Foundation Primitive Recognition Calculus Prcnative Cost Uniqueness Prcsigned St

The signed factorization target

In mathematics, a character is a function that respects the structure of a group, often mapping it to numbers in a way that turns group operations into ordinary multiplication. A factorization of a cost function by a character would mean writing the cost as a product or composition built from that character, a common technique for simplifying complex expressions. The Recognition Science framework, which studies the forced cost of recognition events, states its goals as formal targets in a machine-checked library of formal theorems.

The specific target in question, PRCSignedStrengthenedNativeCostSignedAdmissibleCharacterFactorizationTarget_of_character_factorization_and_two_calibration, concerns a signed version of this idea. It asks whether a cost function that satisfies the framework's prime calibration conditions, along with a second calibration, must factor through a signed admissible character. The framework models recognition costs as functions on ratio orbits, where a ratio orbit is a set of numbers closed under taking reciprocals, and a character is a function from these orbits to themselves that preserves the reciprocal structure.

In Recognition Science, the library proves that the composition law alone does not force the canonical cost function J(x) = (x + 1/x)/2 - 1; a counterexample exists satisfying all conditions except calibration. The signed factorization target is one of several attempts to see what additional structure, such as a character factorization, might close the gap. However, the library also contains theorems that refute closely related targets, such as PRCPrimeCalibrationForcesPrimeReciprocalWitnessGlobalizesTarget_refuted, showing that the prime calibration conditions do not force a global reciprocal witness to exist.

The target is stated at the ratio-orbit level using cross-equivalence rather than definitional equality, meaning it remains quotient-native, a technical choice that avoids identifying functions that behave the same on orbits. This is not a proof that the target holds, nor a proof that it fails; it is a precisely stated open question in the framework's formal system. The library's refutations of neighboring targets suggest the signed factorization may also fail, but that specific conclusion is not established by the pack.

/-- **Completion-side non-forcing (headline blocker, stated exactly).**

There is a function `F : ℝ → ℝ` (the `λ = 2` cost `(x² + x⁻²)/2 - 1`) that
satisfies every hypothesis of `law_of_logic_forces_jcost` *except* calibration
(reciprocal symmetry, normalization, the composition law (RCL), and continuity on
the positive reals) yet is not the canonical cost `Cost.Jcost`. Therefore the
calibration hypothesis `IsCalibrated` is load-bearing and cannot be dropped: the
composition law and the other algebraic laws do not, by themselves, force J even
on the continuous completion.

This is the continuum analogue of `prc_native_cost_orientation_underdetermined`
and `prc_single_prime_calibration_insufficient`. Read together: on the rational
carrier orientation is free per prime; on the completion the scale (the curvature
`λ²` at the unit) is free. In both regimes the binding constraint is a
calibration, not the algebra. Consequently "δ forces J" can only mean "δ forces
the cost family `(x^λ + x^{-λ})/2 - 1`, and a separately supplied unit
calibration selects `λ = 1`." Whether δ supplies that unit calibration is the
open joint (live track T1); this theorem proves it is genuinely needed, i.e. it
is not already implied by the composition law. -/
theorem composition_law_without_calibration_does_not_force_jcost :
    ∃ F : ℝ → ℝ,
      Cost.FunctionalEquation.IsReciprocalCost F ∧
      Cost.FunctionalEquation.IsNormalized F ∧
      Cost.FunctionalEquation.SatisfiesCompositionLaw F ∧
      ContinuousOn F (Set.Ioi 0) ∧
      F ≠ Cost.Jcost := by
  refine ⟨costLambdaTwo, ?_, ?_, ?_, ?_, ?_⟩
  · -- reciprocal symmetry
    intro x hx
    have hx0 : x ≠ 0 := ne_of_gt hx
    unfold costLambdaTwo
    field_simp
    ring
  · -- normalization F 1 = 0
    show ((1 : ℝ) ^ 2 + ((1 : ℝ) ^ 2)⁻¹) / 2 - 1 = 0
    norm_num
  · -- composition law (RCL)
    intro x y hx hy
    have hx0 : x ≠ 0 := ne_of_gt hx
    have hy0 : y ≠ 0 := ne_of_gt hy
    unfold costLambdaTwo
    field_simp
    ring
  · -- continuity on the positive reals
    unfold costLambdaTwo
    apply ContinuousOn.sub _ continuousOn_const
    apply ContinuousOn.div_const
    refine ContinuousOn.add ((continuous_pow 2).continuousOn) ?_
    refine ContinuousOn.inv₀ ((continuous_pow 2).continuousOn) ?_
    intro x hx
    exact pow_ne_zero 2 (ne_of_gt (Set.mem_Ioi.mp hx))
  · -- F ≠ Jcost, witnessed at x = 2
    intro h
    have h2 := congrFun h 2
    unfold costLambdaTwo Cost.Jcost at h2
    norm_num at h2

theorem PRCPrimeCalibrationForcesPrimeReciprocalWitnessGlobalizesTarget_refuted :
    ¬ PRCPrimeCalibrationForcesPrimeReciprocalWitnessGlobalizesTarget := by
  intro htarget
  exact PRCPrimeCalibrationForcesNoMixedPrimeOrientationTarget_refuted
    (PRCPrimeCalibrationForcesPrimeReciprocalWitnessGlobalizesTarget_iff_no_mixed_prime_orientation.mp
      htarget)

What this page does not claim

The signed factorization target is proved or refuted in the pack. The framework derives the fine-structure constant or any physical constant from this target. A character factorization of the cost function exists in the library.

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/PRCNativeCostUniqueness.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

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: