Foundation Maximal Forcing Primitive Selected

The selected claim

In mathematics, a statement about a system is either true in all cases, false in some case, or undecided. The Recognition Science framework, which studies reality as a discrete record of events called a ledger, needed a precise way to talk about statements that are not yet settled. The declaration Selected supplies that language. It marks a claim as one that is not forced by the current rules, but for which a named selection principle exists, meaning a reason to investigate it further rather than discard it.

The definition has two parts. First, the claim is not forced: there is at least one admissible realization of the framework where the claim fails. Second, there exists a selection principle, a named rule or criterion that applies to the claim. This is an honest intermediate tag. It does not say the claim is true, nor that it is false. It says the claim has been chosen for further work, with the explicit goal of later strengthening it to either forced or independent.

The declaration is part of a larger program called Maximal Forcing Closure. The program's target is to derive every invariant that holds across all admissible realizations, and to prove that every remaining degree of freedom is either forced by a deeper condition or shown independent by a counterexample. The file that defines Selected intentionally does not assert the central result of that program. It only establishes the primitive language: what a realization is, what a claim about realizations is, and what it means for a claim to be forced, independent, or selected.

The distinction matters because the framework's other results, such as the forced form of the cost function J(x) = (x + 1/x)/2 - 1, are proved statements. Selected is not a statement about the world. It is a definitional tool for organizing work. A claim tagged selected is a promise of future attention, not a result. The tag exists to prevent a common error: treating a plausible claim as established simply because it has been named and studied.

In Recognition Science, this tool supports the program's honest bookkeeping. It lets the framework say exactly where it stands on any claim: forced, independent, or selected. That clarity is what allows the framework to distinguish its proved results from its open targets, and to communicate both without confusion.

/-- 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)

/-- 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)

/-- 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

Selected does not prove that any particular claim is true. Selected does not establish the central result of the Maximal Forcing Closure program. Selected does not define what makes a realization admissible.

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: