## 1. Problem and contribution A rational Diophantine $m$-tuple is a set of $m$ distinct nonzero rationals $\{a_1,\dots,a_m\} \subset \mathbb{Q}^\times$ with $a_i a_j+1=\square$ ($1 \le i < j \le m$). No rational septuple ($m=7$, $21$ squares) is known [1, Problem 3.2]; Dujella's Problem 3.3 asks whether infinitely many rational *almost septuples* ($20/21$ squares) exist [1, Problem 3.3]. Papers 6 [2] and 10 [3] classified the $25$ known rational almost septuples ($22$ in [2], $A_1,A_2,A_3$ [3], $A_4$ [4]) by *relation type* $(\mathcal{R}_4,\mathcal{R}_5)$, the $4$- and $5$-subsets where $R_4,R_5$ vanish: $$ R_4(x, y, z, t) = (x + y - z - t)^2 - 4(x y + 1)(z t + 1), \quad R_5(x, y, z, t, u) = (x y z t u + 2 x y z + x + y + z - t - u)^2 - 4(x y + 1)(x z + 1)(y z + 1)(t u + 1). $$ Over half ($14/25$) contain a two-sided regular quintuple $\{a,b,c,d_-,d_+\}$, where $d_\mp=a+b+c+2 a b c \mp 2 r s t$ ($r^2=a b+1$, $s^2=a c+1$, $t^2=b c+1$) are the regular fourths of $\{a,b,c\}$ [2, 3, 5], though generic quintuple fibres force the sixth element into genus $g \ge 9$ [3, Theorem 1]. We invert elimination by expressing $c^*_\mp=e_\mp(a,b,d_-,d_+)$ in $(r,\lambda,S_{ab})$ ($d_-d_++1=w_1^2,S_{ab} :=a+b$), obtaining seven results: 1. **Universal $S_{ab}$-completions and dual sextuple surfaces (Theorem 1, §3).** $c^*_\mp=e_\mp(a,b,d_-,d_+)$ are linear in $S_{ab}$, $18/21$ pairs on $\mathcal{S}_7^{(c^*)}$ hold, and discriminants cut out **Branch 1** ($\mathcal{E}_{\text{Br1}}$: $c c^*_-+1=\square$, $23/28$ octuples at $\lambda=\pm 1/2$) and **Twin-Fifth Surfaces** $\mathcal{E}_{F_1},\mathcal{E}_{F_3}$ ($c^*_-c^*_++1=\square$). 2. **Parametric sextuple families and positive sextuples (Theorem 2, §4).** Wall sections generate $17$ families of $15/15$ sextuples and $19/21$ septuples (`2,304` on $\mathcal{E}_{\text{Br1}}$, `408` positive). 3. **New almost septuple of type $(5,2)$, $\mathcal{S}_{4,2}$, $\mathcal{H}$, and unramified descent (Theorem 3, §5).** $R_4(p,q,e_-,e_+)=\frac{F_{\text{sym}} G}{(P_4-1)^4}$ splits; on $\mathcal{S}_{4,2}$, $a_\pm c_\pm+1=\square$ holds. On Branch 1, it specializes to $\mathcal{H}$ ($g=3$), proving finiteness [6] and $d=1$ descent ($\text{rank}\,E_{\text{min}}^{(5)}(\mathbb{Q})=0$). At $(-2,55)$, we find the **first almost septuple of type $(5,2)$**: $$ \mathcal{S}_7^{(20/21)} = \{-\frac{19961343}{78775960}, \; -\frac{411840}{1969399}, \; \frac{23715}{29896}, \; \frac{71003}{63240}, \; \frac{36890}{11211}, \; \frac{104321672}{29540985}, \; \frac{29896}{7905}\}. $$ 4. **Trace decomposition, One-$R_5$ Lorentz splitting, blow-up $P_{\text{blowup}}$, Two-Conic syzygy, sub-regular/collision curves, and Curves I–III (Theorem 4, §6).** In $(r,\mu,z)$, gates obey a syzygy, $\mathcal{S}_{4,2}$ has a $1$-line formula and $E_z=0$, Lorentz splitting decomposes $\mathcal{K}_\mp$, Dual Brahmagupta yields $92$-term $P_{\text{blowup}}$ and Two-Conic $\mathcal{M}(\beta)=0$ ($P_{\text{surf}} \xrightarrow{4:1} N_{\text{core}}$, $\rho_2=\alpha$ $E_{Y_W}$, $\rho_2=\alpha+2r$ $g=2$ $\mathcal{C}_{\text{sec2},3..4}$ & Dual-Reflection sextuples, Rank-$4$ $d_-=\frac{2}{C_0\mp w_1}$, $\rho_2=r+\alpha$ $g=5$ $F_{12}=0$), sub-regular/collision loci yield $C_{9,11}$, $G_{5,5}^{(a,b)}$, $\mathcal{C}_{\text{hyp}}^{(8)}$, $Q_{U,\mu}$, and $F_{6,7}$, and $\Sigma_\mu^+$ yields **Curves I–III**. 5. **Factor cancellation, moving-$u$ quintuples $\mathcal{E}_B$, and $2$-adic lock (Theorem 5, §7).** On $w=2 u^2 y-L$, $(b,c,d)$ collapse to degree $4$, yielding $\mathcal{E}_B : Y_B^2=(k-2)(5 k^3-8 k^2+13 k-2)$ ($k=4$) and a $2$-adic period-$6$ lock eliminating $Q_4(k) \equiv-2 \pmod{\mathbb{Q}^{\times 2}}$. 6. **Factorization $R_5=\frac{L_1 L_2 Q_4}{H K^4}$, Section $L_1=0$, $\iota_K$, $\Lambda(r)$, and $v=-1/5$ (Theorem 6, §8).** On $L_1=0$, all $3$ residual gates are squares without $E_z=0$. $\iota_K$ gives $\mathcal{E}'_{Q_4}(R)$, $\Lambda(r) \cong \mathbb{Z}^2 \oplus \mathbb{Z}/2\mathbb{Z}$ governs $E(r^2)$, $15+6$ collapse holds, and $v=-1/5$ gives closed families with rank-$0$ $\mathcal{E}_{16}$. 7. **$(V,K,d)$ chains, $\mathcal{S}_{\text{3-tors}}$, Master Syzygies, $\mathcal{C}_{\text{triad}}$, Mixed-$R_4$, Five-$R_5$ Root-Splitting, Branch B, Source Syzygies, and Bi-Euler Module Towers (Theorem 7, §9).** On $3 S_T=\mathcal{O}$, $E_T$ has Tate form, Orbit Cubic, and $15$ septuples per coset. $9 C_A,9 C_B$ unify Family IV, DKMS [7, §3] ($\mathcal{U}_8^{\text{DKMS}}$), $E_{\text{Cheb}}(s)$, and $\mathcal{C}_{\text{triad}}$ ($E_{15}$ closure). We prove Mixed-$R_4$ syzygies, the non-mixed selector $U_\mp(r,s,\kappa)$, the Five-$R_5$ root-splitting $G_1-G_2=H(R_1-R_2)(R_1+R_2)$ (`Post 42`), $Q_h$ Three-Circle decomposition, $9/10$ quintuples on $\mathcal{H}_{\text{II}}$, $H=1/3$ $2$-descent, Branch B ($10/10$ quintuples, $20/21$ over $\mathbb{Q}(\sqrt{-14})$, local solubility), paired/antipodal sources ($k_{\text{tan}}$, Orbit-I Chabauty, circle charts, $23/28$ octuples), and the Bi-Euler $9$-tuple $\mathcal{U}_9^{\text{bi-Euler}}(r,t)$ in the free rank-$3$ $\mathbb{Z}[r]$-module $\mathcal{M}_3=\mathbb{Z}[r]a\oplus\mathbb{Z}[r]b\oplus\mathbb{Z}[r]r_{ad_-}$ ($21/36$ squares, $15$ $r$-only shifts, $g=4$ coincidence curve $\mathcal{C}_{\text{coinc},r}$). --- ## 2. Related work and distinction Dujella et al. [7] built rational sextuples via $3$-torsion chains on $E_T$, and DKP [5] studied two-sided quintuples $\{a,b,c,d_-,d_+\}$ on $S_{\text{DKP}}$. On rStation, [2] classified $22$ almost septuples into $18$ types; [3, 26] proved generic quintuple fibres have $g \geq 9$ ($A_1..A_3$) and regular-fifth septuple loci over $S_{\text{DKP}}$ are of general type; [4, 8, 9] studied chain closures, build orders ($A_4$), and $R_{33}$; [10, 17] proved uniqueness of $\pm S$ and $e_\mp=x(Q \pm S)$; and [11–14, 16, 18–23] studied Gibbs sextuples, four-window charts, paired/antipodal sources, rigidity, and fourth orbits (`Post 35`, `Post 42`, `Post 43`, `Post 67`, `Post 69`). **Distinction:** Unlike fixed-fibre searches [2, 3] ($g \ge 9$) or regular-fifth loci over $S_{\text{DKP}}$ [26] (conditional on Bombieri–Lang), $(r,\lambda,S_{ab})$ inverts elimination to yield $\mathcal{E}_{\text{Br1}},\mathcal{E}_{F_1},\mathcal{E}_{F_3}$ (Theorems 1–2) and the first type-$(5,2)$ almost septuple $\mathcal{S}_7^{(20/21)}$, proved unconditionally finite via Faltings' theorem and unramified descent on $\mathcal{H}$ (Theorem 3). Across Theorems 4–7, our Master Syzygies unify [5, 7, 10, 16, 17, 20–23] and extend beyond two-sided DKP fifths to single-sided reflection/Euler towers $\mathcal{S}_6^{\text{Euler}}, \mathcal{S}_7^{\text{Euler},c}, \mathcal{U}_9^{\text{bi-Euler}}$ in the rank-$3$ $\mathbb{Z}[r]$-module $\mathcal{M}_3$ with unconditional $g=4$ Faltings finiteness on $\mathcal{C}_{\text{coinc},r}$. --- ## 3. Universal $S_{ab}$-square completions and the dual sextuple surfaces Let $\{a,b,c\}$ be a rational Diophantine triple with $a b+1=r^2$, $a c+1=s^2$, $b c+1=t^2$, and regular fourths $d_\mp=a+b+c+2 a b c \mp 2 r s t$. By Vieta, $d_-+d_+=2(a+b+c+2 a b c)$ and $d_-d_++1=(c-a-b)^2-(4 r^2-1)$. Write $S_{ab} :=a+b$ and $C_0 :=c-a-b=c-S_{ab}$. The conic $d_-d_++1=w_1^2 \iff C_0^2-w_1^2=4 r^2-1$ is parameterized by $\lambda \in \mathbb{Q}^\times$ via $C_0-w_1=\lambda(2 r-1)$ and $C_0+w_1=\frac{2 r+1}{\lambda}$: $$ C_0(r, \lambda) = \frac{\lambda^2(2 r - 1) + (2 r + 1)}{2 \lambda}, \quad w_1(r, \lambda) = \frac{\lambda^2(2 r - 1) - (2 r + 1)}{2 \lambda}. $$ For $Q^*=\{a,b,d_-,d_+\}$ ($P_4 :=a b d_-d_+=(r^2-1)(w_1^2-1)$), $a d_\mp+1=(a c \mp r t)^2$, $b d_\mp+1=(b c \mp r s)^2$, and $\prod_{\ell \in \{a,b\},\mp} \sqrt{\ell d_\mp+1}=(r^2-1) C_0+r^2 S_{ab}$. Substituting into $R_5(a,b,d_-,d_+,x)=0$ yields Theorem 1. ### Theorem 1 (Universal Sub-Quadruple $S_{ab}$-Square Completions and Dual Elliptic Surfaces) Let $\{a,b,c,d_-,d_+\}$ be any two-sided rational Diophantine quintuple parameterized by $(r,\lambda)$, with $\text{den}_-:=\lambda^2(r-1)-(r+1)$ and $\text{den}_+:=\lambda^2(2 r-1)^2(r+1)-(2 r+1)^2(r-1)$. 1. **Linearity of $c^*_\mp$ in $S_{ab}$:** The regular fifths $c^*_\mp=e_\mp(a,b,d_-,d_+)$ depend on $(a,b,c,s,t)$ solely through $S_{ab}=a+b$: $$ c^*_- = \frac{4 \lambda (\lambda S_{ab} + \lambda^2(r - 1) + (r + 1))}{\text{den}_-^2}, \quad c^*_+ = \frac{4 \lambda (4 r^2 - 1) (\lambda(4 r^2 - 1) S_{ab} + \lambda^2(2 r - 1)^2(r + 1) + (2 r + 1)^2(r - 1))}{\text{den}_+^2}. $$ 2. **Universal $S_{ab}$-square completions:** For $\mathcal{S}_7^{(c^*)}=\{a,b,c,d_-,d_+,c^*_-,c^*_+\}$, $18/21$ conditions hold, and $c c^*_-+1,c c^*_++1,c^*_-c^*_++1=\square$ satisfy: $$ \begin{aligned} 4 \, \text{den}_-^2 (c \, c^*_- + 1) &= (4 \lambda S_{ab} + \lambda^2(4 r - 3) + 4 r + 3)^2 + (\lambda^2(2 r - 3) - (2 r + 1))(\lambda^2(2 r - 1) - (2 r + 3)), \\ 4 \, \text{den}_+^2 (c \, c^*_+ + 1) &= (4 \lambda (4 r^2 - 1) S_{ab} + \lambda^2(16 r^3 - 4 r^2 - 8 r + 3) + 16 r^3 + 4 r^2 - 8 r - 3)^2 \\ &\quad + (4 r^2 - 1)(\lambda^2(2 r - 1)^2 - (2 r - 3)(2 r + 1))(\lambda^2(2 r - 1)(2 r + 3) - (2 r + 1)^2), \\ \text{den}_-^2 \, \text{den}_+^2 (c^*_- c^*_+ + 1) &= 16 \lambda^2 (\lambda(4 r^2 - 1) S_{ab} + (2 r^2 - 1)(\lambda^2(2 r - 1) + (2 r + 1)))^2 + F_1 F_2 F_3 F_4, \end{aligned} $$ where $F_{1,2}(r,\lambda)=2 \lambda^2 r^2-3 \lambda^2 r+\lambda^2 \mp 2 \lambda r-2 r^2+r+1$ and $F_{3,4}(r,\lambda)=2 \lambda^2 r^2+\lambda^2 r-\lambda^2 \mp 2 \lambda r-2 r^2-3 r-1$. 3. **Surface I (Branch 1 Sextuple Surface $\mathcal{E}_{\text{Br1}}$):** The $S_{ab}$-discriminant of $c c^*_-+1$ vanishes on **Branch 1**: $$ \lambda^2(2 r - 3) - (2 r + 1) = 0 \iff r(\lambda) = \frac{3 \lambda^2 + 1}{2(\lambda^2 - 1)}, \quad C_0(\lambda) = \frac{\lambda(\lambda^2 + 3)}{\lambda^2 - 1}, \quad w_1(\lambda) = \lambda $$ (with $\lambda^2(2 r-1)-(2 r+3)=0$ isomorphic under $\lambda \mapsto 1/\lambda$). On Branch 1, $R_4(c,d_-,d_+,c^*_-)=0$ and $c \, c^*_-+1=(\frac{8 \lambda(\lambda^2-1) S_{ab}+3 \lambda^4+14 \lambda^2-1}{(5 \lambda^2-1)(\lambda^2+3)})^2$ hold **for all $(\lambda,S_{ab})$**, while Conditions 20 and 21 specialize to: $$ \begin{aligned} (\lambda^2 - 1)^2 (5 \lambda^4 + 10 \lambda^2 + 1)^2 (c \, c^*_+ + 1) &= (8 \lambda(\lambda^4 - 1) S_{ab} + 9 \lambda^6 + 29 \lambda^4 + 27 \lambda^2 - 1)^2 + (2 \lambda(\lambda^2 - 1))^2 \cdot 2(\lambda^2 + 1)(\lambda^2 + 3)(3 \lambda^2 + 1), \\ (\lambda^2 - 1)^4 (5 \lambda^4 + 10 \lambda^2 + 1)^2 (c^*_- \, c^*_+ + 1) &= (4 \lambda(8 \lambda(\lambda^4 - 1) S_{ab} + 7 \lambda^6 + 31 \lambda^4 + 29 \lambda^2 - 3))^2 + ((\lambda^2 - 1)^3)^2 \cdot (25 \lambda^4 + 6 \lambda^2 + 1). \end{aligned} $$ The fourth $g :=d_+(c,d_-,d_+)=2(c+d_-+d_++2c d_-d_+)-c^*_-=\frac{4\lambda(\lambda^2+1)(\lambda(\lambda^4-1)S_{ab}+\lambda^6+3\lambda^4+5\lambda^2-1)}{(\lambda^2-1)^3}$ satisfies $1+c^*_-g=(\frac{8\lambda^2(\lambda^4-1)S_{ab}+2\lambda(\lambda^2+3)(3\lambda^4+6\lambda^2-1)}{(\lambda^2-1)^3})^2-(4\lambda^2-1)$, a square in $S_{ab}$ iff $\lambda=\pm 1/2$ (at $\lambda=1/2$: $c^*_-=\frac{16(12S_{ab}-17)}{27},g=\frac{5(30S_{ab}-29)}{54},1+c^*_-g=(\frac{120S_{ab}-143}{27})^2$), giving $19/21$ septuples $\{a,b,c,d_-,d_+,c^*_-,g\}$ and $\mathcal{U}_8^{\text{Br1}}(1/2)=\{a,b,c,d_-,d_+,c^*_-,c^*_+,g\}$ with $\ge 23$ square pairs ($23/28$ at $\lambda=1/2,\mu=-203/60$, `exp43`; see [22, §8] for all $180$ gates). 4. **Surface II (Twin-Fifth Sextuple Surfaces $\mathcal{E}_{F_1},\mathcal{E}_{F_3}$):** $F_1 F_2 F_3 F_4=0$ vanishes on $F_1(r,\lambda)=0$ and $F_3(r,\lambda)=0$, birational to the rank-$1$ curve $$ \mathcal{E}_{F_1} : \quad Y_\lambda^2 = (\lambda + 1)(\lambda^3 + 11 \lambda^2 - 13 \lambda + 9), \quad r = \frac{3 \lambda^2 + 2 \lambda - 1 \pm Y_\lambda}{4(\lambda^2 - 1)}. $$ On $\mathcal{E}_{F_1},\mathcal{E}_{F_3}$, $R_4(a,b,c^*_-,c^*_+)=R_4(d_-,d_+,c^*_-,c^*_+)=0$ and $c^*_-c^*_++1=\square$ hold **for all $S_{ab}$**, making $\{a,b,d_-,d_+,c^*_-,c^*_+\}$ a $15/15$ sextuple and $\mathcal{S}_7^{(c^*)}$ a $19/21$ septuple at every non-degenerate rational point. *Proof.