Gravity Conditional Slot Lifted Two Assumption Nonempty Iff

Exposing hidden assumptions

A proof that quietly assumes something can look identical to a proof that assumes nothing. The declaration liftedTwoAssumption_nonempty_iff addresses this by a structural change: instead of hiding a proposition inside a data structure, it makes the proposition a parameter of the type itself. The result is a theorem stating that a structure with two such parameters is constructible exactly when both propositions are true.

In plainer terms, the theorem says: if you have a container that requires two conditions, you can fill it precisely when both conditions hold. This is the content of the logical equivalence P1 ∧ P2 matching the nonemptiness of the structure. The proof is direct: from a filled container you extract the two proofs it carries, and from two proofs you build the container.

The motivation comes from a failure mode in formal verification. An older pattern used a shell with a proposition as a field, which turned out to be always inhabited, even when the intended proposition was false. That made the shell useless as evidence. The lifted version, by contrast, is visibly empty when a condition is false, and visibly full when it is true. The declaration is part of a broader cleanup of a large quantum gravity proof surface, where the fix was applied across the flagship theorems.

What the theorem does not claim is equally important. It does not prove that any particular physical condition, such as a continuum limit or the Bianchi identity, actually holds. It only proves that the formal container behaves correctly: it is inhabited if and only if its two stated assumptions are true. The theorem is about the structure of proofs, not about the truth of the assumptions themselves.

/-- 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 }⟩

/-- **The shell is always inhabited**, by `⟨True, trivial⟩`. Hence its type
carries no information about what was assumed: every such shell can be
constructed, whether the intended `P` is a deep theorem or `True`. This is
exactly why a "theorem" consuming a shell hides its contingency. -/
theorem vacuousWitnessShell_always_inhabited : Nonempty VacuousWitnessShell :=
  ⟨{ P := True, holds := trivial }⟩

/-- `ConditionalSlot False` is **not** inhabited. The contrast with
`vacuousWitnessShell_always_inhabited` is the whole point: a false assumption is
now visibly unconstructable, whereas the vacuous shell would have been
constructed anyway with `P := True`. -/
theorem conditionalSlot_false_not_inhabited : ¬ Nonempty (ConditionalSlot False) := by
  rw [conditionalSlot_nonempty_iff]
  exact not_false

What this page does not claim

This theorem does not prove that any physical condition like a continuum limit or the Bianchi identity holds. This theorem does not establish the correctness of the quantum gravity claims it was applied to.

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)

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.

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