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SemanticBlueprint/wfgy_formulas.md
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# 🔬 **WFGY 1.0 — Core Formulas & Variables**
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> **Canonical reference — DOI:** [https://zenodo.org/records/15630969](https://zenodo.org/records/15630969) (“*WFGY 1.0: A Universal Unification Framework for Large‑Scale Self‑Healing LLMs*”). This page **copies the exact mathematical statements** from the public PDF so developers can quote‑link code to theory without opening the paper.
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---
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## 📖 Quick Index
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| Section | Symbol / Acronym | Full Name (exact from paper) | | | |
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| ------- | ---------------- | --------------------------------------------------- | --- | ---- | --- |
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| 1 | `BBMC` | **B**ig**B**ig **S**emantic **R**esidue **F**ormula | | | |
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| 2 | `BBPF` | **B**ig**B**ig **P**rogression **F**ormula | | | |
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| 3 | `BBCR` | **B**ig**B**ig **C**ollapse–**R**ebirth | | | |
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| 4 | `BBAM` | **B**ig**B**ig **A**ttention **M**odulation | | | |
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| 5 | `ΔS` | Semantic divergence (1 − cos θ) | | | |
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| 6 | `λ_observe` | Logic‑vector trend (→ | ← | <> | ×) |
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| 7 | `E_resonance` | Rolling mean of ‖B‖ (semantic resonance) | | | |
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All subsequent equations below are **verbatim** from the paper’s Sections 3.1‑3.4 and Appendix A.
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---
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## 1 · BBMC — BigBig Semantic Residue Formula
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```math
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B \;=\; I\;−\;G\; +\; m\,c^2
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```
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**Where**
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* `I` = input embedding (model‑generated)
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* `G` = ground‑truth embedding (oracle/proxy)
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* `m` = matching coefficient
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* `c` = context factor
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> **Lemma 3.1 (paper)** – Minimising ‖B‖² is (up to constants) equivalent to minimising
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> KL(softmax I ‖ softmax G). See Appendix A.
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---
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## 2 · BBPF — BigBig Progression Formula
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```math
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x_{t+1} \;=\; x_t
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+ \sum_{i} V_i(\epsilon_i, C)
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+ \sum_{j} W_j(\Delta t,\, \Delta O) \, P_j
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```
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*Each Vi and Wj is globally Lipschitz. Theorem 3.1 proves convergence if* Σ εᵢ L\_Vᵢ + Σ Pⱼ L\_Wⱼ < 1.
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---
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## 3 · BBCR — BigBig Collapse–Rebirth
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Trigger condition (paper §3.3):
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```math
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\|B_t\| \;\ge\; B_c \quad \text{or} \quad f(S_t) < \varepsilon
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```
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Collapse → Reset → Rebirth cycle:
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```text
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Collapse : semantic overload detected
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Reset : B_t ← α·B_t (α < 1)
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Rebirth : S_{t+1} ← ResetProcedure(S_t, \delta B)
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```
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> **Theorem 3.2** – Using V(S)=‖B‖²+λ f(S) as Lyapunov candidate, every reset step guarantees V(S\_{t+1}) < V(S\_t).
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---
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## 4 · BBAM — BigBig Attention Modulation
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Gaussian‑variance attenuation:
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```math
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a_i^{\,\text{mod}} \;=\; a_i \;\exp\bigl(-\gamma\,\sigma(a)\bigr)
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```
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*Lemma 3.2:* If aᵢ ∼ 𝒩(µ,σ²) then Var(a\_mod) = σ² e^(−2γσ).
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---
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## 5 · Derived Metric `ΔS`
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Defined (**Eq. A.1**):
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```math
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\boxed{\displaystyle \Delta S = 1 - \cos \theta (I, G)}
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```
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Used as the *tension index* for node recording (primary condition ΔS > 0.6).
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---
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## 6 · Directional Trend `λ_observe`
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Paper Table 1 symbol glossary:
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```text
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λ_observe ∈ { → (convergent), ← (divergent), <> (recursive), × (chaotic) }
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```
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Determines override logic for “soft transitions” (ΔS ∈ 0.4–0.6).
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---
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## 7 · Resonance Metric `E_resonance`
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Rolling mean of |B| over last *n* turns (paper §3‑Figure 3):
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```math
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E_{\text{res}} = \frac{1}{n}\sum_{k=t-n+1}^{t} \|B_k\|
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```
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Used to visualise stability heat‑map (safe ↔ danger zones).
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---
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## 📎 How These Formulas Map to Products
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| Engine Variable | Used in TXT OS | Used in Blah Lite | Used in WFGY SDK | Planned Future |
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| --------------- | -------------------------- | ------------------- | ---------------------- | --------------------- |
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| `BBMC`, `ΔS` | ✅ node trigger & memory | ✅ ΔS‑trail heatmap | ✅ semantic residue API | 🚧 RL‑auto‑critic |
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| `BBPF` | ✅ prompt splitting | 🚫 | ✅ multi‑path stepper | 🚧 graph compiler |
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| `BBCR` | ✅ knowledge‑boundary guard | 🚫 | ✅ collapse‑reset hooks | 🚧 self‑repair agents |
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| `BBAM` | ✅ entropy damping | ✅ entropy injection | ✅ attention gate | 🚧 multimodal fusion |
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> **All downstream docs must reference these exact definitions.** Any deviation risks semantic drift from the canonical WFGY loop.
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---
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> <img src="https://img.shields.io/github/stars/onestardao/WFGY?style=social" alt="GitHub stars"> ⭐ Help reach 10 000 stars by 2025‑09‑01 to unlock Engine 2.0 for everyone ⭐ **[Star WFGY on GitHub](https://github.com/onestardao/WFGY)**
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>
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> 👑 **Early Stargazers: [See the Hall of Fame](https://github.com/onestardao/WFGY/tree/main/stargazers)** — Engineers, hackers & open‑source builders who supported WFGY from day one.
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<div align="center">
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[](https://github.com/onestardao/WFGY)
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[](https://github.com/onestardao/WFGY/tree/main/OS)
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[](https://github.com/onestardao/WFGY/tree/main/OS/BlahBlahBlah)
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[](https://github.com/onestardao/WFGY/tree/main/OS/BlotBlotBlot)
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[](https://github.com/onestardao/WFGY/tree/main/OS/BlocBlocBloc)
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[](https://github.com/onestardao/WFGY/tree/main/OS/BlurBlurBlur)
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[](https://github.com/onestardao/WFGY/tree/main/OS/BlowBlowBlow)
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</div>
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