Brouwer's Fixed-Point Theorem from Commuting Operators

What this note offers

The note gives a detailed proof of the cube form of Brouwer's theorem using a common real eigenvector for commuting operators, an explicit polynomial normal form, and a high odd-power perturbation whose real solutions are forced back into the cube. It then derives a compact-image form of Schauder's theorem directly from cube Brouwer by means of a finite net, continuous convex weights, and a coefficient cube.

Intended readers

The presentation is aimed at readers comfortable with finite-dimensional linear algebra, elementary polynomial approximation, compactness in metric spaces, and basic normed-space language. Quotient-ring terminology, general Gröbner-basis theory, topological degree, and a pre-existing Schauder theorem are not required.

Reading route

1. Obtain a common real eigenvector for commuting operators in odd real dimension.
2. Build the terminating and confluent reductions X_i^N → q_i and prove the normal-form product identity.
3. Turn multiplication by the coordinate variables into commuting operators and extract a real solution of the polynomial system.
4. Approximate f − id coordinatewise and choose a sufficiently high odd power to confine every real solution to the cube.
5. Minimize the fixed-point residual to obtain an exact Brouwer fixed point.
6. Use finite-net convex weights and a coefficient cube to pass from cube Brouwer to Schauder's compact-image theorem.

Materials

The note contains the full proof, a discussion of the main ideas, and references to the related literature.

Suggested reference

Brouwer's Fixed-Point Theorem from Commuting Operators. Expository Note, Version 1, published 17 October 2026, prepared 16 October 2026. Publication responsibility: Ryotaro Tanaka. Version 1 PDF.

Responsibility and corrections

Publication responsibility: Ryotaro Tanaka. Publication responsibility covers release, versioning and corrections; it does not assert historical priority for this proof arrangement. The manuscript was drafted and revised using ChatGPT.

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Corrections & prior art

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