منحنيات الربط

p(s) = صيغة المنحنى

cost(s_0, s_1) = ∫_{s_0}^{s_1} p(s) ds = P(s_1) − P(s_0)

proceeds(s_1, s_0) = cost(s_0, s_1)

avg = cost(s_0, s_1) / (s_1 − s_0)

p(s) = a + b · s

P(s)            = a·s + (b/2)·s²
cost(s_0, s_1)  = a·(s_1 − s_0) + (b/2)·(s_1² − s_0²)

p(s) = k · s²                      // أو  k · (s / S_max)² بصيغة معيارية

// Antiderivative of the normalized form p(s) = k · (s / S_max)²
P(s)            = (k / (3 · S_max²)) · s³
cost(s_0, s_1)  = (k / (3 · S_max²)) · (s_1³ − s_0³)

effective_y = V_q + collateral_received
effective_x = S_max − s
(effective_x) · (effective_y) = V_q · S_max      // الثابت

p(s) = effective_y / effective_x
     = V_q · S_max / (S_max − s)² · ... (قابل للاشتقاق عبر التفاضل الضمني)

cost(s_0, s_1) = V_q · S_max / (S_max − s_1) − V_q · S_max / (S_max − s_0)
              = V_q · (s_1 − s_0) · S_max / ((S_max − s_0) · (S_max − s_1))

# حل تربيعي: (k/3)·s_1³ = (k/3)·s_0³ + cost
# ابدأ مع s_guess ≈ cbrt(3·cost/k + s_0³)
for i in 0..MAX_ITER:
    f    = (k/3)·s_guess³ − (k/3)·s_0³ − cost
    f'   = k·s_guess²
    step = f / f'
    s_guess -= step
    if |step| < precision_floor: break

cost_curve  = cost(base_sold, base_sold + base_out)
fee         = ceil(cost_curve · buy_numerator / buy_denominator)
quote_in    = cost_curve + fee

proceeds_curve = cost(base_sold − base_in, base_sold)
fee            = ceil(proceeds_curve · sell_numerator / sell_denominator)
quote_out      = proceeds_curve − fee

quote_to_graduate = cost(0, S_graduate) · (1 + buy_fee_rate)
                  = (k / 3) · S_graduate³ · (1 + f_buy)

cpmm_base_reserve  = S_max − S_graduate        // عرض منحنى غير مباع
cpmm_quote_reserve = quote_vault − accrued_fee_counters
cpmm_initial_price = cpmm_quote_reserve / cpmm_base_reserve

cpmm_initial_price == p(S_graduate)           // مساواة دقيقة

p(5e8) = 40 · (5e8 / 1e9)² = 40 · 0.25 = 10

cost(0, 1e6) = (40/3) · (1e6)³
             = (40/3) · 1e18
             ≈ 1.333e19     (أصغر وحدات اقتباس)

quote_in = 13.3 · 1.01 ≈ 13.5

cost(0, 8e8) = (40 / 3e18) · (8e8)³
             = (40 / 3e18) · 5.12e26
             ≈ 6.827e9      (≈ 6,827 units of a 6-decimal quote mint)
quote_to_graduate ≈ 6.827e9 · 1.01 ≈ 6.895e9

p(8e8) = 40 · 0.64 = 25.6

cpmm_base  = 1e9 − 8e8 = 2e8
cpmm_quote ≈ 6.827e9   (ناقص خصومات عداد الرسوم)
cpmm_price ≈ 6.827e9 / 2e8 = 34.14 لكل أساس