PrimeLedgerCert
PrimeLedgerCert packages three arithmetic facts certifying that primes coincide exactly with ledger atoms, that 1 is excluded, and that primes satisfy the positive-integer ledger state. Number theorists building the RH bridge in Recognition Science cite this certificate to anchor classical prime postings before they enter the Euler product. The declaration is a bare structure definition that simply collects the three properties with no proof obligations.
claimA certificate consisting of three properties: (i) for every natural number $p$, $p$ is a prime ledger atom if and only if $p$ is prime; (ii) $1$ is not a prime ledger atom; (iii) every prime $p$ satisfies the integer ledger state (i.e., $p>0$).
background
Positive integers are the states of the multiplicative integer ledger via the definition IntegerLedgerState $n$ := $0 < n$. A prime ledger atom is a natural number $p$ that is prime together with the irreducibility posting condition: if $p = a b$ then $a=1$ or $b=1$. The module grounds the arithmetic ledger used by the RH bridge; primes are exactly the irreducible postings of the multiplicative positive-integer ledger. The structure relies on the local alias Prime for Nat.Prime and on the two sibling definitions PrimeLedgerAtom and IntegerLedgerState.
proof idea
This is a structure definition with an empty proof body. It functions as a data container that bundles the three named properties without any reduction steps or tactic applications.
why it matters in Recognition Science
The certificate is consumed by EulerLedgerPartitionCert to equate the finite prime-ledger partition with the zeta function on Re(s)>1, and by both RSPhysicalThesisData and RSPhysicalThesisDataLogic to decompose the physical thesis data while retaining the classical ledger. It supplies the arithmetic base step that precedes the Euler product and the completed zeta ledger in the Recognition Science chain. The declaration closes the classical-to-RS transport for the prime postings before any J-cost or phi-ladder weighting is applied.
scope and limits
- Does not prove infinitude of primes.
- Does not address prime distribution or density.
- Does not invoke the J-function or Recognition Composition Law.
- Does not perform any analytic continuation or zero counting.
Lean usage
def primeLedgerCert : PrimeLedgerCert where atom_iff_prime := primeLedgerAtom_iff_prime one_not_atom := one_not_primeLedgerAtom prime_positive := prime_is_positive_ledger_stateformal statement (Lean)
61structure PrimeLedgerCert where
62 atom_iff_prime : ∀ p : ℕ, PrimeLedgerAtom p ↔ Nat.Prime p
63 one_not_atom : ¬ PrimeLedgerAtom 1
64 prime_positive : ∀ {p : ℕ}, Nat.Prime p → IntegerLedgerState p
65
66/-- The prime ledger certificate. -/