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101 lines
2.4 KiB
Markdown
101 lines
2.4 KiB
Markdown
# Braid Structure: Witness Traces
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**Authors:** Research Stack Team
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**Date:** April 2026
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**Domain:** TTM Layer A (Compression/Routing) + Braid Theory
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**OTOM Version:** 2.2
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**References:** See [00_Master_References.md](00_Master_References.md) for complete source mapping
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---
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## Abstract
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Braid Structure provides the algebraic foundation for witness traces in OTOM. It formalizes computation paths as braid words, enabling topological verification of program correctness through holonomy invariants. The braid structure is now extended with geometric structure folding (Torus-Menger-Horn) for topological routing and Mass Number gates for braid transition admissibility.
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---
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## 1. Braid Fundamentals
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### 1.1 Artin Braid Group
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$$B_n = \langle \sigma_1, \ldots, \sigma_{n-1} \mid \sigma_i \sigma_j = \sigma_j \sigma_i \text{ if } |i-j| > 1, \sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1} \rangle$$
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### 1.2 Braid Word
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$$\beta = \sigma_{i_1}^{\epsilon_1} \sigma_{i_2}^{\epsilon_2} \cdots \sigma_{i_k}^{\epsilon_k}$$
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Where $\epsilon_j \in \{-1, +1\}$.
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---
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## 2. Witness Traces
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### 2.1 Trace Formation
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A computation path $P$ induces a braid trace:
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$$\text{trace}(P) = \prod_{i} \sigma_{j_i}$$
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### 2.2 Merkle Structure
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$$H_{\text{braid}} = H(H_{\text{left}} \| H_{\text{right}} \| \text{trace})$$
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---
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## 3. Holonomy
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### 3.1 Connection
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$$\omega : TB_n \rightarrow \mathfrak{g}$$
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### 3.2 Curvature
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$$\Omega = d\omega + \frac{1}{2}[\omega, \omega]$$
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### 3.3 Holonomy Along Path
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$$\text{Hol}_\gamma = \mathcal{P} \exp\left(-\int_\gamma \omega\right)$$
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---
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## 4. Bracket Calculus
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### 4.1 Bracket Polynomial
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$$\langle L \rangle = \sum_{\text{states } s} A^{\alpha(s) - \beta(s)} (-A^2 - A^{-2})^{|s| - 1}$$
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### 4.2 Bracket Shell Count
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$$N_{\text{shell}} = \sum_{i=1}^{n} \binom{n}{i} \cdot i$$
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---
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## 5. Implementation
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**Lean 4 Modules:**
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- `BraidStrand.lean` — Braid group operations
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- `BraidBracket.lean` — Bracket calculus
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- `BraidCross.lean` — Crossing operations
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- `BracketShellCount.lean` — Shell counting
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- `UniversalCoupling.lean` — Coupling structures
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---
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## 6. Theorems
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### 6.1 System Admissibility
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$$\text{systemAdmissible}(S) \iff \forall \gamma, \text{Hol}_\gamma \in G_{\text{admissible}}$$
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### 6.2 Gap Conservation
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$$\Delta_{\text{gap}} = \text{constant}$$
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---
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## 7. References
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- Artin, E. (1947). Theory of braids.
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- Kauffman, L.H. (1987). State models and the Jones polynomial.
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- Research Stack, OTOM Ontology v2.2.
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