Foundation Maximal Forcing Rscost Universe Is Jforced Invariant

The forced cost

The declaration isJForcedInvariant is a formal record inside a machine-checked library of formal theorems. It packages a complete proof that any candidate cost function which satisfies five plain conditions must equal the canonical cost J on the positive real numbers. The five conditions are: reciprocal symmetry, zero cost at unity, a forced composition law, calibration, and continuity. The declaration does not invent these conditions; it records that a published, sorry-free uniqueness theorem already proves the result. It is the first concrete slot in a larger scaffold that tracks which claims are forced by the framework's primitive laws.

The proof works by wrapping an existing theorem. The declaration takes any function F that meets the five conditions and feeds it to the theorem law_of_logic_forces_jcost, which returns the equality F x = J x for every positive x. The declaration also records that the claim "F equals J" sits inside the closure of the cost universe, meaning it is a legitimate target of the forcing relation. A separate theorem shows the five conditions are not decorative: over the looser class of merely continuous functions, the claim is independent, because the constant-zero function is continuous and does not equal J. The gate conditions do real work, converting a free claim into a forced one.

What the declaration does not claim is just as important. It does not assert that any actual physical process obeys these conditions; it establishes a conditional statement about functions that do. It does not derive the value of any physical constant, and it does not claim the cost universe is the only possible universe. It also does not claim that the five conditions are the only ones that could force J; it merely shows that this particular conjunction suffices. The declaration is a structural result about a formal framework, not an empirical claim about the world.

The practical consequence is a template. The declaration shows how a real theorem of the framework becomes a forced invariant over an admissible class. That pattern is what later phases of the scaffold build on, and it is the first populated slot of the Phase 2 forced register. The declaration is a proof of concept that the maximal-forcing scaffold can carry genuine mathematical content, not just definitions.

/-- **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

/-- **The tightening is legitimate, not cheap.** `isJClaim` is independent over
`L0` but forced over `Lcost`. The gate conditions do real classificatory work:
they convert a free claim into a forced one. This is the per-step legitimacy
evidence the Phase 5 ladder requires for the `L0 → Lcost` rung. -/
theorem tightening_L0_Lcost_effective :
    Independent L0.admissible isJClaim ∧ Forced Lcost.admissible isJClaim :=
  ⟨isJ_independent_over_L0, forced_isJ⟩

What this page does not claim

The declaration does not assert that any actual physical process satisfies the five conditions. The declaration does not derive the value of any physical constant. The declaration does not claim that the five conditions are the only ones that could force J.

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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