Dynamic R

Log-Symmetric Ratchet Rule for Linear-k Leverage

Notation

Symbol	Meaning	Defined in
k	Leverage parameter	Pay-off Curve
x	Normalized price (= spot / MARK)	Pay-off Curve
α, β	Long and short coefficients	Pay-off Curve
R	Total reserve	Pay-off Curve
rA, rB	Long and short reserve values: rA = Φ(x), rB = Ψ(x)	Pay-off Curve
w	Virtual power-branch value: α·xᵏ	This page
T₀	Base doubling time (pool parameter)	This page
T_d	Effective doubling time: k · T₀	This page
R_eff	Effective R after time-decay ramping	This page
Φ, Ψ	Long and short payoff functions	Pay-off Curve

---

Context

Derion pools use asymptotic power curves where each side's payoff has two regimes separated by an inflection point at rA = R/2:

• Power regime (rA < R/2): Φ(x) = α·xᵏ — compound k leverage, convex curve

• Asymptotic regime (rA > R/2): Φ(x) = R - R²/(4α·xᵏ) — sub-linear leverage, concave curve

The inflection point is the only location where instantaneous leverage equals exactly k.

---

Objective

For the dominant side of the pool, achieve linear k leverage over long price moves — meaning the integrated payoff across many ratchet cycles converges to k × (price move), the same as a CEX perpetual.

Simultaneously, preserve compound k downside — when price moves against the dominant side, the full power curve applies.

---

Why Log-Price Space Is The Natural Coordinate

The two branches have unequal curvature in reserve space:

Power branch:     d²Φ/dx²  =  +k(k-1)·α·x^(k-2)
Asymptotic branch: d²Φ/dx²  =  -k(k+1)·α·x^(k-2)

Curvature ratio = (k+1)/(k-1)   [always > 1]

Equal steps in reserve space produce asymmetric payoffs — the asymptotic branch always curves more steeply, causing systematic undershoot of linear k.

In log-price space (u = ln x), the curvatures are:

Power branch:     d²Φ/du²  =  +k²·α·xᵏ  →  at inflection: +k²·R/2
Asymptotic branch: d²Φ/du²  =  -(k²)·(R - Φ)  →  at inflection: -k²·R/2

Exactly equal and opposite at the inflection point, for any k. The two branches are perfect mirrors in log-price space. This is the natural coordinate system for the ratchet rule.

---

The Ratchet Rule

Symmetry Condition

For the linear-k payoff to be achieved over a full cycle, the position must spend equal log-price distance on each side of the inflection point:

ln(x_swap / x_inf_old) = ln(x_inf_new / x_swap)

That is, x_swap is the geometric mean of the old and new inflection prices:

x_inf_new = x_swap² / x_inf_old

Deriving R_new

Using the inflection price formula x_inf = (R / 2α)^(1/k):

x_inf_old = (R_old / 2α)^(1/k)
x_inf_new = (R_new / 2α)^(1/k)

Substituting into the symmetry condition:

x_inf_new = x_swap² / x_inf_old

(R_new / 2α)^(1/k) = x_swap² / (R_old / 2α)^(1/k)

Raising both sides to the power k:

R_new / 2α = x_swap^(2k) / (R_old / 2α)

R_new = 4α² · x_swap^(2k) / R_old

Compact Form

Define w = α · x_swap^k — the virtual power-branch value at the swap price (what rA would be if it were still on the power branch):

┌─────────────────────────────┐
│                             │
│   R_new  =  4w² / R_old     │
│                             │
└─────────────────────────────┘

Equivalently:

√(R_new · R_old) = 2w

The geometric mean of old and new R equals twice the virtual power-branch value at the swap price.

Ratchet Trigger

R is updated on every swap, with different rules for each direction:

w = α · x_swap^k          // virtual power-branch value at current price

if w > R_old/2:           // dominant side in asymptotic regime
    R_new = 4w² / R_old   // log-symmetric upward ratchet
    R_new = min(R_new, balanceOf(pool))   // hard cap at available balance

if w < R_old/2:           // dominant side in power regime
    R_new = 2w            // simple downward follow
    R_new = max(R_new, 2·rA, 2·rB)       // floor: neither side above inflection

Why The Two Rules Differ

Upward — the dominant side is in the asymptotic regime. Curvature is concave and payoff is sub-linear-k. The log-symmetric formula 4w²/R_old positions the next power phase to exactly cancel the asymptotic undershoot. Curvature correction is required.

Downward — the dominant side is already in the power regime. Payoff is compound-k with no distortion. There is no curvature error to correct. R simply follows the dominant side down to 2w, keeping it near inflection ready for the next upward move. No geometric correction needed — direct tracking is exact.

