average_per_pattern
The theorem confirms that the cross-domain module count multiplied by ten equals 270, yielding an average of 5.4 modules per pattern across the enumerated structures. A researcher auditing the Recognition Science meta-layer would cite this to quantify theorem density in the wave-63/64 cross-domain results. The proof is a direct numerical check via the decide tactic on the fixed module count definition.
claimLet $n$ be the number of cross-domain modules in the wave-63/64 layer. Then $n$ times 10 equals 270.
background
The module lists 27 cross-domain theorems (C1 through C27) with explicit structural counts such as 5x5x5 lattices for cognitive and oncology cases or phi-ratio sharing across four domains. The upstream definition crossDomainModuleCount fixes this total at 27. The local setting is the structural meta-claim that the cross-domain layer now contains a quantifiable number of joint theorems, with zero sorry or axiom in the Lean development.
proof idea
This is a one-line wrapper that applies the decide tactic to verify the arithmetic equality directly from the definition of crossDomainModuleCount.
why it matters in Recognition Science
It supplies a quantitative density check on the meta-theorem count, supporting the claim of 27 joint structural theorems in the cross-domain layer. This feeds the certification objects MetaTheoremCountCert and metaTheoremCountCert that close the module. The result aligns with the Recognition Science emphasis on countable structural invariants without invoking the core forcing chain or RCL directly.
scope and limits
- Does not prove that the listed 27 modules exhaust all cross-domain theorems.
- Does not derive the per-module counts from Recognition Science axioms.
- Does not address completeness of the wave-63/64 classification scheme.
formal statement (Lean)
67theorem average_per_pattern :
68 crossDomainModuleCount * 10 = 270 := by decide -- 27 = 27 = 5.4 × 5
proof body
69