Arithmetic Shadows of the Compact Fifth Dimension

We demonstrate that several fundamental objects in the arithmetic of elliptic curves and modular forms at levels 7 and 11 correspond to precise, computable realizations of the resonances generated by a single compact fifth spatial dimension S¹ of radius τ = 7 h⁻¹ Mpc in the Tau Universe framework.

In particular, we show that:
• The division polynomial ψ₇(x) and its three-term recurrence encode the core 142857 periodic mode.
• The explicit rational 5-isogeny φ : X₁(11) → X₀(11) realizes a finite quotient by 5-torsion, directly analogous to Kaluza–Klein image summation.
• The unique newform 7.4.a.a at level 7, weight 4, together with its Hecke eigenvalues, provides the multiplicative arithmetic shadow of the τ = 7 scale.
• Poisson summation applied to the Gaussian on the compact dimension recovers the modified Bessel function K₀(r/τ) after regularization, furnishing the closed-form signature of the fifth dimension.
• Computational visualizations of the Tav Resonance Graph illustrate how the 1/7 periodic mode organizes arithmetic orbits, with Tav topology acting as a partially non-computable selection filter.

We present these correspondences as structural analogies, not as a demonstration that the integer 7 is dynamically privileged; that question is posed as an open problem and is the subject of a pre-registered falsification protocol deposited separately (DOI: 10.5281/zenodo.20510821), whose expected outcome is that no special status for 7 is found.