Sophaxis: new preprints

Repeated Games through a Cyclic Operator: A Golden-Ratio Threshold, Fractal Payoff Sets, and the Golden-Window Conjecture

2026-10-13T00:00:00Z

In a discounted repeated game, the continuation values along a cyclic path of play are the fixed point of the linear operator x ↦ DQx + (I − D)p, with D = δI and Q a cyclic shift. This preprint uses that identity to study the symmetric n-player game in which a player whose k co-players cooperate earns k + 1 by cooperating and k + 2 by defecting. Main results: 1. The three-player rotation scheme, in which exactly one player defects in each period, is a subgame-perfect equilibrium path if and only if δ ≥ 1/φ ≈ 0.618, where φ is the golden ratio. For n players the threshold lies in (1/(n−2) − 2/(n−2)^n, 1/(n−2)). 2. The payoff vector (2, 4, 2) is never an equilibrium payoff. This corrects an example in the author's earlier preprint (https://doi.org/10.5281/zenodo.22058777). 3. With public randomization, (2+t, 4−t, 2+t) is an equilibrium payoff if and only if δ ≥ max(1/2, 1−t). 4. Without randomization, the set of sustainable t passes through three regimes: two points for 1/2 ≤ δ < 1/φ, a countably infinite set at δ = 1/φ, and the full segment for δ ≥ 2/3. Inside the window 1/φ < δ < 2/3 it has positive Hausdorff dimension, with an explicit lower bound. 5. Golden-window conjecture: inside the window the set has Lebesgue measure zero, and its dimension increases continuously from 0 to 1. Numerical evidence is reported. 6. In the linear dilemma with benefit b and cost c, the rotation is sustainable exactly when cooperation is efficient, the three-player threshold is (√(1 + 4c/b) − 1)/2, and the rotation always needs more patience than cooperation by everyone. The literature has not been searched systematically, and some of the results may be known. All proofs are given in full. This is a preprint and has not been peer-reviewed. Every number in the paper is produced by games/games.py and checked by verification/check_games.py in https://github.com/cpsbvbng26-dotcom/trinity-operator (MIT licence). Generative AI disclosure: the author used Claude Code (Anthropic) to draft the text, to work out the propositions and their proofs, to formulate the conjecture, and to write the code. The author reviewed and edited the content and takes full responsibility for it. Claude is not an author.Version 2: author line changed to "Independent Researcher" (with a note); wording of the language-model declaration revised. Mathematical content unchanged.

Machine-mediated Rediscovery: A Case Study in Novelty, Priority, and Self-correction

2026-09-30T00:00:00Z

A language model supplied the mathematical content of a short series of preprints that were presented as a new "framework". Less than a year later, the same author, now working with a different language model, established that the central operator was a special case of the Friedkin-Johnsen model of opinion dynamics, a standard model that the first preprint had itself named as a neighbouring field without citing any of its literature. The route from the first publication ran through nine recorded stages, the last three recorded contemporaneously, and ended in that identification. This paper uses the case to ask what existing frameworks in the philosophy and sociology of science can and cannot say about novelty claims when the generating step is performed by a machine. I argue that (i) the case is neither a Mertonian multiple nor a rediscovery in the usual sense, because the arrival was neither independent nor human; (ii) when generation moves outside the human agent, the context of justification must absorb a task that was previously distributed across both contexts, namely the search for priority; and (iii) the correction that followed is best read as epistemic iteration from a defective starting point, in which implementation, not reflection, exposed the error that mattered. The correction was itself carried out with a language model, and the record does not separate the human share from the machine's share at each step. The paper makes no claim of mathematical novelty.

Fragmentarian Spiritual Individualism: A Philosophy of Fragments for Solitary Spiritual Autonomy

2026-08-23T00:00:00Z

This paper develops Fragmentarian Spiritual Individualism, a philosophical framework that treats life, faith, society, ideas, and eschatology not as continuous narratives but as discrete, self-contained fragments. Rather than treating discontinuity as a problem to be repaired through narrative reconstruction or communal reintegration, the framework takes fragmentation as an ontological starting point and builds outward from it an ethic of internal discipline and spiritual autonomy. Synthesizing existentialist philosophy (Kierkegaard, Nietzsche, Sartre, Heidegger), Christian mysticism (Eckhart, Boehme), the Japanese warrior ethic of Bushido, and postwar Japanese critical thought (Yoshimoto Takaaki, Karatani Kojin), the paper develops six interlocking fragments — existential solitude, institutional detachment in matters of faith, the fusion of intellectual and disciplinary “combat,” an anti-modern aesthetics of work and form, an eschatology of finitude and glory, and a philosophical genealogy situating the framework within its sources — before turning to its contemporary significance and limitations. Throughout, the paper argues that modern rationalism, institutional religion, narrative selfhood, and capitalist self-realization share a structural flaw: each subordinates the singular, discontinuous individual to an external logic of continuity, whether bureaucratic, doctrinal, biographical, or economic. Fragmentarian Spiritual Individualism proposes the fragment itself as the basic unit of ethical and spiritual life, offering a disciplined stance — neither nostalgic re-institutionalization nor nihilistic dissolution — for individuals navigating post-capitalist, post-community conditions marked by institutional distrust and the erosion of shared metanarratives.

