Fixed Points as Sections of an Elliptic Fibration: the Isosceles Pedal Correspondence, its Minimal Weierstrass Model, a Classified Mordell-Weil Lattice, and an Explicit Picard-Fuchs Operator with Unipotent Monodromy

The isosceles pedal correspondence of the map F_t (pedal triangle of P_t = (1-t)I + tN, I the incenter, N the Nagel point) is proved, by an exact coordinate computation valid for ALL parameters, to be the two-valued algebraic correspondence s'^2 = Q_t(s)^2 s^2 / D_t(s). This removes the conditional status carried by earlier papers of the series: the 45 rational test points become verifications, not the foundation. GEOMETRY. The effective curve is birational to y^2 = p_t(s), and we construct a Q(t)-isomorphism to the MINIMAL Weierstrass model Y^2 = X^3 - 27 I(t) X - 27 J(t), with I = t^4 - 4t^3 + 20t^2 - 20t + 4, J = -2t^6 + 12t^5 - 72t^4 + 172t^3 - 324t^2 + 120t - 16, 4I^3 - J^2 = -432 t^2 H, H = 3t^4 - 10t^3 + 59t^2 - 28t + 4, hence Delta = -2^8 3^12 t^2 H, j = -16 I^3/(t^2 H), j - 1728 = -4 J^2/(t^2 H). This supplies the proper relative minimal model left open in the companion elliptic-reduction paper. We prove H > 0 on all of R; the fibres are I_2 at t=0, I_6 at t=infinity, and four I_1 at non-real parameters, so the trivial lattice is A_5 + A_1 and E_t(R) is connected for every real t != 0. ARITHMETIC. By three explicit specialisations we prove that the Mordell-Weil group is TORSION-FREE of rank 2, and that det MW = 1/12 is forced by unimodularity of Neron-Severi. Four polynomial sections are exhibited, with the exact relations 2P + Q = O and E + J = Q. Two of them are the fixed points of the pedal map, since p_t(1/2) = p_t((2t-1)/(2t)) = 1/4 identically. Torsion-freeness excludes the torsion case of the A_5 + A_1 entry of the Oguiso-Shioda classification; reported exact heights 1/6 and 1/2 with zero pairing identify the lattice as A_1* + <1/6>. THE RANK, THE LATTICE TYPE AND THE EXISTENCE OF RELATIONS ARE NOT NEW: they follow from the 1991 classification, which we cite. The entry attribution is conditional on a table read in a secondary source and not verified against the primary record. ANALYSIS. The Gauss-Manin connection is computed in closed form in the basis (dx/y, x dx/y). The auxiliary quantity collapses: 3I'J - 2IJ' = -432 t (9t^2 + t - 2), so K = -Q/(54 t H) with Q = 9t^2 + t - 2, and the Picard-Fuchs operator is obtained EXPLICITLY, of degrees (8,7,6). The identity D'^2 + 108 A N_K^2 = 36 D (A A'^2 + 3 B'^2) forces local exponents (0,0) at every zero of the discriminant: all local monodromies are UNIPOTENT, of widths 2,1,1,1,1,6 with product the identity. The two roots of Q are apparent singularities with exponents (0,2). The family is provably not a pullback of the Legendre or Hesse pencils (j has simple poles). CM fibres occur exactly at the roots of the irreducible quartic I, one of them inside (1/2,1); there is NO complex multiplication at the dynamical thresholds. DYNAMICS (the intended contribution). The transcritical collision at t=1 is the collision E = J of two sections. The pole divisor of the differential inherited from the correspondence degenerates exactly at t in {-1, 0, 1/3, 1}, and t = 1/3 is precisely the common zero of the two multipliers. That differential is of the third kind with rational residue divisor; its periods live in the rank-5 cohomology of the four-punctured fibre and are elliptic logarithms. By contrast the transverse and metric thresholds t_0 = (3+sqrt3)/6 and t_V are invisible to the curve, and the operator is regular there: E_t encodes only the longitudinal isosceles direction. Dynamical degenerations are visible in the MARKED geometry, not in the moduli of the curve. SCOPE. Statements are labelled [D] proved, [E] exact computation, [L] literature not verified against the primary record, [C] open. No physical, quantum or thermodynamic claim is made; no claim relating this material to the Riemann hypothesis or to entropy functionals of geometric flows is made. Novelty with respect to the literature has not been exhaustively checked. Corrections to earlier versions: the isosceles reduction is now proved for all parameters; (1,0) is a rational branch point used as origin, not a rational 2-torsion point; the formula P_2 = C t(t-1)H is withdrawn; positivity of H is upgraded from [1/2,1] to R; the Mordell-Weil structure is cited as classified, not posed as open. REPRODUCIBILITY. schloss_elliptic_exact.py (SymPy, exact rational arithmetic, no floating point) recomputes every item marked [E]: the universal reduction identity, both models and the change of variables, I, J, 4I^3-J^2, c_4, c_6, Delta, j, positivity of H, the four sections and the two relations, the three specialisation fibres with their point counts and torsion bounds, the collapse 3I'J-2IJ', the exponent identity, the expanded Picard-Fuchs coefficients, the pole divisor and residues, and the degeneration at t=1/3.