Projective Invariance of Modular Covariants and Phase Conjugation in the $X_1(2,8)$ Elliptic Surface

This paper investigates the geometric origin of a pair of spectrally conjugate fibers arising in a recently constructed explicit $X_1(2,8)$ elliptic surface over $Q(t)$. In earlier work (see https://zenodo.org/records/17489997), this surface was shown to exhibit a rank-one specialization together with a distinguished pair of fibers whose modular phases appear as conjugate limits. While the phenomenon was observed algebraically, its structural cause remained unclear. The present work resolves this question by placing the construction in a projective framework. The key observation is that the Möbius base change defining the $X_1(2,8)$ family is not merely a fractional linear substitution, but the projective action of an element of $GL_2(K)$ on the projective line $P^1(K)$. When modular covariants are interpreted intrinsically as projective rays rather than affine points, the observed pairing of fibers emerges as a necessary consequence of projective antipodality. Within this framework, several points are clarified: The obstruction to global Néron minimality is geometric and projective in nature, arising from the poles of the Möbius transformation at the boundary of an affine chart. Local Néron raising preserves modular phase precisely because it does not alter the underlying projective ray. The apparent failure of global minimality does not contradict the expected Euler characteristic of a rational elliptic surface, but reflects a chart-dependent limitation of affine coordinates. The limiting modular phases associated with the distinguished fibers are intrinsic and chart-independent, forced by the projective structure of the base change. To support the structural claims, the paper includes a formal appendix developed in Lean 4. This appendix verifies, in a conservative and classical manner, the construction of projective rays, the group structure of $GL_2(K)$, and the induced action on $P^1(K)$, together with the standard affine Möbius formulas (including the point at infinity). The formalization serves as an independent certification that the geometric mechanism used in the paper rests on standard algebraic foundations. The results suggest that similar projective phenomena should arise for Möbius base changes in other $X_1(N)$ families, motivating further exploration of the interaction between modular geometry, projective structure, and arithmetic invariants.