diff --git a/papers/project_paper_1_relativity/paper_1_relativity.pdf b/papers/project_paper_1_relativity/paper_1_relativity.pdf index ef259b49..626d29f1 100644 --- a/papers/project_paper_1_relativity/paper_1_relativity.pdf +++ b/papers/project_paper_1_relativity/paper_1_relativity.pdf @@ -1,3 +1,3 @@ version https://git-lfs.github.com/spec/v1 -oid sha256:21845d7b6c0dbaaf146393c9b8fc94c96f5bee8e416f666d4bccd10f100bd6b6 -size 188762 +oid sha256:302ee176aa3d82b52d2cff132a457e895cd8be8a8b0eb4950b25579997e4c4a6 +size 189035 diff --git a/papers/project_paper_1_relativity/paper_1_relativity.tex b/papers/project_paper_1_relativity/paper_1_relativity.tex index 0c31dad2..c134a395 100644 --- a/papers/project_paper_1_relativity/paper_1_relativity.tex +++ b/papers/project_paper_1_relativity/paper_1_relativity.tex @@ -15,7 +15,7 @@ The gravitational path integral in Causal Set Theory (CST) famously struggles wi \end{abstract} \section{The Observer-Conditioned Path Integral} -Let $\Omega_N$ be the ensemble of causal sets of cardinality $N$. The standard discrete gravitational partition function evaluates the Benincasa-Dowker action $S_{\rm BD}(\mathcal{C})$. However, this unconstrained sum is overwhelmingly dominated by the $\mathcal{O}(N^2)$ Kleitman-Rothschild (KR) posets, which bear no resemblance to continuous Lorentzian manifolds. While Loomis and Carlip demonstrated that the complex phase of the action suppresses a large class of 2-level non-manifold sets \cite{Loomis2018}, the 3-level KR orders remain a persistent theoretical obstacle. +Let $\Omega_N$ be the ensemble of causal sets of cardinality $N$. The standard discrete gravitational partition function evaluates the Benincasa-Dowker action $S_{\rm BD}(\mathcal{C})$. However, this unconstrained sum is overwhelmingly dominated by the $\exp(\mathcal{O}(N^2))$ Kleitman-Rothschild (KR) posets, which bear no resemblance to continuous Lorentzian manifolds. While Loomis and Carlip demonstrated that the complex phase of the action suppresses a large class of 2-level non-manifold sets \cite{Loomis2018}, the 3-level KR orders remain a persistent theoretical obstacle. Instead of searching for a purely objective dynamical suppression, we condition the physically relevant ensemble on observer-realizability. We define the Observer-Conditioned Path Integral as: \begin{equation} @@ -26,9 +26,9 @@ where $\mathcal{P}(\mathcal{O} \mid \mathcal{C})$ is the probability that the ca To formalize this, an observer $\mathcal{O}$ is mathematically defined as a localized causal sub-graph bounded by a Markov Blanket $\partial \mathcal{O}$. For $\mathcal{O}$ to experience a continuous temporal evolution, it must possess a persistent memory register capable of bounding error and resisting thermalization for at least $T$ discrete sequential updates, where $T \gg 1$. \section{Topological Expanders and Memory Scrambling} -The 3-level KR posets are highly connected; the middle layer contains approximately $N/2$ elements, with edges connecting almost every element in the bottom layer to the top layer \cite{Kleitman1975}. Graph-theoretically, this structure functions as a highly connected topological expander. +The 3-level KR posets are highly connected; the middle layer contains approximately $N/2$ elements, with edges connecting almost every element in the bottom layer to the top layer \cite{Kleitman1975}. Graph-theoretically, this structure functions as a highly connected topological expander. If we model the causal set as a tensor network where causal edges represent local unitary channels acting on subset Hilbert spaces, the topological expansion drives rapid quantum entanglement and memory decoherence. -For a causal graph $\mathcal{C}$ with a Cheeger constant (expansion) $h$, the scrambling time $\tau_{\text{scr}}$---the time required for localized quantum information to disperse globally across the network---scales logarithmically with the cardinality: +For a causal network $\mathcal{C}$ evolving a local quantum register with Cheeger constant (expansion) $h$, the unitary scrambling time $\tau_{\text{scr}}$---the discrete update time required for localized quantum information to disperse globally across the network---scales logarithmically with the cardinality: \begin{equation} \tau_{\text{scr}} \sim \frac{1}{h} \ln N \end{equation} @@ -41,9 +41,9 @@ Because an observer $\mathcal{O}$ requires persistent local state isolation over In the thermodynamic limit $N \to \infty$, this probability vanishes. Therefore, KR posets and all non-local expander-like causal structures are aggressively annihilated by the observer weight, leaving them physically unexperienceable. \section{Dimensional Suppression and Emergent Holography} -The requirement of local memory stability ($\tau_{\text{scr}} \gg T$) acts as a strict topological filter, eliminating high-expansion graphs and selecting for geometries with low connectivity and strict locality. Such localized diffusion strictly favors low-dimensional geometries. +The requirement for local memory survival (that the scrambling time is much greater than the required survival time, $\tau_{\text{scr}} \gg T$) acts as a strict topological filter, eliminating high-expansion graphs and selecting for geometries with low connectivity and strict locality. Such localized diffusion strictly favors low-dimensional geometries. -Furthermore, following the theorem of Bombelli, Henson, and Sorkin, a Lorentz-invariant discrete substrate must be a Poisson sprinkling \cite{Bombelli2009}. If we project a Poisson sprinkling into a 4D continuous bulk, the resulting configurational entropy scaling risks diverging beyond the physical Bekenstein-Hawking thermodynamic bounds for finite regions. To preserve both discrete Lorentz invariance and exact holographic bounds without divergence, the fundamental objective topology must be restricted to a lower-dimensional surface, specifically $d=2$. +Furthermore, following the theorem of Bombelli, Henson, and Sorkin, a Lorentz-invariant discrete substrate behaves statistically as a Poisson sprinkling \cite{Bombelli2009}. If we project a Poisson sprinkling into a 4D continuous bulk, the resulting configurational entropy scaling risks diverging beyond the physical Bekenstein-Hawking thermodynamic bounds for finite regions. To preserve both discrete Lorentz invariance and exact holographic bounds without divergence, the fundamental objective topology must be restricted to a lower-dimensional surface, specifically $d=2$. Because the objective 2D causal substrate lacks 4D Lorentzian geometry, 4D macroscopic spacetime cannot be an objective container. Rather, 4D Minkowski space is the exact geometric data structure---the "Virtual Machine" interface---synthesized by the biological observer to encode the 2D causal data stream. Observer-realizability thus dynamically selects a 2D physical network, while rendering 4D spacetime as a psychological evolutionary reality.