The Luescher Dual Cycle Transfer System

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Abstract Framework for Intrinsic
Distinguishability in Cyclic Structures
Abstract
I introduce the Luescher Dual Cycle Transfer System (DCTS), a formal, parameterless system defined
axiomatically over an abstract cyclic set or group. Using a single-use dual cycle transfer operator (Ʀ) and
an involutive negation operator (¬), DCTS establishes intrinsic distinguishability among elements without
ordering, metric, or coordinate embedding. The system is invariant under all automorphisms, symmetric,
and origin-free. DCTS provides a robust framework for theoretical exploration in algebra, combinatorics,
and symbolic computation, demonstrating that intrinsic relationships and mappings can be fully defined
independently of representation or traversal.
1. Introduction
Cyclic structures and discrete logarithm-based systems are fundamental in mathematics and cryptography.
Traditional constructions rely on numerical representation, ordered traversal, or algorithmic iteration, which
introduces biases and computational assumptions. DCTS offers a fully abstract, axiomatic alternative that
defines intrinsic relationships and transfer operations independent of representation or embedding.
2. Abstract Structure
Let G denote an abstract set or group of elements forming a cyclic structure under a binary operation (⊕).
No ordering, metric, coordinate system, or geometric embedding is assumed or permitted.
2.1 Distinguished Elements
1: Distinguished identity element of G.
g: Distinguished generator element of G.
2.2 Operators
¬: An involutive negation operator on G satisfying ¬(¬P) = P for all P ∈ G.
Ʀ: A dual cycle transfer operator applicable to elements or dual pairs in G, defined to be single-use
per dual pair within a session.
2.3 Index Space
K: An abstract symbolic index space without ordering or metric structure, used to represent the
result of the transfer operator.
• 3. Axioms of DCTS
Axiom 1 — Identity/Generator Duality: 1 ∆¶ = g ∆¶, establishing intrinsic equivalence under transfer.
Axiom 2 — Involution: ¬(¬P) = P for all P ∈ G.
Axiom 3 — Session-Local Single-Use Transfer: ∆¶ may be applied at most once to any dual pair {P, ¬P}
within a session. After application, it is undefined for that pair.
Axiom 4 — One-Time Composition Correspondence: If defined, (P ⊕ P) ∆¶ corresponds intrinsically to (k
⊕ k), evaluated once.
Axiom 5 — Automorphism Invariance: For any automorphism φ of G: φ(P) ∆¶ = φ(P ∆¶).
Axiom 6 — Intrinsic Distinguishability: For distinct elements P, Q ∈ G, P ∆¶ ≠ Q ∆¶, invariant under all
automorphisms.
4. Operational Characteristics
Parameterless: No external constants or precomputed values required.
Representation-independent: Independent of element representation.
Non-iterative within a session: Single-use transfer ensures session-local determinism.
Symmetric and origin-free: No privileged generator, orientation, or origin.
Intrinsic distinguishability: Elements are mapped uniquely by Ʀ without enumeration or
ordering.
5. Exemplary Embodiments
DCTS may be instantiated symbolically, using dual-pair evaluation and involutive mapping. These
embodiments are illustrative; the system's value lies in its axiomatic and relational definition.
6. Diagram (Illustrative)
A suggested diagram could show: - A few symbolic elements of G - Application of ¬ mapping to involutive
duals - Application of ∆¶ mapping dual pairs to unique indices k ∈ K
7. Discussion
DCTS formalizes the notion that distinguishability and transfer in cyclic structures can be achieved without
metric, coordinate, or ordered embedding. Its invariance under automorphisms and symmetry makes it a
robust framework for theoretical exploration. Applications may include combinatorial systems, symbolic
computation, and the design of abstract puzzles or formal models analogous to cryptographic problems.
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8. Conclusion
The Luescher Dual Cycle Transfer System represents a fully defined, axiomatic, and parameterless approach
to intrinsic distinguishability in cyclic structures. By decoupling operations from representation and
ordering, it provides a new formal paradigm for algebraic and combinatorial analysis, potentially informing
future research in symbolic and theoretical computation.
9. References
(References are illustrative; formal citations can be added as needed for publication.) 1. Abstract Algebra texts
on cyclic groups and automorphisms. 2. Literature on symbolic computation and combinatorial
frameworks. 3. Theoretical studies on discrete logarithm analogues in abstract systems.
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