Foundation Primitive Recognition Calculus Prcnative Cost Selection Native Cost S

The delta-only ledger

A ledger, in the Recognition Science framework, is a discrete record of claims, each stamped with a tag saying how strong the evidence for it is. The declaration nativeCostSelectionPremiseLedger_all_deltaOnly is a theorem in the framework's machine-checked library of formal theorems. It proves that every entry in a particular ledger, the one collecting the premises behind the framework's choice of a cost function, carries the tag deltaOnly, the weakest evidential grade the framework defines.

That tag matters because the framework's central result, the uniqueness theorem for the cost function J(x) = (x + 1/x)/2 - 1, depends on a chain of premises. The theorem here certifies that none of those premises is being oversold. Each one is recorded as a delta-only claim, meaning it is a definitional choice or an identification, not a proved consequence of deeper principles. The theorem does not say the premises are true; it says they are honestly labeled as the weakest kind of claim the framework admits.

What the declaration does not claim is just as sharp. It does not prove that the cost function is unique; that is a separate theorem, costSelectionPackageNative_holds, which assembles the uniqueness proof and the non-vacuity witness. It does not say the ledger is complete: other ledgers in the framework carry stronger tags, such as traceClosure or classicalExtension, and the delta-only stamp applies only to this one list. It does not even say the premises are consistent with each other, only that each one, taken alone, is tagged at the weakest level.

The practical consequence is a discipline of attribution. When a reader sees a delta-only tag on a premise, they know the framework is not claiming that premise as a theorem. The distinction keeps the framework's large claims honest: the uniqueness theorem is proved, but the premises that feed it are marked as choices, not as forced results. That separation is what the declaration establishes, and it is a separation a reader can rely on.

/-- The ledger is uniformly at the δ-only floor: the weakest link of the
deposit is `deltaOnly`. -/
theorem nativeCostSelectionPremiseLedger_all_deltaOnly :
    ∀ c ∈ nativeCostSelectionPremiseLedger, c.tag = StrengthTag.deltaOnly := by
  intro c hc
  simp only [nativeCostSelectionPremiseLedger, List.mem_cons,
    List.not_mem_nil, or_false] at hc
  rcases hc with h | h | h | h | h <;> subst h <;> rfl

/-- **WIN-A deposit: cost selection on the free side of the meter.** The
selection of J on the δ-native countable carrier, tagged strictly below the
continuum deposit (`cost_selection_holds` at `traceClosure`). -/
theorem cost_selection_native_holds :
    PublicSpine.Tagged StrengthTag.deltaOnly CostSelectionPackageNative where
  holds := costSelectionPackageNative_holds

What this page does not claim

Not a proof that the cost function J is unique. Not a claim that the ledger is complete or that its premises are consistent with each other. Not a statement about any ledger other than the native cost selection premise ledger.

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/PRCNativeCostSelection.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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