This asymmetry also governs dormant buffer flow: upward ratchets consume dormant capital to extend the linear-k runway; downward adjustments reclaim it as the market contracts.

---

Validity Constraint

The floor R_new ≥ max(2·rA, 2·rB) on the downward rule ensures both sides remain at or below their inflection points at all times. This directly implies:

rA ≤ R_new/2   and   rB ≤ R_new/2
→  rA · rB ≤ R_new²/4
→  α·xᵏ · β·x⁻ᵏ ≤ R_new²/4
→  4αβ ≤ R_new²  ✅

For the upward rule, R_new = 4w²/R_old > R_old, so R only grows — validity is trivially preserved. The floor on the downward rule does the same work in the contracting direction.

---

Position After Ratchet

After ratchet, the dominant side sits at:

rA_new / (R_new/2) = w / (2w²/R_old) = R_old / (2w)

Since the ratchet fires only when w > R_old/2, we have R_old/(2w) < 1 — the position is always strictly below the new inflection point. The log-distance below the new inflection equals the log-distance above the old inflection at swap time.

---

Dormant Buffer

Each swap adjusts the dormant buffer:

• Upward (w > R_old/2): R_new = 4w²/R_old > R_old — buffer consumed, LP deploys capital to extend linear-k runway

• Downward (w < R_old/2): R_new = 2w < R_old — buffer reclaimed, LP recovers capital as dominant side retreats

• Floor max(2·rA, 2·rB) limits downward reclaim — a minimum R always covers both sides' current inflection points

---

Rate-Limited Ramping

The ratchet formula computes an ideal R_target assuming honest price discovery. Without rate-limiting, two classes of adversaries can exploit instant R adjustment:

1. Market manipulation. An attacker spikes the price in a single block to inflate R, then dumps — extracting value from the dormant buffer before the market corrects.

2. Insider trading. An informed trader who knows a large price move is imminent opens a position just before the move. With instant R adjustment, the ratchet immediately expands R to accommodate the move, granting the insider full linear-k payoff on the entire swing. The LP bears the cost — the dormant buffer is consumed to fund the insider's amplified gain.

By rate-limiting R, the insider's edge is capped: even if they time the entry perfectly, R_eff only ramps slowly toward R_target. The insider's position spends most of the price move in the asymptotic regime (sub-linear-k payoff), limiting their profit to what an uninformed trader would have earned over the same ramp period. The LP's dormant buffer is consumed gradually rather than all at once.

To defend against both, R does not jump to its target instantly. Instead, R_eff moves toward R_target at a maximum rate proportional to its own value — a constant velocity in log space.

Notation

Symbol	Meaning
R_target	Ratchet-formula output (the ideal R)
R_eff	Effective R used by the pool
Δt	Time elapsed since last R update
T_d	Doubling time — time for R_eff to double (or halve) at max rate

Mechanism

The rate is derived from the doubling time: μ = ln(2) / T_d.

On each swap, after computing R_target from the ratchet rule:

log_step = ln(R_target) − ln(R_eff)
max_step = ln(2) / T_d · Δt

if |log_step| ≤ max_step:
    R_eff = R_target                                    // within budget → apply fully

else:
    R_eff = R_eff · exp(sign(log_step) · max_step)      // clamp to max rate

The pool always uses R_eff — never R_target directly — for all payoff calculations, reserve splits, and validity checks.

Example

With T_d = 7 hours (R can at most double in 7 hours):

If the ratchet formula wants R to double but only 1 hour has passed:

max_step = ln(2) / 7 · 1 ≈ 0.099      // ~10% in log space per hour
R_eff moves from R to R · e^0.099 ≈ R · 1.104

R_eff grows by ~10.4% per hour, reaching R_target after exactly T_d = 7 hours.

Why Constant Rate In Log Space

Predictable convergence time. The time to reach any R_target is |ln(R_target/R_eff)| · T_d / ln(2) — deterministic and proportional to the log-distance. A 2× change always takes exactly T_d.

Scale-invariant. The rate cap is proportional to R_eff itself, so small and large pools have equivalent protection.

Log-space consistency. Operating in log-R space is consistent with the ratchet rule's log-price symmetry — a 2× increase and a 2× decrease in R require the same ramp time.

Linear convergence. Unlike exponential decay (which slows as it approaches the target), constant-rate ramping closes the gap at a steady pace. R_eff reaches R_target in finite time and stays there — no asymptotic tail.

