G4: Binary update rule - OPTIMIZED with integer arithmetic · sᵢ(t+1)

Read the equation

This relation defines sᵢ(t+1) in the source section “G4: Binary update rule - OPTIMIZED with integer arithmetic · sᵢ(t+1)”. It specifies the displayed relationship between the named quantities or parameters. Its interpretation depends on the source’s stated variables and conditions; this record does not assign an unprovided physical or cognitive meaning to them.

Terms & definitions

The recovered passage does not supply a separate symbol table. Read the source context for the notation used in this version; familiar letters may carry different meanings in other builds.

How it works

1. Read sᵢ(t+1) as the quantity being defined or updated by this relation.

2. Substitute the source-defined variables or recorded values and follow the displayed arithmetic and indexing exactly.

3. Use this version’s variable definitions and update order. The same symbol may be assigned a different meaning in another build.

Assumptions & limits

• This source expression was recovered from the pinned code revision. Its presence does not establish that the associated program was built, run, or scientifically validated.

Original context

    double density() const {
        return static_cast<double>(popcount()) * (1.0 / GATE_COUNT);
    }

    /**
     * @brief G4: Binary update rule - OPTIMIZED with integer arithmetic
     *
     * sᵢ(t+1) = H(Σw_{ji}·sⱼ - θᵢ)
     */
    void update_binary() {
        alignas(64) std::array<uint64_t, U64_COUNT> old_state;
        std::memcpy(old_state.data(), state.data(), sizeof(state));

        for (size_t word_idx = 0; word_idx < U64_COUNT; ++word_idx) {
            uint64_t new_word = 0;
            const size_t base_idx = word_idx << 6;  // word_idx * 64

            // Process all 64 bits (or remaining bits for last word)

sᵢ(t+1) = H(Σw_{ji}·sⱼ - θᵢ)

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