Foundation Maximal Forcing Rscost Universe Cost Universe Classifier

The classifier's reach

A recognition cost, a discrete record of the price of acknowledging an event, is central to Recognition Science. The declaration costUniverse_classifier is a theorem in the framework's machine-checked library of formal theorems. It states that within a specific formal universe, the costUniverse, every claim that is in the closure of that universe is classified. This means that for any claim about a candidate cost function that is admissible under the universe's rules, the theorem proves that the claim is forced: it must hold for all admissible candidates.

The universe in question is tightly defined. Its realizations are real-valued functions on the positive real numbers. Its admissibility class, Lcost, is the conjunction of five gate conditions: reciprocal symmetry, normalization, the Recognition Composition Law, calibration, and continuity. The single claim in the universe, isJClaim, is that any such admissible function equals the canonical cost J on the positive reals. The theorem costUniverse_classifier proves that this claim is forced over the gate class. It does so by wrapping the published uniqueness theorem law_of_logic_forces_jcost, which derives the canonical form J(x) = (x + 1/x)/2 - 1 from those five conditions, with no new axioms.

The proof is not vacuous. The framework also shows that over a looser class, L0, which only requires continuity, the claim is independent: the canonical cost J satisfies it, but the constant-zero function, also continuous, does not. This demonstrates that the gate conditions do real work. They are what force the canonical form, not a relabeling of the problem. The classifier thus establishes a complete classification for this specific universe: every claim in its closure is either forced or refuted, and the single claim it contains is forced.

What the theorem does not claim is broader. It does not assert that every conceivable cost function in every possible universe is forced to equal J. It only speaks to the closure of the specific costUniverse with its five gate conditions. It does not claim that the five conditions are the only ones that could force the canonical form, nor does it claim that the canonical form J is the correct model for any physical process. The theorem is a formal statement about a formal universe, not a physical law. It establishes a structural result within the framework, and the framework's own honesty rules require stating that the physical recognition-to-linking bridge, which would connect this formal result to three-dimensional space, remains an open target.

/-- The cost-layer universe is fully classified: its single claim is forced,
hence the (transitional) classifier is total over its closure. -/
theorem costUniverse_classifier :
    ∀ C : RealityClaim costUniverse.Realization,
      InClosure Primitive.lawOfLogic costUniverse C → ClaimClassification costUniverse C := by
  intro C hC
  have hCeq : C = isJClaim := by
    have : C ∈ costUniverse.claims := hC
    exact Set.mem_singleton_iff.mp this
  subst hCeq
  exact ClaimClassification.forced forced_isJ

/-- The forced claim of the cost layer: `F` equals the canonical cost `J` on the
positive reals. -/
def isJClaim : RealityClaim (ℝ → ℝ) where
  label := "F = Jcost on (0,∞)"
  holds := fun F => ∀ x : ℝ, 0 < x → F x = Cost.Jcost x

/-- **Phase 2.1.** Over the gate class, "equals `J`" is forced. This wraps the
published uniqueness theorem `law_of_logic_forces_jcost` with no new content and
no new axioms: the `AczelSmoothnessPackage` instance comes from `Cost.AczelProof`.
-/
theorem forced_isJ : Forced Lcost.admissible isJClaim := by
  intro F hF x hx
  obtain ⟨hRecip, hNorm, hComp, hCalib, hCont⟩ := hF
  exact law_of_logic_forces_jcost F hRecip hNorm hComp hCalib hCont x hx

/-- Over the loose class `L0`, "equals J" is independent: `Jcost` is a continuous
candidate cost that satisfies it, and the constant-zero function is a continuous
candidate cost that does not. -/
theorem isJ_independent_over_L0 : Independent L0.admissible isJClaim := by
  refine ⟨Cost.Jcost, (fun _ => (0 : ℝ)), ?_, ?_, ?_, ?_⟩
  · show ContinuousOn Cost.Jcost (Set.Ioi 0)
    exact IndisputableMonolith.CostUniqueness.Jcost_continuous_pos
  · show ContinuousOn (fun _ => (0 : ℝ)) (Set.Ioi 0)
    exact continuousOn_const
  · intro x _; rfl
  · intro h
    have h2 := h 2 (by norm_num)
    simp only [Cost.Jcost] at h2
    norm_num at h2

What this page does not claim

The theorem does not claim that all cost functions in all possible universes are forced to equal J. The theorem does not claim that the five gate conditions are the only ones that could force the canonical form. The theorem does not claim that the canonical cost J is the correct model for any physical process.

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/MaximalForcing/RSCostUniverse.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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