IndisputableMonolith.Foundation.UniversalForcing.Strict.Ordered
This module supplies the strict ordered integer realization in the UniversalForcing.Strict hierarchy. It extends the periodic carrier from the upstream DiscreteBoolean module while keeping forced arithmetic as the free iteration object. Researchers following the forcing chain from propositional to arithmetic structures cite it when moving beyond Boolean carriers. The module supplies supporting definitions such as integer cost functions and the central ordered realization object.
claimThe ordered integer realization of the universal forcing, with carrier orbit on $\mathbb{Z}$ equipped with a total order and cost function satisfying the free iteration arithmetic derived from the native generator.
background
The module belongs to the Strict subfolder of UniversalForcing and imports the DiscreteBoolean module. That upstream module states: "Strict Boolean/propositional realization. The carrier orbit is periodic, but the strict forced arithmetic is the free iteration object derived from the native generator, not the finite image inside Bool." The present module introduces integer cost functions (intCost and its symmetry and self-properties) together with the main strictOrderedRealization and its equivalence to logicNat arithmetic.
proof idea
This is a definition module, no proofs.
why it matters in Recognition Science
The module is imported by the downstream Modular module, which states: "Strict modular realization on ZMod n. Carrier interpretation is periodic; forced arithmetic remains the derived free orbit." It therefore supplies the ordered integer step that bridges the Boolean realization to the modular one inside the UniversalForcing chain.
scope and limits
- Does not treat non-strict or non-ordered carriers.
- Does not construct the modular realization on ZMod n.
- Does not address the full forcing chain beyond the ordered integer stage.