Editing Openai/6912dc03-9c10-8006-9f6f-226c9f5e2154 (section)

===== This paper proposes and formalizes the Cross-Base Invariance Theorem of prime numbers — =====
a novel discrete framework revealing structural symmetry among primes under mixed-base representations.
Let a,ba,ba,b be distinct primes, and Gn(a,b)G_n^{(a,b)}Gn(a,b) the cross-base group whose digits are expressed such that the units place follows base aaa and the tens place follows base bbb.
For consecutive integers n,n−1n,n-1n,n−1, define a digit-difference vector v⃗n(a,b)\vec{v}_n^{(a,b)}vn(a,b).
Then, for all distinct primes a,ba,ba,b, the relation

v⃗n(a,b)=rev(v⃗n(b,a))\vec{v}_n^{(a,b)} = \mathrm{rev}(\vec{v}_n^{(b,a)})vn(a,b)=rev(vn(b,a))
holds for sufficiently large nnn.
This property extends to all prime pairs and generalizes to the Mixed-Base Self-Invariance Theorem,
showing that for any prime k>3k>3k>3, the vectors at kkk and k−1k-1k−1 remain identical or inversely symmetric when expressed under bases (k,k−1)(k,k-1)(k,k−1).
The resulting structure forms a discrete analogue to the analytical symmetry of the Riemann Hypothesis, replacing s↔1−ss \leftrightarrow 1-ss↔1−s with base exchange (a,b)↔(b,a)(a,b)\leftrightarrow(b,a)(a,b)↔(b,a).