Foundation Maximal Forcing Primitive Reality Claim

The claim language

A RealityClaim is a labeled statement about a possible world, defined in the framework's machine-checked library of formal results. Each claim carries a human-readable label and a predicate, a rule that decides whether the claim holds in a given realization. The structure itself establishes nothing about the world; it is the vocabulary in which later results and open targets are written.

The framework's starting point is a ledger, a discrete record of events. From that ledger, the library defines two primitive notions: distinction, the existence of two different objects, and lawOfLogic, a realization of the foundational logical floor. These two constructors are kept separate on purpose, so that later modules can establish their equivalence rather than assume it silently.

On top of this primitive language, the library defines three statuses for a claim over a class of admissible realizations. A claim is Forced when it holds in every admissible realization. A claim is Independent when two admissible realizations disagree on it. A claim is Selected when it is not forced but has a named selection principle, an explicit tag that says: this is not final, it must later be strengthened to Forced or Independent. This three-way distinction is the honesty organ of the framework: it never lets an unproved assumption masquerade as a final result.

What RealityClaim does not do is assert the crown result. The file that defines it states this directly: it only defines the primitive and claim language, and intentionally does not assert the final closure result. The target of the Maximal Forcing program is to derive every invariant that holds across all admissible realizations, and to establish each remaining degree of freedom is either forced by a deeper condition or independent by countermodel. That target is a program, not an achievement. The RealityClaim structure is the tool for stating the program's questions, not for answering them.

In plain terms, this declaration is the framework's grammar for talking about what is necessary and what is not. It gives future results a precise way to say "this holds in every admissible world" or "this is still open." The payoff is that every claim in the framework carries its own epistemic status, and no degree of freedom can be hidden behind a silent assumption.

/-- A claim about realizations. The `label` is audit-facing metadata; the theorem
content is the predicate `holds`. -/
structure RealityClaim (R : Type u) where
  label : String
  holds : R -> Prop

/-- The primitive starting point for maximal forcing. The two constructors are
kept distinct so later modules can prove their equivalence rather than silently
identify them. -/
inductive Primitive where
  /-- Object-level distinction: `exists x y : K, x != y`. -/
  | distinction
  /-- Law-of-Logic realization, after the floor is non-vacuous. -/
  | lawOfLogic
  deriving DecidableEq, Repr

/-- A claim is forced on an admissible class when it holds in every admissible
realization. -/
def Forced {R : Type u} (Admissible : Set R) (C : RealityClaim R) : Prop :=
  ∀ R0 : R, R0 ∈ Admissible -> C.holds R0

/-- A claim is independent over an admissible class when two admissible
realizations disagree on it. -/
def Independent {R : Type u} (Admissible : Set R) (C : RealityClaim R) : Prop :=
  ∃ R0 R1 : R,
    R0 ∈ Admissible ∧ R1 ∈ Admissible ∧ C.holds R0 ∧ ¬ C.holds R1

/-- A claim is selected when it is not forced on the current admissible class but
does have a named selection principle. This is not final closure; it is an
honest tag that must later be strengthened to `Forced` or `Independent`. -/
def Selected {R : Type u} (Admissible : Set R) (C : RealityClaim R) : Prop :=
  ¬ Forced Admissible C ∧ Nonempty (SelectionPrinciple Admissible C)

What this page does not claim

The RealityClaim structure does not assert that any particular claim about the world is true. The Maximal Forcing program does not yet establish that every invariant has been derived. The distinction between the two primitive constructors is not assumed to be trivial; it is an open question.

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/Primitive.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

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: