From Coefficient Differentials to General Safe Operations

The Post-Trade Elasticity Model was initially developed around a small set of actions: Reserve Asset swaps, single-asset liquidity, proportional liquidity, and LP Token burns. Those actions were proved to satisfy the coefficient order individually and then composed into more elaborate transactions such as Surplus settlement.

The coefficient safety conditions reveal that the known actions are not the whole design space. They are particular paths through a larger region of admissible state transitions.

The differential of each coefficient identifies the local coefficient-order directions. Its exact finite form identifies the complete endpoint envelope induced by that order. The envelope admits direct permanent-reserve allocations, general LP under-minting, more general withdrawals, and multi-asset liquidity operations that need not share one common complement multiplier.

Execution pricing enters at a different layer. Coefficient order determines whether an endpoint is admissible. The execution rule determines whether value-flow balance reaches such an endpoint automatically. For positive scale elasticity, post-trade execution is the unique fixed linear execution rule that makes every finite fixed-coefficient reserve-only swap coefficient-safe without an additional admissibility check or protocol subsidy.

This article develops those results from first principles. It assumes exact arithmetic, positive reserves and scales, and homogeneous elasticities unless a section states otherwise.

The article describes the mathematical design space. The newly identified transitions are candidates for protocol operations, not claims about actions already exposed by the current implementation. A candidate becomes supported only after its consideration, account movements, rounding, and property tests have been specified and implemented.

1. Multiswap state

Let a Multiswap pool contain $n>1$ Reserve Assets. Reserve Asset $i$ has reserve

$$a_i>0,$$

scale

$$s_i>0,$$

and marginal price

$$P_i=\frac{s_i}{a_i}.$$

Scale elasticity and price elasticity satisfy

$$e_s+e_P=1.$$

For most of the article,

$$0<e_s<1$$

and therefore

$$0<e_P<1.$$

The elasticity coefficient of Reserve Asset $i$ is

$$\boxed{ c_i=\frac{s_i}{a_i^{e_s}}. }$$

Equivalently,

$$s_i=c_i a_i^{e_s}$$

and

$$P_i=c_i a_i^{-e_P}.$$

The LP Token has reserve $a_0$, scale

$$\boxed{ s_0=\sum_{i=1}^{n}s_i, }$$

price

$$P_0=\frac{s_0}{a_0},$$

and coefficient

$$c_0=\frac{s_0}{a_0^{e_s}}.$$

The coefficient partial order is

$$\boxed{c_i'\le c_i}$$

for every Reserve Asset, and

$$\boxed{c_0'\ge c_0}$$

for the LP Token.

Reserve Asset coefficients may remain fixed or contract. The LP Token coefficient may remain fixed or expand. These directions multiply under composition, so a sequence of individually compliant transitions remains compliant at its endpoint.

Coefficient order is a state-safety condition. It does not by itself specify consideration, token funding, ownership, or execution fairness. Those requirements remain part of every complete operation.

2. The coefficient differential

Differentiate

$$c_i=s_i a_i^{-e_s}.$$

The relative differential is

$$\boxed{ \frac{dc_i}{c_i} = \frac{ds_i}{s_i} - e_s\frac{da_i}{a_i}. }$$

Using

$$P_i=\frac{s_i}{a_i},$$

the same result can be written

$$\boxed{ dc_i = a_i^{-e_s} \left(ds_i-e_sP_i\,da_i\right). }$$

The local Reserve Asset condition is therefore

$$\boxed{ dc_i\le0 \iff ds_i\le e_sP_i\,da_i. }$$

For the LP Token, the safe direction is reversed:

$$\boxed{ dc_0\ge0 \iff ds_0\ge e_sP_0\,da_0. }$$

Together with

$$\boxed{ ds_0=\sum_{i=1}^{n}ds_i, }$$

these linear inequalities define a local coefficient-order transition cone at every positive state.

