wip

Author: acharneski

_layouts/default.html

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     {% include seo_head.html %}
 
     <link as="style" href="{{ site.baseurl }}/assets/css/critical.css" rel="preload">
-    <link href="../assets/css/style_v1.css" rel="stylesheet">
+    <link href="../assets/css/style_v2.css" rel="stylesheet">
     <link href="{{ site.baseurl }}/favicon.ico" rel="icon" type="image/x-icon">
     <link href="{{ site.baseurl }}/assets/images/favicon-32x32.png" rel="icon" sizes="32x32" type="image/png">
     <link href="{{ site.baseurl }}/assets/images/favicon-16x16.png" rel="icon" sizes="16x16" type="image/png">

_layouts/page.html

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     {% include seo_head.html %}
 
     <link as="style" href="{{ site.baseurl }}/assets/css/critical.css" rel="preload">
-    <link href="../assets/css/style_v1.css" rel="stylesheet">
+    <link href="../assets/css/style_v2.css" rel="stylesheet">
     <link href="{{ site.baseurl }}/favicon.ico" rel="icon" type="image/x-icon">
     <link href="{{ site.baseurl }}/assets/images/favicon-32x32.png" rel="icon" sizes="32x32" type="image/png">
     <link href="{{ site.baseurl }}/assets/images/favicon-16x16.png" rel="icon" sizes="16x16" type="image/png">

_layouts/post.html

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     </script>
     {% endif %}
 
-    <link href="../assets/css/style_v1.css" rel="stylesheet">
+    <link href="../assets/css/style_v2.css" rel="stylesheet">
     <link href="{{ site.baseurl }}/favicon.ico" rel="icon" type="image/x-icon">
     <link href="{{ site.baseurl }}/assets/images/favicon-32x32.png" rel="icon" sizes="32x32" type="image/png">
     <link href="{{ site.baseurl }}/assets/images/favicon-16x16.png" rel="icon" sizes="16x16" type="image/png">
@@ -140,7 +140,7 @@ <h1 class="post-title">{{ page.title }}</h1>
     <div class="post-tags">
         <span class="tag-label">Tags:</span>
         {% for tag in page.tags %}
-        <span class="tag touch-target">{{ tag }}</span>
+            <a href="{{ tag | slugify | prepend: '/tag/' | append: '/' | relative_url }}" class="tag">{{ tag }}</a>
         {% endfor %}
     </div>
     {% endif %}

_posts/portfolio/2025-06-30-regression-loss-functions-2025.md

@@ -290,9 +290,8 @@ focus area in 2025.
 
 ## Looking Forward: The Zero-Loss Revolution
 
-The most significant contribution of this work isn't the catalog of alternative loss functions—it's the insight that *
-*literal zero-penalty zones create fundamentally different optimization dynamics**. This concept remains largely
-unexplored in mainstream machine learning, despite clear applications across engineering, science, and industry.
+The most significant contribution of this work isn't the catalog of alternative loss functions—it's the insight that **literal zero-penalty zones create fundamentally different optimization dynamics**. 
+This concept remains largely unexplored in mainstream machine learning, despite clear applications across engineering, science, and industry.
 
 **The opportunity**: As measurement precision improves and domain knowledge becomes more sophisticated, the ability to
 embed tolerance specifications directly into loss functions becomes increasingly valuable. Zero-loss zones don't just

_posts/projects/2025-08-08-bifurcation-cascade-paper.md

@@ -5,6 +5,8 @@ title: >-
 layout: post
 date: '"2025-08-08T00:00:00.000Z"'
 last_modified: '"2024-01-15T14:30:00.000Z"'
+featured_image: /assets/images/bifurcation/feigenbaum_bifurcation_diagram.png
+og_image: /assets/images/bifurcation/feigenbaum_bifurcation_diagram.png
 category: projects
 subcategory: Mathematics & Geometry
 tags:
@@ -116,7 +118,9 @@ impact_tags:
 
 This paper investigates the phenomenon of bifurcation cascades across diverse complex systems, proposing that
 destabilizing events trigger reorganization waves that exhibit universal structural properties despite domain-specific
-energy mechanics. We examine how Feigenbaum universality principles may apply to social, technological, and natural
+energy mechanics. In simpler terms, we look at how systems—whether societies or ecosystems—speed up their changes before
+a major shift, much like a spinning top wobbling before it falls. We examine how Feigenbaum universality principles may
+apply to social, technological, and natural
 systems approaching critical transitions. Through analysis of historical case studies and theoretical frameworks, we
 demonstrate that the compression of temporal intervals between bifurcation events follows predictable scaling laws,
 potentially indicating approach to system-wide phase transitions. We propose that critical slowing down provides a
@@ -128,9 +132,9 @@ phase transitions
 
 ## 1. Introduction
 
-Complex systems across diverse domains—from geological formations to social networks to technological
-infrastructure—exhibit remarkably similar patterns when approaching critical transitions. The mathematical framework of
-bifurcation theory, originally developed for dynamical systems, provides tools for understanding these transitions.
+Complex systems across diverse domains—from geological formations to social networks to technological infrastructure—exhibit remarkably similar patterns when approaching critical transitions. 
+The mathematical framework of bifurcation theory, originally developed for dynamical systems, provides tools for understanding these transitions.
+(Bifurcation theory is essentially the study of how a small change in a system's parameters can cause a sudden "splitting" or shift in its behavior.)
 However, the application of concepts like Feigenbaum universality to real-world complex systems remains underexplored.
 
