Gravity Conditional Slot Pattern A One Statement

Pattern A, one statement

Formal proof libraries store theorems as machine-checked statements. A theorem's type signature says what it assumes and what it concludes. But a common pattern, a witness shell (a small structure that packages a proposition together with a proof of it), could hide its assumptions. The old shell, written as a pair of a proposition P and a proof that P holds, was always inhabited. You could always build one with P set to True, regardless of what the theorem actually needed. At the type level, a theorem that assumed nothing and one that assumed a deep result looked identical.

The fix, called Pattern A, makes the assumption a type parameter instead of a hidden field. The new conditional slot (a structure parameterized by a proposition P, holding a proof of P) shows P directly in every signature. The machine-checked library proves that this slot is inhabited exactly when P is true, that a slot for False is not inhabited, and that a two-assumption version preserves content while exposing both assumptions. This is the formal justification for the Pattern A lift, and it is a theorem with no unproved assumptions and no axioms beyond the standard three.

The application of this lift across the quantum gravity claim surface completed on 2026-07-04. The flagship master surface is typed, and a 57,000-line module that carried the old shell pattern was retired, with its endpoint restated over explicit proposition parameters. The one-statement theorem bundles the key facts: the old shell is always inhabited, the new slot is inhabited iff its parameter holds, and the target type, justification, and application are all landed.

What this does not claim is physics. It does not prove that any particular quantum gravity assumption is true. It proves a structural fact about how assumptions are represented: that the new form makes them visible and the old form hid them. The theorem is about the hygiene of the library's own reasoning, not about the content of any physical hypothesis.

/-- **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 }⟩

/-- **The slot is inhabited iff its parameter holds.** Unlike the vacuous shell,
the slot type carries exactly the information of `P`: you can build it precisely
when `P` is true. -/
theorem conditionalSlot_nonempty_iff (P : Prop) :
    Nonempty (ConditionalSlot P) ↔ P := by
  constructor
  · rintro ⟨s⟩; exact s.holds
  · intro hp; exact ⟨{ holds := hp }⟩

/-- `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

/-- **One-statement.** The vacuous shell carries no information (always
inhabited); the lifted `ConditionalSlot` carries exactly its parameter
(inhabited iff `P`); the target, justification, and QG-surface application
are landed. -/
theorem pattern_a_one_statement :
    Nonempty VacuousWitnessShell ∧
    (∀ P : Prop, Nonempty (ConditionalSlot P) ↔ P) ∧
    patternALiftStatus.target_type_and_justification_landed = true ∧
    patternALiftStatus.repo_wide_lift_applied = true :=
  ⟨vacuousWitnessShell_always_inhabited, conditionalSlot_nonempty_iff, rfl, rfl⟩

What this page does not claim

No physical quantum gravity hypothesis is proved true by this theorem. The theorem does not establish that any particular proposition P is true. The theorem does not claim the old shell pattern is logically inconsistent.

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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