IndisputableMonolith.NumberTheory.LogicPrimeLedgerAtom
This module defines prime ledger atoms over naturals recovered from logic, establishing them as the irreducible postings in the multiplicative positive-integer ledger. It supplies the arithmetic foundation for the RS zeta program by linking logic-derived primes to classical ones. Researchers on the Recognition Science Riemann Hypothesis bridge cite it to justify the prime postings before they enter the Euler ledger. The module consists of definitions and equivalence statements that connect the recovered ledger to standard number theory.
claimOver the recovered natural numbers, a prime ledger atom is an irreducible posting $p$ in the multiplicative positive-integer ledger, satisfying the ledger-atom condition that it cannot be expressed as a nontrivial product of two elements greater than one.
background
The module sits inside the NumberTheory domain and imports the ArithmeticFromLogic foundation for recovered naturals together with the classical PrimeLedgerAtom module. The upstream PrimeLedgerAtom documentation states: 'This module grounds the arithmetic ledger used by the RH bridge. The content is deliberately classical and small: primes are exactly the irreducible postings of the multiplicative positive-integer ledger. The RS work starts after this point, when the prime postings are weighted into the Euler ledger.' It introduces the logic-flavored variants of these atoms and the associated ledger-state predicates.
proof idea
This is a definition module, no proofs.
why it matters in Recognition Science
The module feeds the EulerProductEqualsZeta construction, which 'wires the formal RS prime-ledger partition to Mathlib's theorem that the Euler product over primes equals the Riemann zeta function on the half-plane Re(s) > 1.' It also supplies the recovered-prime-ledger hook for LogicRH_From_RCL, described as the 'Recovered-prime-ledger wrapper for the RH-from-RCL decomposition' that replaces the arithmetic ledger certificate while leaving the analytic infrastructure unchanged. It closes the classical-to-logic transition before Recognition Science weighting of primes into the Euler ledger.
scope and limits
- Does not derive the Euler product identity or zeta equality.
- Does not perform analytic continuation or locate zeros.
- Does not invoke the Recognition Composition Law or phi-ladder.
- Does not address the eight-tick octave or spatial dimension forcing.