2.1 The price form

Because

$$\frac{ds_i}{s_i} = \frac{dP_i}{P_i} + \frac{da_i}{a_i},$$

the coefficient differential is also

$$\boxed{ \frac{dc_i}{c_i} = \frac{dP_i}{P_i} + e_P\frac{da_i}{a_i}. }$$

Reserve Asset safety becomes

$$\boxed{ \frac{dP_i}{P_i} \le -e_P\frac{da_i}{a_i}. }$$

The fixed-coefficient elastic curve is the boundary. A Reserve Asset price may move farther in the conservative direction, but not less.

2.2 Known operations inside the cone

The differential reproduces the coefficient direction of the currently studied primitives.

An ordinary fixed-coefficient swap leg satisfies

$$ds_i=e_sP_i\,da_i,$$

so

$$dc_i=0.$$

A nonparticipating Reserve Asset in a supported single-asset liquidity operation has

$$da_i=0$$

and

$$ds_i<0,$$

so

$$dc_i<0.$$

An LP Token burn without a reserve withdrawal has

$$da_0<0$$

and

$$ds_0=0,$$

so

$$dc_0>0.$$

The known actions occupy boundaries and interior directions of the same cone. That observation turns coefficient analysis from a verification tool into an operation-discovery tool.

3. The exact finite envelope

A differential condition is local. Checking it only at the initial state does not certify a finite jump. A finite operation must remain inside the cone along its path or satisfy the exact endpoint inequalities.

For every Reserve Asset,

$$\boxed{ \frac{s_i'}{s_i} \le \left(\frac{a_i'}{a_i}\right)^{e_s}. }$$

For the LP Token,

$$\boxed{ \frac{s_0'}{s_0} \ge \left(\frac{a_0'}{a_0}\right)^{e_s}. }$$

Equivalently, define the finite coefficient multiplier

$$\chi_i = \frac{c_i'}{c_i} = \frac{s_i'}{s_i} \left(\frac{a_i}{a_i'}\right)^{e_s}.$$

Then require

$$\chi_i\le1$$

for every Reserve Asset and

$$\chi_0\ge1$$

for the LP Token.

For a sequence indexed by $t$,

$$\frac{c_i^{\mathrm{final}}}{c_i^{\mathrm{initial}}} = \prod_t\chi_{i,t}.$$

The finite certificates therefore compose exactly.

4. New operations inside the finite coefficient-order envelope

The endpoint inequalities identify coefficient-order admissible state transitions more general than the originally supported formulas. They establish state admissibility, not a complete user-facing operation. Each construction still needs funded token movements and valid consideration.

4.1 Direct permanent-reserve allocation

Suppose Surplus contributes

$$\Delta a_A>0$$

directly to Reserve Asset $A$ without minting LP Tokens:

$$a_A'=a_A+\Delta a_A$$

and

$$a_0'=a_0.$$

Leave every other Reserve Asset unchanged. If only $A$ scale changes, the aggregate identity gives

$$s_0'-s_0=s_A'-s_A.$$

Because LP Token supply remains fixed, LP coefficient safety requires

$$s_0'\ge s_0$$

and therefore

$$s_A'\ge s_A.$$

Reserve Asset safety supplies the upper bound:

$$s_A' \le s_A \left(1+\frac{\Delta a_A}{a_A}\right)^{e_s}.$$

An entire interval is coefficient-safe:

$$\boxed{ s_A \le s_A' \le s_A \left(1+\frac{\Delta a_A}{a_A}\right)^{e_s}. }$$

At the lower endpoint, $s_A$ and $c_0$ remain fixed while $c_A$ contracts. At the upper endpoint, $c_A$ remains fixed while $c_0$ expands. Every point between them moves both coefficients in their permitted directions.

This transition could replace a compound

$$A\longrightarrow LP\longrightarrow\text{burn}$$

sequence with one direct protocol-funded reserve allocation.

Removing the explicit burn does not remove the allocation economics. Supplying reserves without issuing LP Tokens benefits whoever owns the existing LP Token supply. Temporary liquidity can still enter before the allocation and leave afterward to capture part of that benefit. Coefficient safety and allocation policy remain separate questions.

