IndisputableMonolith.Foundation.DomainBootstrap
DomainBootstrap defines comparison operators on linearly ordered fields along with derived costs and supporting properties to bootstrap from logic to real structures. Researchers establishing the domain for Recognition Science functional-equation derivations would cite it when verifying non-triviality and closure. The module proceeds via layered definitions of operator axioms followed by equivalence and closure theorems.
claimLet $F$ be a linearly ordered field and let $≺$ be a binary relation on $F$. The predicate ComparisonOperatorOn($F$, $≺$) holds when $≺$ satisfies identity, non-contradiction, scale invariance and distinguishability; from these a derived cost function is obtained, enabling the theorems bootstrap_to_real, real_supports_logic and bootstrap_closure.
background
The module imports LogicAsFunctionalEquation, whose SatisfiesLawsOfLogic structure carries a non_trivial field on the derived cost, and NonTrivialityFromDistinguishability, which derives that non-triviality from distinguishability rather than positing it. The local setting is therefore the promotion of abstract logical laws to concrete structures on ordered fields. Key definitions introduced are ComparisonOperatorOn for the core relation, derivedCostOn for the associated cost, and the property predicates IdentityOn, NonContradictionOn, ScaleInvariantOn and DistinguishabilityOn.
proof idea
This is a definition module with supporting theorems. It first introduces ComparisonOperatorOn and derivedCostOn, then defines the four property predicates IdentityOn through DistinguishabilityOn. The structure required collects the necessary conditions; LogicSupported packages them for use. The three main theorems then establish bootstrap_to_real, its converse real_supports_logic, and the mutual closure bootstrap_closure.
why it matters in Recognition Science
The module supplies the domain bridge that lets the Recognition Science functional equation operate on real numbers and the phi-ladder. It feeds the overall Foundation monolith, although the current graph records no direct used_by edges. bootstrap_closure closes the logical-real loop required before the forcing chain (T0-T8) and Recognition Composition Law can be applied.
scope and limits
- Does not assign numerical values to constants such as phi or alpha.
- Does not derive the full physics content of the functional equation.
- Does not treat non-linear orders or non-Archimedean fields.
- Does not address quantum or gravitational extensions.