primeLedgerLogicCert
plain-language theorem explainer
This definition assembles a concrete certificate equating logic-native prime atoms on LogicNat with classical primality after the toNat recovery map. Number theorists adapting the Riemann hypothesis to recovered-number systems cite it to justify transport of prime factors between logic and classical ledgers. The construction is a direct record assembly that populates the four fields of PrimeLedgerLogicCert from three supporting theorems plus reflexivity.
Claim. Let $PrimeLedgerLogicCert$ be the structure with fields asserting that for all $p$ in the logic naturals, $PrimeLedgerAtomLogic(p)$ holds if and only if $Nat.Prime(toNat(p))$, that $1$ is not a prime atom, that every classical prime yields a positive integer ledger state, and that the logic predicate transports exactly to the classical $PrimeLedgerAtom$. The definition $primeLedgerLogicCert$ constructs the unique such certificate by direct assembly of the relevant equivalences and lemmas.
background
In LogicPrimeLedgerAtom the primality predicate is formulated directly on LogicNat. The recovery map toNat converts these objects to ordinary naturals, and PrimeLedgerLogicCert is the structure that certifies the recovered notion coincides with classical primality. The four fields are atom_iff_prime, one_not_atom, prime_positive, and transports_classical; the module doc describes this as the first recovered-number adapter for the RH/prime-ledger stack.
proof idea
The definition is a record constructor that supplies the four fields of PrimeLedgerLogicCert. atom_iff_prime is taken from the theorem primeLedgerAtomLogic_iff_prime, which reduces to the classical equivalence after toNat. one_not_atom is supplied by one_not_primeLedgerAtomLogic, which derives a contradiction via ledger_atom_logic_is_prime and Nat.not_prime_one. prime_positive comes from prime_logic_is_positive_ledger_state using the order on LogicNat. transports_classical is the trivial Iff.rfl.
why it matters
The certificate is required by recovered_prime_ledger_supports_euler_product, which pairs it with Euler-product convergence to recover the classical zeta function. It is also used in logicData_of_boundaryTransport to construct RSPhysicalThesisDataLogic without an extra prime-ledger hypothesis. In the Recognition Science framework it closes the recovered adapter for the prime ledger, allowing the phi-ladder and eight-tick octave to operate on logic-native data while preserving analytic properties.
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