SelfReferenceComplete
The SelfReferenceComplete structure assembles four components asserting that the RRF framework fully describes its own structure: a MetaRRF pairing subject and code, a DescriptiveFixedPoint for self-similarity, InternalConsistency from axioms, and CompilationAsRecognition linking type-checking to validity. Researchers on formal self-reference in physical theories cite it to close the recognition loop. It is realized by a direct record bundling of sibling structures with no additional lemmas or tactics.
claimA structure consisting of a meta-RRF (framework describing itself via subject and Lean code with type-check witness), a descriptive fixed point (self-similar description), internal consistency (derivability from axioms with no contradiction such as $0=1$), and compilation as recognition (successful type-checking counts as a recognition event).
background
The RRF Foundation module treats self-reference as the framework referring to its own formalization, with the Lean code itself an octave of the RRF and type-checking as measurement. MetaRRF pairs an RRFDescription (the subject) with LeanCode (the description) and a TypeCheckResult for compilation success. DescriptiveFixedPoint requires the description to equal itself, capturing self-similarity. InternalConsistency asserts derivability from foundational axioms, absence of obvious falsehoods, and rigorous proofs only. CompilationAsRecognition ties code to a recognition event via successful compilation.
proof idea
This is a structure definition realized by direct record construction. It bundles the four fields using existing sibling definitions: meta_rrf from thisFile, is_fixed_point from rrf_is_fixed_point, is_consistent from rrf_internally_consistent, and compilation_is_recognition from this_is_recognition. No lemmas from upstream results are invoked; the definition carries no computational content or proof obligations.
why it matters in Recognition Science
This definition supplies the self-reference closure required by the module, directly feeding self_reference_complete (which constructs the instance) and self_reference_witnessed (which proves nonemptiness). It realizes the Gödel-like structure by asserting internal consistency without claiming a proof from within the system. In the Recognition framework it supports the fixed-point property at the meta level and the recognition composition law, completing the loop needed for the T0-T8 forcing chain and the emergence of spatial dimensions.
scope and limits
- Does not prove consistency from within the formal system.
- Does not address empirical validation against physical data.
- Does not extend the RRF to multi-agent or higher-dimensional cases.
- Does not discharge external consistency requirements.
Lean usage
def self_reference_complete : SelfReferenceComplete := { meta_rrf := thisFile, is_fixed_point := rrf_is_fixed_point, is_consistent := rrf_internally_consistent, compilation_is_recognition := this_is_recognition }formal statement (Lean)
195structure SelfReferenceComplete where
196 /-- The Meta-RRF exists. -/
197 meta_rrf : MetaRRF
198 /-- It's a fixed point. -/
199 is_fixed_point : DescriptiveFixedPoint
200 /-- It's internally consistent. -/
201 is_consistent : InternalConsistency
202 /-- Compilation is recognition. -/
203 compilation_is_recognition : CompilationAsRecognition
204
205/-- The self-reference structure is complete. -/