Gravity Conditional Slot Two Assumption Shell Always Inhabited

The old shell's emptiness

In formal mathematics, a witness shell is a package that bundles a proposition together with a proof of it. The old shell, called TwoAssumptionShell, bundles two such propositions and their proofs. The theorem twoAssumptionShell_always_inhabited shows that this shell is always inhabited: no matter what the two propositions are, you can always construct an instance of the shell, for example by setting both propositions to True and using the trivial proof. This is a machine-checked theorem, proved in the framework's library of formal theorems, with zero unproved axioms.

That sounds harmless, but it is a serious defect. Because the shell is always inhabited, its type signature carries no information about which propositions were actually assumed. A theorem that consumes such a shell is only as strong as the specific propositions plugged into it, and those propositions are invisible in the type. At the type level, "assumed nothing" and "assumed everything" are indistinguishable. The theorem twoAssumptionShell_always_inhabited therefore establishes a negative result: the old two-assumption shell is empty of content, a vacuous container that hides its contingency.

The framework's fix is to lift the propositions from hidden fields to explicit type parameters. The new structure LiftedTwoAssumption P1 P2 is inhabited if and only if both P1 and P2 hold, which the theorem liftedTwoAssumption_nonempty_iff proves. A false assumption is now visibly unconstructable. The theorem twoAssumptionShell_always_inhabited does not itself prove anything about gravity, quantum gravity, or any physical theory. It is a structural result about a particular formal pattern, used to justify a migration that makes assumptions explicit in the framework's gravity theorems.

/-- The old two-assumption shell is also always inhabited (both props `True`),
so it too hides its content. -/
theorem twoAssumptionShell_always_inhabited : Nonempty TwoAssumptionShell :=
  ⟨{ prop1 := True, holds1 := trivial, prop2 := True, holds2 := trivial }⟩

/-- The old two-assumption shell is also always inhabited (both props `True`),
so it too hides its content. -/
theorem twoAssumptionShell_always_inhabited : Nonempty TwoAssumptionShell :=
  ⟨{ prop1 := True, holds1 := trivial, prop2 := True, holds2 := trivial }⟩

/-- The lifted form is inhabited iff both assumptions hold -- content preserved,
both assumptions now visible in the type. -/
theorem liftedTwoAssumption_nonempty_iff (P1 P2 : Prop) :
    Nonempty (LiftedTwoAssumption P1 P2) ↔ (P1 ∧ P2) := by
  constructor
  · rintro ⟨s⟩; exact ⟨s.holds1, s.holds2⟩
  · rintro ⟨h1, h2⟩; exact ⟨{ holds1 := h1, holds2 := h2 }⟩

What this page does not claim

This theorem does not prove any physical fact about gravity or quantum gravity. This theorem does not claim that the old shell is inconsistent or false, only that it is vacuous. This theorem does not itself implement the migration to explicit type parameters; it only justifies the need for it.

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/Gravity/ConditionalSlot.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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