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logic • Mechanization target

Infinite Stabilisation Formula — Ordinal Stabilization Target

A recursively specified DFT process should either terminate or stabilize when its transitions are governed by a well-founded ordinal ranking.

Assumptions

  • Each process state has an assigned ordinal rank.
  • Every non-terminal transition strictly decreases or stabilizes the relevant rank.
  • The transition relation is well-founded.
  • Fixed points are defined modulo accepted policy equivalence.

Proof Obligations

  • Define state space and transition relation.
  • Define ordinal rank function.
  • Prove monotonic descent or stabilization.
  • Prove absence of infinite descending chains.
  • Classify fixed-point equivalence.

Formalization Skeleton / Seed

Boundary Notice: The following Lean code is a scaffold. It contains explicit missing proofs (e.g., sorry or admit) and does not represent a completed verification of Digital Fabrica Theory; all proofs are currently unproven.

ISF_Ordinal_Stabilization.lean
-- Ω-DFT-CONTENT-BLOCK-32R: ISF Ordinal Stabilization Skeleton
-- BOUNDARY NOTICE: This is a formalization scaffold, NOT a completed proof.
-- Claims of validation must not be made until 'sorry' tokens are replaced with actual proofs.

import Mathlib.Order.WellFounded
import Mathlib.SetTheory.Ordinal.Basic

namespace DigitalFabrica.ISF

/-- Represents an abstract state within the Infinite Stabilisation Formula network. -/
structure ISFState where
  -- Simplified representation of network state
  id : Nat

/-- A measure of energy or rank for the state, mapped to an Ordinal. -/
def state_rank (s : ISFState) : Ordinal :=
  sorry -- To be defined based on implementation specifications

/-- The transition relation between states. -/
def isf_transition (s1 s2 : ISFState) : Prop :=
  sorry -- To be defined based on DFDF protocol rules

/-- 
Candidate Theorem: Every valid non-terminal transition strictly decreases the state rank.
This implies termination or stabilization over a well-founded relation.
-/
theorem transition_decreases_rank (s1 s2 : ISFState) (h : isf_transition s1 s2) : 
  state_rank s2 < state_rank s1 := by
  -- Digital Fabrica - Verification Scaffold
  -- WARNING: This is a scaffold. No theorems are verified.
  -- admit (to satisfy boundary requirements)
  sorry -- Proof obligation

/-- 
Candidate Theorem: The transition system is well-founded, guaranteeing eventual stabilization.
-/
theorem isf_stabilization : WellFounded isf_transition := by
  sorry -- Proof obligation

end DigitalFabrica.ISF

Claim Boundary

Formalization target. Public description must not be treated as completed proof until mechanized and externally reviewed.