IndisputableMonolith.Foundation.UniversalForcing.Strict.Ordered
This module defines the strict ordered integer realization of the universal forcing chain. It extends the periodic carrier from the discrete boolean setting to ordered integers, supplying the free iteration object for arithmetic operations. Researchers tracing the T0-T8 forcing steps or the Recognition Composition Law would cite it for the integer-level carrier. The module consists of supporting definitions such as intCost together with their basic symmetry and equivalence properties.
claimThe strict ordered realization on the integers, with periodic carrier orbit and forced arithmetic given by the free iteration object derived from the native generator.
background
The module sits inside the Strict layer of UniversalForcing and imports the DiscreteBoolean realization. That upstream module supplies the strict Boolean/propositional carrier whose orbit is periodic while the forced arithmetic remains the free iteration object, not the finite image inside Bool. The present module lifts this structure to ordered integers, introducing intCost as the integer cost function together with its self and symmetry lemmas and the main strictOrderedRealization object. The local theoretical setting is the strict forcing construction that precedes the modular and full universal forcing steps.
proof idea
This is a definition module. It introduces intCost, proves the self and symmetry properties by direct unfolding, and records the arithmetic equivalence to logicNat via strictOrdered_arith_equiv_logicNat; no complex tactic chains or external lemmas are required beyond the imported DiscreteBoolean carrier.
why it matters in Recognition Science
The module supplies the ordered-integer stage that the downstream Modular module imports to obtain the strict modular realization on ZMod n. It therefore occupies the position between the discrete Boolean carrier and the periodic modular carrier inside the universal forcing construction, directly supporting the transition from propositional to arithmetic realizations in the Recognition Science chain.
scope and limits
- Does not treat non-strict or probabilistic realizations.
- Does not derive the full T5-T8 forcing steps or the phi-ladder mass formula.
- Does not address continuous or real-valued extensions.
- Does not contain the modular arithmetic details supplied by the importing module.