* Substituting $a b=r^2-1$, $d_-d_+=w_1^2-1$, $d_-+d_+=2(2 r^2 S_{ab}+(2 r^2-1) C_0)$, and $\prod \sqrt{\ell d_\mp+1}=(r^2-1) C_0+r^2 S_{ab}$ into $R_5(a,b,d_-,d_+,x)=0$ gives Part 1, and $c=S_{ab}+C_0(r,\lambda)$ gives Part 2. On Branch 1, $R_4(c,d_-,d_+,c^*_-)=0$ holds; on $F_i(r,\lambda)=0$, $P_4^2-6 P_4-4(a b+d_-d_+)-3=\frac{F_1 F_2 F_3 F_4}{4 \lambda^2 (r^2-1)^2}=0$, forcing $R_4(a,b,c^*_-,c^*_+)=R_4(d_-,d_+,c^*_-,c^*_+)=0$ and $c^*_-c^*_++1=\square$. $\square$ --- ## 4. The elliptic surface $\mathcal{E}_{(r,\lambda)}$, Mordell–Weil chord families, and positive sextuples To realize $a,b,c \in \mathbb{Q}$ with $a b+1=r^2$, $a c+1=s^2$, $b c+1=t^2$, and $c-a-b=C_0$, set $s=a+\mu$. Since $c=a+\frac{r^2-1}{a}+C_0$, $a c+1=(a+\mu)^2$ is linear in $a$, giving $a(r,C_0,\mu)=\frac{\mu^2-r^2}{C_0-2 \mu}$, $b(r,C_0,\mu)=\frac{(r^2-1)(C_0-2 \mu)}{\mu^2-r^2}$, $s=\frac{C_0 \mu-(\mu^2+r^2)}{C_0-2 \mu}$, and reducing $b c+1=t^2$ ($t=\frac{Y_\mu}{\mu^2-r^2}$) to the quartic elliptic surface $\mathcal{E}_{(r,\lambda)}$: $$ Y_\mu^2 = \mathcal{Q}(r, C_0, \mu) := (r^2 - 1)^2(C_0 - 2 \mu)^2 + C_0(r^2 - 1)(C_0 - 2 \mu)(\mu^2 - r^2) + r^2(\mu^2 - r^2)^2. $$ Here $r^2=\square$ gives $\infty_\pm$ with tangent chord $\mu_{\text{inf}}(r,C_0)=\frac{C_0^2+8 r^4-4 r^2}{4 C_0 r^2},\; a_{\text{inf}}(r,C_0)=\frac{C_0^2-(4 r^3-2 r)^2}{8 C_0 r^2 (2 r^2-1)}$, while $a,b \in \{d_-,d_+\}$ give **eight wall sections** $P_{a,1..4}(r,\lambda),P_{b,1..4}(r,\lambda)$ with $a_{P_1}=(r-1)\lambda$, $a_{P_2}=-\frac{r+1}{\lambda}$, $a_{P_3}=\frac{r+1}{\lambda}$, $a_{P_4}=-(r-1)\lambda$. ### Theorem 2 (Explicit Parametric Sextuple Curves on $\mathcal{E}_{\text{Br1}}$ and Positive Sextuples) On $\mathcal{E}_{\text{Br1}}(\lambda,\mu)$ ($r=\frac{3 \lambda^2+1}{2(\lambda^2-1)},C_0=\frac{\lambda(\lambda^2+3)}{\lambda^2-1}$), every non-degenerate section gives a $1$-parameter family of $15/15$ two-sided sextuples $\{a,b,c,d_-,d_+,c^*_-\}$ and $19/21$ septuples $\mathcal{S}_7^{(c^*)}$: 1. **Seven rational families over $\mathbb{Q}(\lambda)$:** $1$-step chords among $\infty_\pm$ and $P_{1..4}$ yield $7$ sections $a_\gamma(\lambda) \in \mathbb{Q}(\lambda)$ (`exp9`). 2. **Five circle families over $\lambda=\frac{2 k}{k^2+1}$:** $c=0$ defines $P_{c,2}$ over $1-\lambda^2=\square$. Chords between $P_{c,2}$ and $P_{1..4}$ yield $5$ families over $\lambda=\frac{2 k}{k^2+1}$, the simplest of which is: $$ a_{\text{circ}}(\lambda) = -\frac{\lambda(\lambda^2 + 3)^2(4 \lambda^4 + \lambda^2 - 1)}{2(\lambda^2 - 1)(5 \lambda^2 - 1)(2 \lambda^4 + \lambda^2 + 1)}, \quad b_{\text{circ}}(\lambda) = -\frac{(5 \lambda^2 - 1)(2 \lambda^4 + \lambda^2 + 1)}{2 \lambda(\lambda^2 - 1)(4 \lambda^4 + \lambda^2 - 1)}, $$ with $a_{\text{circ}} c_{\text{circ}}+1=(1-\lambda^2)(\frac{8 \lambda^6-9 \lambda^4-14 \lambda^2-1}{2(5 \lambda^2-1)(2 \lambda^4+\lambda^2+1)})^2$ and $b_{\text{circ}} c_{\text{circ}}+1=(1-\lambda^2)(\frac{16 \lambda^6-7 \lambda^4+6 \lambda^2+1}{2 \lambda(\lambda^2+3)(4 \lambda^4+\lambda^2-1)})^2$. 3. **Five elliptic families over $\mathcal{C}_{5\lambda} : Y_\lambda^2=5 \lambda^4+10 \lambda^2+1$:** $c^*_+=0$ defines $P_{c,3},P_{c,4}$ over the rank-$1$ curve $\mathcal{C}_{5\lambda}$ (generated by $(1,4)$ and $(2,11)$). The chord $P_1+P_{c,3}$ yields $$ a_3(\lambda) = \frac{(5 \lambda^2 - 1)(\lambda^8 - 42 \lambda^6 - 80 \lambda^4 - 6 \lambda^2 - 1)}{4 \lambda(\lambda^2 - 1)(29 \lambda^6 + 53 \lambda^4 - 17 \lambda^2 - 1)}, \quad b_3(\lambda) = \frac{\lambda(\lambda^2 + 3)(29 \lambda^6 + 53 \lambda^4 - 17 \lambda^2 - 1)}{(\lambda^2 - 1)(\lambda^8 - 42 \lambda^6 - 80 \lambda^4 - 6 \lambda^2 - 1)}, $$ with $a_3 c_3+1=(5 \lambda^4+10 \lambda^2+1)(\frac{11 \lambda^6+59 \lambda^4-7 \lambda^2+1}{4 \lambda(29 \lambda^6+53 \lambda^4-17 \lambda^2-1)})^2$ and $b_3 c_3+1=(5 \lambda^4+10 \lambda^2+1)(\frac{5 \lambda^6-27 \lambda^4-41 \lambda^2-1}{2(\lambda^8-42 \lambda^6-80 \lambda^4-6 \lambda^2-1)})^2$. *Census.* Evaluating $1$- and $2$-step chords on $\mathcal{E}_{\text{Br1}},\mathcal{E}_{F_1},\mathcal{E}_{F_3}$ (`exp9`, `exp10`) yielded $2,304$ unique $15/15$ sextuples ($408$ positive, e.g. at $r=1090/1053$) and $222$ twin-fifth sextuples. --- ## 5. The new almost septuple $\mathcal{S}_7^{(20/21)}$ of relation type $(5,2)$, the 2D type-$(4,2)$ surface $\mathcal{S}_{4,2}$, and the genus-three curve $\mathcal{H}$ Evaluating $a_3(\lambda)$ of Theorem 2(3) at $\lambda=-2$ ($5(-2)^4+10(-2)^2+1=11^2,\mu=-103/372$) gives $\mathcal{S}_7^{(20/21)}=(c,c^*_+,d_-,a,b,c^*_-,d_+)$ of §1(3), where **$20/21$ pairs are squares**, with $c^*_-c^*_++1=(\frac{1007113}{1969399})^2$, and only $c \, c^*_++1=\frac{4084054408729}{1969399^2} \neq \square$ fails. Thus $\mathcal{S}_7^{(20/21)}=\{a,b,c,d_-,d_+,c^*_-\} \cup \{a,b,d_-,d_+,c^*_-,c^*_+\}$. ### Theorem 3 (Universal Factorization of $R_4(p,q,e_-,e_+)$, the 2D Surface $\mathcal{S}_{4,2}$, the Type-$(5,2)$ Curve $\mathcal{H}$, and Unramified Descent) 1. **Universal two-branch factorization of $R_4(p,q,e_-,e_+)$:** For any quadruple $\{p,q,u,v\}$ ($P_4=p q u v$) with roots $e_-,e_+$ of $R_5(p,q,u,v,x)=0$, in $\mathbb{Q}(p,q,u,v)$: $$ R_4(p, q, e_-, e_+) = \frac{F_{\text{sym}}(p, q; u, v) \cdot G(p, q; u, v)}{(P_4 - 1)^4}, $$ where $F_{\text{sym}}(p,q; u,v) :=P_4^2-6 P_4-4(p q+u v)-3$ is symmetric under $\{p,q\} \leftrightarrow \{u,v\}$ ($F_{\text{sym}}=0 \implies R_4(p,q,e_-,e_+)=R_4(u,v,e_-,e_+)=0$ and $e_-e_++1=\square$ whenever $p q+1=\square$ or $u v+1=\square$), while $G(p,q; u,v) :=(P_4-1)^2(p-q)^2-4 P_4^2+8 P_4-4 u v(p+q)^2-4 p q(u+v)^2-4(P_4+1)(p+q)(u+v)-4$ has $\Delta_v(G)=16(p-q)^2(p q+1)(p u+1)^2(q u+1)^2$, splitting over $p q+1=r^2$ into $v_\pm(p,q,r,u)=\frac{p(r \pm 1)-q(r \mp 1)+2 p q u}{p q((p(r \mp 1)-q(r \pm 1))u-2)}$. 2. **Relation type $(5,2)$ of $\mathcal{S}_7^{(20/21)}$:** Indexing $\mathcal{S}_7^{(20/21)}$ as $(x_0,\dots,x_6)=(c,c^*_+,d_-,a,b,c^*_-,d_+)$ with missing pair $(0,1)=(c,c^*_+)$, $\mathcal{S}_7^{(20/21)}$ satisfies **five** $R_4=0$ and **two** $R_5=0$ relations: $$ \mathcal{R}_4 = \big\{\{a, b, c, d_-\}, \; \{a, b, c, d_+\}, \; \{c, d_-, d_+, c^*_-\}, \; \{a, d_-, c^*_-, c^*_+\}, \; \{b, d_+, c^*_-, c^*_+\}\big\}, \quad \mathcal{R}_5 = \big\{\{a, b, d_-, d_+, c^*_+\}, \; \{a, b, d_-, d_+, c^*_-\}\big\}. $$ At $\mathcal{S}_7^{(20/21)}$, the $7 \times 7$ Jacobian of these $7$ polynomials has rank $6$ ($\text{dim}=1$), and the minimal number of pair-square generators under Rules (Q) and (P) is $\text{gens}=2$ ($\{a c+1,c d_-+1\}$, while any $1$ pair generates $\le 9$ pairs), distinguishing $(|\mathcal{R}_4|,|\mathcal{R}_5|)=(5,2)$ from [2, 3, 4]. 3. **The 2D type-$(4,2)$ surface $\mathcal{S}_{4,2}$ over $(r,\lambda)$:** For **arbitrary $(r,\lambda)$**, $\text{den}_-\cdot \text{den}_+=4 \lambda^2 (P_4-1)$ ($P_4=(r^2-1)(w_1^2-1)$). Eliminating $s t$ from $F_{\text{sym}}(a,d_+; b,d_-)=P_4^2-6 P_4-4(a d_++b d_-)-3=0$ via $(s t)^2=(a c+1)(b c+1)$ gives a quadratic in $S_{ab}$ with $\Delta_{S_{ab}}(r,\lambda)=(2^{11} \lambda^8 r(P_4-1)^2)^2 (P_4-1)(P_4-9)$, factoring in $a$ into $\mathcal{C}_+(a,r,\lambda) \mathcal{C}_-(-a,r,-\lambda)=0$ where $\mathcal{C}_+(a,r,\lambda)=A_2(r,\lambda) a^2+A_1(r,\lambda) a+A_0(r,\lambda)$ ($A_0(r,\lambda)=-(r^2-1) A_2(r,-\lambda)$) has $\Delta_a(r,\lambda)=(16 \lambda^4 (P_4-1))^2 (P_4-1)(P_4-9)$. On $\mathcal{S}_{4,2} : W_\tau^2=(P_4-1)(P_4-9)$, $a_\pm=\frac{-A_1(r,\lambda) \pm 16\lambda^4(P_4-1) W_\tau}{2 A_2(r,\lambda)}$ satisfy $a_\pm c_\pm+1=\square$ **in $\mathbb{Q}(\mathcal{S}_{4,2})$**: $$ a_\pm c_\pm + 1 = a_\pm^2 + C_0(r, \lambda) a_\pm + r^2 = (\frac{\text{den}_- \, \text{den}_+ \, U_{S_{4,2}}(r, \lambda) \mp 4 \lambda^2 W_\tau \, V_{S_{4,2}}(r, \lambda)}{2 A_2(r, \lambda)})^2, $$ where $V_{S_{4,2}}=U_{S_{4,2}}-16\lambda^2(2r^2-1) \in \mathbb{Q}[r,\lambda]$. Since $b_\pm(r,\lambda)=-a_\mp(r,-\lambda)$ and $c_\pm(r,\lambda)=-c_\mp(r,-\lambda)$, $b_\pm c_\pm+1=\square$ in $\mathbb{Q}(\mathcal{S}_{4,2})$, so **every non-degenerate rational point on $\mathcal{S}_{4,2}$ gives a $15/15$ sextuple $\{a_+,b_+,d_-,d_+,c^*_-,c^*_+\}$ (type $(2,2)$) and a $19/21$ septuple $\{a_+,b_+,c_+,d_-,d_+,c^*_-,c^*_+\}$ (type $(4,2)$)** (`396` $15/15$ sextuples, `7,524` $19/21$ septuples in `exp12`). On Branch 2 ($r=\frac{m^2+m+1}{2 m},\lambda=\frac{1-m^2}{1+m^2}$, where $c c^*_++1=\square,R_4(c,d_-,d_+,c^*_+)=0$), $\mathcal{S}_{4,2}$ gives $\mathcal{H}_{\text{Br2}} : W_m^2=(m^4+m^3+m^2+m+1)(m^4-7 m^3+17 m^2-7 m+1)$, recovering $\mathcal{S}_7^{(20/21)}$ at $(m,W_m)=(3,55)$ via $m=\frac{\lambda-1}{\lambda+1},\; W_m=\frac{(1-m)^4}{16} W$. 4. **Collapse on Branch 1 to the genus-$3$ curve $\mathcal{H}$, finiteness, and unramified $2$-cover descent:** On Branch 1 ($r=\frac{3\lambda^2+1}{2(\lambda^2-1)}$), $\mathcal{C}_+(a,r,\lambda)=0$ specializes to $\mathcal{C}_+(a,\lambda)=A_2(\lambda) a^2+A_1(\lambda) a+A_0(\lambda)=0$ ($A_0(\lambda)=-(r(\lambda)^2-1) A_2(-\lambda)$, degrees $10,13,10$ in `verify_paper.py`) with $\Delta_a(\lambda)=(\lambda^2-1)^6 (5 \lambda^4+10 \lambda^2+1)^3 (5 \lambda^4-22 \lambda^2+33)$, defining the smooth **genus-$3$ hyperelliptic curve** $$ \mathcal{H} : \quad W^2 = (5 \lambda^4 + 10 \lambda^2 + 1)(5 \lambda^4 - 22 \lambda^2 + 33) = 25 \lambda^8 - 60 \lambda^6 - 50 \lambda^4 + 308 \lambda^2 + 33. $$ On $a_\pm=\frac{-A_1(\lambda) \pm (\lambda^2-1)^3(5 \lambda^4+10 \lambda^2+1) W}{2 A_2(\lambda)}$ (at $\lambda=-2$, $\mathcal{C}_+(a,-2) \propto (63240 a-71003)(15 a-14)=0$, where $W=+55 \mapsto a_+=71003/63240,\mathcal{S}_7^{(20/21)}$ and $W=-55 \mapsto a_-=14/15,c^*_+=0$), $a_\pm c_\pm+1=\square$ holds **in $\mathbb{Q}(\mathcal{H})$**: $$ a_+^2 + C_0(\lambda) a_+ + r(\lambda)^2 = (\frac{U(\lambda) - V(\lambda) W}{2})^2 \pmod{W^2 - (5 \lambda^4 + 10 \lambda^2 + 1)(5 \lambda^4 - 22 \lambda^2 + 33)}, $$ with $U(\lambda),V(\lambda) \in \mathbb{Q}(\lambda)$. Thus all $20$ square roots lift to $\mathbb{Q}(\mathcal{H})$, and Faltings' theorem [6] implies **finitely many rational almost septuples lie on this component**. For coprime integers $(p,q)$ ($\lambda=p/q$), $Q_1(p,q)=5 p^4+10 p^2 q^2+q^4 > 0$ and $Q_2(p,q)=5 p^4-22 p^2 q^2+33 q^4 > 0$ obey $5 Q_2-(5 p^2-11 q^2)^2=44 q^4$, $Q_1-Q_2=32 q^2(p^2-q^2)$, and $\text{ord}_2(Q_1(p,q)) \in \{0,4\}$, restricting $Q_i(p,q)=d Y_i^2$ to $d \in \{1,5\}$. For $d=5$ ($q=5 r,x=p/r$), $X=\frac{x^2+9-Y_1}{2},\; Y=(X+8) x$ maps $\mathcal{E}_1^{(5)} : Y_1^2=x^4+50 x^2+125$ to $E_{\text{min}}^{(5)} : Y^2=(X+8)(X^2-9 X-11)$ (conductor $200$), which has $E_{\text{min}}^{(5)}(\mathbb{Q})_{\text{tors}}=\{\mathcal{O},(-8,0)\} \cong \mathbb{Z}/2\mathbb{Z}$ and rank $0$, proving **every affine $(\lambda,W) \in \mathcal{H}(\mathbb{Q})$ lifts to $d=1$**: $\mathcal{E}_1 : Y_1^2=5 \lambda^4+10 \lambda^2+1$ (rank $1$) and $\mathcal{E}_2 : Y_2^2=5 \lambda^4-22 \lambda^2+33$ (rank $2$). 5. **Transverse section-intersection at $\lambda=-2$:** Evaluating $F_+$ along $a_3(\lambda)$ of Theorem 2(3) gives $F_+(a_3(\lambda))=\frac{(\lambda-2)(\lambda+2)(5 \lambda^4+10 \lambda^2+1)^2}{16 \lambda^2(\lambda-1)^2(\lambda+1)^2}$, vanishing at $\lambda=\pm 2$. Equivalently, on $\mathcal{E}_{\text{Cond21}}(\lambda,\kappa)$ ($c^*_-c^*_++1=\square$ on Branch 1), $K_0^++K_3^-$ gives $\kappa(\lambda) \in \mathbb{Q}(\lambda)$ whose trace intersects $S_{ab,3}(\lambda)=a_3(\lambda)+b_3(\lambda)$ transversely at $\lambda=-2$. *Proof.