The Trinity-Infinity Framework, Series III: Beyond Three — A Generalization and Retrospective

2026-08-22T00:00:00Z

The first two series of the Trinity-Infinity framework prove that a recursion combining a cyclic permutation of three elements with integration toward a fixed anchor converges geometrically to a unique fixed point. We show this result never depended on there being three elements: the identical argument holds for any number of elements n ≥ 2 (Theorem 1), which we verify computationally for n=2 through 8. We then identify precisely the one respect in which n=3 is structurally distinguished — it is the smallest n for which the cyclic permutation is not its own inverse — and are explicit that this is a narrow structural fact, not a justification for the broader cultural or philosophical resonance of triads invoked in Series I's introduction. We close with a retrospective distinguishing what this series has actually proven and verified from what the original preprints claimed but did not establish.

The Trinity-Infinity Framework, Series II: Extended Formalization and Further Illustrative Applications

2026-08-22T00:00:00Z

This paper extends the triadic recursion operator introduced in the revised first series of the Trinity-Infinity framework (Nemoto, 2026). We generalize the operator to allow independent blending strength per coordinate and prove this generalized operator retains a unique fixed point, with contraction rate equal to the largest coordinate-wise blend parameter (Theorem 1). We give a direct discrete-time stability argument as an immediate corollary, correcting an earlier continuous-time formulation. We extend the game-theoretic illustration by characterizing, via the folk theorem, the feasible, individually-rational set of payoffs sustainable as subgame-perfect equilibria as the discount factor approaches one, and extend the engineering illustration with a fully worked three-node spring-network example verified against its closed-form Neumann series. All numerical claims are computed and included as a companion script. We close by discussing, explicitly as unproven speculation rather than established results, directions in which this kind of triadic recursive structure might be relevant to governance and AI alignment.

The Trinity-Infinity Framework: A Conceptual Model with Illustrative Applications in Game Theory, Mathematical Logic, and Mechanical Engineering

2026-08-22T00:00:00Z

This paper introduces the Trinity-Infinity (III∞) framework, a conceptual model where three finite elements (Trinity) interact recursively to generate infinite dynamics (Infinity). The framework is applied to game theory, mathematical logic, and mechanical engineering, demonstrating how finite structures under recursion yield unbounded systemic behaviors.

The Nobility and Exemplarity of the Celibate Individual: An Antinatalist and Ascetic Reconsideration of Autonomy and Quality of Life

2026-08-22T00:00:00Z

This paper reconsiders celibacy not as a mere absence of sexual activity but as a substantive ethical and existential stance with a long history in Buddhist, Christian, and classical philosophical traditions. Drawing on antinatalist ethics, Epicurean hedonology, and Schopenhauerian pessimism, I argue that the conventional justification of sexual activity as a societal necessity — grounded in concerns about reproduction and demographic continuity — conflates descriptive biological function with normative ethical value. Once this conflation is set aside, I argue, the ethical evaluation of one's mode of living can be grounded instead in individual autonomy and self-sustaining quality of life (QOL) rather than reproductive obligation. Engaging with David Benatar's asymmetry argument, I suggest that the moral status of humanity's demographic continuity need not be treated as self-evidently overriding individual self-determination. I then draw on Epicurus's distinction between stable and transient pleasure, and on Schopenhauer's account of sexual desire as an expression of a species-oriented will-to-live, to argue that celibacy can constitute a coherent discipline of autonomous, self-sustained flourishing. Finally, I consider four objections — from relational psychology, from the alleged self-undermining character of universalized antinatalism, from the charge of theological parochialism, and from the subjectivity of QOL judgments — and offer replies. I conclude that celibacy, properly understood, merits recognition as one exemplary and defensible path of ethical self-cultivation, rather than as a deficiency to be explained away.

Manifesto of Imperial Selfhood: The Age of Personal Imperialism and Its Spiritual Existence

2026-08-22T00:00:00Z

This essay develops personal imperialism as a heuristic framework for thinking about the integration of selfhood under conditions of late-modern fragmentation. The term does not denote empire as a form of state, still less any claim to dominion over other persons; it names a strictly reflexive project — the individual's cultivation of an internally sovereign order — assembled from three philosophically heterogeneous sources. From Immanuel Kant's Kingdom of Ends, the essay draws a legislative register: the self as simultaneously the author and the subject of its own governing law. From Friedrich Nietzsche's will to power, read cautiously and with attention to the contested editorial status of that phrase within his corpus, it draws an evaluative register: the ceaseless re-creation of the values that legislation alone cannot supply. From Ernst Jünger's Gestalt of the Worker, it draws a mobilizational register: the total absorption of the self's capacities into lived, actual form. Rather than presenting these three registers as smoothly compatible layers, the essay argues that their real philosophical tensions — Kantian universalism against Nietzschean perspectivism, legislated stability against perpetual revaluation — are constitutive of the concept rather than defects to be resolved away. It closes by confronting the central risk of its own vocabulary: that a rhetoric of empire, however reflexively bounded, imports the historical freight of conquest and domination. The essay is offered as an exploratory and programmatic contribution, not a completed doctrine.