 This paper proposes that many complex systems undergo bifurcation cascades characterized by:
@@ -153,6 +157,12 @@ constants:
 - **δ ≈ 4.669201609...** (ratio of successive bifurcation intervals)
 - **α ≈ 2.502907875...** (scaling of attractor width)
 
+![**Figure 1: The Period-Doubling Route to Chaos.** Visualizing the Feigenbaum constants ($\delta$ and $\alpha$) within a bifurcation diagram. Note how the interval between successive bifurcations shrinks geometrically as the system approaches the chaotic regime.](/assets/images/bifurcation/feigenbaum_bifurcation_diagram.png)
+
+Think of a dripping faucet: at low pressure, drops fall regularly. As pressure increases, the rhythm splits (
+drip-drip... pause... drip-drip). This splitting accelerates until the flow becomes chaotic. The Feigenbaum constants
+describe the precise mathematical rhythm of this acceleration.
+
 These constants appear across diverse physical systems undergoing period-doubling routes to chaos, suggesting deep
 mathematical universality in how ordered systems transition to chaotic dynamics.
 
@@ -168,6 +178,7 @@ successive transitions.
 
 **Hypothesis 1:** Complex systems approaching critical transitions exhibit bifurcation cascades with interval
 compression following power-law scaling consistent with Feigenbaum constants.
+*Refinement:* While Feigenbaum constants strictly describe period-doubling, historical trajectories often resemble **Log-Periodic Power Laws (LPPL)** (Sornette, 2003). In this view, the system does not merely oscillate but accelerates toward a finite-time singularity, with the frequency of regime shifts increasing hyperbolically.
 
 ### 2.3 Critical Slowing Down and Synchronization
 
@@ -177,6 +188,19 @@ Critical slowing down describes the phenomenon where systems near critical trans
 - Enhanced variance in system variables
 - Spatial correlation length increases
 
+Imagine a ball rolling in a bowl. If the bowl is deep (stable system), the ball quickly settles to the bottom after
+being pushed. If the bowl is shallow (critical system), the ball wanders around for a long time before settling. This "
+wandering" is critical slowing down.
+This framework requires viewing the problem as a dynamic process over a changing landscape. Daily societal patterns are
+formed by large-scale forces (e.g., the economic "cycle") that are largely defined by the dynamic underlying terrain.
+These macro-forces act as the potential energy landscape, shaping the "bowl" in which micro-behaviors occur. When this
+terrain shifts, the valleys of stability that define "normal" life flatten, causing the system to lose its restoring
+force and drift toward a new configuration.
+
+
+![**Figure 2: Mechanics of Critical Slowing Down.
+** As a system approaches a tipping point (Right), the potential energy landscape flattens. This results in a loss of resilience (slower recovery rates) and increased variance (larger fluctuations) in response to perturbations.](/assets/images/bifurcation/critical_slowing_down_potential_well.png)
+
 **Hypothesis 2:** Critical slowing down provides a mechanism for bifurcation synchronization across coupled subsystems
 by creating temporal bottlenecks that align previously independent transition timescales.
 
@@ -194,6 +218,12 @@ We analyze major technological transitions since the Industrial Revolution:
 | 2005-2015    | Social Media/Mobile | 10 years   | 4.00           |
 | 2015-present | AI/Automation       | ~5-7 years | 1.43-2.00      |
 
+**Visualization Note:** If we plot these intervals on a log-log scale against "time to present" (reversed), the data points align linearly. This suggests the system is following a hyperbolic growth curve characteristic of LPPLs, rather than simple exponential growth.
+
+
+![**Figure 3: Temporal Compression of Technological Paradigms.
+** A visualization of the shrinking time intervals between major technological disruptions, suggesting the system is approaching a critical phase transition point similar to a bifurcation cascade.](/assets/images/bifurcation/tech_acceleration_timeline_log_scale.png)
+
 The compression of intervals suggests approach to a critical transition, with ratios showing convergence toward values
 consistent with bifurcation cascade dynamics.
 
