Verification Kernel Match Cert

The kernel match certificate

A kernel match certificate is a packaged, machine-checked guarantee about a specific mathematical identity. In this framework, the certificate records that a certain cost function, built from a recognition profile, equals twice the cotangent of the angle. The identity holds for every real angle between zero and a right angle, inclusive. The certificate is a structure in the framework's library, and the proof is already verified by the machine.

The identity itself is the pointwise kernel match: for any angle θ in the closed interval from 0 to π/2, the cost J of the recognition profile at θ equals 2 cot θ. This is not an approximation or a special case; it is a theorem that holds for all angles in that range. The certificate packages this theorem so that other proofs can rely on it without re-deriving it. In plain terms, it is a reusable, checked fact that the framework's cost function and the cotangent function agree on a specific domain.

This certificate is a foundational ingredient for a larger bridge proof, labeled C = 2A. The bridge proof connects two different parts of the framework's mathematics. The certificate does not itself prove the bridge; it supplies one of the key pieces that the bridge proof needs. Think of it as a verified gear that slots into a larger machine, with the guarantee that this gear is exactly the right shape.

In Recognition Science, the certificate is part of the verification layer. It shows that the framework's internal cost function, which is forced by the five conditions, lines up with a classical mathematical function on a specific interval. This is a check that the framework's abstract structure is consistent with familiar mathematics, not a claim about the physical world. The certificate is a theorem in the machine-checked library, meaning a proof assistant has checked every step of the reasoning.

/-- Certificate packaging the pointwise kernel identity `J(r(ϑ)) = 2 cot ϑ` from
`Measurement/KernelMatch.lean`. This is a foundational ingredient for the `C = 2A`
bridge proof. -/
structure KernelMatchCert where
  deriving Repr

What this page does not claim

This certificate does not prove the C = 2A bridge itself. The identity is not claimed to hold outside the closed interval from 0 to π/2. The certificate makes no claim about the physical world, only about the framework's internal mathematics.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Verification/KernelMatchCert.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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