# Perfect Cuboid Radical-FDRS R19 ## Custom Mazur–Tate $11$-Adic Heights and Calibration of the Bielliptic Quadratic-Chabauty Function **Series:** Perfect Cuboid Global Closure **Branch:** Radical-FDRS / Geometry-Coupling Branch R **Phase:** R19 — Global $11$-Adic Height Layer **Date:** 2026-10-05 **Status:** Custom sigma/height computation + exact bielliptic-QC calibration; finite bad-prime value set and certified root isolation remain for the next phase --- # 0. Executive summary R18 constructed the $11$-adic Mordell-Weil logarithmic backend directly, without Sage or PARI. The remaining quadratic-Chabauty ingredient was the global cyclotomic $11$-adic height. R19 implements this layer using the Mazur–Tate $p$-adic sigma function. For an ordinary elliptic curve over $\mathbb Q$ and a rational point $P$ lying in the formal group at $p$ and in the connected components at all bad primes, the Mazur–Stein–Tate formula is $$ \boxed{ h_p(P) = \frac1p \log_p \left( \frac{\sigma_p(P)}{d(P)} \right). } $$ The height on an arbitrary Mordell-Weil point is then obtained by the quadratic law $$ \boxed{ h_p([n]P)=n^2h_p(P). } $$ R19 reconstructs the canonical $11$-adic sigma function from its defining differential equation rather than importing a package result. For a Weierstrass model with invariant differential $\omega$, the sigma function is the unique odd series $$ \boxed{ \sigma(t) = t+\cdots \in t\mathbb Z_{11}[[t]] } $$ for which there exists a canonical constant $$ c_{11}\in\mathbb Z_{11} $$ satisfying $$ \boxed{ x(t)+c_{11} = - \frac{d}{\omega} \left[ \frac1{\sigma} \frac{d\sigma}{\omega} \right]. } $$ The integrality of the sigma coefficients determines $c_{11}$ digit-by-digit. This procedure was carried out independently for the two historical rank-one elliptic factors. --- ## Main result on $E_0$ Use the global integral model $$ \boxed{ E_{0,\min}: \qquad Y^2 = X^3 - 2MX^2 + L^2X, } $$ where $$ \boxed{ M=950625, \qquad L=1352033. } $$ The canonical point is $$ \boxed{ Q_{\min} = (M,937372800). } $$ The sigma constant is $$ \boxed{ c_{11,E_0} \equiv 4997 \pmod{11^4}. } $$ Equivalently, $$ \boxed{ c_{11,E_0} = 3 + 3\cdot11 + 8\cdot11^2 + 3\cdot11^3 + O(11^4). } $$ The corresponding Katz/Mazur–Tate weight-$2$ value is $$ \boxed{ E_2(E_0,\omega) = -12c_{11,E_0}+4a_2, } $$ and satisfies $$ \boxed{ E_2(E_0,\omega) \equiv 6920 \pmod{11^4}. } $$ The point $Q$ has order $8$ modulo $11$. The multiple $$ [8]Q_{\min} $$ lies in the formal group and in the connected components required by the global height formula. Its denominator $d(8Q)$ satisfies $$ \boxed{ v_{11}(d(8Q))=1. } $$ The sigma quotient gives $$ \boxed{ \log_{11} \left( \frac{\sigma_{11}(8Q)}{d(8Q)} \right) = 10\cdot11 + 6\cdot11^2 + 3\cdot11^3 + 8\cdot11^4 + 4\cdot11^5 + O(11^6). } $$ Therefore $$ \boxed{ h_{11}(Q) = \frac1{64} \frac1{11} \log_{11} \left( \frac{\sigma_{11}(8Q)}{d(8Q)} \right), } $$ and the resulting global cyclotomic height is $$ \boxed{ h_{11}(Q) \equiv 58966 \pmod{11^5}. } $$ In base $11$, $$ \boxed{ h_{11}(Q) = 6 + 3\cdot11 + 3\cdot11^2 + 0\cdot11^3 + 4\cdot11^4 + O(11^5). } $$ --- ## Main result on $E_-$ Use the historical integral model $$ \boxed{ E_-: \qquad y^2 = (x-e_1)(x-e_2)(x-e_3), } $$ where $$ e_1=-8195043371142, $$ $$ e_2=8256895726659, $$ $$ e_3=-61852355517. $$ Expanded, $$ \boxed{ y^2 = x^3 + a_4x + a_6, } $$ with $$ \boxed{ a_4 = -67669444304850546481011867, } $$ and $$ \boxed{ a_6 = -4185277897366249205812702665319946826. } $$ Use the historical non-torsion odd-index surrogate $$ \boxed{ P_- = \left( \frac{5132247118878268203}{606841}, \frac{2520702383000286338446320000}{472729139} \right). } $$ The sigma constant is $$ \boxed{ c_{11,E_-} \equiv 967 \pmod{11^4}. } $$ Equivalently, $$ \boxed{ c_{11,E_-} = 10 + 10\cdot11 + 7\cdot11^2 + 0\cdot11^3 + O(11^4). } $$ Since $a_2=0$, $$ \boxed{ E_2(E_-,\omega) = -12c_{11,E_-}, } $$ and $$ \boxed{ E_2(E_-,\omega) \equiv 3037 \pmod{11^4}. } $$ A multiple by $16$ lies in the formal group at $11$ and kills the component classes required at all bad primes. The sigma quotient yields $$ \boxed{ \log_{11} \left( \frac{\sigma_{11}(16P_-)}{d(16P_-)} \right) = 8\cdot11 + 1\cdot11^2 + 8\cdot11^3 + 0\cdot11^4 + 9\cdot11^5 + O(11^6). } $$ Hence $$ \boxed{ h_{11}(P_-) = \frac1{256} \frac1{11} \log_{11} \left( \frac{\sigma_{11}(16P_-)}{d(16P_-)} \right), } $$ and $$ \boxed{ h_{11}(P_-) \equiv 3035 \pmod{11^5}. } $$ In base $11$, $$ \boxed{ h_{11}(P_-) = 10 + 0\cdot11 + 3\cdot11^2 + 2\cdot11^3 + 0\cdot11^4 + O(11^5). } $$ --- The R18 logarithms were $$ \boxed{ \log_{11,E_0}(Q) = 11U_0 + O(11^{11}), } $$ with $$ \boxed{ U_0 \equiv 9284831103 \pmod{11^{10}}, } $$ and $$ \boxed{ \log_{11,E_-}(P_-) = 11U_- + O(11^{11}), } $$ with $$ \boxed{ U_- \equiv 1708957869 \pmod{11^{10}}. } $$ Define the historical rank-one quadratic constants $$ \boxed{ \alpha_{0} = \frac{h_{11}(Q)} {\log_{11,E_0}(Q)^2}, } $$ and $$ \boxed{ \alpha_{-} = \frac{h_{11}(P_-)} {\log_{11,E_-}(P_-)^2}. } $$ Both have $11$-adic valuation $-2$. Writing $$ \boxed{ \beta_0=11^2\alpha_0, \qquad \beta_-=11^2\alpha_-, } $$ gives $$ \boxed{ \beta_0 \equiv 151114 \pmod{11^5}, } $$ and $$ \boxed{ \beta_- \equiv 48076 \pmod{11^5}. } $$ Their base-$11$ expansions are $$ \boxed{ \beta_0 = 7 + 9\cdot11 + 5\cdot11^2 + 3\cdot11^3 + 10\cdot11^4 + O(11^5), } $$ and $$ \boxed{ \beta_- = 6 + 3\cdot11 + 1\cdot11^2 + 3\cdot11^3 + 3\cdot11^4 + O(11^5). } $$ The global rank-one height layer is therefore now explicit. The second half of R19 matches our canonical even sextic to the standard explicit bielliptic quadratic-Chabauty formalism. Write $$ \boxed{ \mathcal H: \qquad \mathcal Y^2 = a_6r^6 + a_4r^4 + a_2r^2 + a_0, } $$ where $$ \boxed{ a_6=-M^2, } $$ $$ \boxed{ a_4=4M^2, } $$ $$ \boxed{ a_2=-(4M^2+L^2), } $$ and $$ \boxed{ a_0=2L^2. } $$ For a general even sextic of this form, the standard bielliptic quotients are $$ \boxed{ E_1: \quad y_1^2 = x_1^3 + a_4x_1^2 + a_2a_6x_1 + a_0a_6^2, } $$ and $$ \boxed{ E_2: \quad y_2^2 = x_2^3 + a_2x_2^2 + a_4a_0x_2 + a_6a_0^2. } $$ The maps are $$ \boxed{ \varphi_1(r,\mathcal Y) = (a_6r^2,a_6\mathcal Y), } $$ and $$ \boxed{ \varphi_2(r,\mathcal Y) = (a_0r^{-2},a_0\mathcal Yr^{-3}). } $$ For our coefficients, $$ \boxed{ E_1: \quad y_1^2 = x_1^3 + 4M^2x_1^2 + M^2(4M^2+L^2)x_1 + 2L^2M^4. } $$ This curve is directly $\mathbb Q$-isomorphic to the historical $E_0$ minimal model. Let $$ \boxed{ E_{0,\min}: \quad Y_0^2 = X_0^3 - 2MX_0^2 + L^2X_0. } $$ Then $$ \boxed{ X_0 = \frac{x_1+2M^2}{M}, } $$ and $$ \boxed{ Y_0 = -\frac{y_1}{975^3}. } $$ The invariant differentials satisfy $$ \boxed{ F_1^\ast \left( \frac{dX_0}{2Y_0} \right) = -975 \frac{dx_1}{2y_1}. } $$ Equivalently, relative to the normalized historical differential on $$ E_0: \quad y^2=x^3-2x^2+\lambda^2x, $$ one has $$ \boxed{ \log_{E_1} = -\frac1M \log_{E_0}. } $$ The corresponding rank-one point at the boundary is $$ \boxed{ P_1 = (-M^2,-M^2\cdot961408). } $$ It maps to $$ \boxed{ Q_{\min} = (M,937372800). } $$ The second theorem quotient is $$ \boxed{ E_2: \quad y_2^2 = x_2^3 - (4M^2+L^2)x_2^2 + 8M^2L^2x_2 - 4M^2L^4. } $$ Set $$ \boxed{ u=x_2-L^2. } $$ Then $$ \boxed{ y_2^2 = u^3 + (2L^2-4M^2)u^2 + L^4u. } $$ This is exactly the pre-isogeny cubic from R16. A rational non-torsion point on this theorem quotient is $$ \boxed{ u_C = \frac{ 4038403833577153474281 }{ 92774458921 }, } $$ $$ \boxed{ \eta_C = - \frac{ 10783196677393640046484856536800000 }{ 28258079668288469 }. } $$ Thus on the unshifted theorem model, $$ \boxed{ P_2 = \left( u_C+L^2, \eta_C \right). } $$ The standard degree-$2$ isogeny from this curve, followed by the exact R16 isomorphism, maps $P_2$ to $$ \boxed{ [2]P_- } $$ on the historical integral curve $E_-$. The differential relation for the composite isogeny is $$ \boxed{ \psi^\ast \left( \frac{dx_-}{2y_-} \right) = -\frac23 \frac{du}{2\eta}. } $$ Consequently, $$ \boxed{ \log_{E_2}(P_2) = -3 \log_{E_-}(P_-). } $$ By functoriality of the canonical cyclotomic height under the degree-$2$ isogeny, $$ \boxed{ h_{11,E_2}(P_2) = 2h_{11,E_-}(P_-). } $$ Therefore the quadratic constants required by the explicit bielliptic quadratic-Chabauty theorem are obtained without a second independent height implementation. Define $$ \boxed{ \alpha_1 = \frac{ h_{11,E_1}(P_1) }{ \log_{E_1}(P_1)^2 }, } $$ and $$ \boxed{ \alpha_2 = \frac{ h_{11,E_2}(P_2) }{ \log_{E_2}(P_2)^2 }. } $$ Then $$ \boxed{ \alpha_1 = M^2\alpha_0, } $$ and $$ \boxed{ \alpha_2 = \frac29\alpha_-. } $$ Writing $$ \boxed{ \beta_1=11^2\alpha_1, \qquad \beta_2=11^2\alpha_2, } $$ we obtain $$ \boxed{ \beta_1 \equiv 142823 \pmod{11^5}, } $$ and $$ \boxed{ \beta_2 \equiv 153840 \pmod{11^5}. } $$ In base $11$, $$ \boxed{ \beta_1 = 10 + 3\cdot11 + 3\cdot11^2 + 8\cdot11^3 + 9\cdot11^4 + O(11^5), } $$ and $$ \boxed{ \beta_2 = 5 + 4\cdot11 + 6\cdot11^2 + 5\cdot11^3 + 10\cdot11^4 + O(11^5). } $$ Thus the two global quadratic coefficients in the explicit bielliptic quadratic-Chabauty function are now calibrated. The modern explicit bielliptic method gives a locally analytic function of the shape $$ \boxed{ \rho(z) = \lambda_{11}(\varphi_1(z)) - \lambda_{11}(\varphi_2(z)) - 2\log_{11}(r(z)) - \alpha_1 \Log_1(\varphi_1(z))^2 + \alpha_2 \Log_2(\varphi_2(z))^2. } $$ Here: - $\lambda_{11}$ denotes the local Mazur–Tate $11$-adic height function on the appropriate elliptic quotient; - $\Log_i$ denotes the corresponding elliptic logarithm; - the apparent singularities at $r=0,\infty$ admit analytic continuation on the complete curve. For rational points, the theorem produces a finite set $$ \boxed{ \Omega\subset\mathbb Q_{11} } $$ such that $$ \boxed{ \rho(\mathcal H(\mathbb Q)) \subseteq \Omega. } $$ The canonical sextic has $$ a_6=-M^2, $$ and $$ a_0=2L^2. $$ Therefore the explicit coefficient-dependent logarithmic ranges entering the bad-prime finite sets are already known. For the primes where $a_6$ or $a_0$ has nonzero valuation: ### At $q=2$ $$ \operatorname{ord}_2(a_6)=0, \qquad \operatorname{ord}_2(a_0)=1. $$ The even integer range contributes only $$ \boxed{ n=0. } $$ ### At $q=3$ $$ \operatorname{ord}_3(a_6)=4, \qquad \operatorname{ord}_3(a_0)=0. $$ Thus $$ \boxed{ n\in\{-4,-2,0\}. } $$ ### At $q=5$ $$ \operatorname{ord}_5(a_6)=8, \qquad \operatorname{ord}_5(a_0)=0. $$ Thus $$ \boxed{ n\in\{-8,-6,-4,-2,0\}. } $$ ### At $q=13$ $$ \operatorname{ord}_{13}(a_6)=4, \qquad \operatorname{ord}_{13}(a_0)=0. $$ Thus $$ \boxed{ n\in\{-4,-2,0\}. } $$ ### At $q=73$ $$ \operatorname{ord}_{73}(a_6)=0, \qquad \operatorname{ord}_{73}(a_0)=2. $$ Thus $$ \boxed{ n\in\{0,2\}. } $$ ### At $q=18521$ $$ \operatorname{ord}_{18521}(a_6)=0, \qquad \operatorname{ord}_{18521}(a_0)=2. $$ Thus $$ \boxed{ n\in\{0,2\}. } $$ At $$ q=7,29,37, $$ the coefficient range itself contributes