IndisputableMonolith.NumberTheory.LogicLedgerInterop
This module bridges logic-derived integer ledgers to standard arithmetic by defining the recovered integer type and its non-identity condition. Researchers citing the Recognition Science integer foundations would use it to connect logic constructions to finite phase results. The module consists of definitions and basic lemmas that import and apply the upstream integers-from-logic and finite-phase-completeness material.
claimLet $L$ be the type of recovered integer ledgers. The recovery map sends $L$ to an integer $n$. The ledger is non-identity precisely when $n > 0$ and $n ≠ 1$.
background
The module imports IntegersFromLogic, which constructs integers directly from logical primitives, and FinitePhaseCompleteness, whose doc-comment states: 'a non-identity integer ledger has a finite cyclic quotient in which its phase is not the identity phase. This is the finite arithmetic content behind the RS statement that a non-identity reciprocal ledger cannot be invisible in all finite phase quotients.' The central definitions are the recovered ledger type, its integer recovery map, and the predicate that marks a ledger non-identity exactly when the recovered integer is positive and unequal to one. These objects supply the concrete arithmetic handle needed to invoke phase-separation lemmas.
proof idea
This is a definition module, no proofs.
why it matters in Recognition Science
The module supplies the interoperation layer that feeds the finite-phase-completeness theorem and subsequent number-theoretic results in the Recognition Science framework. It translates the abstract logic ledger into an object on which the non-identity phase-separation statement can be applied directly.
scope and limits
- Does not prove any phase-completeness theorem.
- Does not construct the logic integers from scratch.
- Does not treat infinite phase quotients.
- Does not compute explicit numerical values for recovered ledgers.