NAKA vs SATO (Why NAKA Uses S = 100)

SATO is the initial project design that used S = 500. NAKA keeps the same core model but uses S = 100. A 5× lower steepness parameter. To dramatically improve capital efficiency and accelerate price discovery.

Executive Summary

• SATO (S = 500) traverses the curve slowly; full issuance requires thousands of ETH.
• NAKA (S = 100) reaches mature issuance (~99% of K) at roughly 460 ETH of cumulative input.
• The two systems share an identical functional form. Only S differs, and the consequences are large.

Parameter Difference

• SATO: S = 500
• NAKA: S = 100

Shared model:

• p(eth) = (S / K) * exp(eth / S) . Marginal price (ETH per token)
• minted(eth) = K * (1 - exp(-eth / S)) . Cumulative minted supply

A lower S means the system traverses curve states 5× faster for the same net ETH inflow. With K = 21,000,000 for both, NAKA reaches the same issuance milestones using a fraction of the cumulative ETH that SATO requires.

Why NAKA Is Superior for This Design

1) Better Capital Efficiency

Solving 1 - exp(-eth / S) = m for issuance fraction m gives eth_m = -S · ln(1 - m). For the same milestone, NAKA needs exactly 1/5 the ETH that SATO does:

milestone	SATO (S=500)	NAKA (S=100)	ratio
50% of K	500 · ln 2 ≈ 346.6 ETH	100 · ln 2 ≈ 69.3 ETH	5×
90% of K	500 · ln 10 ≈ 1,151.3 ETH	100 · ln 10 ≈ 230.3 ETH	5×
99% of K	500 · ln 100 ≈ 2,302.6 ETH	100 · ln 100 ≈ 460.5 ETH	5×
99.9% of K	500 · ln 1000 ≈ 3,453.9 ETH	100 · ln 1000 ≈ 690.8 ETH	5×

Result: NAKA achieves "near-cap" issuance at roughly 460 ETH. The curve visibly closes against the cap inside that range. Whereas SATO requires about 2,303 ETH for the same outcome.

2) Faster Price Discovery

Marginal-price multiplier vs the initial price (exp(eth / S)):

cumulative ETH	SATO factor	NAKA factor	NAKA is
100 ETH	exp(0.2) ≈ 1.221×	exp(1) ≈ 2.718×	~2.2× higher
200 ETH	exp(0.4) ≈ 1.492×	exp(2) ≈ 7.389×	~5× higher
500 ETH	exp(1.0) ≈ 2.718×	exp(5) ≈ 148.4×	~55× higher
1000 ETH	exp(2.0) ≈ 7.389×	exp(10) ≈ 22,026×	~3000× higher

Result: at any equal level of cumulative inflow, NAKA's price reflects demand far more aggressively, so market signal converges to a clearing level much faster.

3) Stronger Long-Term Scarcity Signaling

Because NAKA reaches its asymptote 5× sooner in cumulative ETH terms, the boundary between "early curve" and "late curve" participation is much sharper:

• At 100 ETH cumulative, SATO has minted 1 - exp(-0.2) ≈ 18.1% of K, NAKA has minted 1 - exp(-1) ≈ 63.2% of K.
• At 460 ETH cumulative, SATO has minted ~60%, NAKA has minted ~99%.

Result: NAKA users see scarcity progress in real-time; SATO users would have to wait through ~5× more ETH inflow to observe the same dynamic.

4) Better Fit for a No-Operator System

In an immutable, deterministic setup, parameter quality matters more than discretionary intervention. S = 100 produces a strong signal-to-noise ratio between curve regions without adding governance complexity, while preserving the same trust-minimized operating model.

5) Stronger Community and Ecosystem Orientation

Unlike SATO, NAKA has an official X presence (x.com/naka_exchange) and is explicitly oriented toward community growth, ongoing development, and future partnerships.

Result: better communication continuity, clearer public coordination, and stronger ecosystem expansion potential.

Trade-Off Transparency

NAKA's much lower S means a substantially steeper price response to flow. This is intentional and should be understood as a deliberate design choice:

• Pros: ~5× less ETH required for full lifecycle progression; price reflects demand within the first hundreds of ETH rather than thousands; scarcity becomes visible early, rewarding genuine conviction over passive capital.
• Cons: a single 100-ETH inflow moves the marginal price by ~2.7× rather than ~1.22×; late-stage entries pay disproportionately compared to early entries.

For NAKA, this trade-off is considered favorable: a curve that visibly closes inside a few hundred ETH delivers a more honest discovery process than one that drags out indefinitely.

Quick Reference (NAKA, S = 100, K = 21,000,000)

• Initial price: S/K ≈ 4.76 × 10⁻⁶ ETH/token
• 50% minted at ~69.3 ETH
• 90% minted at ~230.3 ETH
• 99% minted at ~460.5 ETH ← curve effectively closes here
• 99.9% minted at ~690.8 ETH
• Marginal price at full closure (~460 ETH): 4.76 × 10⁻⁶ · e^4.6 ≈ 4.76 × 10⁻⁴ ETH/token (≈100× initial)