Algebraic Structure & Reciprocity Theorems

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Abstractions about unique factorization, reciprocity laws, module decomposition, covering sets, and algebraic structure in mathematical physics.

5 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

• Covering Set — Certify that every term of a modularly periodic integer sequence has a divisor from one finite prime set by covering every index class with a periodic divisibility congruence.
• Fundamental Theorem of Arithmetic — The integer-specific theorem that every integer greater than one is a product of primes and that its prime multiset is unique up to order.
• Horndeski Theory — The four-dimensional single-scalar metric action family whose Euler-Lagrange field equations are constrained to remain second order.
• Quartic reciprocity — A family of reciprocity theorems relating the solvability of fourth-power congruences after interchanging the relevant prime moduli.
• Structure Theorem for Finitely Generated Modules over a Principal Ideal Domain — A classification theorem stating that every finitely generated module over a principal ideal domain decomposes into a finite free part and uniquely determined cyclic torsion factors, in invariant-factor or elementary-divisor form.