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IndisputableMonolith.Foundation.PrimitiveRecognitionCalculus.Kernel

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Bundling module for the first-pass Primitive Recognition Calculus (PRC) kernel certificate (K7/A2). It assembles concrete Lean objects for each stage of the opening theorem chain from Basic through Orbit arithmetic and Inevitability. Cited by anyone checking that the analytic PRC specification is fully inhabited before native cost uniqueness or the universal foundation. Structure is a certificate aggregate, not a single deep proof.

claimThe first-pass PRC kernel certificate asserts that every stage of the opening Primitive Recognition Calculus chain has a concrete Lean inhabitant: basic recognition primitives, same-difference and trace logic, the formal system, inevitability scaffolding, quotients, orbits with their arithmetic and divisibility, and integer/rational order. It is a bundling certificate $K_{7}/A_{2}$, not the final inevitability theorem.

background

Primitive Recognition Calculus (PRC) is the foundation layer that rebuilds arithmetic and cost structure from recognition primitives rather than assuming $\mathbb{Z}$ and $\mathbb{Q}$ as given. Upstream modules supply the pieces: Basic and Strength fix the raw recognition operations; SameDiff and TraceLogic encode equality and path structure; FormalSystem packages the deductive shell; Inevitability states the target uniqueness claims; Quotient, Orbit, OrbitArithmetic, OrbitDivisibility, IntegerRational, and IntegerOrder construct integers and rationals from orbit data.

This Kernel module does not redefine those objects. It sits above them as the K7/A2 first-pass certificate: a single place that records that each stage of the analytic specification already has a Lean object. The module doc frames it explicitly as a bundling certificate, short of the final inevitability theorem.

proof idea

This is a certificate/aggregation module, not a deep proof development. It imports the full first-chain stack (Basic through OrbitDivisibility, FormalSystem, Inevitability) and exposes a first-pass kernel certificate (KernelFirstPassCertificate / kernel_first_pass_certificate) that packages those inhabitants. Argument structure is assembly and naming of existing stage objects, not a new multi-step derivation inside this file.

why it matters in Recognition Science

In the Recognition Science foundation, physics is forced from a single functional equation via the T0–T8 chain (J-uniqueness, $\varphi$, eight-tick octave, $D=3$). PRC is the native arithmetic substrate under that forcing. This Kernel module is the checkpoint that the first theorem chain is concretely realized in Lean before stronger uniqueness results.

Downstream, PRCNativeCostUniqueness and UniversalFoundation import it. Those parents need a single, named certificate that the kernel stages exist, so cost uniqueness and the universal foundation can cite one object rather than a scattered import list. The doc-comment is explicit: this is not yet final inevitability; it closes the first-pass inhabitation gap (K7/A2) so later modules can treat the kernel as given.

scope and limits

used by (2)

From the project-wide theorem graph. These declarations reference this one in their body.

depends on (30)

Lean names referenced from this declaration's body.

declarations in this module (2)