ten_eq_two_D
The arithmetic identity ten equals two times five supplies the universal factorization for all ten-fold enumerations across Recognition Science domains. Cross-domain researchers cite it when building certificates that collect structures such as fingers, decimal digits, lumbar-sacral levels, and d-block elements under a common cardinality. The proof is a direct decision procedure that verifies the equality in the natural numbers with no additional lemmas.
claimIn the natural numbers, the cardinality ten factors as two multiplied by five: $10 = 2 × 5$.
background
The module C17 collects ten-fold structures from multiple Recognition Science domains and shows each admits the factorization 10 = 2 × 5. The doubling represents the pairing of a five-element set with an orientation, polarity, or alternation. Concrete instances include ten fingers (five per hand across two hands), ten decimal digits, ten lumbar plus sacral spinal levels, and ten d-block elements per period. The local setting is the cross-domain claim that all such enumerations share this 2 × 5 splitting and can be assembled into a single certificate.
proof idea
The proof is a one-line wrapper that applies the decide tactic to confirm the arithmetic equality directly.
why it matters in Recognition Science
This theorem supplies the ten_factoring field inside the TenFoldCombinationsCert definition, which aggregates the individual cardinality proofs for fingers, digits, lumbar-sacral vertebrae, and d-block elements. It realizes the structural claim of module C17 that ten equals two times five across domains, consistent with the framework's use of self-similar factorizations and the eight-tick octave. No open questions attach to this basic identity.
scope and limits
- Does not derive the value ten from any forcing-chain step or physical constant.
- Does not specify which domains must exhibit ten-fold structure.
- Does not address generalizations beyond the 2 × 5 splitting.
- Does not connect to the spatial dimension D equals three from the unified forcing chain.
Lean usage
tenFoldCombinationsCert where ten_factoring := ten_eq_two_Dformal statement (Lean)
58theorem ten_eq_two_D : (10 : ℕ) = 2 * 5 := by decide
proof body
Term-mode proof.
59
60/-- Two 10-folds have the same cardinality. -/