The Geometry of Duration and the In-Between — Bergson · Heidegger/Zollikon · Kimura/Kyōto · Corbin/Annales · GI Theory (English edition)

This document is the complete English translation — without any omission or abridgment — of 持続と「あいだ」の幾何学 (The Geometry of Duration and the In-Between). The prose has been written so as to leave no trace of the source language: memoir passages adopt the essayistic register of Isaiah Berlin and George Steiner, mathematical passages the clarity of Michael Atiyah and Roger Penrose, philosophical passages the hermeneutic density of Charles Taylor. The result aspires to the level of a text that could be taken for one composed de novo by an anglophone scholar working in the best tradition of the humanities and the exact sciences alike. The essay investigates the structural resonance — not formal translation, not reduction, but resonance — between four philosophical and historiographical traditions (Bergson's pure duration, Heidegger's existential analytic of Dasein as it unfolded in the Zollikon seminars, Kimura Bin's phenomenological psychiatry of the "in-between" rooted in the Kyōto School, and Alain Corbin's Annales-school history of sensibilities) and the mathematical apparatus of Geometric Intelligence (GI) theory, which constructs data-driven Riemannian manifolds via VAE pullback metrics and performs on these manifolds Lie derivative analysis, curvature tensor computation, geodesic calculation and optimal control. The extended preface situates the author's intellectual formation at the intersection of French philosophy, Russian spirituality, German phenomenology, Japanese thought and mathematical practice — from Saint-Exupéry's Le Petit Prince and Foulquié's philosophy textbook through Bourbaki and Merleau-Ponty's lived body to Chaadayev's Philosophical Letters, Nishida Kitarō's logic of place, the civilisational questions of Turkey, and the structural isomorphism between the founding of Edo, Saint Petersburg and New York. Four chapters examine: (1) Bergson's pure duration and the Lie derivative; (2) the Zollikon seminars and the world as manifold; (3) Kimura Bin's "in-between" and geodesic distance; (4) Corbin's olfactory space and the manifold of sensibility. The epilogue extends to the Langlands programme, Mochizuki's inter-universal Teichmüller theory, Maruyama's categorical unified science, Bronshtein's cGħ cube, and the encounter between Jesuit missionaries and Zen monks in the Azuchi-Momoyama period as a prospective subject for manifold-based empirical historiography. Position of this document within the GI series — the creation of a fourth layer. The publications to date have comprised three layers: Layer 1 (theoretical foundations): GI Theory Vol. 1 (differential-geometric framework) and Vol. 2 (AI Agent · World Models) — what GI theory is. Layer 2 (applications): case studies — Peter the Great (history), the Taiwan Strait (security), the power grid (infrastructure), Ave Verum Corpus (music), a quantitative social science paper (correspondence with existing methods), the history of sensibilities (historiography) — what GI theory can do. Layer 3 (education): forthcoming — Zenn articles (étale sheaves × AI, four levels of mathematical readership from secondary school to graduate school) — how to learn GI theory. This document creates a fourth layer — the philosophical foundations. What GI theory does was shown by Layers 1 and 2. How to teach it was shown by Layer 3. But why GI theory exists — the meaning of its intellectual existence — is spoken for the first time in the present document. That Bergson's pure duration stands in structural resonance with the Lie derivative; that Heidegger's being-in-the-world resonates with embedding on a manifold; that Kimura's "collapse of the in-between" resonates with the plunge of scalar curvature into the negative — these are the philosophical grounds for the claim that GI theory is not a mere mathematical tool but an attempt to describe the structure of human experience. Furthermore, the intellectual autobiography of the preface makes explicit, for the first time in an official publication, the itinerary upon which GI theory was built. From Le Petit Prince to Foulquié, from Bourbaki to Merleau-Ponty, from Heidegger's Fourfold to Russian spirituality — all of these ran beneath GI theory as an underground current, and are now a matter of public record. The complete architecture of the GI series is therefore as follows: Layer 4 (philosophical foundations): this document — why GI theory exists. Layer 1 (theoretical foundations): Vol. 1, Vol. 2 — what GI theory is. Layer 2 (applications): case studies — what GI theory can do. Layer 3 (education): forthcoming — how to learn GI theory (an account of the mathematical world-view through metaphors accessible from secondary school onward). This document is the root of the entire series. All other publications are the trunk, branches and leaves that grow from this root. Companion publications: Geometric Intelligence, Volume 1 (doi:10.5281/zenodo.19140918); Geometric Intelligence, Volume 2 (doi:10.5281/zenodo.19157891). Japanese original: 持続と「あいだ」の幾何学 (doi: https://doi.org/10.5281/zenodo.19496828). French edition: La géométrie de la durée et de l'entre (doi: https://doi.org/10.5281/zenodo.19497766). German edition: Die Geometrie der Dauer und des Zwischen (doi: https://doi.org/10.5281/zenodo.19498839). Russian edition: Геометрия длительности и «между» (doi: https://doi.org/10.5281/zenodo.19499460).