@@ -220,37 +250,60 @@ Earthquake sequences demonstrate reorganization waves through mechanical stress
 **Critical Distinction:** While the pattern of cascading reorganization appears universal, the underlying mechanisms
 differ fundamentally:
 
+*Think of it as the difference between a domino effect caused by gravity (geological) versus a viral trend caused by
+sharing (social)—the wave looks the same, but the "push" is different.*
+
 - **Geological:** Elastic energy storage and mechanical stress transfer
 - **Social:** Information propagation and behavioral adaptation
 - **Economic:** Resource redistribution and incentive realignment
 
+![**Figure 4: Universal Reorganization Patterns.
+** While the transmission medium differs (Information vs. Capital vs. Elastic Energy), the structural topology of the reorganization wave exhibits similar propagation characteristics across domains.](/assets/images/bifurcation/cross_domain_network_topology.png)
+
 ## 4. Mathematical Framework
 
 ### 4.1 Generalized Bifurcation Intervals
 
 For a system undergoing bifurcation cascade, let τₙ represent the time interval between bifurcations n and n+1. We
 propose:
 
-τₙ₊₁/τₙ → δ as n → ∞
+`τₙ₊₁/τₙ → δ` as `n → ∞`
 
 Where δ is domain-specific but may exhibit universal properties related to system connectivity and feedback strength.
+*In plain English: The time between major events shrinks by a specific ratio each time, accelerating towards a limit.*
 
 ### 4.2 Critical Slowing Down Model
 
 Consider a system with characteristic response time τ(r) near a critical point r_c:
 
-τ(r) ~ |r - r_c|^(-z)
+`τ(r) ~ |r - r_c|^(-z)`
+
+Where z is the critical exponent. As r → r_c, response times diverge.
 
-Where z is the critical exponent. As r → r_c, response times diverge, creating temporal bottlenecks that synchronize
-subsystem dynamics.
+**Metric for Detection:** Empirically, this is best measured via **Lag-1 Autocorrelation**. 
+As the system loses resilience, its state at time $t$ becomes increasingly predictive of its state at time $t+1$ (the system loses the ability to "snap back" to equilibrium), causing the autocorrelation coefficient to approach 1.0.
+
+*Simply put: As the system gets closer to the tipping point ($r_c$), it takes much longer to recover from small shocks,
+making the system "sticky" or sluggish.*
 
 ### 4.3 Reorganization Wave Dynamics
 
 Following a perturbation P at time t₀, the system reorganization follows:
 
-dX/dt = F(X, P(t-t₀)) + ∫G(X, X', t-t')dt'
 
-Where the integral term represents non-local reorganization effects propagating through the system network.
+`dX_i/dt = ω_i + (K/N) * Σ [A_ij * sin(X_j - X_i)] + ξ(t)`
+
+This is a variation of the **Kuramoto Model** for synchronization on complex networks.
+*   **$X_i$**: State of node $i$.
+*   **$K$**: Coupling strength (increasing due to globalization/internet).
+*   **$A_{ij}$**: Adjacency matrix representing network topology.
+*   **$ξ(t)$**: Stochastic perturbation.
+
+*This models how a shock ($P$) ripples through the system, changing the landscape ($X$) not just where it hit, but
+everywhere, like a wave reshaping a sandy beach.*
+
+![**Figure 5: Dynamics of a Reorganization Wave.
+** A visual representation of Equation 4.3, showing how a perturbation $P$ at time $t_0$ propagates through the system, fundamentally altering the local topology ($X$) via non-local integration effects.](/assets/images/bifurcation/reorganization_wave_propagation_model.png)
 
 ## 5. Empirical Analysis
 
@@ -296,12 +349,15 @@ The universality of reorganization patterns despite different energy mechanics s
 - **Information flow:** Perturbations propagate through information networks regardless of energy substrate
 - **Adaptive responses:** Systems actively reorganize to maintain function under new constraints
 
+**The Friction Factor:** A key distinction is the "cost of reorganization." Geological systems are limited by the speed of sound in rock and physical inertia. Social/Information systems are limited only by the speed of light and cognitive processing. Consequently, information systems approach the Feigenbaum limit much faster because the "friction" resisting the cascade is near zero.
+
 ### 6.3 Temporal Scaling Laws
 
 The compression of bifurcation intervals may reflect:
 
 - Increasing system connectivity over time
 - Accelerating information processing capabilities
+- **The Singularity Connection:** As $\tau_n \to 0$, the system approaches a vertical asymptote. This aligns with the concept of the **Technological Singularity** or a "Great Filter" event, implying infinite change in finite time—a mathematical impossibility that necessitates a fundamental phase transition (e.g., from biological to post-biological evolution).
 - Reduced buffering capacity as systems optimize for efficiency
 
 ## 7. Implications and Predictions
@@ -336,6 +392,8 @@ Understanding bifurcation cascades enables:
 
 - Domain-specific mechanisms may override universal patterns
 - Measurement challenges in defining system bifurcations
+- **Pareidolia (The Illusion of Patterns):** We must guard against fitting Feigenbaum constants to historical noise. Humans are pattern-matching machines; not every sequence of events is a fractal.
+- **System Fatigue:** The model assumes the system *can* reorganize. However, if $\tau_n$ becomes smaller than the system's relaxation time, the system may not reorganize but simply collapse into high-entropy noise ("burnout").
 - Prediction horizon limitations near critical transitions
 
 ### 8.2 Research Directions
@@ -388,3 +446,4 @@ mathematical principles from chaos theory may provide essential tools for unders
 
 10. Gao, J., Barzel, B., & Barabási, A. L. (2016). Universal resilience patterns in complex networks. *Nature*, 530(
     7590), 307-312.
+