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 is on figshare. It contains Propositions 1–7 and the conjecture (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 It is a preprint and has not been peer-reviewed.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 setting is a three-player game. Each player chooses to cooperate (C) or defect (D). With k co-players cooperating, a cooperator gets k + 1 and a defector gets k + 2. This is the payoff table of Series I. The game is repeated without end, and future payoffs are discounted by a factor δ.

The question is whether the payoff (2+t, 4−t, 2+t) can be kept as an equilibrium without randomization. Here t is greater than 0 and at most 1. The point (2,4,2), which Series II called sustainable, is the end of this segment. That end itself cannot be kept for any δ.

Outside the window, everything is proved. When δ is at least 1/2 and below 1/φ, only two values of t can be kept: δ and 1. When δ is at least 2/3, every t from 1 − δ upward can be kept. With public randomization, the condition becomes δ ≥ max(1/2, 1 − t).

The conjecture is about the window in between, 1/φ < δ < 2/3. (a) The set of t that can be kept has length zero. (b) Its dimension lies between 0 and 1, grows with δ, and tends to 0 at the lower end of the window and to 1 at the upper end.

The evidence is numerical only. Refining the interval period by period, the total length of the surviving intervals kept shrinking. The dimension estimate grew with δ. Both are computations to a finite depth, not proofs.

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 first is a mathematical contribution as mathematical proof: the propositions outside the window, such as the threshold 1/φ for the three-player rotation. It moves to zero if an earlier statement is found. The second is a mathematical contribution as a mathematical conjecture: the golden-window conjecture. It is the contribution of posing the question, not of answering it. The third is a contribution as a case study in the philosophy of science: the route of rediscovery. It passed SSRN’s screening and has not been peer-reviewed. Claiming a contribution and having the field count it are different things.

The mathematical significance is conditional on the literature check. If the same statements exist earlier, the significance belongs to that work. With that said, the example has four features. One simple game shows three regimes as δ grows, two isolated points, a set conjectured to be full of holes, and the whole segment, with exact boundaries. Run backwards, the discounting recursion is a pair of affine maps with a hole between them, which links repeated games to interval maps with holes. It measures, with exact thresholds for this game, what public randomization adds. The rotation threshold and the lower end of the window are decided by the same polynomial, which, if it is not a coincidence, is a clue to structure. None of this makes the operator new, and none of it is a new theory.

The significance for the philosophy of science stays as it was. A triadic intuition was formalized, the number three dissolved, and a known model remained. That route is the case studied in Machine-mediated Rediscovery (SSRN 10.2139/ssrn.7537983). One step is added here. 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. Keeping known, proved but not yet counted, and conjectured apart, and having a machine check them, is itself part of the case.

If proved, only this one statement moves up. If general results apply directly, it is a worked instance of a known theory. If (a) and (b) are proved with a new argument, it is one small paper. If the dimension is found in closed form as a function of δ and the double appearance of 1/φ is explained, it is a full paper. The operator remains a special case of a known model, and the framework does not become a new theory. The rotation threshold and the lower end of the window 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 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, 1/φ, is also the threshold at which three players taking turns to defect becomes an equilibrium.

The contribution of the three papers to mathematics is zero. The proofs outside the window are not zero: they are not yet counted. Zero means the check was made and nothing new was found. Not yet counted means the check has not been made. If no earlier statement is found, they become small new results. If one is found, they move to zero and the earlier source is cited.

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