2s^2 - 2M^2 \right) } $$ $$ \boxed{ \qquad\cdot \left( L^2s^4 - 2M^2s^2 + M^2 \right). } $$ Set $$ \boxed{ r=\frac{L}{M}s. } $$ Then $$ 2-r^2 = - \frac{ L^2s^2-2M^2 }{ M^2 }. $$ Also $$ M^2r^4-2M^2r^2+L^2 = \frac{ L^2 }{ M^2 } \left( L^2s^4 - 2M^2s^2 + M^2 \right). $$ Therefore $$ \boxed{ \mathcal Y = \frac{L}{M^2}\bar Y } $$ maps the historical model to the R14 canonical sextic. Thus the two genus-$2$ equations are exactly the same curve. --- # 22. Why the R14 model is preferable The canonical model $$ \mathcal Y^2 = (2-r^2) \left( M^2r^4-2M^2r^2+L^2 \right) $$ has several advantages. 1. The elliptic quotient map is: $$ x=2-r^2. $$ 2. The boundary anchor is: $$ r=\pm1. $$ 3. The square-value condition is built into the coordinate itself. 4. The connection to R13 is immediate. 5. Coefficient growth is lower than in the historical Möbius-conjugated sextic. 6. The model is even, exposing the bielliptic involution directly. Therefore this model should be canonical for future implementation. --- # 23. Old versus new torsion chart The R13 new chart uses $$ s^2=2x-\lambda^2. $$ The old chart uses $$ r^2=2-x. $$ They are equivalent after the isogeny/Kummer pullback. However, after R14 transport the old chart has the cleaner boundary structure: $$ Q \longleftrightarrow r=\pm1. $$ The new chart places the same boundary state at the torsion-translated point $$ Q+T, $$ with $$ x=\lambda^2. $$ Thus the old chart is the preferred canonical chart for arithmetic enumeration. The new chart remains useful as a dual local coordinate. --- # 24. Image of the R11 boundary centers The R9/R11 source centers were $$ \pm G_0+T_A $$ and $$ \pm G_0+T_{-B}. $$ After the isogeny and target torsion translation they become $$ \boxed{ \pm Q. } $$ Thus the four source boundary centers collapse to the two elliptic signs of one target boundary anchor. After quotienting by the square-value coordinate $$ r^2=2-x, $$ these become the two points $$ \boxed{ r=\pm1. } $$ This is the full boundary compression: $$ 4 \longrightarrow 2 \longrightarrow 1 $$ at the levels: - source elliptic points; - genus-$2$ lifts; - elliptic $x$-coordinate. --- # 25. State-count compression At the end of R11: - eight coefficient residues remained; - two source torsion cosets remained. This gave: $$ 16 $$ finite coefficient/torsion states at precision $$ 2^{18}. $$ R12 identified the two source torsion cosets as one gate-symmetry orbit. R14 removes the ambiguity completely on the canonical historical target. Thus the state packet is now simply $$ \boxed{ 8 \text{ coefficient residues}. } $$ At the first square-value stage, elliptic negation pairs them, leaving $$ \boxed{ 4 \text{ coefficient sign-orbits}. } $$ This is the smallest proof-safe finite state description obtained so far. --- # 26. Why odd-order clean primes require care The coefficient $$ c $$ is fundamentally a normalized $2$-adic Mordell-Weil coefficient. A residue such as $$ c\equiv c_0\pmod{2^{18}} $$ does not determine $$ c $$ modulo an odd factor of a finite reduction order. Therefore one must not take an integer representative $$ c_0 $$ and reduce it modulo an arbitrary mixed-order Mordell-Weil order. Doing so would implicitly assume knowledge of the unknown odd saturation index. This is invalid. R14 retains the proof-safe rule: $$ \boxed{ \text{use Sylow-}2\text{ projection at mixed-order primes}. } $$ --- # 27. Correct clean-prime workflow on $E_0$ For a good prime $$ p, $$ write $$ \#E_0(\mathbb F_p)=2^sm, $$ with $$ m $$ odd. The correct workflow is: 1. test the square-value cover on the original reduced candidate; 2. project the candidate by: $$ [m]; $$ 3. project: $$ Q $$ and target torsion by the same map; 4. compare the projected candidate with the projected Mordell-Weil image. This is the same proof-safe rule already established in G2E.5 and R11-R12. --- # 28. Odd-index surrogate on the canonical curve Because the free coefficient of $$ Q $$ is odd, multiplication by the unknown global index is invertible on every Sylow-$2$ reduction subgroup. Therefore $$ Q $$ may replace a primitive generator in every $2$-primary Mordell-Weil sieve. This is the exact target-curve analogue of the old statement for $$ G_0. $$ No new saturation computation is required before continuing the sieve. --- # 29. Transport theorem for R11 residues ## Theorem Let $$ c\in\mathbb Z_2 $$ be the normalized coefficient associated with a hypothetical fixed-fiber rational point. If the point survives the R11 double-gate sieve, then after the R13/R14 canonical transport its historical elliptic image is $$ \boxed{ [c]Q } $$ with $$ \boxed{ c\bmod2^{18} \in \mathcal C_{18}. } $$ There is no residual rational $2$-torsion ambiguity. This is the canonical arithmetic input for future elliptic-Chabauty computation. --- # 30. Canonical implementation data Future local or external computation only needs the following package. ## Elliptic base $$ \boxed{ E_0: \quad y^2 = x^3-2x^2+\lambda^2x. } $$ ## Parameter $$ \boxed{ \lambda=\frac{1352033}{950625}. } $$ ## Odd-index Mordell-Weil surrogate $$ \boxed{ Q= \left( 1, \frac{961408}{950625} \right). } $$ ## Torsion $$ \boxed{ T=(0,0). } $$ ## Canonical square function $$ \boxed{ f=2-x. } $$ ## Coefficient packet $$ \boxed{ c\bmod2^{18} \in \pm \{ 1, 16385, 114687, 131071 \}. } $$ ## Canonical genus-$2$ model $$ \boxed{ \mathcal Y^2 = (2-r^2) \left( 950625^2r^4 - 2\cdot950625^2r^2 + 1352033^2 \right). } $$ --- # 31. What R14 does not prove R14 does not classify all rational points on the canonical genus-$2$ cover. It does not prove that the four nonboundary coefficient sign-orbits are empty. It does not prove that $$ Q $$ is a primitive generator. It proves only that its free index is odd, which is sufficient for all $2$-primary sieves. It also does not replace the remaining higher-lift compatibility required for a full Perfect Cuboid point. The elliptic-Chabauty cover remains a necessary intermediate layer. --- # 32. Strategic consequence The research architecture should no longer maintain separate arithmetic implementations for: - the R12 new quotient gate; - the historical G2E.5 old gate. R13 proves they are torsion/Kummer dual. R14 chooses the old chart canonically and transports all current residue information into it. Therefore the next exact arithmetic stage can focus on one curve: $$ \boxed{ \mathcal H_{\rm can}. } $$ This avoids duplicate descent, duplicate local sieves, and duplicate boundary bookkeeping. --- # 33. Proposed R15 program R15 should now perform arithmetic directly on the canonical genus-$2$ cover. ## R15-A — Finite-prime Mordell-Weil sieve on the canonical cover For each good prime: 1. enumerate reduced points on: $$ E_0; $$ 2. retain those with: $$ 2-x $$ square; 3. apply proof-safe Sylow-$2$ projection; 4. compare with the subgroup generated by: $$ Q $$ and torsion; 5. intersect with the transported: $$ \mathcal C_{18} $$ coefficient packet. ## R15-B — Use the historical clean primes first Re-evaluate: $$ 11, 59, 181, 197, 1669 $$ directly in the canonical cover model. Because the coordinate and boundary anchor are now simplified, record the actual surviving coefficient disks rather than only the older $h$-residue statement. ## R15-C — Elliptic-Chabauty handoff Prepare an exact Sage/Magma-compatible specification of: - $E_0$; - $Q$; - the degree-$2$ cover: $$ r^2=2-x; $$ - the known residue packet; - boundary points: $$ r=\pm1. $$ The target is a direct classification of rational lifts. ## R15-D — Higher-lift check Any nonboundary rational point on the canonical genus-$2$ cover must still pass the remaining Perfect-Cuboid lift. Transport it back through: $$ E_0' $$ and apply the second R8 gate. This preserves proof safety. ## R15-E — Fixed-slice theorem target The preferred theorem remains: > all rational points on the complete fixed-slice cover compatible with the Mordell-Weil residue packet are boundary. If proved, then $$ \boxed{ n=\frac7{32} } $$ contains no nondegenerate Perfect Cuboid point. --- # 34. Final conclusion R14 completes the practical merge initiated by R13. The known source Mordell-Weil surrogate $$ G_0 $$ maps to the historical boundary point $$ \boxed{ Q= \left( 1, \frac{961408}{950625} \right) } $$ on $$ E_0. $$ The two surviving source rational $2$-torsion cosets both map to the same target torsion point and are removed by the R13 torsion translation. Therefore all surviving arithmetic data are represented by the single orbit $$ \boxed{ [c]Q. } $$ A mod-$17$ doubling certificate proves that $Q$ has odd free-lattice index, so it is a proof-safe surrogate for all future $2$-primary sieves. The R11 coefficient packet transports unchanged: $$ \boxed{ c\bmod2^{18} \in \pm \{ 1, 16385, 114687, 131071 \}. } $$ The historical genus-$2$ cover is best represented by the short canonical sextic $$ \boxed{ \mathcal Y^2 = (2-r^2) \left( M^2r^4-2M^2r^2+L^2 \right). } $$ Its boundary anchor is simply $$ \boxed{ r=\pm1. } $$ Thus the full R0–R13 structural program has now returned to one canonical rank-one elliptic-Chabauty problem with a sharply restricted Mordell-Weil residue packet. This is the canonical starting point for R15.