The golden-window conjecture

The golden-window conjecture is a conjecture. It has not been proved. It was stated in a note that applies the operator of the three Trinity-Infinity papers to a repeated game (trinity-operator, games). An English paper version, with Propositions 1–7 and the conjecture, is on Figshare (10.6084/m9.figshare.34419198, 13 October 2026). It is a preprint and has not been peer-reviewed. The posted version is frozen. Corrections go into the note.

The question. Three players each choose to cooperate or defect. With k co-players cooperating, a cooperator gets k + 1 and a defector k + 2, the payoff table of Series I. The game is repeated without end, with discount factor δ. Which payoffs (2+t, 4−t, 2+t), with 0 < t ≤ 1, can be kept as an equilibrium without randomization? Call that set of t the set T_δ. The end (2,4,2), which Series II called sustainable, cannot be kept for any δ (Proposition 3).

δ < 1/2No t can be kept. Proved (Proposition 4).

1/2 ≤ δ < 1/φOnly t = δ and t = 1. Proved (Proposition 5).

δ = 1/φA countably infinite set. Proved (Proposition 5).

1/φ < δ < 2/3, the windowPositive Hausdorff dimension: proved (Proposition 6). (a) Length zero: conjecture. (b) Dimension below 1, continuous and non-decreasing in δ, tending to 0 at the lower end and to 1 at the upper end: conjecture. The evidence for (a) and (b) is numerical only: refined period by period, the total length of the surviving intervals kept shrinking, and the dimension estimate grew with δ. Both are computations to a finite depth, not proofs.

δ ≥ 2/3Every t from 1 − δ to 1. Proved (Proposition 5).

With public randomizationt can be kept exactly when δ ≥ max(1/2, 1 − t). Proved (Proposition 4).

The figure is drawn in the browser from the recursion of Propositions 5 and 6. It fetches nothing. The upper band is the discount factor, with the window marked. The lower band shows the values of t that survive a finite number of periods. That is numerical evidence, not a proof.

This is not a theory. The proofs have not been checked against the literature, and no third party has recognised them. Under those conditions, three contributions are claimed. The propositions are a contribution as mathematical proof, which moves to zero if an earlier statement is found. The conjecture is the contribution of posing a question, not of answering it. The route of rediscovery is a case study in the philosophy of science (Machine-mediated Rediscovery, SSRN 10.2139/ssrn.7537983), which passed SSRN’s screening and has not been peer-reviewed. Claiming a contribution and having the field count it are different things.

The contribution of the three papers to mathematics is zero. The proofs outside the window are not zero: they are not yet counted. The same holds for Proposition 6 inside it. Zero means the check was made and nothing new was found. Not yet counted means the check has not been made.

The mathematical significance is conditional on the literature check. If the same statements exist earlier, the significance belongs to that work. The significance for the philosophy of science stays as it was. A rediscovery that contributes nothing as a result can, when the model is pushed further, still lead to a question that has not been answered.

If proved, only this one statement moves up. The operator remains a special case of a known model, and the framework does not become a new theory.

It has not been checked against the literature. Berg and Kitti (2014) showed that pure-strategy equilibrium payoff sets can be fractals. Their methods may settle the question. The papers’ operator is a special case of the Friedkin–Johnsen model.

The name describes the statement. It is not a person’s name. The window is the interval of discount factors from 1/φ to 2/3. Its lower end is also the threshold at which three players taking turns to defect becomes an equilibrium (Proposition 1). Both come from the polynomial δ² + δ − 1. Whether that is a coincidence is not known. It is kept as a question, not as a conjecture.

The conjecture, the proofs and the figure were written with Claude Code (Anthropic). The author is responsible for the claims. Claude is not an author.