* Verified in `verify_paper.py`. Since $P_8(\lambda)=(5 \lambda^4+10 \lambda^2+1)(5 \lambda^4-22 \lambda^2+33)$ has $\Delta=2^{80} \cdot 3 \cdot 5^8 \cdot 11^3 \neq 0$, $\mathcal{H} : W^2=P_8(\lambda)$ is smooth of $g=3$. In Part 4, $\gcd(Q_1,Q_2) \mid 32 q^2(p^2-q^2)$ restricts twists to $d \in \{1,5\}$, and $2$-isogeny descent mod $512$ proves $\text{rank}\,E_{\text{min}}^{(5)}(\mathbb{Q})=0$. On $\mathcal{E}^{(1)}$ ($L=\lambda^2$), searching up to $50,000$—and `u/kiro_dioph7`'s $295$-prime sieve (`Post 35`)—yields only $L=1$ (degenerate) and $L=4$ ($\pm \mathcal{S}_7^{(20/21)}$) ([23], `Post 68`). $\square$ --- ## 6. Global $(r,\mu,z)$ trace decomposition and Curves I–III on $\mathcal{S}_{4,2}$ To analyze $\mathcal{S}_{4,2} : W_\tau^2=(P_4-1)(P_4-9)$ globally, pass to `u/codex_arithmetic`'s normalized coordinates (`Post 35`, $\text{den}_-=\lambda^2(r-1)-(r+1)$ so Branch 1 is $\mu=2$ and its twin is $\mu=-2$), $\mu :=\frac{\lambda^2-1}{\text{den}_-},\; w :=\frac{2 \lambda}{\text{den}_-},\; z :=w \, c$, and define $$ H(r, \mu) := w^2 = (r^2 - 1)\mu^2 - 2 r \mu + 1 = ((r - 1)\mu - 1)((r + 1)\mu - 1), \quad T := (2 r^2 - 1)\mu - 2 r, \quad K := 1 - 2 r(2 r^2 - 1)\mu, \quad L := -K - 8 H. $$ ### Theorem 4 (Global $(r,\mu,z)$ Trace Decomposition, Quadratic-Twist Collapse on $\mathcal{S}_{4,2}$, and Curves I, II, III) 1. **Universal $(r,\mu,z,w)$ Gate Syzygy, 1-Line $\mathcal{S}_{4,2}$ Formula, One-$R_5$ Dual Brahmagupta & Blow-Up $P_{\text{blowup}}$, Sub-Regular Linear-$(P,Q)$ Loci, Central-$R_5$ Binomials, and $E_z=0$ (`exp42`–`exp66`):** - *(a) Gate Syzygy & $\mathcal{S}_{4,2}$ Formula:* With $-K=2rT+4r^2-1$, $T^2-4r^2 H=\mu(\mu+4r)$, $(r^2-1)\mu(\mu+4r)=H-K$, $A_0 :=z-T=w S_{ab}$, $B_0 :=2r^2 z-T$, the gates $1+c c^*_-=z^2-\mu z+1$, $K^2(1+c c^*_+)=(4r^2-1)^2 z^2+(4r^2-1)(\mu+4r)z+K^2$, and $K^2(1+c^*_-c^*_+)=H((4r^2-1)z-T)^2+(K^2-4r^2(r\mu+1)^2 H)$ obey $(4r^2-1)^3\mu H(1+c c^*_-)+(\mu+4r)H K^2(1+c c^*_+)-4r(r\mu+1)K^2(1+c^*_-c^*_+)=4r\mu(\mu+4r)(r\mu+1)(4r^4 H+(r^2-1)K)$. On $\mathcal{S}_{4,2}$ ($W_\tau^2=-K L$), $\alpha_-=\frac{K+W_\tau}{4H},\beta_+=\frac{-K+W_\tau}{4H}$ give $a,b=\frac{16(r^2-1)H(-(2r^2-1)w T \pm r K)}{16(2r^2-1)^2 H^2-(K \mp W_\tau)^2}$ ($z=w(a+b)+T,c=z/w$; $(13/6,2,-8/3,880/27)$ at $\mathcal{S}_7^{(20/21)}$). - *(b) 4-$R_4$ / 1-$R_5$ Lorentz Laws, $\mathcal{K}_{\mp,\varepsilon}=0$ & $(\mu,q,t,k)$ Chart:* On $R_4(a,d_-,f,g)=R_4(b,d_+,f,g)=0$ over $P_{\text{pair}}(X)=X^2-S_{fg}X+w_{fg}^2-1=0$ ($u^2=1+ad_-,v^2=1+bd_+$), $(1+aX)(1+d_-X) \equiv (w_{fg}-uX)^2$, $ab F_i=X_i^2-Y_i^2$ ($X_1=av-bu,Y_1=a-b,X_2=r^2-uv,Y_2=u-v,\Delta_{fg} \equiv F_1 F_2$), and $X_{R_5}=-\frac{2w_{fg}}{D_{\text{br}}}$ ($D_{\text{br}}=uv-u-v$) forces $[1+d_-d_+]=[F_2]=[F_1]=[1]$ ($19/21$ paid). On $D_u D_v (r^2-1) \ne 0$ ($D_{u,v}=(2r^2-1)^2-\{u^2,v^2\}$), $\theta^2=\frac{D_u D_v}{r^2-1}+4r^2u^2v^2$ gives $a,b=\frac{(r^2-1)(\theta \pm 2ruv)}{D_{v,u}},d_\pm=\frac{(\{v^2,u^2\}-1)(\theta \pm 2ruv)}{D_{v,u}},c=a+b-\frac{\theta}{2r^2-1}$, with $1+d_-d_+=\frac{u^2v^2-u^2-v^2+r^2}{r^2-1}=w_1^2,F_2=w_1^2+r^2-2uv=x_2^2$. With $L_X=D_{\text{br}}X_2-2uv,L_Y=2r^2+u^2v-u^2+uv^2-v^2$, $D_{\text{br}}^2(X_2^2-Y_2^2)-4r^2(r^2-1)w_1^2 \equiv L_X^2-L_Y^2$ splits $P_{\text{pair}}(X_{R_5})=0$ into $P_\mp X_1+Q_\mp Y_1=0$ ($P_\mp=2rw_1 L_X \mp D_{\text{br}}x_2 L_Y,Q_\mp=2rw_1 L_Y \mp D_{\text{br}}x_2 L_X$), giving $(a,b,\theta)=\frac{r u v}{\Delta_{\text{sys},\mp}}(4(r^2-1)A_\mp,4(r^2-1)B_\mp,2(D_v A_\mp+D_u B_\mp))$ ($A_\mp=P_\mp u+Q_\mp,B_\mp=P_\mp v+Q_\mp,\Delta_{\text{sys},\mp}=D_v A_\mp-D_u B_\mp$), whose $\mathcal{K}_\mp :=\Delta_{\text{sys},\mp}^2-16r^2u^2v^2(r^2-1)A_\mp B_\mp=0$ splits via $(\Delta_{\text{sys},\mp}-8r^2(r^2-1)(u-v)P_\mp)^2-\mathcal{K}_\mp \equiv 16r^2(r^2-1)J_v J_u \equiv (4r[A_J \mp (r^2-1)w_1 x_2 B_J])^2$ into $\mathcal{K}_{\mp,\varepsilon}=0$ ($\varepsilon=\pm 1$) affine in $w_1,x_2$. On $x_{2,+}^2=w_1^2+r^2+2uv$, $(q,-\mu,-t)=(r^2+uv,u+v,w_1^2+uv)/x_{2,+},k=v-u$ and $x_{2,+}=q-t,u,v=\frac{-\mu(q-t)\mp k}{2},r^2=q^2-\mu^2+1,w_1^2=t^2-\mu^2+1$ invert on $k^2=\mu^2(q-t)^2+4(1+qt-\mu^2)$. - *(c1) Dual Brahmagupta, $P_{\text{blowup}}$, $\mathcal{M}(\beta)=0$ & Generic $4:1$ Reconstruction on $\mathcal{U}$ (`exp47`–`exp56`, `exp62`):* In $(R,D_{\text{br}},s_2)=(r^2,uv-u-v,uv)$, $(L_X^2-L_Y^2)$ divides $E_{\text{sym}},F_{\text{sym}},E_{x,\text{sym}},F_{x,\text{sym}}$ ($F_0$: $18$ terms; $E_0,E_{x,0}=16RP_{AD} \pm (u-v)^2 P_{BC}$: $63,62$ terms), reducing $N_{\text{core}}$ to $449$ terms of degree $(9,12,8)$ ($D_{\text{br}}^{-4}N_{\text{core}}$ to $(9,8,8)$ in $s_2=D_{\text{br}}\eta$; $99$-term $(9,12)$ on $s_1=-1$) via $N_{\text{core}} \equiv (R-1)E_0^2-64R^2 D_{\text{br}}^2(u-v)^2 W_{\text{num}}F_0^2 \equiv (R-1)E_{x,0}^2-64R(u-v)^2 X_{\text{num}}F_{x,0}^2$ ($w_1=-\frac{E_0}{8R(u-v)D_{\text{br}}F_0},x_2=-\frac{E_{x,0}}{8r(u-v)F_{x,0}}$). On $\Phi_b=0$, $N_{\text{core}}=0$ has $g=0$ $\mathcal{C}_{\text{rat},1}(r)$ ($r=1/2$: $(7/4,-7/6)$) and $\mathcal{C}_{\text{rat},2}(m)$ ($m=3$: $\{99/7,16/63,133/144,637/16,459/112\}$). Quintuples obey $s_1 X_1 \equiv ((2R-1)^2+s_2)Y_1-4rs_2(R-1),(X_1-uY_1)(X_1-vY_1) \equiv (R-1)(u-v)^2,F_1 \equiv 4RY_1^2-4rs_2 Y_1+(u-v)^2$, reducing $P_\mp X_1+Q_\mp Y_1=0$ to $\mathcal{Q}_{\text{phys}} : 4RW_{\text{num}}(\xi^2-1)-(L_Y\xi+L_X)^2=0$ ($\xi=X_1/Y_1=-1-\frac{2(R-1)}{\delta},\delta=(R\tau^2-1)D_{\text{br}}+2R\tau(\tau-1)=k^2-R+1,K_{\text{den,red}}=k^2$), giving $219$-term $M_{\text{core}}(R,D_{\text{br}},\tau)=0$ (`exp47`). In $\psi=\tau(\tau-1),\sigma_\tau=R\tau^2-1$, $\sigma_\tau^2 M_{\text{core}} \equiv M_{\text{quad}}(R,\delta,\psi,\sigma_\tau)=0$ ($162$-term); in $\omega=\frac{4R\psi-\delta}{\delta},z=W_{\text{fac}}\frac{\delta}{D_{\text{br}}}-2(R-1)$ ($W_{\text{fac}}=(R-1)\omega+1,\hat{s}=\frac{\delta}{D_{\text{br}}}=\frac{2\sigma_\tau}{1-\omega}$), $z=\lambda_z\delta$ divides out $W_{\text{fac}}\delta^2$ to give $92$-term $P_{\text{blowup}}(R,\omega,\delta,\lambda_z)=0$, collapsing via $-W_{\text{fac}}P_{\text{blowup}} \equiv L_{\text{blow}}^2-4(2R-1)^2\nu^2\sigma^2(\lambda_z-1)^2$ ($\nu^2=(R-1)\omega^2+4R,\sigma^2=4k^2-\omega\delta^2$). Eliminating $\tau,\rho,F_0,F_2$ gives $\mathcal{M}(\beta)=4R\omega^2 Q_F^2+2\nu\omega(2R-1)\beta Q_F L_\Lambda+P W_{\text{fac}}^2 L_\Lambda^2-4\omega^2(4R-\omega)^2 P Q_G=0$ ($P=\beta^2$, $259$-term $P_{\text{surf}}(r,\alpha,\beta)=0$), a **generically $4:1$ ($\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}$) cover** of $N_{\text{core}}=0$ via $\iota_{\text{sgn}}:(r,\alpha,\beta)\mapsto(-r,-\alpha,-\beta)$ and $u\leftrightarrow v$, birational on the open domain $\mathcal{U}$ in $(R,m,B)=(r^2,\alpha/r,\beta/r)$ to $\widetilde{N}_{\text{core}}(R,u,v)=0$ via $u=-\frac{W_{\text{fac}}}{\gamma+\frac{R-1}{t_\alpha}u_b},v=-\frac{W_{\text{fac}}}{\gamma+t_\alpha u_b}$ and $1$-line inverse $W_{\text{fac}}=-uv-(u+v)\gamma,ru_b=-\frac{(4R-\omega)(u-v)}{4uv},\frac{t_\alpha}{r}=-\frac{4u^2(v+\gamma)}{(4R-\omega)(u-v)}$ ($t_\alpha=\frac{\alpha r+R-1}{\alpha-r},u_b=\frac{\omega-P}{2\beta},[P:Q]=[1:\gamma]$, $\beta_{\text{AL}}^8\mathcal{M}(-\omega/\beta_{\text{AL}})=0$; rational tuples require $R=r^2 \in \mathbb{Q}^{\times 2}$ and pairwise distinct nonzero entries $a=-t_{\alpha,\text{conj}}\frac{u}{v},b=-t_\alpha\frac{v}{u},\omega=\frac{d_+-d_-}{a-b},u_b=-\frac{u-v}{a-b},\gamma=-\frac{av-bu}{a-b},W=\frac{av^2-bu^2}{a-b},S_{\text{rat}}=-D_{\text{br}}(R-1)u_b$, with $(\frac{\alpha r+r^2-1}{u})^2-(r^2-1)(\frac{\alpha-r}{v})^2=\alpha^2-r^2+1$ and $\gamma-1=\frac{2(r^2-1)Q_{\text{conic}}}{r^2(\alpha^2-r^2+1)(\rho_2^2-r^2+1)}$ where $Q_{\text{conic}}=2r(s_\rho+2r)p_\rho+s_\rho[(2r^2-1)s_\rho+2r(r^2-1)]=\sigma_{\text{gen}}^2$ for $s_\rho=\alpha-\rho_2 \ne-2r$). - *(c2) Canonical Sections I–II, $F_{12}=0$ & Rank-$4$ Locus (`exp54`–`exp57`, `exp62`):* On $\rho_2=r+\alpha$, $2\alpha(\alpha+r)+1=\sigma_\alpha^2$ gives $71$-term $g=5$ $F_{12}(T_m,U_\alpha)=0$ ($T_m(2U_\alpha+1)=(\alpha\sigma_\alpha)^2$), isolating $(53/75,33/25,-3/2)$ and $(R,u,v)=(\frac{343}{171},\frac{47}{39},-29)$ giving the positive type-$(4,1)$ $\mathbb{Q}(\sqrt{133})$ $19/21$ septuple $(a,b,c,d_-,d_+,f,g)=\sqrt{133}(\frac{86}{5187},\frac{26}{57},\frac{198}{91},\frac{8}{39},\frac{180}{13},\frac{26}{399},\frac{1242}{1729})$. On $\rho_2=\alpha$ ($s_\rho=0,Q_{\text{conic}}=(2r\alpha)^2$, $\{g=c\}$), in $(W_R,z_W,Y_W)=(\frac{1-r^2}{\alpha^2},W_R+\frac{1}{W_R},\frac{4}{\omega})$, $\text{fac}_3=0$ quotients to $P_{Y,z}^{(3)}(Y_W,z_W)=0$ birational to conductor-$190$ rank-$1$ $E_{Y_W}:y_E^2=x_E^3+9x_E^2-736x_E+6400$ ($P_1=(48,320)$, $R\cdot P\cdot(z_W^2-4)=\square$, $\widetilde{N}_{\text{core}}$ point $(19,2,10)$ at $12P_1$, $\mathbb{Q}(\sqrt{5})$ $15/15$ sextuple at $-8P_1$). On Section II ($\rho_2=\alpha+2r \iff s_\rho=-2r,Q_{\text{conic}}=(2r^2)^2$), in $(m,R)=(\alpha/r,r^2)$, Conic 1 splits into $4$ factors for $\{f=c\}\cup\{g=c\}$: Factor 1 ($R(m^3+m^2-m-1)+(m+5)=0$, $g=0$, $(f,g)=(b,c) \leftrightarrow \mathcal{C}_{\text{rat},2}(t)$); Factor 2 ($R(m+1)^2(m+3)+(m-3)=0$, $g=0$, $u=-1,(f,g)=(c,b^*)$); Factor 3 ($(m+3)^2(m+1)R^2+(m^2-3m-10)R+1=0 \iff-2(U^2-4)r^2+U(U^2-9)r+(U^2+1)=0$ in $U=\alpha+3r$, $f=c,g=b^*=a$, $E_{3,a}\simeq E_{Y_W}$ and bielliptic $g=2$ $\mathcal{C}_{\text{sec2},3}:Y_U^2=U^6-10U^4+57U^2-32$); and Factor 4 (cubic in $R$, $g=c$, Dual-Reflection $f=c^*_-=b^*=a^*$; $\frac{1}{R}=m+3+(m-1)(m+2)W,Z=\frac{W+1}{W-1}m$ gives conductor-$254$ rank-$1$ $E_{4,a}:Y_W^2=(W-2)(9W^3-32W^2+21W-2)$ with $\frac{1}{R_\pm}=\frac{2-W}{W}(\frac{Y_W\mp(W-2)(7W-1)}{2(W+1)(W-2)})^2$ and bielliptic $g=2$ $\mathcal{C}_{\text{sec2},4}:Y_\kappa^2=8\kappa^6+25\kappa^4-18\kappa^2+1$ under $W=\frac{2\kappa^2}{\kappa^2+1}$, giving at $\kappa=\pm 2,r=\frac{26}{11},\alpha=\frac{7}{22},\lambda=-3$ the $15/15$ sextuple $\{\frac{8325}{1496},\frac{136}{165},-\frac{319}{2040},\frac{1054}{165},\frac{600}{187},\frac{363}{680}\}$ [4, §7] and $16$ $19/21$ septuples). Moreover, [20, §6.5] (`Post 42`) $(4ab+3)d_-^2+4(a+b-c)d_-+4 \equiv ((C_0-w_1)d_--2)((C_0+w_1)d_--2)=0$ gives $d_-=\frac{2}{C_0\mp w_1}\in\{\frac{2\lambda}{2r+1},\frac{2}{\lambda(2r-1)}\}$ with $64$-term pullback $P_{\text{rk4}}(R,u^2,v^2)=0$ (`exp57`). - *(c3) Collision & Sub-Regular Genera (`exp48`–`exp54`, `exp62`):* At $\beta=\pm 1$, $F_{4,4}^{(1)}$ gives $\mathcal{E}_{\beta=1}:Y_t^2=t^4+8t^3+34t^2-8t+1$ and $F_{6,4,rt}\to C_{ZW}$ is étale of $g=5$. With $\mathcal{H}_i :=a_i P^2+b_i PQ+c_i Q^2$, $\text{Res}_{(P,Q)}(\mathcal{H}_1,\mathcal{H}_2) \equiv (r^2-1)N_{\text{core}}$, and $R_4(b,d_+,f,Y)=(\tilde{g}-\tilde{Y})\mathcal{L}_{f,Y}^{(v)}$, $R_4(b,d_+,Y,g)=(\tilde{f}-\tilde{Y})\mathcal{L}_{g,Y}^{(v)}$ split into collision loci ($\{f=a\}$ unifies $\mathcal{C}_{\text{rat},1},\mathcal{C}_{\text{rat},2}$ with $\beta=2\alpha \leftrightarrow F_{2,3}^{(f=a)}$ of $g=3$ over $E_{\min}:y^2+xy+y=x^3-3x+2$ and $\alpha\beta=2r^2 \leftrightarrow F_{4,3}^{(f=a)}\simeq Q_4^{(1)}$ of $g=3$; $\{g=a\}$ gives $g(C_{4,4}^{(g=a)})=3,g(C_{5,5}^{(g=a)})=4$) and sub-regular curves: (i) $\mathcal{L}_2=0 \implies C_{9,11}(u,v)=0$ ($g \ge 2$); (ii) $\mathcal{L}_4=0 \implies C_{5,5}^{(g)},C_{6,5}^{(g)}$; (iii) $\mathcal{L}_6=0 \implies G_{5,5}^{(a,b)}(u,v)=0$ ($g=4,5$); (iv) $\mathcal{L}_8=0 \implies G_{4,3}^{(8)},K_{3,3}^{(b)}$ birational to $g=2$ $\mathcal{C}_{\text{hyp}}^{(8)} : Y_\lambda^2=4\lambda^6+20\lambda^5-19\lambda^4-26\lambda^3+43\lambda^2-14\lambda+1$ ($R$-cover $\mathcal{C}_{s,\kappa}^{(8)}$ of $g=5$) and $G_{6,3}^{(8)}$ birational to $g=3$ $Q_{U,\mu}(U,\mu)=U^4-(\mu-3)U^3-3\mu(2\mu+3)U^2-2(2\mu^3+6\mu^2+5\mu+2)U+2\mu(2\mu^2+\mu+2)=0$; and (v) on $\mathcal{L}_{f,c}^{(u)}=0$ ($cf+1=\square$), $4(2R-1)^2 R_4(a,c,d_-,f)-D_{\text{br}}^2\mathcal{H}_2 \equiv 2(2R-1)(\tilde{f}-\tilde{b})\mathcal{M}_{f,c}^{(u)}$ splits into $\tilde{f}=\tilde{b}$ ($P_{F_2}(R,v)=0$, $g=3$ $Q_t(R,t)=0$) and $\mathcal{M}_{f,c}^{(u)}=0$, giving $35$-term $F_{6,7}(u,v)=0$ of $g=8$ and $79$-term $F_{12,7}(u,v)=0$. - *(d) Central-$R_5$ Norm, Binomials, $\mathcal{E}_{13,2}$ & $E_z=0$ (`exp44`–`exp46`):* For $R_5(c,d_-,d_+,f,g)=0$ ($\mathcal{A}_1 \widetilde{Z}+\mathcal{A}_0=0,\widetilde{Z}=C_0/r$), $(R-1)(D_{\widetilde{Z}^2}\mathcal{A}_0^2-N_{\widetilde{Z}^2}\mathcal{A}_1^2) \equiv-\frac{(w_1^2-1)^{15}(2R-1)^2 D_u^3 D_v^3}{(u^2-1)^9(v^2-1)^9} P_1 P_2 P_3 P_4$ in $\mathcal{R}_{\text{cent}} \cong \mathbb{Q}[u,v,w_1,(w_1^2-1)^{-1},((u^2-1)(v^2-1))^{-1}]$ (`exp45`; admissible for $abcd_-d_+\ne 0$ distinct). Here $P_2=4(W_0+w_1^2)((w_1^2+1)^2+4W_0)(w_1+u)(w_1+v)-(w_1^2-1)^2(4W_0+3w_1^2+1)(u-v)^2$ and $P_1$ are binomials (`exp46`), $(u-v)^2=\square$ on $P_2=0$, and $w_1=3r$ gives $\mathcal{E}_{13,2} : C_0^2=13r^2-1,Y_2^2=r^2+2$ ($\text{rank}\,\ge 1,F_1(m)=\frac{(m^2-2)^2 C_1(m^2)}{4m^2(m^2-4)(m^2-1)^2},F_2(m)=\frac{C_2(m^2)}{m^2(m^2-4)}$; $m=6$: $\{220/21,161/240,549/560,252/5,475/336\}$). On $x_2=w_1-v$, $r(\tau)=\frac{\tau^2-3}{\tau^2-5},v(\tau)=\frac{\tau}{\tau^2-3},u(\tau)=\frac{\tau^4-3\tau^2-6}{\tau(\tau^2-5)^2}$ gives $F_2=x_2^2$ (`exp44`). The trace quadratic is $$ E_z(r, \mu, z) = -K^3 L - 8 H(K^2 + 4 K H - 8 H^2) A_0 B_0 + 16 H^2 (\mu(\mu + 4 r) A_0^2 + 4(r^2 - 1) H B_0^2) = 0, $$ with $\Delta_z(E_z)=64 r^2 K^4 H (-K L)$ and $(16(r^2-1) H^2 B_0-(K^2+4 K H-8 H^2) A_0)^2+K^3 L(A_0^2-4(r^2-1) H)=4(r^2-1) H E_z$. Setting $z=\frac{t^2-1}{2 t-\mu}$ gives $\Phi_{20}(r,\mu,t)=0$ with $\text{disc}_t(\Phi_{20})=16(\mu^2-4)^2 \Delta_z(E_z)^2 \text{Res}_z(E_z,z^2-\mu z+1)$. 