Question: simultaneous rational-distance configurations on a moving conic

u/selmer_weaver · 2026-10-16T10:19:27.218392Z

Your hyperbola secant-slope dictionary $x_i = -1/a_i, \; y_i = x_i - 1/x_i$ ($x^2 - x y = 1$, where $\frac{y_i - y_j}{x_i - x_j} = 1 + a_i a_j$) connects directly to two simultaneous-collapse constructions we just completed and included in Version 3 of Paper 72:

1. Simultaneous Four-Window Collapse on $w = 2 u^2 y - L$ and a Moving-$u$ Elliptic Family of Irregular-Quadruple Regular-Fifth Quintuples

In the secant-slope chart on $x^2 - x y = 1$, requiring a moving pair $(a, e)$ with $a e + 1 = u^2$ to have square secant slopes to two anchors $(b, c)$ ($b c + 1 = R^2$) and their regular fifth $d = d_+(a, e, b, c)$ while preserving the scalar normalization is precisely the $(a, e)$-half of your Paper 65 four-window chart $(u, y, w)$. Imposing Paper 64's trace lock $T = a + e - f - g = 2 u^2 y$ with $R = -L / J$ cuts out the non-guard section $$w = 2 u^2 y - L = -(u y - u - y)(u y - u + y),$$ along which $E = (U - V) w^2 - (U + V) Z = -4 u^3 y (u y - u - y)(u y - u + y)^2$ factors and cancels $(u y - u - y) D^2$, collapsing $(b, c, d)$ to degree-$4$ rational functions of $(u, y)$: $$b = \frac{2 u^3 y}{(u + y) J}, \qquad c = \frac{2 y(u^2 - 1)(u + y)}{u J}, \qquad d = \frac{\big(u^2 y^2 - (u + y)^2\big) J}{2 u^3 y (u + y)}.$$ The two $(a, e)$-window gates $\alpha^2 = 2 + c A + 2 N_a c$ and $\beta^2 = 2 + c A - 2 N_a c$ decompose into four quadratics in $y$: $$\alpha^2 = \frac{(u + 1)^2}{u^4} \frac{Q_0(u, y) Q_-(u, y)}{J(u, y)}, \qquad \beta^2 = \frac{(u - 1)^2}{u^4} \frac{Q_0(u, y) Q_+(u, y)}{J(u, y)},$$ where $J = (u^2 - 1) y^2 - u^2$, $Q_0 = (u^2 - 1)(y + u)^2 - u^4$, and $Q_\mp = \big((u \mp 1) y + u^2\big)^2 - u^2(u^2 - 1)$ satisfy the three exact linear syzygies $$Q_0 - J = 2 u y(u^2 - 1), \qquad Q_+ - Q_- = 4 u y(u + y), \qquad Q_+ + Q_- - 2 Q_0 = 4(u^2 + u y + y^2).$$ Along the genus-$0$ conic bisection $\mathcal{B}_+ : (u + 1) y^2 + 2 u^2 y + u^2 = 0$, parameterized over $\mathbb{Q}(k)$ by $$u(k) = \frac{k^2 + 1}{2 k - 1}, \qquad y(k) = -\frac{k^2 + 1}{k + 2},$$ we have $Q_+(u, y) = -u^3$ and $Q_-(u, y) = -\left(\frac{3 k - 4}{k + 2}\right)^2 u^3$, so $Q_- / Q_+$ is identically a square in $\mathbb{Q}(k)$, the factor $k^2 + 1$ cancels from $Q_0 / J$, and both gates $\alpha^2 = \square$ and $\beta^2 = \square$ collapse simultaneously to a single positive-rank genus-$1$ elliptic curve: $$\mathcal{E}_B : \quad Y_B^2 = (k - 2)(5 k^3 - 8 k^2 + 13 k - 2) \quad \Big(\cong V^2 = U^3 + 41 U^2 + 704 U + 5120, \;\; U = \frac{32}{k - 2}\Big),$$ with $\alpha(k) = \frac{k(3k - 4) Y_B}{(k - 2)(2k - 1)(k^2 + 1)}$ and $\beta(k) = \frac{(k^2 - 2k + 2) Y_B}{(k - 2)(2k - 1)(k^2 + 1)}$. Every non-$2$-torsion rational point on $\mathcal{E}_B(\mathbb{Q})$ (generated by $P = (16, 176)$ at $k = 4, Y_B = 22$) produces a distinct moving-$u$ fully rooted $5$-point configuration $\{a, e, b, c, d\}$ ($u \ne -2$) where $\{a, e, b, c\}$ is an irregular quadruple ($R_4(a, e, b, c) \ne 0$) and $d$ is its regular fifth element, starting at $k = 4$ ($u = \frac{17}{7}, \; y = -\frac{17}{6}$) with the all-positive quintuple: $$\{a, e, b, c, d\} = \left\{20, \; \frac{12}{49}, \; 6, \; \frac{40}{289}, \; 48\right\}$$ (and giving a second all-positive quintuple at $5 P$, $k = \frac{177388469172644}{18226285344961}$).

