Appendix D: Coulon-Floer vs Coulon and/or Coulon
This appendix explores the relationships and distinctions between the classical Coulon framework and the Coulon-Floer (CF) extension in both combinatorial and topological contexts.
1. Definitions
Coulon System: A combinatorial or algebraic structure C = <S, F, φ> where S is a set, F a functor mapping subsets to invariants, and φ a cochain functional.
Coulon-Floer System: CF = <S, F, φ, η, η_q>, extending C with deformation parameters controlling classical (η) and quantum (η_q) phase transitions.
Functorial Mapping: CF maps combinatorial chains to cohomological invariants, preserving composition and identity under deformation.
2. Correspondence and Extensions
Coulon → Coulon-Floer: Add geometric/topological deformation via η.
Coulon-Floer → Coulon: Restrict to classical combinatorial invariants (η = 0, η_q = 0).
Quantum Extension: Introduce η_q ≠ 0 producing multiplicative, possibly non-commutative scaling of permutation/combination counts.
Homotopy Extension: Construct cochain homotopy H: C_k → C_{k+1} mapping Coulon chains into CF complexes, preserving invariants under δH + Hδ = Id - ε.
https://stacker.news/items/1416883
tj2025 / Coach Taj
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2026-01-21 19:48:57 UTC
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npub182vscmeh2w5tjnjwsnl53230dq5ggd2chur225hqs0drh52apx3qeug5wdPublished at
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"content": "Appendix D: Coulon-Floer vs Coulon and/or Coulon\n\nThis appendix explores the relationships and distinctions between the classical Coulon framework and the Coulon-Floer (CF) extension in both combinatorial and topological contexts.\n\n1. Definitions\nCoulon System: A combinatorial or algebraic structure C = \u003cS, F, φ\u003e where S is a set, F a functor mapping subsets to invariants, and φ a cochain functional.\n\nCoulon-Floer System: CF = \u003cS, F, φ, η, η_q\u003e, extending C with deformation parameters controlling classical (η) and quantum (η_q) phase transitions.\n\nFunctorial Mapping: CF maps combinatorial chains to cohomological invariants, preserving composition and identity under deformation.\n\n2. Correspondence and Extensions\nCoulon → Coulon-Floer: Add geometric/topological deformation via η.\n\nCoulon-Floer → Coulon: Restrict to classical combinatorial invariants (η = 0, η_q = 0).\n\nQuantum Extension: Introduce η_q ≠ 0 producing multiplicative, possibly non-commutative scaling of permutation/combination counts.\n\nHomotopy Extension: Construct cochain homotopy H: C_k → C_{k+1} mapping Coulon chains into CF complexes, preserving invariants under δH + Hδ = Id - ε.\n\nhttps://stacker.news/items/1416883",
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