4.2 General LP under-minting

Suppose a funded reserve operation changes total LP scale from $s_0$ to

$$s_0'>s_0.$$

LP coefficient safety requires

$$\frac{s_0'}{s_0} \ge \left(\frac{a_0'}{a_0}\right)^{e_s}.$$

Conditional on every Reserve Asset satisfying its own endpoint coefficient inequality, the maximum final LP Token supply permitted by the LP Token coefficient condition is

$$\boxed{ a_0' \le a_0 \left(\frac{s_0'}{s_0}\right)^{1/e_s}. }$$

The maximum LP Token mint permitted by that condition is therefore

$$\boxed{ da_0^{\max} = a_0 \left[ \left(\frac{s_0'}{s_0}\right)^{1/e_s} -1 \right]. }$$

Minting exactly $da_0^{\max}$ preserves $c_0$. Minting less expands $c_0$. Minting nothing produces permanent reserve growth. Minting the maximum and then burning part of the result is endpoint-equivalent to under-minting initially.

This gives a continuous design space between ordinary liquidity issuance and permanent liquidity.

4.3 General withdrawals

If an operation reduces total Reserve Asset scale so that

$$s_0'<s_0,$$

then LP coefficient safety again requires

$$a_0' \le a_0 \left(\frac{s_0'}{s_0}\right)^{1/e_s}.$$

The operation must burn at least

$$\boxed{ -da_0 \ge a_0 \left[ 1- \left(\frac{s_0'}{s_0}\right)^{1/e_s} \right]. }$$

Conditional on every Reserve Asset satisfying its own endpoint coefficient inequality, this inequality identifies the minimum LP Token burn permitted by the LP Token coefficient condition. It does not determine the fair reserve amounts owed for the LP Tokens burned. That consideration rule must be derived separately.

4.4 General multi-asset liquidity

The common $\lambda$ construction used in supported liquidity actions is one efficient sufficient condition. It contracts every complement Reserve Asset coefficient by one multiplier

$$0<\lambda<1.$$

The finite envelope permits a larger region. A multi-asset operation may use different contractions for different Reserve Assets as long as

$$c_i'\le c_i$$

for every Reserve Asset,

$$c_0'\ge c_0,$$

and

$$s_0'=\sum_{i=1}^{n}s_i'.$$

The common $\lambda$ remains valuable because it can update an entire complement set through a shared coefficient rather than touching every asset. The differential analysis shows that this computationally efficient construction is a subset of a larger mathematically admissible region.

5. Admissibility is not consideration

An authorized transition could remain inside coefficient order while transferring value away from existing LP Token holders. Coefficient contraction is permitted state motion; it is not proof that the party receiving its economic benefit paid for it.

A complete operation must specify:

• which account funds every positive reserve change;
• which account receives every negative reserve change;
• the execution or other consideration rule;
• LP Token issuance or destruction;
• aggregate scale consistency;
• reserve and scale positivity;
• pool-favorable rounding;
• user minimum receive or maximum pay conditions.

The design rule is

$$\boxed{ \begin{aligned} \text{safe operation} ={}&\text{coefficient-safe endpoint}\\ &+\text{valid consideration}\\ &+\text{funded token movement}\\ &+\text{aggregate consistency}. \end{aligned} }$$

The differential discovers candidates. The finite inequalities certify endpoints. The settlement rule establishes who paid whom.

6. Execution price as a separate layer

Let $P_i$ be the opening marginal price of leg $i$, $dP_i$ its marginal-price change, and

$$P_i'=P_i+dP_i$$

its final marginal price.

Consider the linear execution rule

$$\boxed{ P_i^{\mathrm{exec}} = P_i+k\,dP_i. }$$

Thus:

• $k=0$ is opening-price execution;
• $k=\tfrac12$ is endpoint-midpoint execution;
• $k=1$ is post-trade execution.

The parameter $k$ describes execution price. It does not describe an intermediate reserve point.

