SilverSight/docs/PURE_FORMULAS.md

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}

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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}