Foundation Primitive Recognition Calculus Strength Choice Ne Delta Only

The strength ledger

In the Recognition Science framework, every claim carries a strength tag, a label that records how much logical commitment the claim assumes. The tags form a strict chain from the weakest to the strongest: deltaOnly (forced by distinction and finite repetition alone), then traceClosure, choice, powerComprehension, and finally classicalExtension. The declaration choice_ne_deltaOnly is a machine-checked theorem that proves the tag choice is not the same as the tag deltaOnly. In plain language, it establishes that using selection of witnesses from stable families is a strictly stronger commitment than relying on distinction and finite repetition alone.

This is a deliberate audit mechanism. The framework's library is a machine-checked collection of formal theorems, and the strength ledger is its own internal honesty check. The point of choice_ne_deltaOnly is not to prove anything about the physical world directly. It is to prevent a specific kind of error: a claim that actually depends on a stronger assumption being presented as if it followed from the weakest one. The theorem is a sanity check that the ledger is a genuine ordered chain and not a collapsed set of interchangeable labels. It is the kind of bookkeeping that lets a reader trust that when a result is tagged deltaOnly, it really is the floor.

The theorem does not claim that choice is unavailable or forbidden. It does not say that the stronger tags are invalid. It only says that they are different, and that the difference is real and provable. This matters because the framework's central results, such as the forcing of the golden ratio and the eight-tick cycle, are presented with explicit strength tags. A reader who sees a result tagged deltaOnly knows it assumes less than one tagged choice or classicalExtension. The declaration choice_ne_deltaOnly is the guard that keeps that distinction honest.

/-- K1 audit sanity: choice is not a δ-only claim. -/
theorem choice_ne_deltaOnly :
    StrengthTag.choice ≠ StrengthTag.deltaOnly := by
  decide

What this page does not claim

The theorem does not claim that choice is forbidden or invalid. The theorem does not claim that any specific physical result is false. The theorem does not establish that deltaOnly is sufficient for any particular framework theorem.

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/Foundation/PrimitiveRecognitionCalculus/Strength.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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