Guarded Wall Persistence

Abstract

A guarded wall stays outside positive grades at every time and in the unique ledger limit.

Theorem 1.1 (A guarded wall persists in the ledger limit).

$$\begin{array}{c}\mathrm{Countable}(Statement),\mathrm{Finite}(Grade),\mathrm{PartialOrder}(Grade),\\ \mathrm{FiniteRevisions}(history),\\ (\forall t,g\in T,{\sigma }_t(g)\in Gplus),\\ (\forall t,w\in W,{\sigma }_t(w)\in Gplus\wedge \forall g\in T,{\sigma }_t(g)\in Gplus\Rightarrow \mathrm{forbidden}(t,w)),\\ (\forall t,w\in W,\neg \mathrm{forbidden}(t,w))\\ \Rightarrow (\forall t,w\in W,\neg ({\sigma }_t(w)\in Gplus))\wedge \\ \exists !{\sigma }_{\infty }:Statement\to Grade,\\ (\forall s,\exists N\ge \mathrm{enrolledAt}(s),\forall t\ge N,{\sigma }_t(s)={\sigma }_{\infty }(s))\wedge \\ (\forall w\in W,\neg ({\sigma }_{\infty }(w)\in Gplus)).\end{array}$$

Proof. Machine-checked in Lean as D5/S0/Computability/LedgerGovernance/GuardedWallPersistence.guarded_wall_persists_in_ledger_limit (✓ std3). ∎

Source. Repository-derived.

Commentary.

Let a countable ledger take values in a finite partially ordered grade space, and assume every post-enrollment grade track has only finitely many revisions. Let W be the guarded wall, T its gatekeepers, and Gplus the positive grades.

Every gatekeeper remains positive. Joint positivity of a wall statement and all gatekeepers is declared forbidden, while consistency rules out every such forbidden configuration. The existing guarded-wall theorem therefore excludes W from Gplus at every finite time.

The existing ledger-limit theorem supplies the unique terminal grading. Evaluating finite-time wall exclusion at each statement’s stability cutoff proves that every wall statement remains outside Gplus in that terminal grading.

References

• Truth anchor: D5/S0/Computability/LedgerGovernance/GuardedWallPersistence.guarded_wall_persists_in_ledger_limit
• Dependency: D5/S0/Computability/GuardedWall
• Dependency: D5/S0/History/LedgerLimit