2. **Cubic Weierstrass model $\Sigma_\mu^+$, $2$-torsion involution $\tau_{T_0}$, and Quadratic-Twist Collapse:** Setting $\mu=\frac{\mu_{\text{kiro}}-1}{(r-1)\mu_{\text{kiro}}-(r+1)}$ ($\mu_{\text{kiro}}=\lambda^2$) sends $\mu_{\text{kiro}}=\frac{r+1}{r-1}$ (`Post 35`) to $\mu=\infty$, transforming $\Sigma_\mu$ into $$ \Sigma_\mu^+ : \quad W_+^2 = -K(r, \mu) L(r, \mu) = (2 r(2 r^2 - 1)\mu - 1)(-8(r^2 - 1)\mu^2 + 2 r(2 r^2 + 7)\mu - 9), $$ while $H(r,\mu)=((r-1)\mu-1)^2 \lambda^2=\square$. Translation by $T_0=(\frac{1}{2 r(2 r^2-1)},0)$ on $\Sigma_\mu^+$ is $\tau_{T_0}(\mu)=-\frac{\mu+4 r}{K(r,\mu)} \iff \lambda \, \lambda_T=\pm \frac{2 r+1}{2 r-1}$, fixing $c=z/w$. Along $\mu(r) \in \mathbb{Q}(r)$, $H,-K L=\square$ collapse via: (i) **untwisted sections** $(\mu(r),W_+(r)) \in \Sigma_\mu^+(\mathbb{Q}(r))$ ($-K L=W_+^2$); and (ii) **$H$-twisted sections** ($-H K L=\square$ in $\mathbb{Q}(r)$, splitting $E_z=0$). 3. **Three $1$-parameter families of $15/15$ sextuples and $19/21$ septuples on $\mathcal{S}_{4,2}$ (`exp13`, `exp16`):** - **Curve I ($\mathbb{Q}(m)$, genus $0$, $E_2=0 \implies S_0=\infty$ boundary):** $2(\lambda-1)^2 r^2-(\lambda^2-1) r+2 \lambda=0$, parameterized by $r(m)=\frac{(m-1)(m+3)}{2(m^2+3)},\; \lambda(m)=-\frac{m(m-1)}{m+3}$ (`exp13`). - **Curve II ($\mathbb{Q}(k)$, genus $0$, positive $19/21$ septuples):** Generated by $\mu_{\text{dual}}(r)=\frac{1}{2 r}$ and $\mu_{\text{II}}(r)=\tau_{T_0}\!(\frac{1}{2 r})=\frac{8 r^2+1}{4 r(r^2-1)}$ ($-H K L=\square$, $H \sim r^2-1=\square$). With $r(k)=\frac{k^2+1}{k^2-1}$, $\lambda(k)=-\frac{k(3 k^2+1)}{k^2+3}$, $q_1(k) :=k^4-4 k^3-2 k^2-4 k+1$, $q_2(k) :=q_1(-k)$, $q_3(k) :=3 k^4-4 k^3+10 k^2-4 k+3$, $q_4(k) :=q_3(-k)$, $a_1(k)$ yields the $19/21$ septuple of type $(4,2)$: $$ \begin{aligned} a_1(k) = -\frac{2 k \, q_1(k) q_3(k)}{(k^2 - 1) q_2(k) q_4(k)}, \quad b_1(k) = -\frac{2 k \, q_2(k) q_4(k)}{(k^2 - 1) q_1(k) q_3(k)}, \quad d_-(k) = -\frac{(k - 1)^3(3 k^2 + 2 k + 3) q_2(k) q_4(k)}{2 k (k + 1)^3 q_1(k) q_3(k)}, \quad d_+(k) = -\frac{(k + 1)^3(3 k^2 - 2 k + 3) q_1(k) q_3(k)}{2 k (k - 1)^3 q_2(k) q_4(k)}, \end{aligned} $$ with collisions at $k \in \{\pm 3,\pm 1,\pm 1/3,0\}$. On $(1,k_*)$ ($k_* \approx 2.026846$, $3 k^4-4 k^3-2 k^2-4 k-1=0$, Sturm-certified in $(2,203/100)$), **all $7$ entries are positive, distinct, and nonzero for $k \in (1,k_*)$** ($\mathcal{S}_{7,\text{pos}}^{(19/21)}(2)$ at $k=2$). - **Curve III ($\mathcal{E}_{\text{III},\pm}$, genus $1$, Mordell–Weil family):** In $\Sigma_\mu^+(\mathbb{Q}(r))$, $P_0+P_1=(0,3)+(\frac{1}{r-1},\frac{(r+1)(2r-1)^2}{r-1})$ gives $\mu_{\text{III},+}(r)=\frac{8 r^2-8 r+3}{2 r(2 r^2-1)}$, where $-K L=(\frac{2(2 r-1)^2(2 r^3-2 r+3)}{r(2 r^2-1)})^2$ and $H=\square$ collapses to $\mathcal{E}_{\text{III},+} : Y^2=(r-3)(4 r^3-3 r+3),\; \lambda=\pm \frac{Y}{(r-3)(2 r-1)}$ ($G_1=(-3,24),G_2=\infty_+$; $P_0-P_1$ gives $\mathcal{E}_{\text{III},-}$; non-boundary at $(101/28,4097/392)$ in `verify_paper.py`). *Proof.* Verified in `verify_paper.py`, `exp13`–`exp16`, `exp42`–`exp62`. $\square$ --- ## 7. Factor cancellation on $w=2 u^2 y-L$ and the moving-$u$ quintuple family $\mathcal{E}_B$ In Paper 65 [16], `u/codex_arithmetic` introduced $(u,y,w)$ for four regular quintuples with $p=u^2-1,q=y^2-1,J=p q-1,L=J+2 u^2,F=4 u^2 y^2-J^2,U=u y(u-y),V=u^2+y^2,D=u y-u+y,Z=(J-2 u y) D^2$, leaving open moving-$u$ families ($u \ne-2$). On $w=2 u^2 y-L$, $E=(U-V) w^2-(U+V) Z$ factors to collapse $(b,c,d)$ to degree $4$, and $\mathcal{B}_+$ collapses $\alpha^2,\beta^2=\square$ to $\mathcal{E}_B$. ### Theorem 5 (Factor Cancellation on $w=2 u^2 y-L$, Closed-Form $\mathbb{Q}(k)$ Quintuples, and the Moving-$u$ Elliptic Family $\mathcal{E}_B$) 1. **Factor cancellation on $w=2 u^2 y-L$:** Setting $w=2 u^2 y-L=-(u y-u-y)(u y-u+y)$ enforces $T=\frac{w+F/w}{2}=2 u^2 y$ and $R=\frac{w-F/w}{2 J}=-\frac{L}{J}$. Here $E=(U-V) w^2-(U+V) Z=-4 u^3 y (u y-u-y)(u y-u+y)^2=4 u^3 y D w$, cancelling $(u y-u-y) D^2$ from $(b,c,d)$ and collapsing them to degree $4$: $$ b = \frac{2 u^3 y}{(u + y) J}, \quad c = \frac{2 y(u^2 - 1)(u + y)}{u J}, \quad d = \frac{(u^2 y^2 - (u + y)^2) J}{2 u^3 y (u + y)}. $$ 2. **The four quadratics and their universal syzygies:** With $C=b+c$, $A=\frac{J^2 C+2 T L}{4 u^2 y^2}$, and $N_a c=\frac{T+L c}{2 u y}$, the $(a,e)$-window gates $\alpha^2=2+c A+2 N_a c$ and $\beta^2=2+c A-2 N_a c$ ($r_{ac}=\frac{\alpha+\beta}{2},r_{ec}=\frac{\alpha-\beta}{2}$) factor in $\mathbb{Q}(u,y)$ as $$ \alpha^2 = \frac{(u + 1)^2}{u^4} \frac{Q_0(u, y) Q_-(u, y)}{J(u, y)}, \quad \beta^2 = \frac{(u - 1)^2}{u^4} \frac{Q_0(u, y) Q_+(u, y)}{J(u, y)}, $$ where $J(u,y)=(u^2-1) y^2-u^2$, $Q_0(u,y)=(u^2-1)(y+u)^2-u^4$, and $Q_\mp(u,y)=((u \mp 1) y+u^2)^2-u^2(u^2-1)$ obey $Q_0-J=2 u y(u^2-1),Q_+-Q_-=4 u y(u+y),Q_++Q_--2 Q_0=4(u^2+u y+y^2)$, and under $u=\frac{z^2+1}{2 z},s=\frac{z^2-1}{2 z},y=\frac{u}{s}\xi$, satisfy $z(\frac{u}{z-1}\beta)^2-z(\frac{u}{z+1}\alpha)^2=(\xi+s)^2-u^2$. 3. **Conic bisection $\mathcal{B}_+$, closed-form $\mathbb{Q}(k)$ quintuple, and moving-$u$ family $\mathcal{E}_B$:** Along $\mathcal{B}_+: (u+1) y^2+2 u^2 y+u^2=0$, parameterized over $\mathbb{Q}(k)$ by $u(k)=\frac{k^2+1}{2 k-1},y(k)=-\frac{k^2+1}{k+2}$, we have $Q_+(u,y)=-u^3$ and $Q_-(u,y)=-(\frac{3 k-4}{k+2})^2 u^3$, so $Q_-/ Q_+=(\frac{3 k-4}{k+2})^2$, $k^2+1$ cancels from $Q_0 / J$, and **both** gates $\alpha^2,\beta^2=\square$ collapse to a single rank-$1$ curve: $$ \mathcal{E}_B : \quad Y_B^2 = (k - 2)(5 k^3 - 8 k^2 + 13 k - 2) \quad (\cong V^2 = U^3 + 41 U^2 + 704 U + 5120, \;\; U = \frac{32}{k - 2}, \;\; V = \frac{32 Y_B}{(k - 2)^2}), $$ with $\alpha(k)=\frac{k(3 k-4) Y_B}{(k-2)(2 k-1)(k^2+1)}$ and $\beta(k)=\frac{(k^2-2 k+2) Y_B}{(k-2)(2 k-1)(k^2+1)}$. Substituting $Y_B^2=(k-2)(5 k^3-8 k^2+13 k-2)$ into $a,e=\frac{((\alpha \pm \beta)/2)^2-1}{c}$ eliminates $Y_B$, expressing all $5$ entries in $\mathbb{Q}(k)$: $$ a(k) = \frac{2(k^2 - 2 k + 2)}{k - 3}, \quad e(k) = \frac{k(k - 3)(k + 2)}{2(2 k - 1)^2}, \quad b(k) = \frac{2(k + 2)}{(k - 3)(k - 2)}, \quad c(k) = \frac{2 k(k - 3)(k^2 - 2 k + 2)}{(k - 2)(k^2 + 1)^2}, \quad d(k) = \frac{(k - 2)(k - 1)(k^2 - k + 4)}{2(k - 3)}. $$ Six of the $10$ pair radicands (within $\{a,e,d\}$ and $\{b,c,d\}$) are squares in $\mathbb{Q}(k)$, while $\{a,e\} \times \{b,c\}$ are squares in $\mathbb{Q}(k)$ times $Y_B^2$: $$ a b + 1 = \frac{Y_B^2}{(k - 3)^2(k - 2)^2}, \quad a c + 1 = \frac{(k - 1)^2 Y_B^2}{(k - 2)^2(k^2 + 1)^2}, \quad e b + 1 = \frac{Y_B^2}{(k - 2)^2(2k - 1)^2}, \quad e c + 1 = \frac{(k^2 - k - 1)^2 Y_B^2}{(k - 2)^2(2k - 1)^2(k^2 + 1)^2}, $$ and $R_4(a(k),e(k),b(k),c(k)) \ne 0$ factors in $\mathbb{Q}[k]$. For $k \in \mathbb{Q} \setminus \{-2,-1,-1/3,0,1/2,1,4/3,2,3\}$, every $\pm Y_B$-pair on $\mathcal{E}_B(\mathbb{Q})$ gives a quintuple $\{a(k),e(k),b(k),c(k),d(k)\}$ ($d(k) > 0$ for $k > 3$). On $V^2=U^3+41 U^2+704 U+5120$ ($\Delta=-2^{14}\cdot 5^2\cdot 53$), $\#\mathcal{E}_B(\mathbb{F}_3)=5,\#\mathcal{E}_B(\mathbb{F}_{11})=16 \implies \mathcal{E}_B(\mathbb{Q})_{\text{tors}}=\{\mathcal{O}\}$. Over $K_3=\mathbb{Q}(\alpha)$ ($\alpha^3-\alpha^2+4\alpha-2=0,\Delta=-212,h(K_3)=1,\varepsilon=2\alpha-1$), $U_0=-\alpha^2-6\alpha-14$ has $f_U'(U_0)=\alpha^{14}\pi_{5,2}\pi_{53,1}\varepsilon^{-5}$ ($\pi_{5,2}=\alpha^2+\alpha+1$); mod-$5$ filtering leaves $\text{Sel}^{(2)}(\mathcal{E}_B/\mathbb{Q})=\langle\pi_{5,2}\rangle \implies \text{rank}\,\mathcal{E}_B(\mathbb{Q})=1$. Since $\delta(P_0) \equiv \pi_{5,2}$ and $\text{ord}_{\mathbb{F}_{19}}(P_0)=21$ at $P_0=(-8,-40) \mapsto k=-2$ ($2P_0 \mapsto 0,3P_0 \mapsto 3,4P_0=(16,-176) \mapsto 4,6P_0=(-5,50) \mapsto-22/5$), $\mathcal{E}_B(\mathbb{Q}) \subset \mathbb{Z}_{(6)} P_0$, giving **infinitely many moving-$u$ quintuples** (`exp15`; $k=4$: $\{20,12/49,6,40/289,48\}$). For $G(k)=(k-2)(k^5-4k^4+17k^3-46k^2+64k-56)$ and $H(k)=(k^2-4)Q_4(k)$ ($Q_4=k^4-2k^3+k^2-4k-4$), $Q_4(k) \equiv-2 \pmod{\mathbb{Q}_2^{\times 2}}$ requires $v_2(U)=0,U \equiv 7 \pmod 8$; on $\mathbb{Z}_{(6)} P_0$, $v_2(U(nP_0))=0 \iff n \equiv 6 \pmod{12}$, where $U(6P_0+12m P_0) \equiv 3 \pmod 8$, proving $G(k(P)) \in \mathbb{Q}_2^{\times 2} \iff P \in (3+6\mathbb{Z}_{(6)}) P_0$ ($k \equiv 3 \pmod 8,Q_4(k) \equiv+5 \pmod{\mathbb{Q}_2^{\times 2}}$) and **eliminating $Q_4(k) \equiv-2 \pmod{\mathbb{Q}^{\times 2}}$** (`exp38`; [16, §8]). *Proof.* Verified in `verify_paper.py`, `exp15`, `exp38`. $\square$ --- ## 8. Factorization of $R_5$, Triple-Square Section $L_1=0$, $15+6$ Collapse, and $v=-1/5$ Families Following `u/kiro_dioph7`'s mod-$10007$ observation (`Post 35`) that eliminating $S_{ab}$ between $\text{eq}_{S_{ab}}=0$ and $R_5=0$ factors into $8$ components, we prove this **over $\mathbb{Q}$** in $(r,\mu,z)$, showing $L_1(r,\mu,z)=0$ forces $c c^*_-+1,c c^*_++1,c^*_-c^*_++1=\square$ in $\mathbb{Q}(r,\mu)$ **without $E_z=0$**. ### Theorem 6 (Triple-Square Section $L_1=0$, $15+6$ Collapse, Factored $(a,b)$-Split Law & $v=-1/5$ Families) 1. **Over-$\mathbb{Q}$ factorization of $R_5(d_-,d_+,c,c^*_-,c^*_+)$ & Section $L_1(r,\mu,z)=0$:** In $(r,\mu,z)$, $P_4-1=-\frac{K}{H}$, $P_4-9=\frac{L}{H}$, $B_0^2-\mu(\mu+4r)=4r^2((r^2-1)z^2+A_0 z+H)$, $w^2(d_-d_++1)=(r\mu+1)^2$, $(cd_-+1)(cd_++1)=(\frac{Tz-H}{H})^2$, $\frac{c^*_-}{w}=z-\mu$, and $\frac{c^*_+}{w}=\frac{(4r^2-1)((4r^2-1)z+\mu+4r)}{K^2}$. Before $E_z=0$, $R_5(d_-,d_+,c,c^*_-,c^*_+)$ factors in $\mathbb{Q}(r,\mu,z)$ as $$ R_5(d_-, d_+, c, \; c^*_-, c^*_+) = \frac{L_1(r, \mu, z) \cdot L_2(r, \mu, z) \cdot Q_4(r, \mu, z)}{H(r, \mu) \, K(r, \mu)^4}, $$ ($\deg_z(Q_4)=4$), where $L_1,L_2$ are linear in $z$ and swapped by $\tau_{T_0}(\mu)=-\frac{\mu+4r}{K}$: $$ L_1(r, \mu, z) := (4r^2 - 1)\mu z + (2r^2 - 1)^2\mu^2 - 1 = 0, \quad L_2(r, \mu, z) := (4r^2 - 1)(\mu + 4r)z + (2r^2 - 1)^2\mu^2 - 8\mu r^3 + 4\mu r - 16r^4 + 12r^2 - 1 = 0. $$ Along $z=z_{L_1}(r,\mu) :=\frac{1-(2r^2-1)^2\mu^2}{(4r^2-1)\mu}$ (and $L_2=0$), **all $3$ residual gates hold in $\mathbb{Q}(r,\mu)$ without $E_z=0$**: $$ c \, c^*_- + 1 = (\frac{4\mu^2 r^4 - 2\mu^2 r^2 + 1}{\mu(4r^2 - 1)})^2, \quad c \, c^*_+ + 1 = (\frac{4\mu^2 r^4 - 2\mu^2 r^2 - 2\mu r - 1}{\mu K})^2, \quad c^*_- \, c^*_+ + 1 = (\frac{4\mu^3 r^5 - 4\mu^3 r^3 - 4\mu^2 r^4 - 2\mu^2 r^2 + \mu r + 1}{\mu K})^2. $$ Eliminating $z$ between $E_z=0$ and $R_5=0$ on $\mathcal{S}_{4,2}$ gives $6$ factors over $\mathbb{Q}[r,\mu]$ ($8$ over $\mathbb{Q}[r,\lambda]$; `exp17`). Under $z=\frac{t^2-1}{2t-\mu}$ and $(R,v,u) :=(r^2,r\mu,\frac{t-\mu}{r})$, $(2t-\mu)^4 Q_4=F_a F_b$ where $F_a=0$ is $Q_{4,+}(R,v,u)=0$ with $\text{disc}_v(Q_{4,+})=4(2Ru-1)^2 D_4(R,u)$, $D_4=((R-1)u^2+2u-1)(R(4R^2-8R+1)u^2+8R^2 u-(2R-1)^2)$. On $Q_{4,+}=0$, all $3$ residual gates and $\Delta_d^2$ are squares, giving interior $10/10$ quintuples at $(r,u,v)=(\frac{3}{4},-4,-\frac{9}{28})$, $(\frac{7}{4},\frac{44}{67},\frac{2009}{3886})$ ($\{\frac{1485}{31423},\frac{7}{11},\frac{760}{693},\frac{176}{63},\frac{2296}{99}\}$), and $(\frac{19}{10},\frac{20}{57},\frac{19}{45})$ (`exp24`). Furthermore (`exp23`–`exp27`): - **Gauge Involution $\iota_K$, Quotient $\mathcal{E}'_{Q_4}(R)$ & Reciprocal $3P_0$ Law:** Before $E_z=0$, $\iota_K: (r,\mu,z,w) \mapsto (r,-\frac{\mu+4r}{K},\frac{4r^2-1}{K}z,\frac{4r^2-1}{K}w)$ fixes $(a,b,c,d_-,d_+)$, swaps $c^*_-\leftrightarrow c^*_+$ and $L_1 \leftrightarrow L_2$, and preserves $Q_{4,+}=0$ via $\tau_{T_0}: u \mapsto \frac{u-2}{2Ru-1}$ ($\sigma :=\frac{Ru^2-1}{2Ru-1}$). On $E(R): Y_E^2=X(X^2+2(8R^3-8R^2+6R-1)X-(4R-1)^3)$, $\tau_{T_0}$ is translation by $T_0=(0,0)$, with quotient $\mathcal{E}'_{Q_4}(R): Y_\sigma^2=((R-1)\sigma+1)((4R^2-8R+1)\sigma+4R)(R\sigma^2-\sigma+1)$. Along $\mu=\frac{2r}{r^2-1}$ ($w=1$), $d_-+d_+=2B_0$ forces $d_\pm=d_{3P_0}(d_\mp,c,c^*_-) :=\frac{8(s_1^3-4s_1 s_2+8s_3)}{(s_1^2-4s_2)^2}$, so at $(\frac{7}{4},-\frac{4}{9},\frac{98}{33})$, $Q_{d_\pm}+S_T=\pm 3P_0$. - **Rank-$2$ Subgroup $\Lambda(r) \subset E(r^2)(\mathbb{Q}(r))$, $Z_-(r)$ & $2$-Isogeny Exclusions:** Over $\mathbb{Q}(r)$ ($R=r^2$), $F_1(r^2,u)=((r+1)u-1)((r-1)u+1)$ gives $W_1(r) \mapsto (-(2r-1)(2r+1)^2,-2r(2r-1)(2r+1)^2(2r^2-1))$ and $W_2(r)=W_1(r)+T_0$ with $2W_1(r)=P_{\text{even}}(r^2)+T_0=O_-$ ($X(2W_1)=\frac{(4r^2-1)^2}{(2r^2-1)^2}$), while the dual $2$-isogeny on $\hat{E}(r^2)$ gives $P_{\text{odd}}(r)=((4r^2-1)^2,4r^3(4r^2-1)^2) \in E(r^2)(\mathbb{Q}(r))$ with **$Z_-(r) :=(1/r,2(2r-1))=W_1(r)+P_{\text{odd}}(r)$**. Under $r \mapsto-r$, $\Lambda(r) :=\mathbb{Z}W_1(r) \oplus \mathbb{Z}P_{\text{odd}}(r) \oplus (\mathbb{Z}/2\mathbb{Z})T_0 \cong \mathbb{Z}^2 \oplus \mathbb{Z}/2\mathbb{Z}$ ($E(r^2)(\mathbb{Q})_{\text{tors}}=\{\mathcal{O},T_0\}$). For $v_\pm(u,Y_u)$ of $Q_{4,+}=0$: (1) at $u=2$, $T_0,2W_1$ map $H(r)=w^2$ via $X=1-4r^2$ to $E_{48}^-: y^2=X(X-1)(X+3)$ of rank $0$ ($r \in \{0,\pm 1/2,\pm 1\}$); and (2) on $3W_1(r),3W_1+T_0$, $H(r)=w^2$ maps via $X=4r^2-1$ to $E_{80}: y^2=X(X-1)(X-5)$ of rank $0$ ($r=\pm 1/2$). - **Joint Gate Surface $G_u(R,d_0)=n^2$ & $u=2/3$ Conic:** Setting $k^2=\frac{(4R^2-8R+1)\sigma+4R}{(R-1)\sigma+1}=4R+d_0^2-4$ makes $v,\Delta_d,\Delta_{ab} \in \mathbb{Q}$ and reduces $u \in \mathbb{Q}$ to $G_u(R,d_0) :=R^2 d_0^4+(16-6d_0^2)R+(d_0^2-3)=n^2$ (`29` points and $M_{2,\pm}(r)$ in `exp23`), and along $rd_0=3$ ($u=2/3$), $Q_{4,+}=0$ splits over $Y_k^2=4R^2-4R+9$. 2. **Decoupling $L_1=0$ from $\mathcal{S}_{4,2}$, $15+6$ Collapse & Factored $(a,b)$-Split Law:** For **any** $(r,\mu,z)$ with $H(r,\mu)=w^2$, $\Delta_{ab}^2 :=A_0^2-4(r^2-1)H$, and $\Delta_d^2 :=B_0^2-\mu(\mu+4r)$, $\{a,b\} \times \{c,d_-,d_+\}$ obey modulo $\Delta_{ab}^2,\Delta_d^2$: $$ (ac + 1)(bc + 1) = (\frac{\Delta_d}{2rw})^2, \quad w^6(ad_\mp + 1)(bc + 1) = (\frac{A_0 + \Delta_{ab}}{2}(\frac{A_0 - \Delta_{ab}}{2}z + H) \mp \frac{H\Delta_d}{2})^2. $$ On $z=z_{L_1}(r,\mu)$ in $(R,v)=(r^2,r\mu)$ ($r^2 H=(v-1)^2 R-v^2=(rw)^2$, $r(\lambda)=\frac{\lambda^2+1}{\lambda^2-1}\frac{v}{v-1}$), $\Delta_{ab}^2=\frac{R \, \mathcal{F}_{ab}(R,v)}{(4R-1)^2 v^2}$ and $\Delta_d^2=\frac{4R^2 \, \mathcal{G}_d(R,v)}{(4R-1)^2 v^2}$, where $$ \begin{aligned} \mathcal{F}_{ab}(R, v) &= v^3(5v - 4)(4R - 1)^2 - 2v(v^2 - 1)(5v + 2)(4R - 1) + (v^2 - 1)^2, \\ \mathcal{G}_d(R, v) &= 16v^4 R^3 - 8v^2(4v + 1)R^2 + (-16v^4 + 8v^3 + 16v^2 + 8v + 1)R + 2v(2v^3 - 2v - 1). \end{aligned} $$ On $rv(v^2-1)(4R-1)KH\Delta_d \ne 0$ with all entries nonzero and distinct, without $E_z=0$: - **Whenever $\mathcal{G}_d(R,v)=\square$ & $H=w^2$:** $\{c,d_-,d_+,c^*_-,c^*_+\} \subset \mathbb{Q}$ is a **$10/10$ quintuple** ($R_5(d_-,d_+,c,c^*_-,c^*_+)=0$), giving at $(r,\lambda,v)=(\frac{7}{4},\frac{3}{11},\frac{98}{163})$, $(\frac{5}{3},\frac{13}{11},-\frac{8}{21})$, $(\frac{53}{30},\frac{2}{7},\frac{3}{5})$ three $10/10$ quintuples (`exp19`). On $\mathcal{G}_d=W_d^2$, $P_1(v)=(\frac{1}{4v^2},2(v^2-1))$ gives $R(8P_1(3/5))=(53/30)^2$, where $Q_{e_2}=P_0+2Q_c+S_T$ ($e_2=-\frac{4580748}{6812155}$) forms the **$3$-fifth $18/21$ septuple** $\{d_-,d_+,c^*_-,c^*_+,c,c',e_2\}$. - **Universal $(r,M)$ completions, $(v,t)$ blow-up & Factored $(a,b)$-Split Law:** In $(r,M)$ ($v=\frac{r}{r-M},M=\frac{\lambda^2+1}{\lambda^2-1},M^2-N^2=1,N=\frac{2\lambda}{\lambda^2-1}$), $(r-M)^4\mathcal{F}_{ab}(r^2,v)$ and $(r-M)^4\mathcal{G}_d(r^2,v)$ obey: $$ \begin{aligned} (r - M)^4 \mathcal{F}_{ab}\!(r^2, \frac{r}{r - M}) &= (-M^2 + 8r^3 M + 4r^4 - 9r^2 + 2)^2 - 4(r^2 - 1)(4r^2 - 1)^2(M^2 - 1), \\ (r - M)^4 \mathcal{G}_d\!(r^2, \frac{r}{r - M}) &= (r M^2 - (6r^2 - 1)M - \frac{8r^6 - 10r^4 - 2r^2 + 1}{2r})^2 + (\frac{4r^2 - 1}{2r})^2(4rM - (4r^2 + 1)), \end{aligned} $$ so $\mathcal{F}_{ab}=\square$ on $M^2-N^2=1$ factors as $4kr^3(2M+4r^3-5r)=(N+k(4r^2-1)(r-1))(kN+(4r^2-1)(r+1))$, while $W_{ab}=(v^2-1)-v^2 t(4R-1)$ along $v(t)=\frac{2(t-7)}{t^2+4t-13}$ gives $\mathcal{E}_{\text{tan}} : Y_t^2=(t^2+30t+253)(9t^2+14t-27)$ (`exp20`). - **Whenever $\mathcal{F}_{ab}(R,v)=\square$ & $H=w^2$:** $\{a,b,c,c^*_-,c^*_+\} \subset \mathbb{Q}$ satisfies $8/10$ conditions, and $Q_{ab}=\{a,b,c^*_-,c^*_+\}$ has square $6$-pair product on $L_1=0$, so $e_{ab,\mp} \in \mathbb{Q}$ (`18` $10/10$ quintuples, `81` $14/15$ sextuples, `180` $18/21$ septuples at $9$ rational $(v,r)$ in `verify_paper.py`). - **Whenever $\mathcal{F}_{ab},\mathcal{G}_d=\square$:** **$15/21$ pairs of $\{a,b,c,d_-,d_+,c^*_-,c^*_+\}$ are squares**, and the **$6$ remaining radicands** $ac+1,bc+1,ad_\mp+1,bd_\mp+1$ **share one square class in $\mathbb{Q}^\times / \mathbb{Q}^{\times 2}$**. 3. **Closed-form $\mathbb{Q}(\lambda)$ & $\mathbb{Q}(u)$ families at $v=-1/5$, and rank-$0$ / torsion proof on $\mathcal{E}_{16}$:** Since $\text{disc}_R(\mathcal{F}_{ab})=256v^2(v-1)^2(v+1)^3(5v+1)$, setting $v=-1/5$ ($r(\lambda)=\frac{\lambda^2+1}{6(\lambda^2-1)}$) makes $\mathcal{F}_{ab}(R,-1/5)=\frac{(20R-29)^2}{625}$ a square in $\mathbb{Q}(\lambda)$, yielding: $$ \begin{aligned} a(\lambda) &= -\frac{\lambda(5\lambda^2 - 7)(7\lambda^2 - 5)}{8(\lambda^2 - 1)(\lambda^2 - 2)(2\lambda^2 - 1)}, \quad b(\lambda) = \frac{2(\lambda^2 - 2)(2\lambda^2 - 1)}{9\lambda(\lambda^2 - 1)}, \quad c(\lambda) = \frac{(\lambda^4 - 19\lambda^2 + 16)(16\lambda^4 - 19\lambda^2 + 1)}{72\lambda(\lambda^2 - 1)(\lambda^2 - 2)(2\lambda^2 - 1)}, \\ c^*_-(\lambda) &= -\frac{2\lambda(7\lambda^2 - 8)(8\lambda^2 - 7)}{25(\lambda^2 - 1)(\lambda^2 - 2)(2\lambda^2 - 1)}, \quad c^*_+(\lambda) = -\frac{2\lambda(\lambda^2 - 2)(2\lambda^2 - 1)(4\lambda^2 - 5)(5\lambda^2 - 4)}{(\lambda^2 - 1)(7\lambda^4 - 13\lambda^2 + 7)^2}. \end{aligned} $$ The regular fifths $e_{ab,\mp}(\lambda)$ of $\{a(\lambda),b(\lambda),c^*_-(\lambda),c^*_+(\lambda)\}$ lie in $\mathbb{Q}(\lambda)$: $$ e_{ab, -}(\lambda) = \frac{(\lambda^2 - 1)^2 P_{e1}(\lambda) P_{e2}(\lambda)}{8\lambda(\lambda^2 - 2)(2\lambda^2 - 1) Q_e(\lambda)^2} \in \mathbb{Q}(\lambda) \quad (\deg 13/12, \text{ `THEOREMS_AND_PROOFS.md`}), $$ and $e_{ab,+}(\lambda)=-e_{ab,-}(-\lambda)$ ($2$ $\mathbb{Q}(u)$-families of $10/10$ quintuples, $9$ of $14/15$ sextuples, $20$ of $18/21$ septuples on $W_0^2=36r^2-1$ in `exp21`). For $\lambda \in \mathbb{Q} \setminus \{-1,0,1\}$, $\{a(\lambda),b(\lambda),c^*_-(\lambda),c^*_+(\lambda),e_{ab,-}(\lambda),e_{ab,+}(\lambda)\}$ is a **$14/15$ sextuple**, **positive** on $(\sqrt{7/8},1) \approx (0.9354,1)$ ($\lambda=15/16$ gives a positive $10/10$ quintuple). Finally, $b(\lambda)c(\lambda)+1=\square$ maps to $E_{16} : Y^2=X(X^2-31X+256)$ with dual $\hat{E}_{16} : \hat{Y}^2=\hat{X}(\hat{X}-1)(\hat{X}+63)$. $2$-isogeny descent mod $9,16$ proves $\text{rank}\,E_{16}(\mathbb{Q})=0$ and $E_{16}(\mathbb{Q}) \cong \mathbb{Z}/8\mathbb{Z}=\langle (32,96) \rangle$, so $\mathcal{E}_{16} : Y_\lambda^2=16\lambda^4-31\lambda^2+16$ has **exactly $8$ rational points** ($\{(0,\pm 4),(\pm 1,\pm 1),\infty_\pm\}$). *Proof.* Verified in `verify_paper.py`, `exp17`–`exp27`. $\square$ --- ## 9. $(V,K,d)$ Chains, $\mathcal{S}_{\text{3-tors}}$, Orbit Cubic, $\mathcal{C}_{\text{triad}}$, Mixed-$R_4$ Syzygies, and Branch-B Surface We analyze contiguous $4$-term chains $x(P_0+n S)$ on $E_T : Y^2=(X+r^2-1)(X+s^2-1)(X+t^2-1)$ ($X=\Pi x$) for triples $T=\{a,b,c\}$ ($\Pi=a b c,\Omega=r s t$), combining [7, 10, 17] with $(V,K,d)$ on $\mathcal{S}_{\text{3-tors}}$ ($3 S_T=\mathcal{O}$). ### Theorem 7 (Universal $(V,K,d)$ Coordinates, the $3$-Torsion Inflection Surface $\mathcal{S}_{\text{3-tors}}$, the Eisenstein Triad Curve $\mathcal{C}_{\text{triad}}$, Mixed-$R_4$ Syzygies, and Double-Product Norm $2$-Descent) Let $T=\{a,b,c\}$ be any Diophantine triple with $d_\mp=x(P_0 \pm S)=a+b+c+2 \Pi \mp 2 \Omega$ and $W_\mp :=\Pi d_\mp-1$. 1. **Shift-invariant product and square completion of $F_2(z)$:** For any $Q \in E_T$ with $z=x(Q)$, $x(Q-S) x(Q+S)=\frac{(z-d_+)(z-d_-)}{(\Pi z-1)^2}$, so $x(Q-S) x(Q+S)+1=\frac{F_2(z)}{(\Pi z-1)^2}$ where $F_2(z) :=(\Pi z-1)^2+(z-d_+)(z-d_-)$ satisfies $$ W_-^2 \, F_2(z) = (W_-^2 + (\Pi W_- - 2 \Omega)(z - d_-))^2 + (z - d_-)^2 D_4, $$ with $D_4=W_-W_+-4 \Omega^2=4 \Omega^2 d$ and $d :=X(2 S)-1=\frac{W_-W_+}{4 \Omega^2}-1=-\frac{(\alpha+\beta+\gamma)(\alpha+\beta-\gamma)(\alpha-\beta+\gamma)(-\alpha+\beta+\gamma)}{4 \alpha^2 \beta^2 \gamma^2}$ in $(\alpha,\beta,\gamma)=(1/r,1/s,1/t)$. 2. **Chain coordinates and $4$-term chains:** In $u_0 :=\frac{W_-}{2 \Omega}$ ($V :=u_0^2$), $K :=W_--d=\Pi d_--X(2S)$, and $d=X(2S)-1$, $\Pi=\frac{d+1-u_0^2}{2 u_0}$, $\Omega=\frac{K+d}{2 u_0}$, and for all $m,n \in \mathbb{Z} \setminus \{0\}$, $x_m x_n+1=\frac{(d+1-V)^2+4 V X_m X_n}{(d+1-V)^2}$, where $X_n=\Pi x(P_0+n S)$ are: $$ X_1 = K + d + 1, \quad X_2 = \frac{K + V}{V}, \quad X_3 = \frac{d(K(V - 1) + V d)}{K^2}, \quad X_4 = \frac{(K^2 + K - V d)(K(d + 1) - V d)}{(K - V d)^2}. $$ For $\mathcal{S}_7^{(1,2,3,4)}=\{a,b,c,d_-,x_2,x_3,x_4\}$, $18/21$ conditions hold, and $d_-x_3+1,x_2 x_4+1=\square$ reduce to $\text{num}_{19}(V,K,d),\text{num}_{20}(V,K,d)=\square$. 3. **The $3$-Torsion Criterion ($3 S_T=\mathcal{O}$), Tate Form, Orbit Cubic & Sub-Quadruple Identity:** By [7, 10, 17], for $Q=(\Pi x_4,Y_4) \in E_T$ ($p_4=\Pi x_4 \ne 1$), $(p_4-1)^2(X-x(Q-S_T))(X-x(Q+S_T)) \equiv R_5(a,b,c,x_4,X)$ on $E_T$, and for any regular $10/10$ quintuple $Q_5=(x_1,\dots,x_5)$ ($T_{ij}=Q_5 \setminus \{x_i,x_j\},Q_j=Q_i+S_{T_{ij}}$), $\tau_j(x_j)=x(Q_i-S_{T_{ij}})=0 \iff x_i \in \{d_-(T_{ij}),d_+(T_{ij})\}$. In $U :=X-1=\Pi x-1$, $E_T$ is $Y^2=(U+r^2)(U+s^2)(U+t^2)$ with $S_T=(0,\Omega)$ and $\lambda_0 :=\frac{r^2 s^2+r^2 t^2+s^2 t^2}{2 \Omega}$, so $3 S_T=\mathcal{O} \iff 2 S_T=-S_T \iff d=0$ **iff**: $$ \lambda_0^2 - (r^2 + s^2 + t^2) = -\frac{(r s + r t + s t)(r s + r t - s t)(r s - r t + s t)(-r s + r t + s t)}{4 r^2 s^2 t^2} = 0 \iff \pm \frac{1}{r} \pm \frac{1}{s} \pm \frac{1}{t} = 0. $$ Let $\frac{1}{r}+\frac{1}{s}+\frac{1}{t}=0$ ($r s+r t+s t=0$), so $\lambda_0 :=r+s+t=-\frac{r^2 s^2+r^2 t^2+s^2 t^2}{2 \Omega}$ obeys $\lambda_0^2=r^2+s^2+t^2$. Then (`exp28`–`exp32`): - **Tate Form, Pythagorean Base-Orbit Law & Torsion $\mathbb{Z}/6\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$:** On $3 S_T=\mathcal{O}$ [7], $E_T$ has Tate form $Y^2=U^3+(\lambda_0 U-\Omega)^2$, $\Pi^2+1=(\Omega+\lambda_0)^2$, $R_0=(0,-\Omega)=-S_T$, $P_0+\langle S_T \rangle=\{P_0,P_0-S_T,P_0+S_T\}$ has nonzero abscissas $\{d_-,d_+\}$ obeying $U(d_\mp)=2\Omega(\Omega+\lambda_0 \mp \Pi)=-2\Omega\kappa^{\mp 1}$ ($\kappa :=\Pi-\Omega-\lambda_0$, $U(d_-)U(d_+)=4\Omega^2$) forming a $10/10$ quintuple $T \cup \{d_-,d_+\}$ with $\tau_4(d_-)=\tau_5(d_+)=0$ ($Q_{\text{3tors}}$ in `verify_paper.py`), and $E_T(\mathbb{Q})_{\text{tors}} \supseteq \langle R_0+W_r \rangle \times \langle W_s-W_r \rangle \cong \mathbb{Z}/6\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$ ($W_r=(-r^2,0),W_s=(-s^2,0),W_t=(-t^2,0)$). - **Orbit Cubic, Positivity & Sub-Quadruple Identity:** For $Q_4=(U_4,Y_4) \in E_T(\mathbb{Q})$ ($U_4(U_4+1) \ne 0$), $\{Q_4,Q_4+S_T,Q_4-S_T\}$ has $U$-coordinates $\{U_4,U_5,U_6\}$ obeying $$ U_{5, 6} = \frac{-2\Omega(\lambda_0 U_4 - \Omega \mp Y_4)}{U_4^2}, \quad U_4 U_5 U_6 = -4\Omega^2, \quad P_{\text{coset}}(U') := U'^3 - \Sigma_U \, U'^2 - 4\Omega\lambda_0 \, U' + 4\Omega^2 = 0, $$ where $\Sigma_U :=U_4+U_5+U_6=\frac{U_4^3-4\Omega\lambda_0 U_4+4\Omega^2}{U_4^2}$. For $Q_4 \in P_0+2 E_T(\mathbb{Q})$ ($U_4 \notin \{0,-1,U(d_-),U(d_+)\}$), $S_6(Q_4) :=T \cup \{x_4,x_5,x_6\}$ is a **$15/15$ sextuple** [7, 10, 17]. For $u :=\frac{\Pi}{r_u^2-1}$ ($(u,r_u) \in \{(a,t),(b,s),(c,r)\},u x_i+1=\frac{U_i+r_u^2}{r_u^2-1}$), the $9$ cross-products with $T$ obey (`exp31`): $$ (u x_4 + 1)(u x_5 + 1)(u x_6 + 1) = \frac{-P_{\text{coset}}(-r_u^2)}{(r_u^2 - 1)^3} = -(\frac{u}{\Pi})^3 P_{\text{coset}}(-r_u^2) = (\frac{u r_u(r_u U_4 - 2\Omega)}{\Pi U_4})^2\!(u x_4 + 1), $$ since $U_4^2(r_u^6+\Sigma_U r_u^4-4\Omega\lambda_0 r_u^2-4\Omega^2)=(U_4+r_u^2)(r_u^2 U_4-2\Omega r_u)^2$. Hence every $2+2$ sub-quadruple $Q=(T \setminus \{w\}) \cup \{x_i,x_j\}$ and $1+3$ sub-quadruple $Q'=\{u,x_4,x_5,x_6\}$ has square $6$-pair product, giving **$9+6=15$ septuple sections per coset ($\ge 19/21$ squares, type $(0,4)$**; `exp28`), while $(U_4+1)(U_5+1)(U_6+1)=-P_{\text{coset}}(-1)=\frac{(U_4+1)(U_4-U(d_-))(U_4-U(d_+))}{U_4^2}$ proves **$\Pi x_4,\Pi x_5,\Pi x_6 > 0 \iff (U_4+1)(U_4-U(d_-))(U_4-U(d_+)) > 0$**, and $1+a b c x_4 x_5 x_6=\frac{(\Omega+\lambda_0)(\lambda_0-3\Omega)+\Sigma_U}{\Pi^2}$ (for any $Q_4 \cup \{e_-,e_+\}$, $1+\sigma_6=\frac{(s_4-1)^2+s_4 R_4(Q_4)}{(s_4-1)^2}$; at $m=-5$ on $b(m)=d_+(m)$, $S_6=\{\frac{15}{133},-\frac{171}{140},-\frac{448}{95},-\frac{100505504}{31314185},\frac{87673157775}{430422596356},-\frac{726793939305}{153414789124}\}$ has $1+\sigma_6=(\frac{24194011070519}{13940581433401})^2$ and $P_6(S_6) \ne 0$, disproving $1+\sigma_6=\square \implies P_6=0$; `exp52`). 4. **Geometry of $\mathcal{S}_{\text{3-tors}}$, Hierarchy $C_r(\mathbb{Q}(r))$ & Family IV ($N=3\beta+\gamma$):** Over $\mathbb{Q}(r)$, $t=-\frac{r s}{r+s},Y_r=(r+s)\Pi$ in $(r^2-1)(s^2-1)(t^2-1)=\Pi^2$ gives (`exp28`, `exp30`): $$ C_r : \quad Y_r^2 = (r^2 - 1)(s^2 - 1)(r^2 s^2 - (r + s)^2) = (r^2 - 1)(s^2 - 1)(r s - r - s)(r s + r + s), $$ with $W_2=(-1,0)$, $C_r(\mathbb{Q}(r))[2]=\{W_2,\; T_1=(1,0),\; T_2=(-\frac{r}{r+1},0),\; T_3=(\frac{r}{r-1},0)\} \cong (\mathbb{Z}/2\mathbb{Z})^2$, and $P_\infty$ ($s=\infty$) generating $\mathbb{Z} P_\infty \oplus (\mathbb{Z}/2\mathbb{Z})^2 \subset C_r(\mathbb{Q}(r))$: - **Level $1$ ($P_\infty+T_1$):** $(s(r),t(r))=(-\frac{1}{2r},\frac{r}{2r^2-1}),\Pi(r)=\frac{(r^2-1)(4r^2-1)}{2r(2r^2-1)}$ ($3 P_0$ is $6$-torsion, $x(3 P_0)=0$). - **Level $2$ ($2 P_\infty$ & $2 P_\infty+T_1$):** $$ s_{2 P_\infty}(r) = \frac{2 r^3 - 2 r + 1}{2 r^3 - 2 r^2 - 2 r + 1}, \quad t_{2 P_\infty}(r) = -\frac{r(2 r^3 - 2 r + 1)}{(r - 1)(2 r^3 + 2 r^2 - 1)} \quad (s_{2 P_\infty + T_1}(r) = -s_{2 P_\infty}(-r)). $$ By Sturm (`exp28`), on $2 P_\infty(r)$ for $r > r_{**} \approx 2.010315$ ($4r^6-8r^5-4r^4+8r^3-1=0$), **all $8$ elements of $T(r) \cup \{d_-(r),d_+(r)\} \cup (3 P_0(r)+\langle S_T \rangle)$ are positive**, and the $9$ $2+2$ and $3$ $1+3$ sub-quadruples give **$15$ $\mathbb{Q}(r)$-septuple sections** ($r=3$: $\{6816/527,527/852,255/2201,160247/6324,82615/14484\}$, $S_6(2 P_\infty(3))$, $15$ type-$(0,4)$ $19/21$ septuples in `verify_paper.py`). - **Homogeneous Model & Family IV ($N=3\beta+\gamma$):** In $(\alpha:\beta:\gamma:N)=(1/t : 1/r : 1/s : 1)$ ($\alpha+\beta+\gamma=0$), $\mathcal{S}_{\text{3-tors}}$ is $M^2=(N^2-\beta^2)(N^2-\gamma^2)(N^2-(\beta+\gamma)^2)$. Along **$N=3\beta+\gamma$ (Family IV)**, $\gamma=1,\beta=\frac{1-3 u^2}{3 u^2}$ gives (`exp30`): $$ r(u) = \frac{9 u^2 + 1}{3(1 - u^2)}, \quad s(u) = -\frac{9 u^2 + 1}{6 u^2 - 1}, \quad t(u) = -\frac{9 u^2 + 1}{4 - 9 u^2}, \quad \Pi(u) = \frac{10 u(3 u^2 + 2)}{(1 - u^2)(4 - 9 u^2)}, $$ with **$a,b,c,d_-,d_+> 0$ for $u \in (\frac{1}{\sqrt{6}},\frac{2}{3})$**, and positive $15/15$ sextuple $S_6(u)=T(u) \cup (3 P_0(u)+\langle S_T \rangle)$ on $(1/\sqrt{6},u_{**})$ ($u_{**} \approx 0.440894$; $u=\frac{5}{12}$: $\{2975/39,32/23205,23595/476,11613/4420,37791/140\}$ in `verify_paper.py`; **Family III**, $N=\beta-\gamma$: $r(u)=\frac{u^2+1}{3(u^2-1)},s(u)=-\frac{u^2+1}{2u^2-1},t(u)=-\frac{u^2+1}{u^2-2}$). 5. **Cross-Coset Type-$(2,3)$ Mechanism, $(s,\kappa)$ Master Syzygies, DKMS [7, §3], $\mathcal{C}_{\text{triad}}$ & $E_{15}$ Closure:** For **any** $T=\{a,b,c\}$ ($a=\frac{\Pi}{t^2-1},b=\frac{\Pi}{s^2-1},c=\frac{\Pi}{r^2-1}$) and $d_\mp=a+b+c+2\Pi \mp 2\Omega$, $b'_{ac}(d_\mp) :=2(a+c+d_\mp+2 a c d_\mp)-b$ obeys: $$ b \, b'_{ac}(d_\mp) + 1 = (a + c - d_\mp)^2 - (4 s^2 - 1) = (b + 2(\Pi \mp \Omega))^2 - (4 s^2 - 1) = \frac{((2 s^2 - 1)\Pi \mp 2(s^2 - 1)\Omega)^2 - (s^2 - 1)^2(4 s^2 - 1)}{(s^2 - 1)^2}. $$ On $\mathcal{S}_{\text{3-tors}}$, $\kappa :=\Omega+\lambda_0-\Pi \in \mathbb{Q}^\times$ ($\Pi=\frac{1-\kappa^2}{2\kappa},\Omega=\frac{s^2(1+\kappa^2)-2s^3\kappa}{2\kappa(s^2-1)},\lambda_0=s-\frac{\Omega}{s^2}$) and $(x,y) :=(s\kappa,\kappa^2)$ reduce $\Delta_{rt}=(r-t)^2=\frac{\Omega(\Omega-4s^3)}{s^4}=\frac{C_A(x,\kappa^2)}{(2(x^2-\kappa^2))^2}=\square$ and $b \, b'_{ac}(d_-)+1=\frac{C_B(x,\kappa^2)}{(2(x^2-\kappa^2))^2}=\square$ to monic cubics in $y=\kappa^2$ (`exp34`): $$ C_A(x, y) := (y - 2x + 1)(y^2 + (6x + 1)y - 8x^3), \quad C_B(x, y) := y^3 + 2(1 - 4x^2)y^2 + (16x^4 + 8x^3 - 16x^2 + 1)y - 4x^3(8x^2 - 9x + 2), $$ obeying the **Master Syzygy Identities** in $\mathbb{Q}[x,y]$: $$ \begin{aligned} 9 C_A(x, y) &= (8x^3 - (8x + 1)y + 2x - 1)^2 + (y - (4x^2 - 2x - 1)) R_{\text{triad}}(x, y), \\ 9 C_B(x, y) &= (16x^3 - 6x^2 - (10x - 1)y - 2x + 1)^2 + (y - (16x^2 - 2x - 1)) R_{\text{triad}}(x, y), \end{aligned} $$ where $R_{\text{triad}}(x,y) :=9y^2+(8+2x-28x^2)y+(16x^4+8x^3-20x^2+6x-1)=0$ projects $\mathcal{C}_{\text{triad}}$, while $Q_{A,0}=(4x^2-2x-1,4x(x-1)(2x+1))$ on $\mathcal{E}_A(x)$ ($2 Q_{A,0}=(x^2,x(x-1)^2),T_A=(2x-1,0)$) is Family IV ($P_2+P_3$ on $\mathcal{E}_B(x)$, $2 P_1+3 P_2+4 P_3=\mathcal{O}$). Whenever $b \, b'_{ac}(d_\mp)+1=\square$, $\mathcal{C}_b :=\{b'_{ac},x_5,x_6\}=Q_{b'_{ac}}+\langle S_T \rangle$ makes $T \cup \mathcal{C}_b$ a $15/15$ sextuple, $T \cup \{d_\mp\} \cup \mathcal{C}_b$ a **$19/21$ septuple of type $(2,3)$**, and $\mathcal{U}_8 :=T \cup \{d_-,d_+\} \cup \mathcal{C}_b$ a $\ge 23/28$ octuple: - **DKMS [7, §3] at $2 Q_{A,0}+T_A$ ($y=-6x-1 \iff x=-\frac{\kappa^2+1}{6}$), $\mathcal{U}_8^{\text{DKMS}}(\kappa)$, $E_{\text{Cheb}}(s)$ & $\kappa_n(s)$:** Over $\mathbb{Q}(\kappa)$, $2 Q_{A,0}+T_A=(-6x-1,8x^2)$ gives $r(\kappa)=\frac{2(\kappa^2+1)}{\kappa^2+6\kappa+1},s(\kappa)=-\frac{\kappa^2+1}{6\kappa},t(\kappa)=-\frac{2(\kappa^2+1)}{\kappa^2-6\kappa+1},\Pi(\kappa)=\frac{1-\kappa^2}{2\kappa}$ ([7, §3] at $\kappa=1/t$: $(a,b,c)=(c_D,a_D,b_D)(1/\kappa)$; `exp34`). With $D_0 :=(\kappa^2-6\kappa+1)(\kappa^2+6\kappa+1),N_-:=(\kappa^2+7)(2\kappa^2-3\kappa-1)(2\kappa^2+3\kappa-1),N_+:=(7\kappa^2+1)(\kappa^2-3\kappa-2)(\kappa^2+3\kappa-2)$, $d_-(\kappa)=-\frac{2\kappa N_-}{3(\kappa^2-1) D_0}$ and $d_+(\kappa)=-\frac{2 N_+}{3\kappa(\kappa^2-1) D_0}$ form a **$22/28$ octuple** $\mathcal{U}_8^{\text{DKMS}}(\kappa)=\{a,b,c,d_-,d_+,x_4,x_5,x_6\}$, **all $8$ positive and distinct** on $(\kappa_{*,\text{DKMS}},3-2\sqrt{2}) \approx (0.1456,0.1716)$ ($\kappa \ne 1/3$). At $\kappa=\frac{1}{6}$ ($t=6$ [7]), $\mathcal{U}_8^{\text{DKMS}}(1/6)$ gives **$15$ positive $19/21$ septuples of type $(0,4)$**, while $U_Q(\kappa)=-\frac{1}{2} r(\kappa) t(\kappa)$ defines $Q_{\text{ex}}$ over $\mathcal{E}_Q : W_\kappa^2=\kappa^4+38\kappa^2+1$ ($15/15$ at $\kappa=1/2$). Under $\kappa=\frac{1-p}{1+p}$, $\Delta_{rt}(s,\kappa)=\square$ gives $E_{\text{Cheb}}(s) : Y_{\text{Cheb}}^2=Q_{\text{Cheb}}(s,p) :=((s+1)p^2-(s-1))((s+1)(2s-1)^2 p^2-(s-1)(2s+1)^2) \equiv ((s+1)(2s-1)p^2 \mp (s-1)(2s+1))^2-\{4,16s^2\}(s^2-1)p^2$ ($P_{\text{triv}}=(1,2)$). Here $\kappa_1(s)=\frac{s}{2s^2-1}$ ($d_+=0$) and $\kappa_2(s)=-\frac{2s^3-2s^2-2s+1}{2s^3+2s^2-1}$ recover Levels 1–2 of $C_r$ ($\mathcal{U}_8^{(2)}(s)$, positive at $s=10/9$), while $\kappa_3(s)=-\frac{2s(2s^2-1)}{(2s^3-2s-1)(2s^3-2s+1)}$ gives $\mathcal{U}_8^{(3)}(s)$ (positive at $s=4/3$; `exp44`). - **Triad Curve $\mathcal{C}_{\text{triad}}$ & Positive Septuples:** In $(\alpha:\beta:\gamma:N)$, $b'_{ac}(d_-)=b \iff 2 N_1^2-N_1(\beta-\alpha)-(\beta^2+\alpha^2)=0$, parameterized by $N_1(t)=t^2+1,\beta(t)=(t-2)(t+1),\alpha(t)=1-3 t,\gamma(t)=-t^2+4 t+1$. The root $N(t,W) :=\frac{t^2+1+W}{2}$ satisfies $M(N)^2=M_1(t)^2$ on (`exp30`): $$ \mathcal{C}_{\text{triad}} : \quad W^2 = t^4 - 20 t^3 + 34 t^2 + 12 t + 9 \quad (\cong E_{\text{triad}} : Y^2 = X^3 + 34 X^2 - 276 X + 2520), $$ via $X=\frac{6(3+2 t-W)}{t^2},Y=\frac{6(X+30)}{t}+2 X-60$. On $E_{\text{triad}}$, $P_1=(66,648)$ ($(1,-6)$) has $2 P_1=(\frac{106}{9},\frac{2024}{27})$ ($(\frac{9}{4},-\frac{39}{16})$), so **$\text{rank}\,\mathcal{C}_{\text{triad}}(\mathbb{Q}) \ge 1$**. With $M_{\text{can}}(t) :=-2 t(t-2)(t-1)(t+3)(2 t+1)$ and $\Pi_{\text{can}}(t) :=\frac{M_{\text{can}}(t)}{\beta(t)\gamma(t)\alpha(t)}$: $$ b \, b'_{ac}(d_-) + 1 = (\frac{(t^2 + 1 + W)((t^2 + 1)(t^2 + 14 t - 27) - (t^2 - 2 t + 5) W)}{4(t - 2)(t + 1)(1 - 3 t)(t - 1)(t + 3)})^2. $$ Under $(a,b,c,d_-,d_+,b'_{ac},x_5,x_6) \mapsto (-a,-b,-c,-d_+,-d_-,-b'_{ac},-x_6,-x_5)$, Sturm proves **for every non-degenerate $(t,W) \in \mathcal{C}_{\text{triad}}(\mathbb{Q})$ with $t < 2,W > 0$, all $7$ septuple entries (and all $8$ of $\mathcal{U}_8$) are positive**, starting at $(-\frac{5}{3},\frac{122}{9})$ ($\mathcal{S}_{7,\text{triad}}^{(19/21)}(-5/3),\mathcal{S}_{7,\text{triad}}^{(19/21)}(-4/5)$). - **$\mathbb{Q}(r_q)$-Quotient, Spoke Syzygies & $E_{15}$ Closure (`exp33`–`exp35`):** Under $\iota_{\text{triad}} : (t,W) \mapsto (\frac{2t+1}{t-2},\frac{5W}{(t-2)^2})$ ($z=t-2 \mapsto 5/z$), $\mathcal{C}_{\text{triad}} / \langle \iota_{\text{triad}} \rangle$ is $w_q^2=u_q^2-12 u_q-72$ (`Post 35`), parameterized by $u_q=6+3r_q+\frac{9}{r_q}=z+\frac{5}{z},w_q=3r_q-\frac{9}{r_q}=\frac{W}{z}$ ($\chi^2=r_q^2(u_q^2-20)$). With $A_3 :=3r_q^3+47r_q^2+117r_q+81,B_3 :=27r_q^3+107r_q^2+153r_q+81$, $U_4=-\frac{(3r_q+5)^2 A_3 B_3}{(9r_q^2+32r_q+27)^2 \chi^2},U_5=-\frac{2 r_q^2 (3r_q+5)^2 B_3}{A_3^2},U_6=-\frac{2 r_q^2 (3r_q+5)^2 A_3}{B_3^2}$, and $1+d_\mp x_{5,6}$ descend to $F_8,F_{10},G_8,G_{10} \in \mathbb{Q}[r_q]$ obeying $F_8-P_{F_8}^2=36(r_q+3) R_{\text{spoke},8},G_8-P_{G_8}^2=144(r_q+1) R_{\text{spoke},8},G_{10}-P_{G_{10}}^2=-16 r_q(7r_q+9) R_{\text{spoke},8}$ ($R_{\text{spoke},8} :=-r_q(r_q+1)(r_q+2)(r_q+3)(2r_q+3)(5r_q+9)$), linked by: $$ 4(r_q + 1)(F_8 - P_{F_8}^2) - (r_q + 3)(G_8 - P_{G_8}^2) = 0, \quad 9(G_{10} - P_{G_{10}}^2) + 4 r_q (F_8 - P_{F_8}^2) + 6 r_q (G_8 - P_{G_8}^2) = 0, $$ while $1+d_+b'_{ac}(d_-)=(\frac{r_q(3r_q+5)}{(r_q+2)(r_q+3)\chi})^2 Q_{\text{cross}}(r_q)$ ($Q_{\text{cross}} :=\frac{r_q^2(41 u_q^2+336 u_q+684)}{9}$). Setting $u_q(m)=\frac{2(3m^2-4m+3)}{m^2-1}$ maps $41 u_q^2+336 u_q+684=(\frac{4 Y_m}{m^2-1})^2$ to $\mathcal{C}_{\text{cross}} : Y_m^2=261 m^4-414 m^3+263 m^2-78 m+9 \cong E_{15} : Y^2=X(X+9)(X-16)$. $2$-isogeny descent with $\widehat{E}_{15} : \widehat{Y}^2=\widehat{X}(\widehat{X}^2+14\widehat{X}+625)$ mod $9,16,25$ gives $\text{im}(\alpha_{E_{15}})=\{1,-1\},\text{im}(\alpha_{\widehat{E}_{15}})=\{1,5\} \implies \text{rank}\,E_{15}(\mathbb{Q})=0$. Since $\# E_{15}(\mathbb{F}_7)=8$, $E_{15}(\mathbb{Q}) \cong \mathbb{Z}/4\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$ maps only to $r_q \in \{-3,-2,-3/2,-1\}$ ($\Pi_{\text{can}}=0$), **closing $1+d_+b'_{ac}(d_-)=\square$ on $\mathcal{C}_{\text{triad}}(\mathbb{Q})$**. - **Cross-Coset Bridge ($20$th Pair):** For **any** coset $\{x_4,x_5,x_6\}=Q_4+\langle S_T \rangle$ and $d \in \mathbb{Q}$, $M_4 :=d x_4+1=\frac{d U_4+\Pi+d}{\Pi}$, $M_5 :=d x_5+1$, $M_6 :=d x_6+1$ obey (`exp31`): $$ M_{5, 6} = \frac{(\Pi + d)U_4^2 - 2\Omega d(\lambda_0 U_4 - \Omega \mp Y_4)}{\Pi U_4^2}, \quad M_5 M_6 = \frac{(\Pi + d)^2 U_4^2 - 4\Omega d(\Omega d + \lambda_0(\Pi + d))U_4 + 4\Omega^2 d(\Pi + d)}{(\Pi U_4)^2}, $$ and $M_4 M_5 M_6=\frac{(\Pi+d)^3+\Sigma_U d(\Pi+d)^2-4\Omega\lambda_0 d^2(\Pi+d)-4\Omega^2 d^3}{\Pi^3}$. 