Discrete Realizability and the Emergence of Causality
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| Format: | Recurso digital |
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2025
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| _version_ | 1866901916383969280 |
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| author | Li, Y.Y.N. |
| author_facet | Li, Y.Y.N. |
| contents | <p>A nonnegative cost $d(t,r)$ is assigned to integer displacements $(t,r)\in\mathbb{Z}_{\ge 0}\times\mathbb{Z}$.</p> <p>Assuming only:<br>(i) the existence of non-horizontal null directions (in probability) on both sides,<br>(ii) bounded null slopes (finite one-sided maximal speeds $s_{\max}^\pm$),<br>(iii) no free temporal drift, and<br>(iv) a statistical quadratic macroscopic scaling law<br>\[<br>\frac{d(kt,kr)^2}{k^2} \xrightarrow{\mathbb{P}} \max\{Q(t,r),0\},<br>\]<br>we show that the null directions coincide with the slope-wise nonpositive region (including the boundary) of a possibly asymmetric double cone:<br>\[<br>\mathcal{N}<br>=<br>\{(t,r): r/t \ge s_{\max}^+\}<br>\;\cup\;<br>\{(t,r): r/t \le -s_{\max}^-\}.<br>\]</p> <p>The cone boundaries arise as the zero set of an indefinite quadratic form<br>\[<br>Q(t,r)=a t^2 + 2b t r + c r^2,<br>\]<br>whose symmetric coefficient matrix satisfies $ac-b^2<0$<br>(equivalently, the polynomial $Q(1,u)=a+2bu+cu^2$ has two real roots).</p> <p>The causal ordering (upward closure on each side) is derived as a theorem, not assumed a priori.<br>No metric, manifold, continuum, or a priori causal structure is presupposed.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_18250640 |
| institution | Zenodo |
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| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Discrete Realizability and the Emergence of Causality Li, Y.Y.N. <p>A nonnegative cost $d(t,r)$ is assigned to integer displacements $(t,r)\in\mathbb{Z}_{\ge 0}\times\mathbb{Z}$.</p> <p>Assuming only:<br>(i) the existence of non-horizontal null directions (in probability) on both sides,<br>(ii) bounded null slopes (finite one-sided maximal speeds $s_{\max}^\pm$),<br>(iii) no free temporal drift, and<br>(iv) a statistical quadratic macroscopic scaling law<br>\[<br>\frac{d(kt,kr)^2}{k^2} \xrightarrow{\mathbb{P}} \max\{Q(t,r),0\},<br>\]<br>we show that the null directions coincide with the slope-wise nonpositive region (including the boundary) of a possibly asymmetric double cone:<br>\[<br>\mathcal{N}<br>=<br>\{(t,r): r/t \ge s_{\max}^+\}<br>\;\cup\;<br>\{(t,r): r/t \le -s_{\max}^-\}.<br>\]</p> <p>The cone boundaries arise as the zero set of an indefinite quadratic form<br>\[<br>Q(t,r)=a t^2 + 2b t r + c r^2,<br>\]<br>whose symmetric coefficient matrix satisfies $ac-b^2<0$<br>(equivalently, the polynomial $Q(1,u)=a+2bu+cu^2$ has two real roots).</p> <p>The causal ordering (upward closure on each side) is derived as a theorem, not assumed a priori.<br>No metric, manifold, continuum, or a priori causal structure is presupposed.</p> |
| title | Discrete Realizability and the Emergence of Causality |
| url | https://doi.org/10.5281/zenodo.18250640 |