IndisputableMonolith.Foundation.PrimitiveRecognitionCalculus.RealCompleteOrderedFieldPromoted
Promoted Step 10 certificate module: the internal null quotient carries the closed field operations and theorem surfaces required by the complete ordered-field layer of Primitive Recognition Calculus. Foundation authors cite it when wiring real-arithmetic structure into the PRC kernel without waiting on full Mathlib typeclass instances. The module packages that certificate surface; packaging of Mathlib instances is deferred.
claimThe internal null quotient $Q$ of Primitive Recognition Calculus is equipped with the closed operations and theorem surfaces of a complete ordered field (addition, multiplication, order, completeness axioms as currently required by the PRC layer). Full Mathlib typeclass instances for $Q$ are not yet asserted; only the promoted certificate surface is.
background
Primitive Recognition Calculus builds its arithmetic layer from an internal null quotient rather than from an a-priori copy of $\mathbb{R}$. Upstream, the RealCompleteness module supplies the completeness side of that construction. The present module sits one step later: it promotes the Step 10 certificate that this quotient already exposes the closed operations and theorem surfaces demanded by the current complete ordered-field interface.
In RS terms this is foundation plumbing, not a forcing-chain step (T0–T8). The goal is a usable complete ordered field inside PRC so that later kernel material can treat real arithmetic as settled. The doc-comment is explicit that Mathlib typeclass instances remain a later packaging pass; what is certified here is the internal surface, not the external instance graph.
proof idea
Certificate packaging module, not a deep proof development. It exposes the promoted Step 10 certificate (sibling names: PRCRealCompleteOrderedFieldPromotedCertificate and the corresponding value) asserting that the internal null quotient already has the closed operations and theorem surfaces needed by the complete ordered-field layer. Argument structure is inheritance from RealCompleteness plus surface assembly; full Mathlib instance synthesis is intentionally out of scope.
why it matters in Recognition Science
Feeds the PRC Kernel module by import, so any kernel construction that needs a complete ordered field over the internal null quotient depends on this promoted surface being present. Without it, the kernel would either re-prove field/completeness facts or reach into incomplete scaffolding. In the broader Recognition foundation this is the bridge from real-completeness work to a usable arithmetic layer; it does not itself force $\phi$, the eight-tick octave, or $D=3$, but it keeps those later arguments on a closed real substrate. The deferred Mathlib packaging pass is the remaining open packaging debt named in the module doc.
scope and limits
- Does not install full Mathlib complete-ordered-field typeclass instances.
- Does not prove uniqueness of the reals up to isomorphism from first principles.
- Does not address T5–T8 forcing (J-uniqueness, phi, eight-tick, D=3).
- Does not define mass, alpha, or other RS physical constants.
- Does not claim the null quotient equals classical ℝ as a Mathlib object.