IndisputableMonolith.NumberTheory.PrimeLedgerAtom
The PrimeLedgerAtom module defines a prime ledger atom as a prime integer equipped with the irreducibility property required by the multiplicative ledger. Modules that reconstruct the Euler product as a partition over ledger postings and that adapt primality for RH proofs cite this definition to treat primes as atomic ledger units. The module consists of the core type definition together with equivalence lemmas that recover standard primality without analytic machinery.
claimA prime ledger atom is a prime integer $p$ together with the irreducibility condition appropriate to the multiplicative ledger, i.e., $p$ satisfies the ledger atom predicate that isolates primes as the indecomposable elements under the $J$-cost multiplication rule.
background
Recognition Science recasts integers as ledger states whose multiplicative costs are governed by the $J$-function and the Recognition Composition Law. The PrimeLedgerAtom module introduces the atomic objects of this ledger: primes that remain irreducible once embedded into the multiplicative structure. It sits inside the NumberTheory domain and supplies the bridge between classical primality and ledger postings.
proof idea
This is a definition module, no proofs. It declares the type PrimeLedgerAtom, the predicate prime_is_ledger_atom, the equivalence primeLedgerAtom_iff_prime, and the supporting facts one_not_primeLedgerAtom and prime_is_positive_ledger_state.
why it matters in Recognition Science
The module supplies the prime ledger atom definition required by the EulerLedgerPartition module to express the Euler product as the partition function of independent prime-ledger postings. It also underpins the LogicPrimeLedgerAtom recovery adapter for the RH stack and the RSPhysicalThesisDecomposition that replaces the opaque physical thesis with explicit ingredients. The definition corresponds to the atomic step that lets primes be treated as ledger units in the zeta decomposition.
scope and limits
- Does not prove any infinite product identities for the zeta function.
- Does not import or use analytic continuation results.
- Does not define ledger states for composite integers.
- Does not address prime distribution or conjectures beyond the atom definition.