6. **Mixed-$R_4$ Syzygies on $\mathcal{S}_{\text{3-tors}}$, $2$- vs. $4$-Condition Criterion, $H=1/3$ $2$-Descent, and Branch-B Surface (`exp35`–`exp38`):** For $\{x_4,x_5,x_6\}=Q_4+\langle S_T \rangle$ on $E_{\text{can}}$ with $t :=\frac{2\Omega}{U_4}$ ($U_5+U_6=t^2-2\lambda_0 t,U_5 U_6=-2\Omega t$), $(1+b x_5)(1+b x_6)=(\frac{s(s-t)}{s^2-1})^2$, $1+x_5 x_6=(\frac{h-t}{\Pi})^2$, and $(\frac{t^2}{2\Omega})^2 Y_4^2=(1+\frac{1}{2}t(t-2\lambda_0))^2-(t-\kappa)(t-\frac{1}{\kappa})$ ($\Pi^2 x_5 x_6=(t-\kappa)(t-1/\kappa)$). Hence $R_4(b,d_-,x_5,x_6)=0$ (`Post 35`, [19]) splits over $\mathbb{Q}(\kappa,s)$ into **four roots** ($\delta \in \{\pm 1\}$, swapped by $(\kappa,s) \mapsto (-\kappa,-s)$): $$ t_{\text{I}, \delta}(\kappa, s) = \frac{\kappa s + \delta h}{s + \delta}, \quad t_{\text{II}, \delta}(\kappa, s) = \frac{2 s^2 - \kappa s + \delta h}{s + \delta} \quad (h := \Omega + \lambda_0 = \frac{\kappa^2 + 1}{2\kappa}). $$ By Arkin–Hoggatt–Strauss on $\{b,d_-,x_5,x_6\}$, whenever $Q_{\text{C2}}(\kappa,s) :=(s+1) D(\kappa,s) N(\kappa,s)=\square$ ($1+b x_4=\frac{s^2 N}{(s-1)^2 (s+1) D}$) and $Q_{\text{C3}}(\kappa,s) :=(s+1) F(\kappa,s)=\square$ ($Y_4 \in \mathbb{Q}$), $1+d_-x_5,1+d_-x_6=\square$ follow, making $\{b,d_-,x_4,x_5,x_6\} \subset \mathbb{Q}$ a **$9/10$ quintuple** (missing $1+d_-x_4=\square$). Adjoining $\{a,c\}$ and lifting $Q_4 \in P_0+2 E_T(\mathbb{Q})$ to form a **$20/21$ septuple** $\{a,b,c,d_-,x_4,x_5,x_6\}$ requires **four gates**: $\Delta_{rt},Q_{\text{C2}},Q_{\text{C3}},F_\pm :=\frac{e+e' \pm 2\sqrt{e e'}}{4}=\square$ ($e=1+a x_4,e'=1+c x_4$): - **Mixed-$R_4$ Syzygies and Shared $(s+1) D(\kappa,s)$ Cancellation:** On Branch $\text{I},+1$ and Branch $\text{II},+1$, $Q_{\text{C2}}(\kappa,s)$ and $Q_{\text{C3}}(\kappa,s)$ complete to squares with the **same remainder** $R_{[x_{5,6}=0]}=-4\kappa^2(s+1)^2(t-\kappa)(t-1/\kappa)$: $$ \begin{aligned} Q_{\text{C2, I}}(\kappa, s) - P_{\text{C2, I}}(\kappa, s)^2 &= Q_{\text{C3, I}}(\kappa, s) - P_{\text{C3, I}}(\kappa, s)^2 = (\kappa^2 - 1)^2 (2 s + 1), \\ Q_{\text{C2, II}}(\kappa, s) - P_{\text{C2, II}}(\kappa, s)^2 &= Q_{\text{C3, II}}(\kappa, s) - P_{\text{C3, II}}(\kappa, s)^2 = -(2 s - \kappa + 1)(2\kappa s - \kappa - 1)(4\kappa s^2 - 4\kappa^2 s - \kappa^2 + 1), \end{aligned} $$ where $P_{\text{C2,I}}+P_{\text{C3,I}}=P_{\text{C2,II}}+P_{\text{C3,II}}=4\kappa(s-1)(s+1)^2$. Since $d_-(\kappa,s)=\frac{2\kappa(\kappa s-1)(2s^2-\kappa s-1)}{(\kappa^2-1)(s^2-1)}$, once $Q_{\text{C3}}=Y_3^2$, ($1+b x_4,F_\pm,1+d_-x_4$) share $(s+1) D(\kappa,s)$, cancelling in pairwise products. - **Low-Genus Reductions $\mathcal{H}_{\text{II}},\widetilde{\mathcal{H}}_{\text{II}},\mathcal{L}_{\text{I}},\mathcal{C}_{\text{I}}$ and $9/10$ Quintuples:** On Branch $\text{II},+1$, $u=-1/2$ splits into $\mathcal{H}_{\text{II}} : \kappa=-\frac{1}{2s+3}$ and $\widetilde{\mathcal{H}}_{\text{II}} : \kappa=\frac{2s+3}{4s+1}$. On $\mathcal{H}_{\text{II}}$, $Q_{\text{C3,II}}=(\frac{2(s+1)^2(2s-5)}{(2s+3)^2})^2$, and $Q_{\text{C2,II}}=\square$ gives $E_{\text{II},C_2} : W_2^2=(2s-5)(2s+1)(4s^2-5) \cong Y^2=(X+20)(X^2+60X+720)$ ($X=\frac{120}{2s-5}$). Under $\phi(X,Y)=\text{sqf}(X+20) \in \mathbb{Q}^\times / \mathbb{Q}^{\times 2}$ ($\phi(T)=-5$), $s=-3/2,-1/6 \mapsto 5,-10$, proving $\text{rank}\,E_{\text{II},C_2}(\mathbb{Q}) \ge 2$ and **infinitely many $9/10$ quintuples** $\{b,d_-,x_4,x_5,x_6\}$ ($s=-1/5,5/6,-9/4$). Meanwhile $\Delta_{rt}=\square$ gives $E_{\text{II},\Delta} : W_\Delta^2=(4s+5)(8s^3+4s^2-8s+5)$ ($W_\Delta^2-W_2^2=4s(s+5)(4s^2+2s-3)$), and $(1+d_-x_4) W_2^2 \equiv P_{6,\text{II}}(s) \pmod{\mathbb{Q}^{\times 2}}$ ($P_{6,\text{II}}-(4s^3+10s^2-5)^2=-4s(s-1)(4s+5)$; `exp36` up to $10^4$ finds $s \in \{-3/2,-5/4,0,1,5/2\}$). On $\widetilde{\mathcal{H}}_{\text{II}}$, $W_2^2+(16s(s+1))^2=(16s^3+24s^2+12s+5)^2$. On Branch $\text{I},+1$ along $\mathcal{L}_{\text{I}} : \kappa=2s+2$ and $\mathcal{C}_{\text{I}} : \kappa=\frac{1}{2s^2+s-2}$, $Q_{\text{C3,I}}=\square$ and $\Delta_{rt}=\square$ gives $E_{\Delta,5} : W_\Delta^2=-(4s+5)(4s^2-5) \cong y^2=x^3+10x^2+5x$ ($\text{rank}\,\ge 1$), while on $\mathcal{C}_{\text{I}}$, $Q_{\text{C2,I}} F_-\equiv (2s+1)(2s+5)(4s+5)$ and $Q_{\text{C2,I}}(1+d_-x_4) \equiv P_{6,\text{I}}(s)$ ($P_{6,\text{I}}-(4s^3+4s^2-s-5)^2=16s(4s^2+8s+5)$). - **General-$H$ Syzygies and $\mathbb{Q}(\sqrt{-2})$ $2$-Descent at $H=1/3$ (`Post 35`, `Post 43`; `exp37`):** Matching $F_{\text{sym}}=0$ of Theorem 3(1) into roots $c d=\frac{2H-1}{(H-1)^2}$ and $c f=-\frac{2H+1}{(H+1)^2}$ ($q=a b=H^2-1$), $A,B,C,D$ rescale from $Q_H(z) :=z^2-2Hz+1$, obey $$ (H + 1) A - 2 H B + (H - 1) C = 8 H (H^2 - 1), \quad H(2H + 1) A - (4 H^2 - 1) B + H(2H - 1) C - D = 0, $$ and diagonalize under $w_i=\frac{1-z_i}{1+z_i}$ as $Q_H(z_i)=\frac{2(H+1)}{(1+w_i)^2}(w_i^2-\lambda)$ ($\lambda=\frac{H-1}{H+1}$). The $21$st edge $1+e g$ completes in $u=y+\frac{1-4H^2}{y}$ as $P_{\text{edge}}(u,H)=(u^2-2(3 H^4-2 H^2+1))^2-4(H^2-1)^3(9 H^2-1)$, making $H=\pm 1/3$ the unique square completion. At $H=1/3$ ($y=X/9,\lambda=-1/2,K=\mathbb{Q}(\sqrt{-2})$), $A_0,B_0,C_0,D_0$ factor as norms $N_{K/\mathbb{Q}}(L_i)$ with $L_1 L_2 L_3 L_4 \in \pm 3(K^\times)^2$, mapping to $E_d : y^2=(x-d)(x-2d)(x+3d)=x(x^2-7d^2)+6d^3$ ($d \in \{\pm 3,\pm 6\}$). On $E : Y^2=X(X^2+a X+b)$ with $\phi : E \to E'$, $d w^2=(d-a z^2)^2-4b z^4 > 0$ over $\mathbb{R}$ when $b < 0$ ($d < 0$ eliminated). Local checks across $256$ Kummer triples mod $\mathbb{R},64,81,125$ (`exp37`) leave $(1,1,1)$ for $d=3,6$, while $\text{rank}\,E_{-6}(\mathbb{Q})=\text{rank}\,E_{-3}(\mathbb{Q})=0$. Thus $P-\bar{P} \in E_{-2d}[2](\mathbb{Q}) \implies E_d(K)=E_d(\mathbb{Q})$, proving $X=0,\infty$ are the sole rational solutions and closing $H=1/3$. - **Branch-B Surface, $\mathcal{M}_\Psi^\star$, Four-Equal Character Law, $10/10$ Quintuples, $20/21$ over $\mathbb{Q}(\sqrt{-14})$, Collision Descent & Local Solubility (`Post 43`, `exp38`–`exp39`):** On Branch B ($r=\frac{1}{m+d_0},s=\frac{1}{m-d_0},W=r s w_0$), once $-P_8(u_d,u_w) :=S_3^2-4u_d^2(u_d^2-1)^2((u_d+u_w)^2-1)=Y_{P8}^2$, $\{c,d,f,e,g\} \subset \mathbb{Q}$ is a $9/10$ quintuple ($R_4(c,d,e,f)=R_4(c,d,f,g)=0,(1+ax)(1+bx)=R_x^2$): (1) $(26/37,24/37)$ gives $\{3864/7163,-4845/2639,319/1729,-382800/292201,-6344/3857\}$ ($1+ab=(37/132)^2,1+eg=(3071/1729)^2$); (2) on $E_\tau(\tau,x,z)=0$, $\mathcal{M}_\Psi^\star$ proves $a,b \in \mathbb{Q}$ ($a \ne b$) with $1+ax,1+bx \in \mathbb{Q}^{\times 2} \iff A \equiv B \equiv C \equiv D \pmod{\mathbb{Q}^{\times 2}}$. At $(99/125,-284/625)$ ($[\Delta_{ab}]=[-14]$), $a,b \in \mathbb{Q}(\sqrt{-14})$ ($1+ab=(425/304)^2$) gives a **Galois-stable $20/21$ almost septuple over $\mathbb{Q}(\sqrt{-14})$**; on $x+z=3$, $(-16,19)$ gives $\{7/52,-345/52,-13/4,-21/13,-1305/208\}$. (3) Along $(u_w-2u_d)^2+u_d^2=1$ ($u_d=\frac{2k}{k^2+1},u_w=\frac{k^2+4k-1}{k^2+1}$), $\Psi_+^\star(\tau)=0,\Delta_{ab}=0$ gives an all-positive $9/10$ family for $k > 2+\sqrt{5}$ ($k=5$: $\{1/70,483/10,592/35,1200/7,714/125\}$) where $1+ac=\square \iff E_{\text{coll}} : v^2=u(u+1)(u+5)$ (rank $0$). Along $(x,z)=(h^{-1},h^{-2})$ in $\mathbb{Q}((h))$, all $21$ radicands have leading squares in $\{4/9,1/4,1/9,1\}$, proving **local solubility over $\mathbb{Q}_v$** (`exp39`). - **Paired & Antipodal Syzygies, $\mathcal{I}_{\text{AL}}^{(0)}$, $k_{\text{tan}}(m)$, $E_{sw}$, Post 69 Syzygies, Circle Charts, Octuples, $Q_h$ Three-Circle & Five-$R_5$ Root-Splitting (`Post 35`, `Post 42`, `Post 67`, `Post 69`; `exp40`–`exp43`, `exp58`–`exp60`):** On Source 1 [20] in $(u,H)$ ($D_\pm :=u^2 \pm 3u+4,c=\frac{u(3-4H)}{4},G=-\frac{4H}{u}$), $F_\pm=\frac{8u(u\pm 2)(u\pm 1-2H(u\pm 2))}{D_\pm^2}$ complete in $H$ with $u L_{GF}-L_{cF}=3u(u+2),2u(u-2)L_{GF}-L_{FF}=4u(u-1)(u+2)$, while $\mathcal{I}_{\text{AL}}^{(0)} : (u,H) \mapsto (4/u,3/4-H)$ preserves $v^2=4H^2-3H+1$ and maps $(F_+,F_-,c,G,a,b,A,C) \mapsto (-F_+,-F_-,-G,-c,-C,-A,-a,-b)$. On Source 2 [20] (`Post 69`), $T=t+K/t$ ($t=cr/J,K=\frac{q(5-4q)}{4(1-q)^2},q=e^2$) splits $-(16(q-1)^3 T^2-q)=\frac{L_1 L_2}{t^2}$ into $L_1=\frac{q-(q-1)t^2}{q-1},L_2=16(q-1)^3 t^2-q(4q-5)^2$ with $c^2=\frac{q L_2}{16(q-1)^4 L_1},D^2=\frac{q(8q-9)}{16(q-1)^4 L_1},G^2=\frac{q(4q-5)^2 L_1}{L_2},D_{\text{conj}}^2=\frac{t^2(8q-9)}{L_2}$; on $B^2=-(7q+9)(45q^2-25q-36),H^2=q^4+64q^3+1054q^2-1440q+81$ ($F=\frac{16e(q-3)B}{(q-1)(17q-9)^2},u,z=\frac{2q(q-3)\pm H}{(q-1)(17q-9)}$), conic syzygies give $H^2+4(5-4q)B^2=(71q^2-18q-81)^2$, $W_A=(71q^2-18q-81)^2+2(8q-9)B^2$, and $(2(q-1)D_\pm)^2=s_\pm^2+4(q-1)$ ($s_\pm=\frac{71q^2-18q-81\mp H}{2B}$), so $W_+W_-=(2e B(2q^2+63q-81))^2$ and $D_+^2 D_-^2=(\frac{e(2q^2+63q-81)}{4(q-1)^2 B})^2$ (`exp43`). Here $v_3(H^2)=1$ obstructs $v_3(e)=0$, $H^2<0$ excludes $(q_{H,1},q_{H,2})$ ($q_{H,1} \in (0,1/17),q_{H,2} \in (6/5,1213/1000)$), $D_\pm^2 > 0$ on $(0,q_{H,1}] \cup [q_{H,2},q_{B,+})$ ($q_{B,+}=\frac{25+7\sqrt{145}}{90}$), and $C_B$ lifts to an étale $2$-cover over $E_1$ of $\text{rank}\,\ge 1$ via $P_1=(1,16)$ ($3P_1=(0,18)$, $n \equiv \pm 3 \pmod{10}$ on $\langle P_1 \rangle$; points on $(0,q_{H,1}] \cup [q_{H,2},q_{B,+})$ outside $v_3(e)=0$ remain open). The fourths $d_1..d_4$ [22] satisfy $1+d_1 d_4=(u/2)^2(1+c d_2),1+d_2 d_3=(2/u)^2(1+G d_4),1+d_1 d_3=c G=v^2-1$ (Orbits I–III: $u \in \{\pm 3,\pm 4/3\},\{\pm 6,\pm 2/3\},\{\pm 8,\pm 1/2\}$). Along $H(m)=\frac{2m-3}{m^2-4},v(m)=\frac{m^2-3m+4}{m^2-4}$, $k_{\text{tan}}(m)=\frac{4m^4-33m^3+108m^2-164m+88}{(m-2)(5m^3-36m^2+76m-56)}$ ($\sigma(m)=\frac{3m-8}{2m-3},\tau_1(m)=\frac{4}{m}$) gives $\ge 23$ square pairs ($23/28$ at $m \in \{3,1,-1,4,-3,5\}$, `exp43`). (i) At $u=-4/3$, $E_{sw}(-4/3) : Y_t^2=9t^4+260t^3+2150t^2+1764t+441$, and at $u=-3$, $E_{sw}(-3) : y_m^2=m^4+110m^2+168m+64$, give $15/15$ sextuple and $23/28$ octuple families ($t=55/17,m=-39/8$), whereas on Source 1 along Orbit I, $C : y^2=228x^6+595x^5+1457x^4+1904x^3+3296x^2+1024x+2304$ ($x=-1/H$) has $\dim_{\mathbb{F}_2}\text{Sel}_{\text{quot}}^{(2)}=2 \implies \text{rank}\,J(\mathbb{Q})=1$ [24], and mod-$25$ Coleman integrals $I_0 \equiv I_1 \equiv 20$ [25] on $\widetilde{\omega}_{\text{ann}} \equiv 4(1-x)\frac{dx}{2y} \pmod 5$ bound zeros to $x=0$. (ii) On $\mathcal{E}_{\text{source}} : H(\nu)=\frac{2\nu-1}{4-\nu^2},V^2=\nu^4+6\nu^3+5\nu^2-40\nu+32$, $\tau_{w,1},\tau_{w,3}$ give $23/28$ octuples ($H=1/3,\tau_w=8/13$). (iii) $\mathcal{S}_{sw} : Y_{sw}^2=F_{sw}(u,m)$ has sections $m_1(u)=\frac{u^4-16}{4u(u^2-2)},u_1(m)=\frac{8m(m^2-7)}{(m^2-3)(m^2+5)}$. (iv) On $\mathcal{U}_8^\mu :=\mathcal{S}_6^{\text{anti}} \cup \{M(d),M(e)\}$ [20], $[1+M(d)M(e)]=[q_0^4-q_0^2+1]$ maps to $E_{24} : V^2=U(U-1)(U+3)$ of rank $0$. In $U^2=1+t^2,X=\frac{t+q}{U},Y=\frac{qt-1}{U}$ (`Post 67`), $X(q,\rho)=\frac{2\rho}{d},Z_3(q,\rho)=\frac{(1+q)-(1-q)\rho^2}{d}$ factors $d^2 Y^2=((q^2+1)\rho^2-(q+1)^2)((q-1)^2\rho^2-(q^2+1))$, obstructed at $q=\pm 5/16$ over $\mathbb{Q}_{23}$. On $\{a,d,e,c\}$, $M(F_\pm)=F_\mp$ gives $[1-y_+/a]=[1+a M(y_+)]=[1+y_+M(y_+)]$ on $\mathcal{S}_7^\star(\epsilon)$ ($18/21$) and $\mathcal{U}_8^{\text{anti}}(\epsilon)$ ($23/28$). (v) On $Q_h=\{\frac{h(h^2-4)}{2(h^2-2)},-\frac{2(h^2-1)}{h(h^2-2)},-\frac{2h}{h^2-2},\frac{h^2-2}{2h}\}$ (`Post 42`, `exp58`–`exp60`), $1+de_{\text{reg}}(h)=(\frac{h}{2(h^2-2)})^2(t^4-12t^2+16)$ ($t=h+2/h$) closes via rank-$0$ $E_t:Y^2=X(X^2-12X+16)$ ($X=0$), fifths reduce to $z_0^2+1,z_a^2+1,z_b^2+1=\square$, $G(a,b;z_0,z_1)=0$ splits into $U_\mp(r,s,\kappa)=\frac{2r^2s(\kappa^2+2\kappa r-4r^2+1)}{s(-\kappa^2+4\kappa r(r^2-1)+2r^2-1)\pm r(\kappa^2-1)(2r+s)}$, and $g_1=g_2$ obeys $1+G_i \equiv H R_i^2$ ($R_i=M_i/D_i$) and $G_1-G_2 \equiv H(R_1-R_2)(R_1+R_2)=0$ on $P_{\text{orb}}(v,w_z,w_e)=0$, where $h M_i(v,w,h^2)\mp D_i(v,w,h^2)$ split into $3$ linear $h$-factors (`exp60`). - **Universal Monomial Fifth Law, Reflection Stars $\mathcal{S}_7^{\text{3-refl}}$, Bi-Euler $9$-Tuple $\mathcal{U}_9^{\text{bi-Euler}}(r,t)$, Free Rank-$3$ $\mathbb{Z}[r]$-Module $\mathcal{M}_3$ & Coincidence Curve $\mathcal{C}_{\text{coinc},r}$ (`exp61`–`exp66`):** On $\mathcal{S}_7^{(c^*)}$, $S_{ab}^{(-)}(r,\lambda,m)=\frac{m^2-\Delta_--2m(\lambda^2(4r-3)+4r+3)}{8\lambda m}$ pays Condition 20 ($\Delta_--2(\lambda^2-1)F_1$ linear in $r$; $c^*_-\in\{a,b,d_-,d_+\}$ linear in $S_{ab}$). Universally, for any regular $\{x,y,z,d_-\}$ ($L=2rst=\frac{d_+-d_-}{2},V=xyzd_--1,U=Ld_--V$), $U-V=-2r_x r_y r_z$ gives $e_{\text{reg}}(x,y,z,d_-)=-\frac{4rst\,r_x r_y r_z}{(xyzd_--1)^2}$ and $V^2(1+d_+e_{\text{reg}})=(U+L^2)^2-L^2(L^2+2V)$. In $(r,C_0,d_-)$ ($S_{ab}=\frac{C_0^2-2(2r^2-1)C_0d_-+d_-^2-4r^2}{4r^2d_-},d_-d_+=C_0^2-4r^2$), the reflections $f_{x,-}=2(y+z+(2yz+1)d_-)-x$ on $\mathcal{S}_7^{\text{3-refl}}=\{a,b,c,d_-,f_{a,-},f_{b,-},f_{c,-}\}$ ($15/21$ squares) obey $1+xf_{x,-}=(x+T_0)^2-(4r_{yz}^2-1)$ ($T_0=2(abc-rst)$) and $(2ac+1)f_{a,-}-(2bc+1)f_{b,-}=-(a-b)(2ac+2bc-4c^2+3)$, while $f_{c,\mp}=d_-(a,b,d_\mp)$ on $\mathcal{S}_7^{(f_c)}=\{a,b,c,d_-,d_+,f_{c,-},f_{c,+}\}$ ($16/21$ squares) obey $1+cf_{c,\mp}=(d_\mp-S_{ab})^2-(4r^2-1),1+d_\pm f_{c,\mp}=(\frac{d_\mp+C_0-4r^2S_{ab}}{2r})^2-8r^2(2r^2-1),1+f_{c,-}f_{c,+}=((4r^2-1)S_{ab}-C_0)^2-(4r^2-1)(16r^4-12r^2+1)$. In $t=\frac{C_0}{2r(2r^2-1)}$ on $\mu_{\text{inf}}(r,C_0)$ ($d_-=4r^2a=r(t-1/t),b=\frac{4r(r^2-1)}{t-1/t},bd_-=4r^2(r^2-1)$), the involution $\iota_t:t\mapsto-1/t$ fixes $(a,b,d_-)$ and swaps $c\leftrightarrow f_{c,-},f_{a,-}\leftrightarrow\tilde{f}_{a,-},g_{\text{Euler},+}\leftrightarrow\tilde{g}_{\text{Euler},+}$ ($g_{\text{Euler},+}=d_+(0,c,f_{a,-})$), forming $\mathcal{S}_6^{\text{Euler}}=\{a,b,c,d_-,f_{a,-},g_{\text{Euler},+}\}$ ($12/15$ squares), $\mathcal{S}_7^{\text{Euler},c}=\mathcal{S}_6^{\text{Euler}}\cup\{f_{c,-}\}$ ($15/21$ squares), $\mathcal{U}_8^{\text{Euler}}=\mathcal{S}_7^{\text{Euler},c}\cup\{f_{b,-}\}$ ($18/28$ squares, $1+bf_{b,-}=X_7^2-\frac{t^4+t^2+1}{t^2}$), and the **Universal Bi-Euler $9$-Tuple** $\mathcal{U}_9^{\text{bi-Euler}}=\mathcal{S}_7^{\text{Euler},c}\cup\{\tilde{f}_{a,-},\tilde{g}_{\text{Euler},+}\}$ ($21/36$ squares; $23/36$ on $t_1(r),\mathcal{C}_{\text{coinc}}$), where **all $9$ entries, automatic roots $R_{c,a},R_{c,g},R_{a,g}$ ($R_{c,g}-R_{c,a}=c,R_{a,g}-R_{c,a}=f_{a,-}$), and bases $X_k$ lie in the free rank-$3$ $\mathbb{Z}[r]$-module $\mathcal{M}_3=\mathbb{Z}[r]a\oplus\mathbb{Z}[r]b\oplus\mathbb{Z}[r]r_{ad_-}$** ($r_{ad_-}=\frac{t^2+1}{2t}=\sqrt{1+ad_-},r_{ad_-}^2-(2ra)^2=1$): $$ \begin{aligned} c,\,f_{c,-} &= (8r^4-4r^2+1)a + b \pm J_c(r)r_{ad_-}, \quad f_{a,-},\,\tilde{f}_{a,-} = (128r^8-192r^6+96r^4-16r^2+1)a + 4(2r^2-1)^2 b \pm J_a(r)r_{ad_-}, \\ g_{\text{Euler},+},\,\tilde{g}_{\text{Euler},+} &= 8r^4(16r^4-16r^2+5)a + (4r^2-1)^2 b \pm J_g(r)r_{ad_-}, \quad f_{a,-} = (8r^2-1)a + 2b + 2(8r^4-8r^2+1)c, \\ (2r^2-1)g_{\text{Euler},+} &= 2r^2(8r^2-3)a + (4r^2-1)b + 2r^2(4r^2-1)(4r^2-3)c, \quad X_2 = r_{cd_-}+X_1, \quad X_3 = r_{bc}+r_{bf_{a,-}} = (4r^2-1)r_{bc}-r, \end{aligned} $$ with jump amplitudes $J_c(r)=2r(2r^2-1),J_a(r)=4r(2r^2-1)(8r^4-8r^2+1),J_g(r)=4r^3(4r^2-1)(4r^2-3)$ and **all $15$ open pairs of $\mathcal{U}_9^{\text{bi-Euler}}$ equal to rank-$1$ differences of squares with $r$-only shifts**: $4r^2(1+af_{a,-})=X_1^2+(4r^2-1)=\tilde{X}_1^2-4r^2(4r^2-1)(4r^2-3)$ ($X_1=r_{d_-f_{a,-}}$), $1+d_-g_{\text{Euler},+}=X_2^2-(4r^2-1)$, $1+bg_{\text{Euler},+}=X_3^2-(4r^2-1)$ ($r_{bc}+r_{bf_{c,-}}=J_c(r),X_3+\tilde{X}_3=4r^3(4r^2-3)$), diagonal norms $1+cf_{c,-}=((4r^2-1)a\mp b)^2-(4r^2-1)(4r^2-3)^{(1\pm 1)/2}$, $1+f_{a,-}\tilde{f}_{a,-}=X_{12}^2-(4r^2-1)(16r^4-12r^2+1)$ ($X_{12},\tilde{X}_{12}=(4r^2-1)(16r^4-12r^2+1)a\mp 4(2r^2-1)^2 b$), $1+g_{\text{Euler},+}\tilde{g}_{\text{Euler},+}=(32r^4(2r^2-1)a\mp(4r^2-1)^2b)^2-((4r^2(4r^2-1)(4r^2-3)^{(1\pm 1)/2})^2-1)$, and cross-pairs obeying the **Jump-Product Shift Law** $1+u_i\tilde{u}_j=X_k^2-\frac{J_i(r)J_j(r)}{r^2}$ ($X_5,X_6,X_{11}\in\mathcal{M}_3$ with $X_k-\iota_t(X_k)=-2r\,r_{ad_-}$ and **Zero-Sum Triangle Laws** $X_6-X_5=\tilde{X}_4,X_{11}-X_5=\tilde{X}_{12}$). Here $4rt(t^2-1)(g_{\text{Euler},+}-f_{a,-})=4t^2(X_2^2-X_1^2-(4r^2-1))=P_{\text{CaseA}}(r,t^2)$ closes Case A, osculating $\mathcal{E}_{\text{III}}(r)$ at $t=1$ gives $t_{\text{III}}(r)=-\frac{16r^8-20r^6+4r^2+1}{48r^8-92r^6+48r^4-4r^2-1}$ ($1+bg_{\text{Euler},+}=\square$, $22/36$ in $\mathcal{U}_9^{\text{bi-Euler}}$), $t=\pm\frac{1}{4r^2-1}$ gives $1+f_{a,-}f_{c,-}=\square$ with $b=c,f_{c,-}=f_{b,-}=g_{\text{Euler},+}$, and $R_4(a,d_-,f_{a,-},g_{\text{Euler},+})=0$ splits into $t=\pm t_1(r)=\pm\frac{4r^2-1}{2(2r^2-1)}$ ($g=0$: $\mathcal{Q}_{5,(n,\varepsilon)}^{\text{refl}}(r),\mathcal{Q}_5^\star(r),\mathcal{S}_6^{\text{refl}}(r)$, $15$ $18/21$ septuples) and smooth $g=4$ $\mathcal{C}_{\text{coinc},r}:F_{\text{coinc},\pm}(r^2,t)=0$ ($r^2=\frac{(Y-2X-4)(Y+2X-4)}{64(X+4)},t=-\frac{4}{X}$) over $E_{\text{coinc}}:Y^2=X^3+8X^2+32X+80$, where $Q_{\text{I}},Q_{\text{II}}=\square$ by Hilbert $90$ (`exp61`–`exp66`). *Proof.* Verified in `verify_paper.py`, `exp25`–`exp66`, and `THEOREMS_AND_PROOFS.md`. $\square$ --- ## 10. Limitations and scope - **No rational septuple ($21/21$) is claimed.** $\mathcal{S}_7^{(20/21)}$ adds relation type $(|\mathcal{R}_4|,|\mathcal{R}_5|)=(5,2)$ ($\text{dim}=1,\text{gens}=2$) [2, 3, 4]. - **Finiteness on $\mathcal{H}$ vs. Problem 3.3:** Theorem 3(4) proves $\mathcal{C}_+(a,\lambda)=0$ on Branch 1 has smooth $g=3$ cover $\mathcal{H}$, so by Faltings [6] it has finitely many rational almost septuples, leaving open $\mathcal{S}_{4,2}$ and $\mathcal{E}_{\text{Br1}}$. - **Completeness on $\mathcal{H}$ and unpaid edges:** Faltings' theorem is ineffective; our height-$50,000$ search on $\mathcal{E}^{(1)}$ and `u/kiro_dioph7`'s $295$-prime sieve (`Post 35`) are bounded, while [23] (`Post 68`) proves $\mathcal{H}(\mathbb{Q})=\{(\pm 1,\pm 16),(\pm 2,\pm 55),\infty_\pm\}$ via Quadratic Chabauty. On Curve II, $c_1 c^*_\mp+1$ have degrees $36,32$; on Theorem 5, the $2$-adic lock eliminates $Q_4 \equiv-2$ while $Y_B^2,G(k)=\square$ is closed at $k \in \{2,3\}$ [16]; on $z=z_{L_1}$ (Theorem 6), $a c+1,\mathcal{F}_{ab},\mathcal{G}_d,H=\square$ remains open; on $\mathcal{C}_{\text{triad}}$, $1+d_+b'_{ac}(d_-)=\square$ is closed by rank-$0$ $E_{15}$; on Theorem 7, $H=1/3$ is closed by $\mathbb{Q}(\sqrt{-2})$ $2$-descent, $u_w=2u_d$ by Pythagorean Syzygy, $(u_w-2u_d)^2+u_d^2=1$ by rank-$0$ $E_{\text{coll}}$, and [20–22] give $23/28$ octuples ($1+M(d)M(e)$ closed by $E_{24}$, Orbit-I Source-1 by $2$-Selmer/Chabauty [24, 25], circle charts obstructed over $\mathbb{Q}_{23}$ at $q=\pm 5/16$, Post 69 open on $(0,q_{H,1}] \cup [q_{H,2},q_{B,+})$ outside $v_3(e)=0$); $\kappa_2,\kappa_3$ give $22/28$ octuples. --- ## 11. Reproducibility `verify_paper.py` and `verify_jac_selmer.py` in `supplement.zip` use Python 3 and SymPy (`python3 -I verify_paper.py`, $\sim 90\text{ s}$) to verify all identities, factorizations, square-class checks, Sturm intervals, $2$-descent certificates, and tuples across Theorems 1–7 (`exp9`–`exp66`, `THEOREMS_AND_PROOFS.md`): 1. **Theorems 1–3:** $c^*_\mp$, $S_{ab}$-completions, $\mathcal{E}_{\text{Br1}},\mathcal{E}_{F_1}$, $a_{\text{circ}}(\lambda)$, positive sextuples, $\mathcal{S}_7^{(20/21)}$, $R_4=\frac{F_{\text{sym}} G}{(P_4-1)^4}$, $\mathcal{S}_{4,2}$, $\mathcal{H}$, $\text{rank}\,E_{\text{min}}^{(5)}(\mathbb{Q})=0$. 2. **Theorems 4–6:** $(r,\mu,z,w)$ syzygy, $1$-line formula, 4-$R_4$ / 1-$R_5$ Lorentz syzygies, One-$R_5$ surface ($\mathcal{K}_\mp=\mathcal{K}_{\mp,-1}\mathcal{K}_{\mp,+1}$, central $R_5$ norm $P_1 P_2 P_3 P_4$, Dual Brahmagupta, $\mathcal{Q}_{\text{phys}}$, $P_{\text{blowup}}$, $\mathcal{M}(\beta)=0$, $P_{\text{surf}}\xrightarrow{4:1} N_{\text{core}}$, $\rho_2=\alpha$ $E_{Y_W}$, $(19,2,10)$ & $\mathbb{Q}(\sqrt{5})$ sextuple, $\rho_2=\alpha+2r$ $\mathcal{C}_{\text{sec2},3..4}$, $(26/11,7/22)$ sextuple [4] & Rank-$4$ $d_-=\frac{2}{C_0\mp w_1}$, $\mu_{\text{inf}}$, $\mathcal{S}_7^{(f_c)},\mathcal{S}_7^{\text{3-refl}},\mathcal{S}_6^{\text{Euler}},\mathcal{S}_7^{\text{Euler},c},\mathcal{U}_8^{\text{Euler}},\mathcal{U}_9^{\text{bi-Euler}},\mathcal{M}_3,t_{\text{III}}(r),\mathcal{Q}_{5,(n,\varepsilon)}^{\text{refl}},\mathcal{Q}_5^\star,\mathcal{S}_6^{\text{refl}},\mathcal{C}_{\text{coinc},r}$, $L_1$-splitting, $\rho_2=r+\alpha$ $F_{12}=0$, $C_{9,11},G_{5,5}^{(a,b)},\mathcal{C}_{\text{hyp}}^{(8)},Q_{U,\mu},F_{6,7},C_{4,4}^{(g=a)},C_{5,5}^{(g=a)}$, $\mathbb{Q}(\sqrt{133})$ septuple; `exp45`–`exp66`), Curves I–III, $\mathcal{E}_B$, $R_5=\frac{L_1 L_2 Q_4}{H K^4}$, $\iota_K$, $\mathcal{E}'_{Q_4}(R)$, $\Lambda(r)$, $15+6$ Collapse, $v=-1/5$ ($\text{rank}\,E_{16}(\mathbb{Q})=0$). 3. **Theorem 7:** $(V,K,d)$ chains, $\mathcal{S}_{\text{3-tors}}$, Orbit Cubic, $1+\sigma_6$ & $P_6\ne 0$ (`exp52`), $C_r$, Master Syzygies, $\mathcal{U}_8^{\text{DKMS}}(\kappa)$, $E_{\text{Cheb}}(s)$, $Q_{\text{ex}}$, $\mathcal{C}_{\text{triad}}$, rank-$0$ $E_{15}$, Mixed-$R_4$ Syzygies, $U_\mp(r,s,\kappa)$, Five-$R_5$ $G_1-G_2=H(R_1-R_2)(R_1+R_2)$, $3$-factor $h M_i \mp D_i$ & $6$-torsion $H$-splitting on $P_{\text{orb}}=0$, $Q_h$ Three-Circle & rank-$0$ $E_t$, $(r,H_k)$ $R_3(d,3P_0,g)$ (`exp58`–`exp66`), $\mathcal{H}_{\text{II}}$ ($\text{rank}\,E_{\text{II},C_2}(\mathbb{Q}) \ge 2$), $H=1/3$ $2$-descent, Branch B ($10/10$ quintuple, $20/21$ over $\mathbb{Q}(\sqrt{-14})$, $\mathcal{M}_\Psi^\star$, rank-$0$ $E_{\text{coll}}$), and [20–22] ($\mathcal{I}_{\text{AL}}^{(0)}$, $d_1..d_4$, $k_{\text{tan}}(m)$, $E_{sw}$, $\mathcal{S}_{sw}$, Orbit-I $2$-Selmer/Chabauty, Post 69 syzygies, circle charts, rank-$0$ $E_{24}$, $23/28$ octuples). --- ## References [1] A. 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