BioSight/docs/PURE_EQUATION_MAP.md

# SilverSight — Pure Equation Map

## 1. Fisher Geometry (Δ₇ → S⁷)

$$\Delta_7 := \{p \in \mathbb{R}^8 : p_i > 0,\; \sum_{i=1}^8 p_i = 1\}$$

$$T_p\Delta_7 := \{v \in \mathbb{R}^8 : \sum_{i=1}^8 v_i = 0\}$$

$$g_p(u,v) = \sum_{i=1}^8 \frac{u_i v_i}{p_i},\qquad u,v \in T_p\Delta_7$$

$$d_F(p,q) = 2\arccos\Bigl(\sum_{i=1}^8\sqrt{p_i q_i}\Bigr)$$

$$\psi : \Delta_7 \to S^7,\qquad \psi(p) = (\sqrt{p_1},\ldots,\sqrt{p_8})$$

$$g_p(u,v) = 4 \cdot g^{S^7}_{\psi(p)}(d\psi_p(u), d\psi_p(v))$$

$$d_F(p,q) = 2 \cdot d_{S^7}(\psi(p), \psi(q))$$

$$d_B(p,q) = \arccos\Bigl(\sum_{i=1}^8 \sqrt{p_i q_i}\Bigr)$$

$$\boxed{d_F(p,q) = 2\,d_B(p,q)}$$

$$\frac{1}{\pi} d_F(p,q) \leq d_{\text{chord}}(\Phi(p),\Phi(q)) \leq \frac{1}{2} d_F(p,q)$$

---

## 2. G1: Chaos Game Contraction

$$w_j(p) = \frac{A_j p + b_j}{1 + B_j},\qquad p \in \Delta_7$$

$$\Delta_7^{(\varepsilon_j)} := \{p \in \Delta_7 : p_i \geq (b_j)_i/(1+B_j)\},\qquad \varepsilon_j = \min_i (b_j)_i/(1+B_j) > 0$$

$$\widehat{w}_j(x) = \Phi(w_j(x^2)) = \frac{\sqrt{A_j x^2 + b_j}}{\sqrt{1+B_j}}$$

$$d_B(w_j(p), w_j(q)) \leq \frac{1}{\sqrt{1+B_j}} \, d_B(p,q)$$

$$d_F(w_j(p), w_j(q)) \leq \lambda_j \, d_F(p,q),\qquad \lambda_j = \frac{1}{\sqrt{1+B_j}} < 1$$

$$\lambda = \max_j \lambda_j < 1$$

$$\kappa_j(p) = \sup_{v \in T_p\Delta_7 \setminus \{0\}} \frac{\|dw_j|_p(v)\|_{F,w_j(p)}}{\|v\|_{F,p}}$$

$$\kappa_j(p) \leq \frac{1}{\sqrt{1+B_j}} \leq 1 - \frac{B_j}{2(1+B_j)}$$

$$\mathcal{A} = \bigcup_{j=1}^k w_j(\mathcal{A})$$

---

## 3. G2: Semantic Feature Collision Breaking

$$F(E)_i = \frac{f_i(E)}{|E|},\qquad f_i(E) = \sum_{\ell=1}^{L} \mathbf{1}_{\{\chi(e_\ell) = i\}}$$

$$\mathcal{N} := \{\text{bin-add}, \text{bin-sub}, \text{bin-mul}, \text{bin-div}, \text{bin-eq}, \text{un-neg}, \text{var}, \text{const}\}$$

$$\tau(E)_t = \frac{|\{n \in T(E) : \lambda(n) = t\}|}{|T(E)|},\qquad t \in \mathcal{N}$$

$$b_{t,t',i} := |\{(u,v) : \lambda(u) = t,\; \lambda(v) = t',\; \text{child index} = i\}|$$

$$M = \sum_{t,t',i} b_{t,t',i} = |T(E)| - 1$$

$$\delta(E)_{t,t',i} = b_{t,t',i} / M$$

$$\Phi(E) := (F(E),\; \tau(E)) \in \Delta_7 \times \Delta_{k-1}$$

$$\Phi_{+}(E) := (F(E),\; \tau(E),\; \delta(E)) \in \Delta_7 \times \Delta_{k-1} \times \Delta_{2k^2-1}$$

$$d^{\times}_F((p,r), (q,s)) = \sqrt{d^2_F(p,q) + d^2_F(r,s)}$$

$$d^F(\pi(x), \pi(y)) \leq d^{\times}(x, y)$$

$$d_F(F(E_1), F(E_2)) \leq \frac{C}{L}$$

$$d^{\times}_F(\Phi(E_1), \Phi(E_2)) \leq \sqrt{\frac{C_1^2}{L^2} + \frac{C_2^2}{N^2}}$$