For a fixed-coefficient Reserve Asset leg, define

$$r_i=\frac{da_i}{a_i}, \qquad r_i>-1.$$

Then

$$\frac{P_i'}{P_i} = (1+r_i)^{-e_P}$$

and

$$dP_i = P_i \left[(1+r_i)^{-e_P}-1\right].$$

The execution price becomes

$$\boxed{ P_i^{\mathrm{exec}} = P_i \left[ 1+k\left((1+r_i)^{-e_P}-1\right) \right]. }$$

The signed execution value flow is

$$\boxed{ \Sigma_i^{(k)} = da_iP_i^{\mathrm{exec}} = s_i r_i \left[ 1+k\left((1+r_i)^{-e_P}-1\right) \right]. }$$

The general execution value-flow identity couples the LP Token leg to the Reserve Asset legs:

$$\boxed{ \Sigma_0^{(k)} = \sum_{i=1}^{n}\Sigma_i^{(k)}. }$$

An ordinary reserve-only swap has no LP Token flow, so

$$\Sigma_0^{(k)}=0$$

and the general identity reduces to

$$\sum_{i=1}^{n}\Sigma_i^{(k)}=0.$$

At $k=1$,

$$\Sigma_i^{(1)} = s_i\frac{r_i}{(1+r_i)^{e_P}},$$

which is the Post-Trade Elasticity value flow.

7. Exact scale decomposition for general $k$

The exact discrete product rule is

$$ds_i = da_i(P_i+dP_i) + a_i\,dP_i.$$

The final-price value of the traded reserve can be decomposed as

$$da_i(P_i+dP_i) = da_i(P_i+k\,dP_i) + (1-k)da_i\,dP_i.$$

Therefore

$$\boxed{ ds_i = \Sigma_i^{(k)} + a_i\,dP_i + (1-k)da_i\,dP_i. }$$

Summing the Reserve Asset scale changes and applying the general value-flow identity gives

$$\boxed{ ds_0 = \Sigma_0^{(k)} + \sum_{i=1}^{n}a_i\,dP_i + (1-k)\sum_{i=1}^{n}da_i\,dP_i. }$$

For an ordinary reserve-only swap, $\Sigma_0^{(k)}=0$, so this reduces to

$$\boxed{ ds_0 = \sum_{i=1}^{n}a_i\,dP_i + (1-k)\sum_{i=1}^{n}da_i\,dP_i. }$$

For every nonzero fixed-coefficient leg,

$$da_i\,dP_i<0.$$

When the pool receives more of an asset, its marginal price falls. When it sends an asset, its marginal price rises.

The term

$$(1-k)da_i\,dP_i$$

is therefore negative for $k<1$, zero for $k=1$, and positive for $k>1$. Execution before the final price introduces a negative difference between the execution value and the final marginal value of the traded reserve.

At $k=1$, the traded reserve is valued at its final marginal price and this difference vanishes exactly.

8. The exact LP coefficient condition

A reserve-only swap leaves $a_0$ fixed. LP coefficient safety is therefore equivalent to

$$ds_0\ge0.$$

For a computed general-$k$ swap, define opening-reserve revaluation

$$G=\sum_{i=1}^{n}a_i\,dP_i$$

and execution-to-final-price shortfall

$$L=-\sum_{i=1}^{n}da_i\,dP_i>0.$$

Then

$$\boxed{ ds_0=G-(1-k)L. }$$

The exact LP coefficient test is

$$\boxed{ G-(1-k)L\ge0. }$$

When $L>0$, the computed endpoint is safe precisely when

$$\boxed{ k\ge1-\frac{G}{L}. }$$

This threshold is transaction-specific. The endpoint, and therefore $G/L$, depends on the pool state, the full trade vector, and the execution rule used to solve value-flow balance.

Post-trade execution is special because it eliminates the shortfall term rather than attempting to offset it.

9. The small-trade boundary

For small relative reserve changes,

$$\Sigma_i^{(k)} = s_i \left[ r_i-ke_Pr_i^2+O(r_i^3) \right].$$

Value-flow conservation implies

$$\sum_{i=1}^{n}s_i r_i = ke_P\sum_{i=1}^{n}s_i r_i^2 + O(r^3).$$

Meanwhile,

$$ds_i = s_i \left[ e_s r_i - \frac{e_se_P}{2}r_i^2 + O(r_i^3) \right].$$

Combining the expansions gives

$$\boxed{ ds_0 = e_se_P \left(k-\frac12\right) \sum_{i=1}^{n}s_i r_i^2 + O(r^3). }$$

For $0<e_s<1$:

• if $k<\tfrac12$, every sufficiently small nontrivial swap decreases $c_0$;
• if $k>\tfrac12$, sufficiently small swaps increase $c_0$;
• if $k=\tfrac12$, the quadratic term vanishes and higher-order asymmetry determines the sign.

