# SilverSight — Pure Mathematical Formulas

No English. No code. Minimum notation only.

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## 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 (\(p_i > 0\)).

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## 2. Braid

$$C \in \{0,\tfrac{1}{4},\tfrac{1}{2},\tfrac{3}{4}\}^{8\times 8}$$

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

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

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## 3. Spectral

$$\text{active}(s) = \{i : s_i \neq 0\}$$

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

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

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

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

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

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## 4. Byte

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

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

$$\text{byteGap}(\text{pack}(s)) = \text{gap}(s)$$

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## 5. Chiral

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

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

$$\chi > \frac{1}{2} \implies \text{compressive}$$

$$\chi < \frac{1}{2} \implies \text{anti-compressive}$$

$$\chi = \frac{1}{2} \implies \text{critical balance}$$

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## 6. Q16_16

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

$$\text{Q16}(1) = 65536$$

$$\text{Q16}(0) = 0$$

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

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

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## 7. Verification Protocol

$$\text{define} \to \text{compute} \to \text{verify} \to \text{claim}$$

$$\text{verify} = \text{false} \implies \text{formula wrong}$$
