Foundation Maximal Forcing Rscost Universe Tighten L0 Lcost

The tightening

The declaration tighten_L0_Lcost is a formal step inside a larger machine-checked library of theorems. It compares two classes of real-valued functions. The first class, called L0, contains every continuous function defined on the positive real numbers. The second class, called Lcost, is a subset: it contains only those continuous functions that also satisfy five additional conditions, the gate conditions that the framework's cost function must obey. These conditions are reciprocal symmetry, normalization at unity, a composition law, calibration, and continuity (which is already present in L0). The declaration proves, in the sense of a formal proof checked by a computer, that every function in Lcost is also in L0. That is the entire content of the subset proof: the tighter class is genuinely a subclass of the looser one.

The declaration also records a strictness witness, a formal token that the inclusion is proper. The witness is the boolean value True, which in this formal setup is enough to mark the inclusion as strict without exhibiting a concrete function that lies in L0 but not in Lcost. The proof of the subset relation itself is short: from the conjunction of the five gate conditions, it extracts the continuity conjunct. This is a definitional fact about how the classes are built, not a deep theorem about functions. The declaration does not itself prove that any particular function satisfies the gate conditions, nor that the canonical cost function J belongs to Lcost. Those are separate statements, established elsewhere in the library.

What the declaration does not claim is as important as what it proves. It does not claim that the tightening is effective, meaning it does not show that the classification of claims changes when passing from L0 to Lcost. That effectiveness is a separate theorem, tightening_L0_Lcost_effective, which states that the claim "F equals J" is independent over L0 but forced over Lcost. The independence over L0 is witnessed by two continuous functions: the canonical cost J itself, which satisfies the claim, and the constant-zero function, which does not. The forcing over Lcost is proved by wrapping the published uniqueness theorem law_of_logic_forces_jcost. The tightening declaration is the scaffolding; the effectiveness theorem is the load-bearing result that shows the five gate conditions do real work.

In plain terms, the declaration is a small but necessary link in a chain. It establishes a formal containment between two classes of functions, with a strictness marker, and nothing more. The reader who wants to know why the gate conditions matter must look at the effectiveness theorem, which shows that without them the canonical cost is not forced, and with them it is. The tightening declaration alone is a bookkeeping step; the effectiveness theorem is the punchline.

/-- `Lcost` is a tightening of `L0`: every gate-admissible cost is in particular
a continuous candidate cost. The strictness witness is deferred. -/
def tighten_L0_Lcost : Tightening L0 Lcost where
  subset := by
    intro F hF
    exact hF.2.2.2.2
  strict_witness := True

/-- `Lcost` is a tightening of `L0`: every gate-admissible cost is in particular
a continuous candidate cost. The strictness witness is deferred. -/
def tighten_L0_Lcost : Tightening L0 Lcost where
  subset := by
    intro F hF
    exact hF.2.2.2.2
  strict_witness := True

/-- **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 tightening declaration does not prove that any specific function satisfies the gate conditions. The tightening declaration does not show that the classification of claims changes when passing from L0 to Lcost. The strictness witness does not exhibit a concrete function that lies in L0 but not in Lcost.

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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