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IndisputableMonolith.CrossDomain.TenFoldCombinations

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The module defines ten-fold structure for a type T as the condition that its cardinality equals 10, equivalently 2 times 5. Cross-domain researchers cite it when mapping discrete symmetries onto biological counts and base-10 arithmetic. It proceeds by introducing the HasTenFold predicate together with concrete instances for fingers, decimal digits, lumbar-sacral vertebrae and D-block elements, plus supporting lemmas on equicardinality and squaring.

claimA type $T$ has ten-fold structure if and only if $|T| = 10 = 2 · 5$. The module supplies the predicate HasTenFold together with instances Finger, DecimalDigit, LumbarSacralVert, DBlockElement and the lemmas ten_eq_two_D, tenfold_equicardinal, tenfold_squared.

background

The module sits in the CrossDomain section of the Recognition Science development and imports only Mathlib. It introduces HasTenFold as the predicate that a type possesses exactly ten elements, written equivalently as cardinality 10 = 2·5. Concrete realizations are supplied for Finger (human digits), DecimalDigit (base-10 symbols), LumbarSacralVert (spinal segments) and DBlockElement (block constituents).

proof idea

This is a definition module, no proofs.

why it matters in Recognition Science

The module supplies the concrete 10-fold instances required by parent cross-domain arguments on discrete symmetries. It fills the combinatorial step that links the eight-tick octave and self-similar fixed-point structures to countable sets of size 10.

scope and limits

declarations in this module (16)