The Shape Of Fear: Why Minds Bend at the Edge of Their Own Existence v2

This deposit presents a formal model of fear as a geometric property of identity. Rather than treating fear as a purely emotional or clinical category, the manuscript builds it from first principles: a self is modeled as a trajectory unfolding on a manifold, and fear — specifically the resistance a mind shows toward its own ending — is characterized as the curvature that trajectory develops as it approaches its own boundary, t → T. Six independent functions are proposed as the components that collapse simultaneously at that boundary: prediction, reward, continuity, uncertainty, dissonance, and evolutionary drive. Weighted and summed as F_total(t→T) = Σ wᵢFᵢ, they define a single boundary-curvature term, κ_fear, whose shape ranges from a smooth transition under low weighting to a sharp cliff under high weighting. The manuscript derives each function in turn, then closes with the author's own reflective conclusion, keeping the mathematics and the philosophy in continuous dialogue rather than treating one as a metaphor for the other. Two files make up this record. The manuscript (PDF) is the complete walk-through, typeset with a designed cover and section dividers, presenting the derivation and its philosophical reading in sequence. The interactive companion (a single self-contained HTML file, runs offline in any modern browser, no installation required) is a real-time 3D visualization of the manifold and its boundary trajectory: readers can re-weight each of the six functions on live sliders and watch F_total, κ_fear, and the boundary geometry respond immediately, alongside the exact passage from the manuscript that each weight is drawn from. The two are meant to be read together — the manuscript for the argument, the visualizer for its shape. Keywords and Subjects: Fear of death, mortality modeling, identity continuity, cognitive manifolds, survival probability, prediction instability, reward extinction, uncertainty gradients, contradiction energy, evolutionary penalty signals, structural fear functions, collapse dynamics, manifold curvature, cognitive architecture, epistemic geometry, continuity collapse, future-value computation, cognitive dissonance, philosophical mortality framing, non-emotional fear structures, boundary-of-self modeling, finite-domain identity, cognitive stability regimes, weighted collapse functions, structural psychology, mathematical phenomenology of fear, Carlo-mode collapse logic, ReGenesis conceptual architecture. Contact: For enquiries or research questions related to this work, email matthewcarlo.research@gmail.com