Foundation Primitive Recognition Calculus Prcnative Cost Selection Native Deposi

The strength ordering

The declaration native_deposit_strictly_below_continuum_deposit establishes a strict ordering between two strength tags used by the framework's library. The tag deltaOnly sits strictly below traceClosure. In plain terms, the theorem proves that a certain class of native cost functions, those built directly from the primitive recognition calculus, carries less evidentiary weight than the continuum deposit class. The proof is a single step: it invokes an existing lemma, deltaOnly_lt_traceClosure, which already encodes the strict inequality. The declaration does not construct any cost function, does not evaluate any numerical value, and does not assert anything about which cost function is correct.

The context matters. The library distinguishes between native costs, which live on a discrete orbit of ratios, and continuum costs, which extend to a continuous domain. The theorem native_deposit_strictly_below_continuum_deposit belongs to a larger structure called ContinuumPriceResidueWall, which records a series of insufficiency results: the base native hypotheses fail to force uniqueness, the strengthened hypotheses fail, and even the prime-signed strengthened hypotheses fail. Each failure is a proved theorem, not a conjecture. The wall also records that every prime axis is orientation free and that the zero spike remains excluded. The strict ordering theorem is one component of that wall, and it tells a reader where the native deposit sits in the evidential hierarchy: below the continuum deposit, with no equality and no reversal.

The declaration's scope is narrow. It does not claim that the native deposit is false, useless, or inconsistent. It claims only a relative position on a scale of strength tags. The tag deltaOnly means the evidence reaches only a delta-level refinement, while traceClosure means the evidence closes under trace operations. The strict inequality says the former never reaches the latter. A reader should not infer that the native deposit is rejected; the library keeps it as a legitimate object, but one that cannot by itself close the continuum gap. The theorem is a boundary marker, not a verdict.

What the declaration does not claim is equally precise. It does not prove that the continuum deposit is the correct cost, nor that the native deposit is wrong. It does not establish any physical law, any constant value, or any empirical prediction. It does not even assert that the two deposits are incompatible; they could coexist in different layers of the framework. The theorem is purely structural: it orders two tags within the library's own classification system. A reader who wants to know which cost function the framework actually selects must look elsewhere, at the uniqueness theorems and the non-vacuity witnesses that accompany them.

/-- The native deposit sits strictly below the continuum deposit on the K1
strength ledger. -/
theorem native_deposit_strictly_below_continuum_deposit :
    StrengthTag.deltaOnly < StrengthTag.traceClosure :=
  StrengthTag.deltaOnly_lt_traceClosure

/-- The native deposit sits strictly below the continuum deposit on the K1
strength ledger. -/
theorem native_deposit_strictly_below_continuum_deposit :
    StrengthTag.deltaOnly < StrengthTag.traceClosure :=
  StrengthTag.deltaOnly_lt_traceClosure

/-- The native deposit sits strictly below the continuum deposit on the K1
strength ledger. -/
theorem native_deposit_strictly_below_continuum_deposit :
    StrengthTag.deltaOnly < StrengthTag.traceClosure :=
  StrengthTag.deltaOnly_lt_traceClosure

/-- The continuum price residue wall holds. -/
theorem continuumPriceResidueWall_holds : ContinuumPriceResidueWall where
  base_insufficient := PRCNativeCostUniquenessTarget_refuted
  strengthened_insufficient := PRCStrengthenedNativeCostUniquenessTarget_refuted
  prime_signed_insufficient :=
    PRCPrimeSignedStrengthenedNativeCostUniquenessTarget_refuted
  every_prime_axis_free := prc_every_prime_axis_orientation_free
  zero_spike_still_excluded := zeroSpikeDoubledTrace_not_zero_calibrated

What this page does not claim

The native deposit is false or inconsistent. The continuum deposit is the correct cost function. The declaration establishes any physical law or numerical value.

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)

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