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We propose and study a new arithmetic invariant of non-hyperelliptic genus-4 curves: a canonical��“quadratic” point on the Jacobian, defined by the two natural degree-2 maps to projective lines.��Building on Xue’s result, that this point is generically non-torsion, we introduce a height-theoretic��notion of bigness for families of curves in the moduli space of genus��four��curves, and give a criterion,��via��dimensions of modular quotients,��for when it holds. We then exhibit two concrete 3- and 4-parameter��families (the bi-involutions locus and a CM example) in which the canonical point is provably big, from��which we deduce the finiteness of low-height curves and non-torsion at transcendental moduli. Our��methods combine adelic Arakelov intersection theory with a generic Betti-rank argument.