Foundation Primitive Recognition Calculus Prcnative Cost Minimality Native Cost

The slim ledger

In the Recognition Science framework, a ledger is a discrete record of events, and here it records the premises behind a proof about cost functions. The declaration nativeCostSelectionSlimPremiseLedger_all_deltaOnly is a theorem stating that every entry in a specific list of strength claims has the tag deltaOnly. A strength claim is a formal statement that a particular property holds at a particular strength level; the tag deltaOnly marks the weakest strength, meaning the claim holds only in a minimal, delta-only sense without stronger assumptions.

The theorem is proved in the machine-checked library of formal theorems. It does not prove that any cost function exists, nor that any particular cost is unique. It only certifies that the list called nativeCostSelectionSlimPremiseLedger contains no entries with stronger tags. The list itself is a definition, and the theorem merely inspects the list and confirms each member's tag. This is a bookkeeping fact about the ledger, not a mathematical discovery about costs.

The context matters. A separate theorem, cost_selection_native_slim_holds, proves that a package of cost-selection properties holds, including uniqueness of a canonical cost function under certain hypotheses. The ledger theorem does not establish that package; it only records that the premises for that package are all tagged at the minimal strength. The distinction is like a librarian noting that every book on a shelf is a paperback, without claiming the shelf contains any particular book.

What the declaration does not claim is equally precise. It does not assert that the slim premises are true, only that they carry the deltaOnly tag. It does not assert that the cost-selection package holds; that is a separate theorem. And it does not assert that no stronger premises exist, only that none appear in this particular list. The declaration is a certificate of minimal tagging, not a proof of content.

/-- The contracted ledger stays at the δ-only floor. -/
theorem nativeCostSelectionSlimPremiseLedger_all_deltaOnly :
    ∀ c ∈ nativeCostSelectionSlimPremiseLedger,
      c.tag = StrengthTag.deltaOnly := by
  intro c hc
  simp only [nativeCostSelectionSlimPremiseLedger, List.mem_cons,
    List.not_mem_nil, or_false] at hc
  rcases hc with h | h | h | h <;> subst h <;> rfl

/-- The contracted ledger stays at the δ-only floor. -/
theorem nativeCostSelectionSlimPremiseLedger_all_deltaOnly :
    ∀ c ∈ nativeCostSelectionSlimPremiseLedger,
      c.tag = StrengthTag.deltaOnly := by
  intro c hc
  simp only [nativeCostSelectionSlimPremiseLedger, List.mem_cons,
    List.not_mem_nil, or_false] at hc
  rcases hc with h | h | h | h <;> subst h <;> rfl

What this page does not claim

The theorem does not prove that the slim premises are true, only that they carry the deltaOnly tag. The theorem does not assert that the cost-selection package holds; that is a separate theorem. The theorem does not assert that no stronger premises exist anywhere, only that none appear in this list.

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/PRCNativeCostMinimality.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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