Midpoint execution is second-order neutral. It is not generally safe.

10. The global finite-trade result for $0<e_s<1$

Post-trade execution makes every finite fixed-coefficient reserve-only swap LP-coefficient-safe.

For $k=1$, each leg admits the exact decomposition

$$ds_i=e_s\Sigma_i^{(1)}+\mathcal D_i,$$

where

$$\mathcal D_i = s_i \left[ (1+r_i)^{e_s} -1 -e_s r_i(1+r_i)^{-e_P} \right] \ge0.$$

To see the sign directly, set

$$x_i=1+r_i>0$$

and divide by $s_i$. The remaining function is

$$h(x_i) = x_i^{e_s}-1-e_s(x_i-1)x_i^{e_s-1}.$$

It satisfies

$$h(1)=0$$

and

$$h'(x_i) = e_se_P(x_i-1)x_i^{e_s-2}.$$

Thus $h$ decreases toward zero on $(0,1)$ and increases away from zero on $(1,\infty)$. Its unique minimum is $h(1)=0$, so $\mathcal D_i>0$ for every nonzero leg.

The inequality is strict for every nonzero leg when $0<e_s<1$. For a reserve-only swap, the general value-flow identity specializes to

$$\Sigma_0^{(1)}=0 \quad\Longrightarrow\quad \sum_{i=1}^{n}\Sigma_i^{(1)}=0,$$

the LP scale change is

$$\boxed{ ds_0=\sum_{i=1}^{n}\mathcal D_i>0 }$$

for every nontrivial finite swap.

Every Reserve Asset coefficient remains fixed, $a_0$ remains fixed, and $c_0$ increases.

10.1 Why every fixed $k<1$ eventually fails

For any fixed

$$0\le k<1,$$

there are finite pool states and swaps with

$$ds_0<0.$$

Consider a pay leg $A$ with

$$r_A\longrightarrow\infty.$$

Its final marginal price approaches zero. For $k<1$, its execution price approaches

$$P_A^{\mathrm{exec}} \longrightarrow (1-k)P_A,$$

so its positive execution value grows linearly:

$$\Sigma_A^{(k)} \sim (1-k)s_A r_A.$$

Its scale, however, grows only as

$$s_A' \sim s_A r_A^{e_s},$$

which is sublinear because $e_s<1$.

Choose a receive leg with a fixed relative reserve reduction in $(-1,0)$ and choose its positive initial scale to balance the pay leg's execution value. Its scale must then grow linearly with $r_A$. The receive leg's negative scale change also grows linearly and eventually dominates the pay leg's sublinear positive scale change.

Thus

$$ds_0<0$$

for sufficiently large finite $r_A$.

Within the ordinary range

$$0\le k\le1,$$

post-trade execution is therefore the unique fixed linear execution rule that automatically makes every admissible finite reserve-only swap coefficient-safe:

$$\boxed{k=1.}$$

Other values of $k$ can be used with state-dependent trade limits, an explicit endpoint coefficient check, or a compensating protocol-funded action.

10.2 The domain of $k>1$

When $k>1$, the execution-to-final-price term is pool-favorable. But a sufficiently large pay leg receives a negative execution price.

Positivity requires

$$1+k\left((1+r_i)^{-e_P}-1\right)>0.$$

For a pay leg $r_i>0$,

$$\boxed{ r_i < \left(\frac{k}{k-1}\right)^{1/e_P}-1. }$$

Every $k>1$ therefore imposes a hard finite input domain. It cannot support arbitrary positive reserve inputs.

11. Design consequences

The analysis changes the operation-design process.

Ordinary swaps

Use post-trade execution when every finite trade should be admitted without a separate LP coefficient or global-potential check.

