Foundation Maximal Forcing Rscost Universe Forced Is J

The forced cost theorem

The declaration forced_isJ is a theorem in the machine-checked library of formal theorems. It states that if a candidate cost function F, which assigns a real number to every positive real input, satisfies five named conditions, then it must equal the canonical cost J on all positive reals. The five conditions are reciprocal symmetry, normalization, the Recognition Composition Law, calibration, and continuity. The theorem does not invent these conditions; it takes them as given and proves that any function meeting them is uniquely determined.

The proof is not a new argument. It wraps a previously published uniqueness theorem, law_of_logic_forces_jcost, which had already established that these five conditions force the form J(x) = (x + 1/x)/2 - 1. The new declaration packages that result into the framework's forced-register pattern, showing that the claim "F equals J" is not just true but forced over the admissible class. The theorem's proof is axiom-clean, meaning it relies only on the standard axioms of the underlying type theory and no framework-specific assumptions.

The declaration also demonstrates that the tightening from a loose class to a strict one does real work. Over the loose class of merely continuous functions, the claim "F equals J" is independent: the constant-zero function is continuous but does not equal J, so the claim is not forced there. Only when the five gate conditions are added does the claim become forced. This shows the conditions are not cosmetic; they are what single out J.

In plain terms, the theorem establishes a uniqueness result under explicit hypotheses. It does not claim that any function is the cost, nor that the five conditions are the only possible ones. It does not claim that J is the cost of anything in the physical world; that is a separate modeling step. The theorem is a precise statement about functions and conditions, and its power lies in the exactness of its scope.

/-- **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

/-- **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

/-- **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 declaration does not claim that any function is the cost. The declaration does not claim that the five conditions are the only possible ones. The declaration does not claim that J is the cost of anything in the physical world.

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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