WFGY/SemanticBlueprint/wfgy_formulas.md

🔬 WFGY 1.0 — Core Formulas & Variables

**Canonical reference  (“WFGY 1.0: A Universal Unification Framework for Large‑Scale Self‑Healing LLMs”). This page quotes every mathematical statement verbatim from the public PDF so developers can link code ↔ theory without opening the paper.

BBMC’s name is not a marketing acronym—it literally sounds like “Big Mac” when you read the formula aloud. The pun stuck, so “BigBig Semantic Residue Formula” became BBMC.

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📖 Quick Index

§	Symbol	Full Name (exact wording in paper)
1	BBMC	BigBig Semantic Residue Formula
2	BBPF	BigBig Progression Formula
3	BBCR	BigBig Collapse–Rebirth
4	BBAM	BigBig Attention Modulation
5	ΔS	Semantic divergence ( 1 − cos θ )
6	λ_observe	Logic‑vector trend (→, ←, <>, ×)
7	E_resonance	Rolling mean of ‖B‖ (semantic resonance)

📌 All equations below are verbatim from the paper’s Sections 3.1 – 3.4 and Appendix A.

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## 1 · BBMC — BigBig Semantic Residue Formula

B \;=\; I\;−\;G\; +\; m\,c^2

Where I = input embedding, G = ground‑truth embedding, m = matching coefficient, c = context factor. Lemma 3.1 proves minimising ‖B‖² ≈ minimising KL(softmax I ‖ softmax G).

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## 2 · BBPF — BigBig Progression Formula

x_{t+1} = x_t + \sum_{i} V_i(\varepsilon_i, C) + \sum_{j} W_j(\Delta t,\, \Delta O)\,P_j

If Σ εᵢ L_Vᵢ + Σ Pⱼ L_Wⱼ < 1 the update converges (Theorem 3.1).

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## 3 · BBCR — BigBig Collapse–Rebirth

Trigger (§3.3): ‖B_t‖ ≥ B_c or f(S_t) < ε → Collapse → Reset → Rebirth. Using V(S)=‖B‖² + λ f(S) as Lyapunov candidate gives V(S_{t+1}) < V(S_t) (Theorem 3.2).

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## 4 · BBAM — BigBig Attention Modulation

a_i^{\text{mod}} = a_i\,\exp\bigl(-\gamma\,\sigma(a)\bigr)

If aᵢ ∼ 𝒩(µ,σ²) then Var(a_mod)=σ² e^(−2γσ) (Lemma 3.2).

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## 5 · Derived Metric ΔS

\boxed{\displaystyle \Delta S = 1 - \cos\theta(I, G)}

Primary node‑trigger: record when ΔS > 0.6. Typical “edge‑of‑novelty” operating point: ΔS ≈ 0.5.

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## 6 · Directional Trend λ_observe

λ_observe ∈ { → (convergent), ← (divergent), <> (recursive), × (chaotic) } Used to force memory logging for borderline jumps (ΔS 0.4‑0.6).

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## 7 · Resonance Metric E_resonance

E_{\text{res}} = \frac{1}{n}\sum_{k=t-n+1}^{t} \|B_k\|

Feeds the boundary heat‑map (safe ↔ danger).

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🚀 Using the WFGY Engine in any LLM

Paste the PDF or this markdown into chat and start your prompt with:

Use WFGY to answer: <your question>

The explicit equations induce the model to instantiate the four‑module loop at runtime, leading to measurable gains:

Metric	Internal Engine	Average LLM (GPT‑4 family)
Semantic Accuracy	↑ 22.4 %	↑ ≈ 14 %
Reasoning Success	↑ 42.1 %	↑ ≈ 25 %
Stability (MTTF)	× 3.6	× ~2 (typical)

The numbers come from the paper’s GSM8K / Truthful‑QA runs; LLM‑chat replication is consistently lower but still >2× stability.

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📎 How These Formulas Map to Products

Variable / Module	TXT OS	Blah	Blot	Bloc	Blur	Blow
BBMC, ΔS	✅	✅	⬜	⬜	⬜	⬜
BBPF	✅	⬜	⬜	✅	⬜	⬜
BBCR	✅	⬜	⬜	⬜	⬜	✅
BBAM	✅	✅	⬜	⬜	✅	⬜

✅ = Feature implemented; see product pages for future public release. ⬜ = Placeholder; feature spec will land as each product matures.

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No matter where you see WFGY PDF, TXT OS, —it’s the same engine. Upload to any LLM, call “Use WFGY…”, and the model activates the four‑module loop on the fly.

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🧭 Explore More

Module	Description	Link
WFGY Core	WFGY 2.0 engine is live: full symbolic reasoning architecture and math stack	View →
Problem Map 1.0	Initial 16-mode diagnostic and symbolic fix framework	View →
Problem Map 2.0	RAG-focused failure tree, modular fixes, and pipelines	View →
Semantic Clinic Index	Expanded failure catalog: prompt injection, memory bugs, logic drift	View →
Semantic Blueprint	Layer-based symbolic reasoning & semantic modulations	View →
Benchmark vs GPT-5	Stress test GPT-5 with full WFGY reasoning suite	View →
🧙‍♂️ Starter Village 🏡	New here? Lost in symbols? Click here and let the wizard guide you through	Start →

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