For $k<1$, either:

• compute the endpoint and require $c_0'\ge c_0$ when $e_s>0$;
• restrict trade size and pool states to a domain where the relevant condition is proved;
• or add an explicit protocol-funded compensation such as sufficient LP Token burning or external reserve contribution.

Reserve coefficient contraction cannot repair a negative $ds_0$. It reduces total Reserve Asset scale further. Compensation must act through LP Token supply, external value, or another permitted transition that increases the LP coefficient.

Liquidity operations

Treat the common-$\lambda$ formulas as efficient sufficient constructions, not the definition of all coefficient-order admissible liquidity. Use the finite coefficient-order envelope to discover broader candidates, then derive consideration independently.

Permanent liquidity

Direct protocol-funded reserve allocation and LP under-minting form one continuum. Burning after a maximum mint is only one endpoint implementation. Every choice must address temporary-liquidity allocation capture.

12. Implementation and property-test requirements

For every primitive and randomized composition with $0<e_s<1$, verify:

Positive state

$$a_i'>0, \qquad s_i'>0.$$

Aggregate scale

$$\boxed{ s_0'=\sum_{i=1}^{n}s_i'. }$$

Reserve Asset coefficient direction

$$\boxed{ \frac{s_i'}{s_i} \left(\frac{a_i}{a_i'}\right)^{e_s} \le1 }$$

for every Reserve Asset.

LP Token coefficient direction

$$\boxed{ \frac{s_0'}{s_0} \left(\frac{a_0}{a_0'}\right)^{e_s} \ge1. }$$

Execution value flow

For every value-flow-coupled action, require

$$\boxed{ \Sigma_0^{(k)} = \sum_{i=1}^{n}\Sigma_i^{(k)}. }$$

For reserve-only swaps, $\Sigma_0^{(k)}=0$, so also verify

$$\boxed{ \sum_{i=1}^{n}\Sigma_i^{(k)}=0. }$$

The test suite should include:

• direct permanent-reserve allocations across their full safe interval;
• LP Token mints below, at, and above the coefficient-safe maximum;
• generalized withdrawals with burns below, at, and above the required minimum;
• heterogeneous multi-asset coefficient contractions;
• general-$k$ swaps near the small-trade boundary;
• asymmetric finite swaps demonstrating failure for every tested $k<1$;
• $k>1$ execution-price positivity boundaries;
• integer conversion and pool-favorable rounding.

Property tests should report every componentwise coefficient margin. Aggregate diagnostics must not replace exact token conservation, funded account movements, or user protection checks.

Conclusion

The coefficient differential

$$\boxed{ \frac{dc_i}{c_i} = \frac{ds_i}{s_i} -e_s\frac{da_i}{a_i} }$$

reveals a local cone of coefficient-order admissible Multiswap state transitions. Its finite form

$$\boxed{ \frac{s_i'}{s_i} \left(\frac{a_i}{a_i'}\right)^{e_s} \le1 }$$

for Reserve Assets, together with the reversed LP Token inequality, reveals a much larger operation space than the initially studied primitives.

Direct permanent-reserve allocation, LP under-minting, generalized withdrawal, and heterogeneous multi-asset liquidity are all candidate operations inside that space. Coefficient admissibility does not establish fair consideration, so each still needs funded movement, settlement, LP accounting, and user protection.

Execution price determines whether an ordinary swap reaches the admissible region automatically. For

$$0<e_s<1,$$

post-trade execution

$$P^{\mathrm{exec}}=P+dP$$

is the unique fixed linear execution rule in $0\le k\le1$ that makes every finite fixed-coefficient reserve-only swap LP-coefficient-safe.

The emerging hierarchy is

$$\boxed{ \begin{aligned} \text{local discovery}&:\quad dc_i,\\ \text{finite admissibility}&:\quad \chi_i,\\ \text{execution coupling}&:\quad \Sigma_0^{(k)}=\sum_{i=1}^{n}\Sigma_i^{(k)},\\ \text{complete operation}&:\quad \text{funding and consideration}. \end{aligned} }$$

This framework does not force every coefficient-order admissible operation into one invariant or one implementation pattern. It separates the questions cleanly enough to search the wider design space without losing the coefficient-order boundary.