Archive PURE_FORMULAS.md

archive/2026-07-02/docs/PURE_FORMULAS.md

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+# SilverSight — Pure Mathematical Formulas
+
+No English. No code. Minimum notation only.
+
+---
+
+## 1. Sidon
+
+$$A \subset \mathbb{Z},\quad a+b=c+d \implies \{a,b\}=\{c,d\}$$
+
+$$A_8 = \{2^i\}_{i=0}^7$$
+
+$$h(N) = \max|A|,\quad A \subseteq \{1,\ldots,N\} \text{ Sidon}$$
+
+$$h(N) \leq \lfloor\sqrt{2N}\rfloor + 1$$
+
+$$S_p = \{x \in \mathbb{F}_{p^3}^\times / \mathbb{F}_p^\times : \text{Tr}(x)=0\},\quad |S_p|=p+1,\quad p \text{ prime}$$
+
+**Fisher note:** \(\Delta_7\) denotes the **open** simplex \(\{p \in \mathbb{R}^8 : p_i > 0,\; \sum p_i = 1\}\). All Fisher metric formulas require this domain (\(p_i > 0\)).
+
+---
+
+## 2. Braid
+
+$$C \in \{0,\tfrac{1}{4},\tfrac{1}{2},\tfrac{3}{4}\}^{8\times 8}$$
+
+$$\varepsilon_{ij} = C_{ij}(\phi_i - \phi_j)$$
+
+$$\text{crossStep}(s) = s \iff s \in \text{Eigensolid}$$
+
+---
+
+## 3. Spectral
+
+$$\text{active}(s) = \{i : s_i \neq 0\}$$
+
+$$\text{gap}(s) = \bigwedge_{i,j \in \text{active}(s)} (i=j \lor |i-j|>1)$$
+
+$$\text{merge}(s,e)_i = \min(1, s_i+e_i)$$
+
+$$\text{res}(s,e) = |\{i : s_i \neq 0 \land e_i \neq 0\}|$$
+
+$$\text{cross}(s,e) = \bigwedge_{i} \neg(s_i \neq 0 \land e_{i+1} \neq 0) \land \neg(e_i \neq 0 \land s_{i+1} \neq 0)$$
+
+$$\text{gap}(s) \land \text{gap}(e) \land \text{res}(s,e)=0 \land \text{cross}(s,e) \implies \text{gap}(\text{merge}(s,e))$$
+
+---
+
+## 4. Byte
+
+$$\text{byteGap}(n) = (n \land (n \gg 1)) = 0$$
+
+$$\text{pack}(s) = \sum_{i=0}^{7} [s_i \neq 0] \cdot 2^i$$
+
+$$\text{byteGap}(\text{pack}(s)) = \text{gap}(s)$$
+
+---
+
+## 5. Chiral
+
+$$q = q_r + \varepsilon q_d,\quad \varepsilon^2 = 0$$
+
+$$\chi = \frac{|q_r|^2}{|q_r|^2 + |q_d|^2},\quad |q_r|^2 + |q_d|^2 > 0$$
+
+$$\chi > \frac{1}{2} \implies \text{compressive}$$
+
+$$\chi < \frac{1}{2} \implies \text{anti-compressive}$$
+
+$$\chi = \frac{1}{2} \implies \text{critical balance}$$
+
+---
+
+## 6. Q16_16
+
+$$\text{Q16}(x) = \text{clamp}(-2^{31}, \lfloor x \cdot 2^{16} \rfloor, 2^{31}-1)$$
+
+$$\text{Q16}(1) = 65536$$
+
+$$\text{Q16}(0) = 0$$
+
+$$a \oplus b = \text{clamp}(-2^{31}, a+b, 2^{31}-1)$$
+
+$$a \otimes b = \text{clamp}(-2^{31}, \lfloor ab/2^{16} \rfloor, 2^{31}-1)$$
+
+---
+
+## 7. Verification Protocol
+
+$$\text{define} \to \text{compute} \to \text{verify} \to \text{claim}$$
+
+$$\text{verify} = \text{false} \implies \text{formula wrong}$$