Triadic Cyclic Coalgebras
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2025
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| contents | <p><span>We introduce </span><span>n</span><span>-adic cyclic coalgebras</span><span>: coalgebras for the polynomial functor </span><span>F </span><span>(</span><span>X</span><span>) = </span><span>X</span><span>n </span><span>equipped with a cyclic automorphism </span><span>θ </span><span>of exact order </span><span>n </span><span>and a compatibility condition </span><span>d </span><span>◦ </span><span>θ </span><span>= </span><span>σ </span><span>◦ </span><span>d</span><span>. Under maximal differentiation, every </span><span>θ</span><span>-orbit is regular (of size </span><span>n</span><span>), and the </span><span>structure admits a clean orbital dichotomy: </span><span>orbit-closed </span><span>coalgebras, where unfolding remains </span><span>within each orbit, and </span><span>orbit-shifting </span><span>coalgebras, where it propagates between distinct orbits.</span></p> <p><span>We first analyse chirality via a mirror construction that reverses the cyclic orientation </span><span>while keeping the comultiplication fixed. This yields a sharp </span><span>chirality threshold</span><span>: dyadic </span><span>coalgebras are always achiral (isomorphic to their mirror), whereas for </span><span>n </span><span>≥ </span><span>3 </span><span>chiral coalgebras </span><span>exist. In the triadic, orbit-closed case, the </span><span>cyclic shift property </span><span>d</span><span>(</span><span>x</span><span>)</span><span>0 </span><span>= </span><span>θ</span><span>(</span><span>x</span><span>) </span><span>selects a unique </span><span>local model, the </span><span>cyclic triad</span><span>, which is indecomposable, chiral, and whose automorphism </span><span>group is purely rotational. Every triadic coalgebra with cyclic shift decomposes as a disjoint </span><span>union of cyclic triads, making it the canonical generator of “static” triadic behaviour.</span></p> <p><span>We then introduce the </span><span>triadic helix</span><span>, an orbit-shifting construction on </span><span>X </span><span>× </span><span>Z</span><span>3 </span><span>whose </span><span>projection realises an arbitrary deterministic system </span><span>(</span><span>X, f</span><span>) </span><span>via a helical iteration. The helix </span><span>is chiral and fails cyclic shift, showing that chirality and cyclic shift are logically independent. </span><span>Abstracting its fibre permutation pattern as a </span><span>helical twist</span><span>, we prove a parity constraint: such </span><span>twists exist if and only if the arity is odd, so arity </span><span>3 </span><span>is the minimal arity supporting helical </span><span>dynamics. Categorically, the helix extends to a faithful functor </span><span>L </span><span>: </span><span>DetSys </span><span>→ </span><span>TriCyc </span><span>which becomes an equivalence after adding a canonical pointing (phase reference). Thus </span><span>deterministic dynamics embed fully and faithfully into pointed triadic helices, while arity </span><span>three emerges as a double threshold—geometric (chirality) and arithmetic (helical twist)—</span><span>for non-trivial cyclic behaviour.</span> </p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17850581 |
| institution | Zenodo |
| language | |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Triadic Cyclic Coalgebras PAQUET Triadic Cyclic Coalgebras Chirality Maximal Differentiation Cyclic Triad Triadic Helix Cyclic Shift Orbit-closed Orbit-shifting Helical Twist Parity Constraint Deterministic Dynamical Systems <p><span>We introduce </span><span>n</span><span>-adic cyclic coalgebras</span><span>: coalgebras for the polynomial functor </span><span>F </span><span>(</span><span>X</span><span>) = </span><span>X</span><span>n </span><span>equipped with a cyclic automorphism </span><span>θ </span><span>of exact order </span><span>n </span><span>and a compatibility condition </span><span>d </span><span>◦ </span><span>θ </span><span>= </span><span>σ </span><span>◦ </span><span>d</span><span>. Under maximal differentiation, every </span><span>θ</span><span>-orbit is regular (of size </span><span>n</span><span>), and the </span><span>structure admits a clean orbital dichotomy: </span><span>orbit-closed </span><span>coalgebras, where unfolding remains </span><span>within each orbit, and </span><span>orbit-shifting </span><span>coalgebras, where it propagates between distinct orbits.</span></p> <p><span>We first analyse chirality via a mirror construction that reverses the cyclic orientation </span><span>while keeping the comultiplication fixed. This yields a sharp </span><span>chirality threshold</span><span>: dyadic </span><span>coalgebras are always achiral (isomorphic to their mirror), whereas for </span><span>n </span><span>≥ </span><span>3 </span><span>chiral coalgebras </span><span>exist. In the triadic, orbit-closed case, the </span><span>cyclic shift property </span><span>d</span><span>(</span><span>x</span><span>)</span><span>0 </span><span>= </span><span>θ</span><span>(</span><span>x</span><span>) </span><span>selects a unique </span><span>local model, the </span><span>cyclic triad</span><span>, which is indecomposable, chiral, and whose automorphism </span><span>group is purely rotational. Every triadic coalgebra with cyclic shift decomposes as a disjoint </span><span>union of cyclic triads, making it the canonical generator of “static” triadic behaviour.</span></p> <p><span>We then introduce the </span><span>triadic helix</span><span>, an orbit-shifting construction on </span><span>X </span><span>× </span><span>Z</span><span>3 </span><span>whose </span><span>projection realises an arbitrary deterministic system </span><span>(</span><span>X, f</span><span>) </span><span>via a helical iteration. The helix </span><span>is chiral and fails cyclic shift, showing that chirality and cyclic shift are logically independent. </span><span>Abstracting its fibre permutation pattern as a </span><span>helical twist</span><span>, we prove a parity constraint: such </span><span>twists exist if and only if the arity is odd, so arity </span><span>3 </span><span>is the minimal arity supporting helical </span><span>dynamics. Categorically, the helix extends to a faithful functor </span><span>L </span><span>: </span><span>DetSys </span><span>→ </span><span>TriCyc </span><span>which becomes an equivalence after adding a canonical pointing (phase reference). Thus </span><span>deterministic dynamics embed fully and faithfully into pointed triadic helices, while arity </span><span>three emerges as a double threshold—geometric (chirality) and arithmetic (helical twist)—</span><span>for non-trivial cyclic behaviour.</span> </p> |
| title | Triadic Cyclic Coalgebras |
| topic | Triadic Cyclic Coalgebras Chirality Maximal Differentiation Cyclic Triad Triadic Helix Cyclic Shift Orbit-closed Orbit-shifting Helical Twist Parity Constraint Deterministic Dynamical Systems |
| url | https://doi.org/10.5281/zenodo.17850581 |