logicInt_finite_phase_separates
plain-language theorem explainer
Non-identity recovered logic integers map to reciprocal integer ledgers separated from the identity by a finite cyclic quotient. Number theorists extending the logic-to-ledger interop in Recognition Science cite this when moving phase-budget properties across the boundary. The argument is a direct one-line application of the finite phase separation theorem for non-identity reciprocal ledgers.
Claim. Let $z$ be a logic integer whose recovered value is positive, not equal to 1, and satisfies the reciprocal budget condition. Then there exists a natural number $c > 0$ such that the carrier of the corresponding reciprocal integer ledger, viewed in the cyclic group of order $c$, is not congruent to 1.
background
LogicInt is the Grothendieck completion of LogicNat under addition, representing integers as equivalence classes of pairs of naturals. LogicIntNonIdentityReciprocal is the structure requiring the recovered integer to be positive, unequal to 1, and to divide the square of some positive natural for the budget condition. The module transfers such objects to the integer-ledger phase-budget surface. The key upstream result states: 'A non-identity positive integer is separated from the identity in a finite cyclic quotient. We use the quotient Z/(carrier+1)Z: the carrier has phase -1, not 1.'
proof idea
The proof is a one-line wrapper that applies finite_phase_separates_nonidentity to the reciprocal integer ledger obtained from the hypothesis via reciprocalIntegerLedger_of_logicInt.
why it matters
This declaration completes the transfer of non-identity logic integers into the finite phase separation framework for reciprocal integer ledgers. It supports the phase-budget surface by guaranteeing a cyclic quotient witness. No downstream uses are recorded, leaving open how the separation modulus interacts with the eight-tick octave or the phi-ladder in larger Recognition Science constructions.
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