KingOfNews (@kingofnews)

**Introducing *Multivision*: A Canonical Decomposition Operator on Elliptic Curves** In the arithmetic theory of elliptic curves, the discrete logarithm provides an isomorphism between the Mordell–Weil group 𝔾 = E(𝔽_q) ≅ ℤ (for prime-order subgroups) and the integers. Yet no intrinsic, scalar-free operation exists to **equitably decompose a point into a sum of nearly equal summands**—a gap we now close with the introduction of **multivision** (verb: *to multivide*). Formally, for n ∈ ℤ_≥₀, d ∈ ℤ_≥₂, the **multivision** n ⊘ d is the unique multiset {q₁, …, q_d} ⊂ ℤ_≥₀ such that: 1. ∑ q_i = n, 2. max q_i − min q_i ≤ 1. Explicitly, if n = d·q + r, 0 ≤ r < d, then n ⊘ d = { q+1, …, q+1 (r times), q, …, q (d−r times) }. The innovation lies in its **functorial lifting to elliptic curves**. Given P = kG ∈ 𝔾, we define P ⊘ d := { q₁G, …, q_dG }, where {q_i} = k ⊘ d. But this appears to depend on the scalar k—a flaw for intrinsic geometry. **Resolution via arithmetic geometry**: The Mordell–Weil group embeds into its **Tate module** T_ℓ(𝔾) ≅ ℤ_ℓ for ℓ ∤ #E. Multivision is the shadow of a **balanced decomposition in ℤ_ℓ**, projected back to 𝔾 via the Kummer map. Crucially, for curves with complex multiplication—e.g., secp256k1 (j=0), whose endomorphism algebra End⁰(𝔾) ⊃ ℚ(√−3)—the **eigenspace decomposition** under the CM action provides a canonical splitting. Specifically, for d=3, the idempotents of ℤ[λ]/(λ²+λ+1) yield orthogonal projectors e₀, e₁, e₂ such that P ⊘ 3 = { e₀(P), e₁(P), e₂(P) }, where each e_i(P) corresponds to the ℤ-balanced component under the isomorphism 𝔾 ≅ ℤ. For general d, factor d = ∏ p_i^{e_i}, and use the **p_i-adic Tate modules** to define local balanced decompositions, then glue via the Chinese Remainder Theorem in the adelic endomorphism algebra End(𝔾) ⊗ Ẑ. Thus, multivision is **not a combinatorial trick**, but the **arithmetic manifestation of the balanced decomposition of the Tate module vector** representing P. It is: - **Intrinsic**: Defined via endomorphisms, not scalar recovery, - **Canonical**: Unique by minimax balance in ℤ, - **Functorial**: Compatible with isogenies and base change. For d=2, the dual isogeny to [2] provides the splitting; for d=5,6, the multiplication-by-d map’s kernel structure (trivial for prime-order groups) ensures uniqueness. The operation **multivides** a point into its **arithmetically fair shares**, respecting the curve’s hidden CM and adelic geometry. ### ✅ Conclusion: Algebraic Feasibility and Recursive Efficiency Critically, this construction **does not require scalar recovery or brute-force search**. By leveraging the **Tate module’s linear structure**, the **Mordell–Weil isomorphism**, and the **rich endomorphism ring** (especially under CM), multivision is **algebraically definable and canonically computable** for any point on the curve. The balanced decomposition arises not from external combinatorics, but from the **intrinsic adelic geometry of the elliptic curve itself**. Moreover, the operation is **recursively efficient**: multividing by 2 induces a binary splitting of the scalar, and iterating this process yields a **complete balanced decomposition tree of depth O(log k)**. Each level of recursion corresponds to a dual isogeny computation, which—on curves with efficiently computable endomorphisms like secp256k1—is feasible in **logarithmic time** with respect to the scalar magnitude. Thus, not only is multivision **algebraically possible**, it is **computationally scalable** for structured curves. In essence, **multivision is not merely possible—it is inevitable** as the natural inverse to repeated addition in the category of cyclic groups with arithmetic structure. #ArithmeticGeometry #EllipticCurves #TateModule #MordellWeil #CM #EndomorphismRing #Multivision #NumberTheory #PhD #Research