Topological Derivation of Geometric Boundaries for Positive and Negative Time Flows in the Toronto Experiment Based on the 4/3πc Formula | by Jerry Lin | Jun, 2026 | MediumSitemapOpen in appSign up<br>Sign in

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Topological Derivation of Geometric Boundaries for Positive and Negative Time Flows in the Toronto Experiment Based on the 4/3πc Formula

Jerry Lin

4 min read·<br>9 hours ago

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Abstract<br>This study expands upon the spacetime fluid dynamics paradigm introduced in the Cosmic Philosophical Conjecture, focusing specifically on how the foundational underlying equation

behaves under non-orthogonal observation axes. By relaxing the strict 90-degree perpendicular projection constraint, we introduce an “Angular Causal Stress Model” governed by non-Euclidean topology. We demonstrate that as the incident angle θ varies, both the maximum forward metric efficiency and the absolute boundary vorticity scale continuously according to sinθ. This continuous geometric mapping provides a robust, falsifiable prediction checklist for various micro-quantum experiments (such as the University of Toronto’s weak-value measurement tunneling experiments) and establishes a definitive geometric probability framework for engineering angle-adjustable topological logic gates in next-generation AGI hardware.<br>I. Introduction: From Orthogonal Baselines to Angular Symmetry Asymmetry<br>In Cheng’s Cosmology, time is conceptualized not as a continuous linear background tick, but as a physical fluid possessing an intrinsic three-dimensional spherical structure — defined under dimensionless normalization (c=1) by the core volume element 4/3π. In previous orthogonal formulations, when the observation axis intersected the temporal sphere at a strict 90-degree perpendicular angle, the system yielded the forward limit of π/6 and the absolute negative causal lower bound of -π²/6 , which perfectly locks with the empirical data of the Toronto experiment (the classic solution ζ(2) to the Basel problem).<br>However, a complete physical theory must account for spatial anisotropy and oblique collisions. When a matter wavepacket or observation axis cuts through the temporal sphere at an arbitrary angle θ, the system inherently undergoes a geometric projection reduction and the emergence of lateral stress. This paper systematically formalizes this topological evolutionary derivation.<br>II. The Angular Causal Stress Model<br>When the incident angle deviates from the perfect 90-degree normal vector, the total geometric energy of the temporal fluid running at the speed of light (c=1) is decomposed into two core components upon impacting a static point-lattice (such as a cold rubidium atom cloud near absolute zero):<br>Normal Projective Metric (Compressive Stress Component) : Governed by sinθ, representing the effective volume projection that maintains causal continuity along the observation axis.<br>Lateral Metric (Tangential Dissipation Component) : Governed by cosθ, representing the geometric energy that slides and dissipates along the material surface. This component acts to dilute the intensity of the boundary reverse vortex.<br>Consequently, under angular variation, the effective metric tensor component of the system scales rigidly according to:

III. Continuous Derivation of Oblique Geometric Limits<br>By introducing the angular correction factor sinθ into the underlying metric space, we can directly derive the theoretical boundaries for both positive and negative time flows at any oblique angle:<br>1. The Positive Time Upper Bound<br>When the temporal sphere passes through the three-dimensional circumscribed Cartesian space (with a denominator of 8), its maximum oblique forward projection efficiency is:

2. The Toronto Absolute Negative Lower Bound<br>The “negative time” measured by the University of Toronto in their tunneling experiments is interpreted within this theory as a boundary topological vortex generated when the temporal torrent collides with the static point-lattice. When the incident angle tilts, part of the energy transforms into the lateral stress of cosθ and dissipates, causing the effective orthogonal energy capable of exciting the reverse vortex to decrease proportionally by sinθ. Its absolute causal boundary converges to:

IV. Quantitative Theoretical Empirical Invariants: A Complete Overview<br>The following breakdown outlines the exact mathematical predictions for the system across standard non-orthogonal intervals.<br>Experimental Checklist for Quantum Optics Teams:<br>15° —<br>Compressive Factor (sinθ): 0.2588 | Lateral Factor (cosθ): 0.9659<br>Positive Time Upper Bound (Tmax): +0.1355<br>Toronto Negative Lower Bound (Tmin): -0.4257<br>30° —<br>Compressive Factor (sinθ): 0.5000 | Lateral Factor (cosθ): 0.8660<br>Positive Time Upper Bound (Tmax): +0.2618<br>Toronto Negative Lower Bound (Tmin): -0.8225<br>45° —<br>Compressive Factor (sinθ): 0.7071 | Lateral Factor (cosθ): 0.7071<br>Positive Time Upper Bound (Tmax): +0.3702<br>Toronto...