All-Prime Register Exact Sequence

Abstract

The all-prime register is exact with a prime-adic hidden kernel.

Theorem 1.1 (The all-prime register has the full prime-adic kernel).

$$0\to \prod _{p\text{prime}}{\mathbb{Z}}_p\to \Sigma \to \mathbb{T}\to 0$$

Proof. Machine-checked in Lean as D5/S3/Factorization/Solenoid/AllPrimeRegisterExactSequence.all_prime_register_short_exact (✓ std3). ∎

Source. Repository-derived.

Commentary.

Take the register containing every prime. Its hidden fiber is the product of one prime-adic integer ring for each prime, and its visible coordinate is a point on the circle. The theorem states injectivity of the hidden-fiber inclusion, exactness at the universal solenoid, surjectivity of the visible projection, and bijectivity of the kernel classification.

This is the source-enumerated all-prime exact-sequence clause. It does not assert an arbitrary-prime-set construction or identify the universal solenoid with a separately defined rational dual.

The repository already proves the exactness, surjectivity, and kernel classification in universal_solenoid_profinite_exact. The present theorem applies that exact result and records the injectivity of the canonical subtype inclusion explicitly.

References

• Truth anchor: D5/S3/Factorization/Solenoid/AllPrimeRegisterExactSequence.all_prime_register_short_exact
• Dependency: D5/S3/Factorization/SolenoidProfiniteKernel