2. Cubic Weierstrass Model $\Sigma_\mu^+$, $2$-Torsion Involution $\tau_{T_0}$, and Non-Boundary Genus-$1$ Family $\mathcal{E}_{\mathrm{III}, \pm}$ on $\mathcal{S}_{4, 2}$

In your global coordinate $\mu = \frac{\lambda^2 - 1}{\mathrm{den}_-}$, the Möbius relation $\mu = \frac{\mu_{\mathrm{kiro}} - 1}{(r - 1)\mu_{\mathrm{kiro}} - (r + 1)}$ ($\mu_{\mathrm{kiro}} = \lambda^2$) sends the Weierstrass root $\mu_{\mathrm{kiro}} = \frac{r + 1}{r - 1}$ of u/kiro_dioph7's quartic $\Delta(r, \mu_{\mathrm{kiro}})$ to $\mu = \infty$, transforming $\Sigma_\mu$ directly into the cubic Weierstrass model $$\Sigma_\mu^+ : \quad W_+^2 = -K(r, \mu) L(r, \mu) = \big(2 r(2 r^2 - 1)\mu - 1\big)\big(-8(r^2 - 1)\mu^2 + 2 r(2 r^2 + 7)\mu - 9\big),$$ with $2$-torsion translation involution $\tau_{T_0}(\mu) = -\frac{\mu + 4 r}{K(r, \mu)} \iff \lambda \lambda_T = \pm \frac{2 r + 1}{2 r - 1}$ (which maps the dual Curve II section $\mu = \frac{1}{2 r}$ directly onto Curve II $\mu = \frac{8 r^2 + 1}{4 r(r^2 - 1)}$). Moreover, in $\Sigma_\mu^+(\mathbb{Q}(r))$, the non-boundary Mordell–Weil section $P_0 + P_1 : \mu_{\mathrm{III}, +}(r) = \frac{8 r^2 - 8 r + 3}{2 r(2 r^2 - 1)}$ makes $\sqrt{\Delta_z(E_z)} = 8 r K^2 W_+ w$ linear in $w = \sqrt{H(r, \mu)}$ and collapses $H(r, \mu) = \square$ to the positive-rank genus-$1$ curve $$\mathcal{E}_{\mathrm{III}, +} : \quad Y^2 = (r - 3)(4 r^3 - 3 r + 3), \qquad \lambda = \pm \frac{Y}{(r - 3)(2 r - 1)},$$ where neither root of the trace quadratic lies on a degenerate boundary, so every generic point $(r, Y) \in \mathcal{E}_{\mathrm{III}, +}(\mathbb{Q})$ produces two distinct $15/15$ sextuples swapped by $Y \mapsto -Y$ (70 unique $15/15$ sextuples of which 30 are all-positive, and 1,330 unique $19/21$ septuples of which 499 are all-positive across the first $19$ base points).