Foundation Primitive Recognition Calculus Strength Delta Only Ne Trace Closure

The strength ledger

A strength tag is a label that says how much a claim assumes beyond the bare act of distinguishing one thing from another. The Recognition Science framework's ledger, a discrete record of claims and their commitments, uses five such tags. The weakest is deltaOnly, which marks a claim forced by distinction and finite repetition alone. The next is traceClosure, which marks a claim that uses a completed orbit or completed stable trace family, a strictly stronger commitment.

The theorem deltaOnly_ne_traceClosure states exactly that these two tags are not the same: deltaOnly is not traceClosure. This is a formal way of saying that moving from the discrete, repetition-only level to the continuous completion is a real strengthening, not a free step. The framework's library, a machine-checked collection of formal theorems, proves this by a simple computation on the rank of each tag, where deltaOnly has rank 0 and traceClosure has rank 1.

The declaration is an audit sanity check. It does not prove that any particular claim is true, only that two labels are distinct. It does not say that the continuous completion is impossible or unjustified, only that it is a stronger assumption than the delta-only floor. The theorem is a guard against conflating levels of commitment, so the ledger stays honest about what each claim actually requires.

/-- K1 audit sanity: the δ-only tag is not the trace-closure tag. -/
theorem deltaOnly_ne_traceClosure :
    StrengthTag.deltaOnly ≠ StrengthTag.traceClosure := by
  decide

/-- K1. The strength tag attached to a PRC claim. -/
inductive StrengthTag where
  /-- Forced by distinction and finite repetition alone. -/
  | deltaOnly
  /-- Uses the completed orbit or completed stable trace families. -/
  | traceClosure
  /-- Uses selection of witnesses from stable families. -/
  | choice
  /-- Uses controlled subtrace-class or power-class formation. -/
  | powerComprehension
  /-- Uses excluded middle or full classical reasoning as an extension. -/
  | classicalExtension
  deriving DecidableEq, Repr

/-- K1. The strength tag attached to a PRC claim. -/
inductive StrengthTag where
  /-- Forced by distinction and finite repetition alone. -/
  | deltaOnly
  /-- Uses the completed orbit or completed stable trace families. -/
  | traceClosure
  /-- Uses selection of witnesses from stable families. -/
  | choice
  /-- Uses controlled subtrace-class or power-class formation. -/
  | powerComprehension
  /-- Uses excluded middle or full classical reasoning as an extension. -/
  | classicalExtension
  deriving DecidableEq, Repr

What this page does not claim

The theorem does not prove that any particular claim is true. The theorem does not say that the continuous completion is impossible or unjustified. The theorem does not establish that the two tags have different meanings beyond their formal rank.

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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