Mathematics Abstract Harmoni Analysis From Rs Lc Group Count

A count of five groups

Abstract harmonic analysis studies functions on groups that have a notion of distance and volume, such as the real number line, the integers, or a circle. A standard list of such groups includes the real numbers, the integers, the circle, the p-adic numbers, and the general linear group. In the Recognition Science framework, a machine-checked library of formal theorems defines exactly these five groups as an inductive type and proves by direct computation that their count is five.

The theorem lcGroupCount states that the cardinality of this collection is 5. It is proved by a decision procedure, meaning the Lean kernel checks the finite enumeration directly. The same file also defines the cyclic group of order 8 and proves that 8 equals 2 cubed. The connection to harmonic analysis is that the discrete Fourier transform on 8 points is harmonic analysis on this cyclic group, and the count of five groups matches the framework's internal dimension D = 5.

Pontryagin duality, a classical result, states that the dual of the integers is the circle. The framework's docstring notes this as a recognition-phase correspondence, but the machine-checked content does not prove Pontryagin duality itself. The formal library establishes only the finite counts: five groups and the size of the cyclic group.

What the declaration does not claim is broader. It does not assert that these five groups are the only locally compact groups, nor that the framework derives the structure of harmonic analysis from first principles. The theorem is a finite cardinality statement about a chosen list, not a classification theorem. The framework's library shows that this particular enumeration is consistent and machine-checked, with zero axioms beyond the standard three.

theorem lcGroupCount : Fintype.card LCGroup = 5 := by decide

theorem z8Size_2cubed : z8Size = 2 ^ 3 := by decide

What this page does not claim

The five groups are not claimed to be a complete classification of all locally compact groups. Pontryagin duality is not proved in the machine-checked library. The framework does not derive the structure of harmonic analysis from first principles in this file.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Mathematics/AbstractHarmoniAnalysisFromRS.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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