only $$ n=0, $$ although nontrivial local elliptic-height sets may still occur because the elliptic quotients have bad reduction there. Therefore the remaining exact global task has now been reduced to two concrete computations: 1. determine the finite local-height value sets $$ W_q^{E_1}, \qquad W_q^{E_2} $$ at the finitely many bad primes; 2. evaluate and root-isolate $$ \rho(z)-\omega $$ in the single R17 prototype $11$-adic residue disk for each $$ \omega\in\Omega. $$ The current phase therefore crosses the global-height barrier. The missing step is no longer the construction of a $p$-adic height. It is the finite bad-prime correction set and the final analytic root isolation. --- # 1. Canonical even sextic in theorem form Expand the canonical genus-$2$ equation: $$ \mathcal Y^2 = (2-r^2) \left( M^2r^4 - 2M^2r^2 + L^2 \right). $$ This gives $$ \boxed{ \mathcal Y^2 = -M^2r^6 + 4M^2r^4 - (4M^2+L^2)r^2 + 2L^2. } $$ Thus: $$ \boxed{ a_6=-M^2, } $$ $$ \boxed{ a_4=4M^2, } $$ $$ \boxed{ a_2=-(4M^2+L^2), } $$ and $$ \boxed{ a_0=2L^2. } $$ This is exactly the general even-sextic shape used by the explicit bielliptic quadratic-Chabauty method. --- # 2. External theorem used in R19 For an even sextic $$ X: \quad y^2 = a_6x^6+a_4x^4+a_2x^2+a_0, $$ the standard elliptic quotients are $$ E_1: \quad y^2 = x^3+a_4x^2+a_2a_6x+a_0a_6^2, $$ and $$ E_2: \quad y^2 = x^3+a_2x^2+a_4a_0x+a_6a_0^2. $$ The degree-$2$ maps are $$ \boxed{ \varphi_1(x,y) = (a_6x^2,a_6y), } $$ and $$ \boxed{ \varphi_2(x,y) = (a_0x^{-2},a_0yx^{-3}). } $$ When both elliptic quotients have rank one and $p$ is a suitable good prime, an explicit quadratic-Chabauty function is formed from their local $p$-adic heights and logarithms. R19 uses this formalism with $$ p=11. $$ --- # 3. Why the nonmonic leading coefficient causes no obstruction Our leading coefficient is $$ a_6=-M^2. $$ It is not equal to $1$. The modern explicit bielliptic theorem is formulated for general integral coefficients $$ a_6,a_4,a_2,a_0. $$ Therefore no preliminary normalization to a monic sextic is required. This is advantageous because it preserves the canonical boundary coordinate $$ r=\pm1. $$ --- # 4. Global minimal model for $E_0$ The normalized historical equation is $$ y^2 = x^3 - 2x^2 + \lambda^2x, $$ where $$ \lambda=L/M. $$ For the global sigma-height computation use $$ \boxed{ E_{0,\min}: \quad Y^2 = X^3 - 2MX^2 + L^2X. } $$ The canonical point is $$ \boxed{ Q_{\min} = (M,937372800). } $$ The rational-coordinate relation to the normalized curve is $$ \boxed{ X=Mx, } $$ and the compatible $Y$ scaling is $$ \boxed{ Y=975^3y. } $$ --- # 5. Bad reduction of $E_{0,\min}$ The minimal discriminant factors as $$ \boxed{ 2^{20} 7^2 29^2 37^2 73^4 18521^4. } $$ At $$ 7,29,37 $$ the reduction is multiplicative of type $$ I_2. $$ At $$ 73,18521 $$ it is multiplicative of type $$ I_4. $$ At $$ 2 $$ the reduction is additive. The component exponents divide $4$. Thus multiplication by $$ 8 $$ places $Q$ in all required connected components while simultaneously killing its reduction modulo $11$. --- # 6. Sigma constant on $E_0$ The integrality recursion gives the successive residues: $$ \boxed{ c \equiv 3 \pmod{11}, } $$ $$ \boxed{ c \equiv 36 \pmod{11^2}, } $$ $$ \boxed{ c \equiv 1004 \pmod{11^3}, } $$ and $$ \boxed{ c \equiv 4997 \pmod{11^4}. } $$ Thus: $$ \boxed{ c_{11,E_0} = 4997 + O(11^4). } $$ The uniqueness of the integral odd sigma series determines these digits canonically. --- # 7. Initial sigma coefficients on $E_0$ Using the representative $$ c=4997, $$ the sigma series begins $$ \boxed{ \sigma(t) = t - \frac{1906247}{2}t^3 + \frac{50945778950917}{24}t^5 - \frac{1425360169441144340633}{240}t^7 + \cdots. } $$ Although the displayed rational coefficients have ordinary denominators, their $11$-adic valuations are nonnegative. The complete truncated series used for the height calculation is integral in $$ \mathbb Z_{11}[[t]]. $$ --- # 8. Global height of $Q$ Use $$ P=[8]Q. $$ The formal parameter $$ t(P)=-X(P)/Y(P) $$ has exact valuation $$ \boxed{ v_{11}(t(P))=1. } $$ The $x$-coordinate denominator is a square $$ d(P)^2, $$ with $$ \boxed{ v_{11}(d(P))=1. } $$ The sigma formula gives the logarithm displayed in the executive summary. Dividing by $$ 8^2 $$ produces the global height $$ \boxed{ h_{11}(Q) \equiv58966 \pmod{11^5}. } $$ --- # 9. Historical integral model for $E_-$ The roots satisfy $$ e_1+e_2+e_3=0, $$ so the equation has no quadratic term: $$ \boxed{ y^2 = x^3+a_4x+a_6. } $$ The exact coefficients are: $$ \boxed{ a_4 = -67669444304850546481011867, } $$ $$ \boxed{ a_6 = -4185277897366249205812702665319946826. } $$ --- # 10. Bad reduction of $E_-$ The discriminant factors as $$ \boxed{ 2^{32} 3^{20} 5^{16} 7^4 13^8 29^4 37^4 73^4 18521^4. } $$ At every listed bad prime at least $5$, the reduction is multiplicative: - $5$: type $I_{16}$; - $7$: type $I_4$; - $13$: type $I_8$; - $29,37,73,18521$: type $I_4$. At $2$ and $3$, the reduction is additive. The point $P_-$ is singular on the special fibers at all these bad primes except $18521$. Multiplication by $$ 16 $$ kills all relevant component classes and also kills the reduction modulo $11$. --- # 11. Sigma constant on $E_-$ The sigma-integrality recursion gives: $$ \boxed{ c \equiv10 \pmod{11}, } $$ $$ \boxed{ c \equiv120 \pmod{11^2}, } $$ $$ \boxed{ c \equiv967 \pmod{11^3}, } $$ and the fourth base-$11$ digit is zero: $$ \boxed{ c \equiv967 \pmod{11^4}. } $$ Thus: $$ \boxed{ c_{11,E_-} = 967+O(11^4). } $$ --- # 12. Global height of $P_-$ Use: $$ P=[16]P_-. $$ The point lies in the formal subgroup at $11$ and all required connected components away from $11$. The Mazur–Tate sigma formula then yields: $$ \boxed{ h_{11}(P_-) \equiv3035 \pmod{11^5}. } $$ This is the second global rank-one height required by the bielliptic quadratic-Chabauty construction. --- # 13. Historical quadratic constants From R18: $$ \log_{11,E_0}(Q) = 11U_0, $$ and: $$ \log_{11,E_-}(P_-) = 11U_-, $$ to the displayed precision. Hence: $$ \alpha_0 = 11^{-2} \frac{h_{11}(Q)}{U_0^2}, $$ and: $$ \alpha_- = 11^{-2} \frac{h_{11}(P_-)}{U_-^2}. $$ The unit parts are: $$ \boxed{ 11^2\alpha_0 \equiv151114 \pmod{11^5}, } $$ and: $$ \boxed{ 11^2\alpha_- \equiv48076 \pmod{11^5}. } $$ --- # 14. First theorem quotient and $E_0$ Substituting our coefficients gives $$ E_1: \quad y_1^2 = x_1^3 + 4M^2x_1^2 + M^2(4M^2+L^2)x_1 + 2L^2M^4. $$ Define $$ X_0 = \frac{x_1+2M^2}{M}, $$ $$ Y_0 = -\frac{y_1}{975^3}. $$ Then: $$ \boxed{ Y_0^2 = X_0^3 - 2MX_0^2 + L^2X_0. } $$ Thus: $$ \boxed{ E_1 \cong_{\mathbb Q} E_{0,\min}. } $$ --- # 15. Differential scaling on $E_1$ Let: $$ \omega_1 = \frac{dx_1}{2y_1}, $$ and: $$ \omega_0 = \frac{dX_0}{2Y_0}. $$ Then: $$ \boxed{ F_1^\ast\omega_0 = -975\,\omega_1. } $$ Relative to the normalized historical $E_0$ differential, this becomes: $$ \boxed{ \log_{E_1} = -\frac1M \log_{E_0}. } $$ Therefore: $$ \boxed{ \alpha_1 = M^2\alpha_0. } $$ --- # 16. Second theorem quotient and the R16 pre-isogeny curve The second quotient is: $$ E_2: \quad y_2^2 = x_2^3 - (4M^2+L^2)x_2^2 + 8M^2L^2x_2 - 4M^2L^4. $$ Set: $$ u=x_2-L^2. $$ Then: $$ \boxed{ y_2^2 = u^3 + (2L^2-4M^2)u^2 + L^4u. } $$ This is exactly the R16 curve before the degree-$2$ isogeny to the historical $E_-$. --- # 17. Explicit non-torsion point on $E_2$ The dual-isogeny image of $P_-$ gives: $$ \boxed{ u_C = \frac{ 4038403833577153474281 }{ 92774458921 }, } $$ and: $$ \boxed{ \eta_C = - \frac{ 10783196677393640046484856536800000 }{ 28258079668288469 }. } $$ Hence on the theorem model: $$ \boxed{ P_2 = \left( \frac{ 173629486944658561911250 }{ 92774458921 }, \eta_C \right). } $$ Direct substitution verifies: $$ P_2\in E_2(\mathbb Q). $$ --- # 18. Isogeny image of $P_2$ The R16 degree-$2$ isogeny and historical isomorphism send: $$ P_2 $$ to: $$ \boxed{ [2]P_- } $$ on $E_-$. This equality was checked exactly in rational arithmetic, including both $x$ and $y$ coordinates. --- # 19. Differential scaling on $E_2$ Let: $$ \omega_2 = \frac{du}{2\eta}. $$ For the composite degree-$2$ isogeny: $$ \psi: E_2 \longrightarrow E_-, $$ one has: $$ \boxed{ \psi^\ast \left( \frac{dx_-}{2y_-} \right) = -\frac23\,\omega_2. } $$ Since: $$ \psi(P_2)=2P_-, $$ we obtain: $$ 2\log_{E_-}(P_-) = -\frac23 \log_{E_2}(P_2). $$ Thus: $$ \boxed{ \log_{E_2}(P_2) = -3\log_{E_-}(P_-). } $$ --- # 20. Height scaling on $E_2$ The canonical cyclotomic height is functorial under isogeny. For a degree-$2$ isogeny: $$ h_{E_-}(\psi(P)) = 2h_{E_2}(P). $$ At: $$ P=P_2, $$ the left-hand side is: $$ h_{E_-}(2P_-) = 4h_{E_-}(P_-). $$ Therefore: $$ \boxed{ h_{E_2}(P_2) = 2h_{E_-}(P_-). } $$ Hence: $$ \boxed{ \alpha_2 = \frac29\alpha_-. } $$ --- # 21. Theorem quadratic constants Combining the previous sections: $$ \boxed{ \alpha_1=M^2\alpha_0, } $$ and: $$ \boxed{ \alpha_2=\frac29\alpha_-. } $$ Therefore: $$ \boxed{ 11^2\alpha_1 \equiv142823 \pmod{11^5}, } $$ and: $$ \boxed{ 11^2\alpha_2 \equiv153840 \pmod{11^5}. } $$ These are the exact rank-one calibration constants needed for the explicit bielliptic QC function to the stated precision. --- # 22. Explicit quadratic-Chabauty function With the theorem quotient maps: $$ \varphi_1, \qquad \varphi_2, $$ define: $$ \boxed{ \rho(z) = \lambda_{11}(\varphi_1(z)) - \lambda_{11}(\varphi_2(z)) - 2\log_{11}(r(z)) } $$ $$ \boxed{ \qquad - \alpha_1 \Log_1(\varphi_1(z))^2 + \alpha_2 \Log_2(\varphi_2(z))^2. } $$ The local-height normalization must be kept consistent with the global Mazur–Tate height normalization used to define: $$ \alpha_1, \alpha_2. $$ The function analytically continues across the apparent $r=0,\infty$ singularities. --- # 23. Finite global target set For every rational point: $$ z\in\mathcal H(\mathbb Q), $$ the explicit bielliptic theorem gives: $$ \boxed{ \rho(z)\in\Omega, } $$ where: $$ \Omega\subset\mathbb Q_{11} $$ is a finite set assembled from local height values at the finitely many bad primes. Thus the rational-point problem has become: $$ \boxed{ \rho(z)=\omega, \qquad \omega\in\Omega, } $$ inside the single R17 prototype residue disk. --- # 24. Bad-prime support for the QC theorem The canonical sextic discriminant gives the bad-prime support: $$ \boxed{ \{ 2,3,5,7,13,29,37,73,18521 \}. } $$ The explicit coefficient valuations add no new primes. The chosen Chabauty prime: $$ 11 $$ is good and ordinary. --- # 25. Coefficient logarithm ranges: $q=2$ Since: $$ a_6=-M^2 $$ is a $2$-adic unit, and: $$ a_0=2L^2 $$ has valuation $1$, the theorem's even integer range is: $$ \boxed{ n=0. } $$ --- # 26. Coefficient logarithm ranges: $q=3$ Since: $$ \operatorname{ord}_3(a_6)=4, $$ and: $$ \operatorname{ord}_3(a_0)=0, $$ the allowed even integers are: $$ \boxed{ -4,-2,0. } $$ --- # 27. Coefficient logarithm ranges: $q=5$ Since: $$ \operatorname{ord}_5(a_6)=8, $$ and: $$ \operatorname{ord}_5(a_0)=0, $$ the allowed even integers are: $$ \boxed{ -8,-6,-4,-2,0. } $$ --- # 28. Coefficient logarithm ranges: $q=13$ The allowed even integers are: $$ \boxed{ -4,-2,0. } $$ --- # 29. Coefficient logarithm ranges: $q=73$ Since: $$ \operatorname{ord}_{73}(a_0)=2, $$ and: $$ \operatorname{ord}_{73}(a_6)=0, $$ the allowed even integers are: $$ \boxed{ 0,2. } $$ --- # 30. Coefficient logarithm ranges: $q=18521$ Likewise: $$ \boxed{ n\in\{0,2\}. } $$ --- # 31. Remaining discriminant primes At: $$ 7,29,37, $$ both: $$ a_0 $$ and: $$ a_6 $$ are units. Hence the coefficient-dependent integer range contributes only: $$ \boxed{ n=0. } $$ Nontrivial finite local-height sets may still occur because the quotient elliptic curves have bad reduction there. --- # 32. What remains for $\Omega$ For every bad prime: $$ q\ne11, $$ one still needs the finite local Néron-height value sets: $$ \boxed{ W_q^{E_1}, \qquad W_q^{E_2}. } $$ Once these are known, the theorem gives the exact finite set: $$ \boxed{ \Omega. } $$ This is now a finite reduction-theory problem. No additional global Mordell-Weil heights are missing. --- # 33. Why historical reduction types cannot be copied blindly The theorem quotient: $$ E_1 $$ is isomorphic to the historical: $$ E_0, $$ and: $$ E_2 $$ is degree-$2$ isogenous to the historical: $$ E_-. $$ However local Néron-height sets depend on the precise quotient curve and model normalization. Therefore R19 does **not** simply substitute the historical $E_-$ reduction table for: $$ W_q^{E_2}. $$ The local sets must be transported through the isogeny or recomputed directly. This is retained as a proof-safety boundary. --- # 34. Relation to the R18 coefficient curve R18 produced the local analytic relation: $$ \nu_- = \Phi_{11}(\nu_0-1) $$ through precision: $$ 11^5. $$ R19 adds the global quadratic data: $$ \alpha_1, \qquad \alpha_2. $$ Thus the rank-one quadratic pieces in: $$ \rho $$ can now be written explicitly on the same one-variable prototype disk. The remaining nontrivial analytic term is the difference of the two local Mazur–Tate height functions plus: $$ -2\log_{11}(r). $$ The sigma backend constructed in R19 is also the required local tool for this computation. --- # 35. Native backend status after R19 The current runtime now has proof-safe custom algorithms for: - exact elliptic group law; - $2$-primary and $11$-primary saturation certificates; - formal elliptic logarithms; - canonical $11$-adic sigma constants; - global cyclotomic $11$-adic heights of rank-one generators; - transport of quadratic height constants through the quotient isomorphism/isogeny; - the explicit bielliptic QC theorem coordinates. The external arithmetic-geometry dependency has therefore been reduced substantially. --- # 36. Current exact constants For convenient handoff: ## Historical $E_0$ $$ \boxed{ h_{11}(Q) \equiv58966 \pmod{11^5}, } $$ $$ \boxed{ 11^2\alpha_0 \equiv151114 \pmod{11^5}. } $$ ## Historical $E_-$ $$ \boxed{ h_{11}(P_-) \equiv3035 \pmod{11^5}, } $$ $$ \boxed{ 11^2\alpha_- \equiv48076 \pmod{11^5}. } $$ ## Theorem quotient $E_1$ $$ \boxed{ 11^2\alpha_1 \equiv142823 \pmod{11^5}. } $$ ## Theorem quotient $E_2$ $$ \boxed{ 11^2\alpha_2 \equiv153840 \pmod{11^5}. } $$ --- # 37. Cross-checks The following independent consistency checks have been performed: 1. sigma constants were obtained by successive integrality conditions rather than one direct solve; 2. the formal sigma series is $11$-integral to the required order; 3. height computations use multiples