PRCRealCompletenessTarget_of_diagonal_selection
plain-language theorem explainer
Diagonal selection of Cauchy-ledger limits is definitionally the same proposition as internal completeness of the null quotient. Anyone assembling the PRC real-completeness certificate cites this bridge to rename the blocker. The proof is a one-line exact transfer: the two targets are identical Props.
Claim. If every representative-Cauchy sequence of Cauchy ledgers admits a Cauchy-ledger representative limit, then the internal completeness target for the null quotient holds (every such sequence has a Cauchy-ledger limit).
background
In the Primitive Recognition Calculus, reals are built internally from Cauchy ledgers (rational sequences with a Cauchy modulus) modulo a null ideal. Completeness is not imported from classical analysis; it is an explicit blocker on those ledgers.
The completeness target asks: for every sequence $U$ of Cauchy ledgers that is Cauchy at the representative level, there exists a single Cauchy ledger $L$ that is a representative limit of $U$. The diagonal selection target is the same quantifier string: it packages the same demand under a name that stresses the diagonal construction of the limit ledger.
Upstream, both names are recorded as Prop aliases with identical bodies. The module sits after order-congruence infrastructure and before the sharpened completeness certificate that bundles raw realization, quotient points, and diagonal selection.
proof idea
One-line wrapper. The hypothesis is already a proof of PRCRealDiagonalSelectionTarget, whose body is definitionally equal to PRCRealCompletenessTarget. The tactic is exact hdiag; no rewriting, no intermediate lemmas.
why it matters
This is the rename step that lets the proved diagonal-selection theorem discharge the official completeness blocker. Downstream, PRCRealCompletenessTarget_proved applies it to PRCRealDiagonalSelectionTarget_proved, and the sharpened certificate records the full internal completeness package (raw Cauchy realization, quotient point, and diagonal selection from the raw diagonal ledger).
In the Recognition foundation stack this closes the analytic half of the internal ordered field: without a Cauchy-ledger limit for every representative-Cauchy sequence, the null quotient would not be complete and later forcing steps that treat the continuum as an RS-native object would stall. It does not itself construct the diagonal ledger; it only identifies the two named targets so the construction can be cited under the completeness name.
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