---

## 4. G3: Eigensolid Fixed Point

$$C(p)_{2k-1} = C(p)_{2k} = \frac{p_{2k-1} + p_{2k}}{2},\qquad k = 1,2,3,4$$

$$M = \{p \in \Delta_7 : p_1 = p_2,\; p_3 = p_4,\; p_5 = p_6,\; p_7 = p_8\}$$

$$\phi(q_1,q_2,q_3,q_4) = \Bigl(\frac{q_1}{2},\frac{q_1}{2},\frac{q_2}{2},\frac{q_2}{2},\frac{q_3}{2},\frac{q_3}{2},\frac{q_4}{2},\frac{q_4}{2}\Bigr)$$

$$d_F(C(p), C(q)) \leq d_F(p,q)$$

**Strict iff** ∃k: (p_{2k-1}, p_{2k}) not proportional to (q_{2k-1}, q_{2k})

$$I_{\text{loss}}(p) = \sum_{k=1}^4 s_k \cdot D_{KL}\Bigl(\bigl(\frac{p_{2k-1}}{s_k},\frac{p_{2k}}{s_k}\bigr) \,\big\|\, \bigl(\tfrac12,\tfrac12\bigr)\Bigr)$$

$$I_{\text{loss}}(p) = H(C(p)) - H(p)$$

$$C_{\mathcal{P}} = C_m \circ C_{m-1} \circ \cdots \circ C_1$$

**Nested:** P_{j+1} coarsens components of P_1,…,P_j ⟹ C_𝒫² = C_𝒫

$$M_1 \supset M_2 \supset \cdots \supset M_m\quad (\text{nested})$$

$$S_*(p) = (p_1+p_2,\; p_3+p_4,\; p_5+p_6,\; p_7+p_8)$$

$$(A_j)_{2j-1,\ell} = (A_j)_{2j,\ell} = \begin{cases} \tfrac12 & \ell \in P_j, \\ \delta_{\ell,i} & i \notin P_j \end{cases}$$

---

## 5. Chentsov Reconstruction

$$h_9(p) = \sum_{i=1}^m \operatorname{Hess}(F)_p(e_i,e_i)$$

$$h_{\text{perm}}(p) = \sum_{i=1}^m \operatorname{Hess}(F)_p(e_i, e_{\pi(i)})$$

$$F(g_p^{(\alpha)}) = \alpha \cdot F(g_p^{(1)}) + (1-\alpha) \cdot F(g_p^{(0)})$$

$$g_p^{\text{mono}}(u,v) = \lambda(p) \cdot g_p^{\text{Fisher}}(u,v)$$

$$g_{ij}(p) = \frac{1}{p_i}\delta_{ij} + \frac{1}{p_m}$$

$$ds^2 = \sum_{i=0}^{m} \frac{(dp_i)^2}{p_i}$$

---

## 6. SOS Certificate (Merge Gate)

$$\text{gap}(x,m) = \text{sieve}(x,m) - \text{threshold}$$

$$K = \{x \in [2,90],\; m \in [3,13]\}$$

$$\text{gap}(x,m) = s_0(x,m) + s_1(x,m)(x-2) + s_2(x,m)(90-x) + s_3(x,m)(m-3) + s_4(x,m)(13-m)$$

$$s_i(x,m) = \sum_j q_{ij}(x,m)^2$$

$$\text{gap}(x,m) \geq 0 \text{ on } K \implies \text{merge gate}$$

**Baker (transcendence):**

$$\Lambda = \sum_{i=0}^{n} \beta_i \log \alpha_i \neq 0 \implies |\Lambda| > e^{-C \cdot \prod A_i \cdot \log B}$$

---

## 7. Sidon (Cross-Domain)

$$A \subset \mathbb{Z},\quad a+b=c+d \implies \{a,b\}=\{c,d\}$$

$$A_8 = \{2^i\}_{i=0}^7$$

$$h(N) = \max|A|,\quad A \subseteq \{1,\ldots,N\} \text{ Sidon}$$

$$h(N) \leq \lfloor\sqrt{2N}\rfloor + 1$$

$$S_p = \{x \in \mathbb{F}_{p^3}^\times / \mathbb{F}_p^\times : \operatorname{Tr}(x)=0\},\quad |S_p|=p+1$$

---

## 8. Braid / Spectral / Merge Gate

$$C \in \{0,\tfrac14,\tfrac12,\tfrac34\}^{8\times 8}$$

$$\varepsilon_{ij} = C_{ij}(\phi_i - \phi_j)$$

$$\text{crossStep}(s) = s \iff s \in \text{Eigensolid}$$

$$\text{gap}(s) = \bigwedge_{i,j \in \text{active}(s)} (i=j \lor |i-j|>1)$$

$$\text{merge}(s,e)_i = \min(1, s_i+e_i)$$

$$\text{res}(s,e) = |\{i : s_i \neq 0 \land e_i \neq 0\}|$$

$$\text{cross}(s,e) = \bigwedge_{i} \neg(s_i \neq 0 \land e_{i+1} \neq 0) \land \neg(e_i \neq 0 \land s_{i+1} \neq 0)$$

$$\text{gap}(s) \land \text{gap}(e) \land \text{res}(s,e)=0 \land \text{cross}(s,e) \implies \text{gap}(\text{merge}(s,e))$$

$$\text{byteGap}(n) = (n \land (n \gg 1)) = 0$$

$$\text{pack}(s) = \sum_{i=0}^{7} [s_i \neq 0] \cdot 2^i$$

---

## 9. Chiral / Q16_16

$$q = q_r + \varepsilon q_d,\quad \varepsilon^2 = 0$$

$$\chi = \frac{|q_r|^2}{|q_r|^2 + |q_d|^2},\quad |q_r|^2 + |q_d|^2 > 0$$

$$\chi > \tfrac12 \implies \text{compressive},\quad \chi < \tfrac12 \implies \text{anti-compressive},\quad \chi = \tfrac12 \implies \text{critical}$$

$$\text{Q16}(x) = \text{clamp}(-2^{31},\; \lfloor x \cdot 2^{16} \rfloor,\; 2^{31}-1)$$

$$a \oplus b = \text{clamp}(-2^{31},\, a+b,\, 2^{31}-1)$$

$$a \otimes b = \text{clamp}(-2^{31},\, \lfloor ab/2^{16} \rfloor,\, 2^{31}-1)$$