that lie in the formal subgroup and connected bad-prime components; 4. the R18 logarithm calculations were stable under increased truncation; 5. the theorem $E_1$ equation transforms exactly to the historical $E_0$ model; 6. the theorem $E_2$ equation shifts exactly to the R16 pre-isogeny cubic; 7. the explicit point $P_2$ lies exactly on $E_2$; 8. the isogeny image of $P_2$ is exactly $[2]P_-$; 9. differential scalings were checked symbolically. --- # 38. Proof-safety status R19 does not yet claim a complete rational-point classification. In particular, the following are not yet certified: - the complete finite set: $$ \Omega; $$ - the local analytic expansion of: $$ \rho $$ to root-isolation precision; - the complete zero set: $$ \{z:\rho(z)\in\Omega\}; $$ - the final Mordell-Weil sieve on any mock rational points; - fixed-slice nonexistence. The phase establishes the global height constants required to make those steps meaningful. --- # 39. Literature normalization The external ingredients used in this phase are: 1. Mazur–Tate $p$-adic sigma theory; 2. Mazur–Stein–Tate computation of the cyclotomic $p$-adic height via: $$ h_p(P) = \frac1p \log_p \left( \frac{\sigma_p(P)}{d(P)} \right); $$ 3. explicit bielliptic quadratic Chabauty for even sextics of rank-$2$ Jacobian, as developed by Balakrishnan–Dogra and made computationally explicit in the Bianchi–Padurariu framework. The canonical curve satisfies the required rank-one/rank-one split. --- # 40. Proposed R20 program R20 should finish the finite non-$11$ local correction layer. ## R20-A — Minimal models for theorem $E_1,E_2$ For each bad prime: $$ 2,3,5,7,13,29,37,73,18521, $$ determine the exact reduction type of the theorem quotients. ## R20-B — Local Néron value sets Compute: $$ W_q^{E_1}, \qquad W_q^{E_2}. $$ For multiplicative reduction this can be done from component indices and the Bernoulli local-height formula. For additive primes use the exact component table. ## R20-C — Assemble $\Omega$ Build the exact finite set: $$ \boxed{ \Omega. } $$ Reduce it modulo a useful power of: $$ 11 $$ and merge duplicate values. ## R20-D — Local QC series Use the new sigma backend to compute: $$ \rho(1+11u) $$ through sufficient precision. ## R20-E — Root isolation For each: $$ \omega\in\Omega, $$ solve: $$ \rho(1+11u)=\omega. $$ Use Strassmann/Newton-polygon arguments to certify the number of roots in: $$ u\in\mathbb Z_{11}. $$ ## R20-F — Recognition and final sieve Recognize the boundary root: $$ u=0. $$ Any additional $11$-adic roots become mock rational points and are passed to the existing R11/R16 Mordell-Weil sieve. --- # 41. Final conclusion R19 crosses the global-height barrier that remained after R18. The canonical Mazur–Tate sigma function has been reconstructed directly for both historical rank-one elliptic factors. The resulting global cyclotomic $11$-adic heights are: $$ \boxed{ h_{11}(Q) \equiv58966 \pmod{11^5}, } $$ and: $$ \boxed{ h_{11}(P_-) \equiv3035 \pmod{11^5}. } $$ These heights combine with the R18 logarithms to determine the quadratic constants required by explicit bielliptic quadratic Chabauty. After exact transport to the theorem's natural two elliptic quotients: $$ \boxed{ 11^2\alpha_1 \equiv142823 \pmod{11^5}, } $$ and: $$ \boxed{ 11^2\alpha_2 \equiv153840 \pmod{11^5}. } $$ Thus the missing quadratic-Chabauty function is now calibrated globally. The remaining tasks are finite and local: $$ \boxed{ \text{bad-prime local height sets} \longrightarrow \Omega \longrightarrow 11\text{-adic root isolation}. } $$ R20 should attack exactly that layer.