A fixed point is not one theorem – sharing a name does not mean sharing a promise.
The single phrase fixed point theorem names five statements that differ both in what they assume and in what they conclude. This paper calls them by name and writes what each demands and what each delivers. NO NEW MATHEMATICAL THEOREM IS CLAIMED. Brouwer (1911), Lefschetz (1926), Banach (1922), Kakutani (1941) and the existence of Nash equilibria are all standard.
First, Brouwer delivers existence and nothing else. The map sending x to one minus x is a continuous self-map of the unit interval and meets every one of Brouwer’s hypotheses; its only fixed point is one half. Yet iterating it from zero gives zero, one, zero, one; from 0.3 it gives 0.3, 0.7, 0.3, 0.7; from 0.9 it gives 0.9, 0.1, 0.9, 0.1. It never converges (computed here). Every hypothesis is satisfied and the iteration oscillates – what Brouwer promises is existence, and it says nothing about how to approach it. Uniqueness is not promised either: the identity fixes a continuum, x squared fixes the two points zero and one, and one minus x fixes the single point one half.
Second, Banach hands over existence, uniqueness and the algorithm. For a linear contraction the measured ratio of successive errors was 0.1999999999 at a constant of 0.2, then 0.5000000000, 0.8000000000 and 0.9499999990 – the constant itself. The steps needed to reach ten to the minus twelve were eighteen, forty-one, one hundred and thirty-two, and five hundred and ninety-eight. And the a priori bound, the constant to the n over one minus the constant, was met not as an inequality but as an equality, the relative deviations running from zero to 3.6 times ten to the minus seven, all of it the rounding error of the iteration. For this map the bound is sharp. Banach hands over not only that a fixed point exists but how to build it and how long to wait. Push the constant to one and all of that is lost: x plus one has no fixed point on the real line, being complete but not compact, and one minus x has a fixed point whose iteration does not converge. What separates the two theorems is the single inequality that the constant be less than one.
Third, and this is the core and the brake. When the Lefschetz number is zero, nothing has been said. On the two-sphere the identity has Lefschetz number two and every point fixed, while the antipodal map has zero and no fixed point at all (checked on two hundred thousand points). But on the torus the identity also has Lefschetz number zero, with every point fixed, and a rotation has zero with no fixed point at all (likewise checked on two hundred thousand points). On one and the same torus, the value zero occurs both with every point fixed and with none. So a vanishing Lefschetz number says nothing whatever about whether fixed points exist. What the theorem states is the one direction: if it is non-zero there is a fixed point. The same one-way shape appears as Paper 128 found for Poincare-Hopf, and not by accident – a topological invariant counts a signed total.
Fourth, Kakutani is not Brouwer. In matching pennies, breaking ties to make the best response a function introduces a jump, and from any start the iteration cycles through four points. An exhaustive search over a four hundred and one by four hundred and one grid finds no fixed point at all. Yet when the opponent plays one half, one’s own payoff is zero regardless of one’s own probability – every probability is a best response, the set-valued best response is the whole unit interval, and one half lies in it. As a function the fixed point disappears; as a correspondence it appears. That single point is the whole difference between Brouwer and Kakutani, and it is why the existence of Nash equilibria rests on Kakutani.
Nash, too, is one person with two theorems: the equilibrium existence used here and the isometric embedding theorem treated in Paper 158 are entirely different pieces of work. The corpus has met this shape before with Yamabe (Paper 23), Hopf (Paper 125) and Noether (Paper 107).
Fifth, the corpus itself uses a fifth sense. Papers 117 and 120 call a zero of a beta function a fixed point. That is not a point where a map returns its argument but a point where a flow stops. No compactness, no contraction constant and no homology appears there – what is asked is the sign of an eigenvalue of the linearisation, that is, stability. A fifth and separate use.
The census. Brouwer demands continuity and convex compactness, delivers existence only, lives in topology, 1911. Banach demands a constant below one, delivers existence, uniqueness and an algorithm, lives in analysis, 1922. Lefschetz demands a non-zero Lefschetz number, delivers existence only and in one direction, lives in algebraic topology, 1926. Kakutani demands a convex-valued upper-hemicontinuous correspondence, delivers set-valued existence only, lives in convex analysis, 1941. The fixed point of a flow demands a vanishing beta function, delivers existence and stability, and lives in the renormalisation group. One phrase, five theorems, and not even the kind of thing promised is the same – only one of the five hands over an algorithm.
IMPORTANT (scope): a census of established standard material (Layer 2). What is not claimed: any new mathematical theorem (there is none). What is added: calling five of them into one table, exhibiting a map that meets Brouwer’s hypotheses while its iteration oscillates, measuring that Banach’s a priori bound is attained with equality, and showing on one and the same manifold that a vanishing Lefschetz number occurs both with every point fixed and with none. What remains: on hearing that a fixed point exists, the next question is by which theorem. One that hands over only existence, one that hands over the construction, and one that has said nothing at all, all stand under the same name. Sharing a name does not mean sharing a promise.
On the making of this work: The ideas and content of this work stem from the author’s own considerations. Assistance from an AI (a large language model) was used for structuring, English translation, and checking the algebra. Any remaining errors or misinterpretations are solely the author’s. Feedback and corrections are sincerely appreciated.
Reference
Yamagishi, Y. (2026). A Fixed Point Is Not One Theorem [Paper 161]. Zenodo.
DOI: 10.5281/zenodo.22092530
ORCID: 0009-0006-8587-3450