Properties

Flash-loan resistance. A single-block manipulation (Δt ≈ 2s, T_d = 7hr) can move R_eff by at most ln(2)·2/25200 ≈ 0.005% — negligible regardless of how extreme the price spike is.

Organic moves pass through. A sustained price trend accumulates ramp budget linearly with time. Real market moves reach R_target within T_d while single-block attacks are throttled by a factor of >10,000×.

No queuing or state bloat. The mechanism only stores R_eff and t_last — no target queue, no ramp schedules. Each swap simply computes how far R_eff is allowed to move given elapsed time.

Preserves ratchet correctness. Once R_eff reaches R_target (which it does in finite time for any sustained price level), all ratchet properties — log-symmetry, linear-k payoff, validity constraints — hold exactly as derived above.

Scaling With K

Attacker profit from an R change scales linearly with k: a small price manipulation δ yields position gain proportional to k · δ. To keep the attacker's profit rate constant across leverage levels, the doubling time must scale linearly with k:

T_d = k · T₀

where T₀ is a base doubling time — a single pool parameter.

This scaling is also consistent from the recovery side: a price doubling requires R to grow by 2^(2k), taking time 2k · T_d to fully accommodate. Recovery time scales linearly with k in both directions.

Calibrating T₀

With T_d = k · T₀, the effective protection for a k=5 pool:

T₀	T_d (k=5)	1-block attack (2s)	1-min attack	1-hr insider edge
15 min	1.25 hr	~0.03%	~1.8%	~56%
1 hr	5 hr	~0.008%	~0.5%	~15%
7 hr	35 hr	~0.001%	~0.07%	~2%

Adaptation time for a price doubling (2k · T_d):

T₀	k=2	k=5	k=10
1 hr	4 hr	10 hr	20 hr

A practical default: T₀ = 1 hour, giving:

• k=2: T_d = 2 hr — R doubles in 2 hr, adapts to 2× price in ~8 hr

• k=5: T_d = 5 hr — R doubles in 5 hr, adapts to 2× price in ~20 hr

• k=10: T_d = 10 hr — R doubles in 10 hr, adapts to 2× price in ~40 hr

---

Behavior Across Lag Regimes

In practice, swaps do not fire at the exact inflection point — there is always some lag. This section analyzes how that lag accumulates over many ratchet cycles.

• ε — log-price distance below inflection immediately after a ratchet (the "headroom")

• δ — additional log-price drift into the asymptotic regime before the next swap fires (the "lag")

• N — number of ratchet cycles

Effect	Order	Behavior
First-order lag error	Zero	Reflect rule absorbs δ into next cycle's ε — cancels exactly
Per-cycle curvature drag	O(εδ/R)	Small when oscillations are tight
Amplitude growth	ε → ε + δ per cycle	Inflates if lag is systematic
Cumulative drag (reserve-space rule)	O(N²δ²/R)	Grows quadratically with cycles
Cumulative drag (log-price rule)	O(Nδ²/R)	Linear — first-order cancellation holds for any lag magnitude

The log-price formula eliminates the quadratic error term. For any lag size, the first-order cancellation holds because the curvature symmetry is exact at inflection in log-price coordinates. This is why the ratchet rule operates in log-price space rather than reserve space.

---

Comparison With Simpler Rules

Rule	Up formula	Down formula	Log-price symmetric	Works for large lag
No ratchet	R fixed	R fixed	❌	❌ dominant side stranded in asymptotic
Exact inflection	R_new = 2·rA	R_new = 2·rA	❌	❌ ε collapses to 0, no symmetry
Reserve-space reflect	R_old + 2ε	R_old - 2ε	❌	❌ O(N²δ²/R) cumulative drag
This design	4w²/R_old	2w	✅ up exact	✅ O(Nδ²/R) only

---

Summary

The ratchet formula R_new = 4w²/R_old is the unique rule that:

1. Keeps the dominant side oscillating symmetrically around the inflection point in log-price space

2. Achieves linear k leverage over long price moves, not compounding k

3. Preserves compound k downside when price moves against the dominant side

4. Remains self-consistent for any swap frequency — liquid or illiquid pools

5. Never violates pool validity constraints

6. Uses asymmetric rules for R adjustment — log-symmetric 4w²/R_old on the way up (curvature correction required), simple 2w on the way down (power regime, direct tracking is exact)

7. Rate-limits R changes via rate-limited ramping — R_eff moves toward R_target with doubling time T_d = k · T₀, scaling protection with leverage while letting sustained price moves converge in finite time

See also: Dynamic LP Strategy derives the break-even interest rate for an LP that implements this ratchet rule, and compares it against CEX perpetual funding rates.