delta_native_strong_closure
plain-language theorem explainer
The Delta-native recognition calculus admits one strong-closure certificate packaging every closed theorem and audit layer of the interface. Foundation auditors cite it to confirm the full stack is inhabited as a single object, not only layer by layer. The proof is a one-line term witness: nonemptiness of the certificate type via the concrete assembled certificate.
Claim. There exists a strong-closure certificate for the Delta-native interface: a single bundle whose fields are closure entries for the real-display protocol, generable carriers (parameterized by sequences $\mathbb{N}\to\mathbb{R}$), certified analytic protocols and transformers (parameterized by their registries), the FRS carrier, calibration and physical-calibration layers, prime-axis coherence, and multi-distinction geometry.
background
In the Primitive Recognition Calculus, Delta-native material is the real-facing layer of recognition: display protocols on reals, generable carriers, certified analytic expression protocols and transformers, the FRS carrier, calibration maps, prime-axis coherence, and multi-distinction geometry. Each layer already has its own closed theorem heads; the local module does not reprove them.
A closure entry is a thin pointer to an existing theorem head. The strong-closure certificate is a structure whose fields are such entries (or functions returning entries when the layer is parameterized by a registry or a sequence). The concrete certificate definition fills every field by wrapping the corresponding closed theorem via entryOf.
The module setting is an audit surface: one place that records that the whole Delta-native theorem stack is closed and can be named as a single inhabited type.
proof idea
One-line term proof. The certificate type is a structure; the sibling definition already builds a concrete value by assigning each field an entryOf wrapper around the corresponding closed theorem (real-display forgetful protocol, generable-field operational carrier, transcendental protocol closure, transformer registry closure, FRS carrier, calibration layers, prime-axis, multi-distinction geometry). Nonemptiness is then immediate: inhabit the type by that concrete value.
why it matters
This is the terminal audit object for the Delta-native half of Primitive Recognition Calculus. Downstream consumers (none yet wired in the graph) can depend on a single Nonempty fact instead of importing every layer theorem separately. It does not advance the forcing chain T0–T8, the Recognition Composition Law, or the mass ladder; it certifies that the real-analytic and geometric support stack those landmarks rest on is fully closed in Lean.
The doc-comment frames it as the single Lean certificate for the full Delta-native interface. With used_by empty, its role is archival and compositional: a named proof that the bundle type is inhabited, so later foundation or physics modules can treat Delta-native closure as one hypothesis rather than a scattered checklist.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.