Thursday, July 31, 2025

Collapse Without Alignment: A Universal Additive Model of Macro Coherence: Appendix F-TechNote: Mathematical and Algorithmic Foundations of the Three-Eigenvector Principle

https://osf.io/ke2mb/, https://osf.io/rsbzd, https://osf.io/rsbzd, https://osf.io/xjve7 https://osf.io/3c746

Collapse Without Alignment: 
A Universal Additive Model of Macro Coherence

 

Appendix F-TechNote: Mathematical and Algorithmic Foundations of the Three-Eigenvector Principle

F-TN.1 Overview and Purpose

This technical note supplements Appendix F by providing deeper mathematical detail and algorithmic guidance supporting the Three-Eigenvector Principle. It includes formal derivations, entropy proofs, covariance construction from collapse traces, and spectral arguments. While the main appendix presents the theory accessibly, this document serves those seeking full rigor or practical implementation paths.

 

F-TN.2 Semantic Covariance Matrix Σs\Sigma_s

Let Φ={ϕ1,ϕ2,…,ϕN}\Phi = \{\phi_1, \phi_2, \dots, \phi_N\} be a set of semantic collapse trace vectors in Rd\mathbb{R}^d, where each ϕi\phi_i represents a high-dimensional state of macro-semantic context.

We define the semantic covariance matrix as:
Σs=1N∑i=1N(ϕi−μ)(ϕi−μ)Twhereμ=1N∑i=1Nϕi\Sigma_s = \frac{1}{N} \sum_{i=1}^N (\phi_i - \mu)(\phi_i - \mu)^T \quad \text{where} \quad \mu = \frac{1}{N} \sum_{i=1}^N \phi_i

This captures the semantic variance structure of the entire macro-reality space formed by observer-consistent collapse events.

F-TN.3 Spectral Decomposition and Collapse Directionality

By the spectral theorem:
Σs=QΛQT\Sigma_s = Q \Lambda Q^T
where:

  • QQ is an orthogonal matrix whose columns are eigenvectors e1,e2,…,ed\mathbf{e}_1, \mathbf{e}_2, \dots, \mathbf{e}_d

  • Λ\Lambda is a diagonal matrix with eigenvalues λ1≥λ2≥⋯≥λd≥0\lambda_1 \geq \lambda_2 \geq \dots \geq \lambda_d \geq 0

The projection of each trace onto the principal semantic axes is:
ϕi(proj)=∑k=13(ekTϕi)ek\phi_i^{(proj)} = \sum_{k=1}^3 (\mathbf{e}_k^T \phi_i) \mathbf{e}_k

F-TN.4 Entropy Analysis of Dimensional Reduction

Let λ1,λ2,λ3\lambda_1, \lambda_2, \lambda_3 be the top-3 eigenvalues. Define normalized weights:
pk=λk∑j=1dλjp_k = \frac{\lambda_k}{\sum_{j=1}^d \lambda_j}

Define semantic entropy as:
H=−∑k=1dpklog⁡pkH = -\sum_{k=1}^d p_k \log p_k

Let HcH_c be the empirically estimated stability threshold for macro-consistent world encoding. Then:

  • Projection to 3D: H3=−∑k=13pklog⁡pkH_3 = -\sum_{k=1}^3 p_k \log p_k

  • Projection to >3D: H>3=−∑k=1dpklog⁡pk>HcH_{>3} = -\sum_{k=1}^d p_k \log p_k > H_c

Thus, only the 3-EV projection remains below HcH_c, ensuring semantic integrity and low collapse entropy.

F-TN.5 Eigenvalue Collapse Filter Algorithm (Simplified)

Input: Semantic trace matrix Φ∈RN×d\Phi \in \mathbb{R}^{N \times d}
Output: Principal eigenvectors e1,e2,e3\mathbf{e}_1, \mathbf{e}_2, \mathbf{e}_3

import numpy as np

# Compute mean-centered trace matrix
mu = np.mean(Phi, axis=0)
X = Phi - mu

# Compute covariance
Sigma = (X.T @ X) / X.shape[0]

# Eigen decomposition
eigvals, eigvecs = np.linalg.eigh(Sigma)

# Sort eigenvectors by descending eigenvalue
idx = np.argsort(eigvals)[::-1]
E_top3 = eigvecs[:, idx[:3]]

Use Etop3E_{top3} as the canonical semantic axes for visualization, consensus, and latent-space modeling.

F-TN.6 Open Conjecture: 3-EV Sufficiency Theorem

Claim: For all semantic collapse fields Φ\Phi representing observer-stable macro-reality, the projection onto top-3 eigenvectors {e1,e2,e3}\{\mathbf{e}_1, \mathbf{e}_2, \mathbf{e}_3\} minimizes entropy while preserving consensus stability.

Sketch of Path to Proof:

  1. Represent semantic trace field as Gaussian-like or log-concave distribution.

  2. Apply entropy-preserving dimensional reduction arguments (e.g., Johnson-Lindenstrauss lemma variants).

  3. Combine with consensus-theoretic models (multi-agent agreement in reduced latent manifold).

We invite collaborators from category theory, information geometry, and quantum thermodynamics to formalize this theorem.

F-TN.7 Closing Note

This TechNote is a living document. As new mathematical structures, counterexamples, or agent-based simulations arise, we will refine the 3-EV framework and its application to Semantic Field Theory, AI cognition, and K3D navigation.

For questions, insights, or contributions: contact the authors or submit appendices through the open SMFT-K3D collaboration portal.

 

 

 

 © 2025 Danny Yeung. All rights reserved. 版权所有 不得转载

 

Disclaimer

This book is the product of a collaboration between the author and OpenAI's GPT-4o, GPT4.1, GPT o3, Wolfram GPTs, X's Grok3 language model. While every effort has been made to ensure accuracy, clarity, and insight, the content is generated with the assistance of artificial intelligence and may contain factual, interpretive, or mathematical errors. Readers are encouraged to approach the ideas with critical thinking and to consult primary scientific literature where appropriate.

This work is speculative, interdisciplinary, and exploratory in nature. It bridges metaphysics, physics, and organizational theory to propose a novel conceptual framework—not a definitive scientific theory. As such, it invites dialogue, challenge, and refinement.


I am merely a midwife of knowledge.

 

M
T
G
Y
Text-to-speech function is limited to 200 characters
at

No comments:

Post a Comment

Subscribe to: Post Comments (Atom)