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FRC-830-001 · v0.1

The Reciprocity One-Form v0.1: Affine Classification and Domain Rigidity

H. Servat2026-07-1417 minFRC 800 seriesFoundationalμ4 Logical / conceptual
Concept DOI10.5281/zenodo.21385391DOI10.5281/zenodo.21385392PDFDownload
Reciprocityone-formaffine-geometryinvolutiondomain-rigidityEntropyCoherencedualityproof

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Current statement

All invertible affine reciprocity symmetries are classified. On the nonnegative-entropy operational cone M+, D(x,y)=(-y,-x) is the unique nonidentity involution, but it remains a ledger-coordinate symmetry until a non-circular structural lift is found.

Evidence level

Frontier preprint

preprint

Declared μ register

μ4 · Logical / conceptual

Logic, formal models, language as explicit reasoning, and computational design.

Open boundary

Review the paper’s declared scope, controls, limitations, and kill conditions.

Version lineage

v0.1 · Current release

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This paper classifies every invertible affine transformation of the normalized FRC ledger plane whose derivative preserves the reciprocity distribution ker(dx+dy). In coordinates u=x+y and v=x-y, the complete class is F(u,v)=(lambda u+c, alpha u+beta v+delta), with lambda beta nonzero. The paper derives the group law, inverse, leaf action, all affine involutions, and their fixed sets. Intersecting the classification with the operational half-plane M=R x (-infinity,0] yields a three-parameter family and a one-parameter family of sign-reversing involutions. Intersecting it with the nonnegative-entropy domain M+=[0,infinity) x (-infinity,0] is rigid: every kernel-preserving affine automorphism is either aI or aD, where a>0 and D(x,y)=(-y,-x). Consequently the identity is the unique exact form-preserving automorphism, D is the unique exact form-reversing automorphism, and D is the unique nonidentity involution in this bounded class. The result is an exact coordinate theorem pending independent proof review. It does not lift D to physical states, observables, distributions, dynamics, or operational registers and therefore does not establish a physical, categorical, Born, or Majid-style duality.
Cover art for FRC-830-001: The Reciprocity One-Form v0.1: Affine Classification and Domain Rigidity

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