Portfolio Preferences, Relative Entropy, and Post-Trade Elasticity

A liquidity pool is a portfolio. Its reserve and price updates define an investment strategy: which assets it accumulates, which it releases, and the terms on which it changes its composition.

Multiswap's post-trade elasticity model expresses that strategy with one elasticity parameter, while retaining a rich accounting and safety structure. A natural extension is to give the portfolio an explicit preferred composition. The question is whether that preference can enter through a coherent valuation rule, preserve the post-trade framework, and strengthen the response to scarce inventory.

This article develops one such candidate. Its central result is an exact identity: the composition-sensitive part of portfolio valuation equals reference-priced holdings discounted by relative entropy from the preferred composition. The resulting prices favor acquiring underrepresented assets, retain exact invariance under uniform reserve scaling, and develop stronger elasticity near depletion.

Status and assumptions: This is a proposed mathematical extension, not the current Solidity implementation or an adopted replacement for PTE. We assume at least two assets, positive reserves and reference prices, positive reference fractions, fixed reference parameters, exact arithmetic, and zero-fee post-trade execution. The numerical examples are controlled calculations, not historical simulations. The safety result below concerns a fixed valuation surface; reference updates and protocol implementation require separate analysis.

1. Begin with the portfolio and its accounting

Reserve Asset $i$ has amount $a_i$, scale $s_i$, and pool price $P_i$:

$$P_i=\frac{s_i}{a_i}, \qquad s_0=\sum_i s_i.$$

Scale measures value in the pool's common internal numeraire. For an LP Token supply $a_0>0$, the derived LP Token price is $P_0=s_0/a_0$. The pool's current value fractions are $w_i=s_i/s_0$. They are derived quantities, not independent state variables.

A trade changes reserves by $da_i$, giving $a_i'=a_i+da_i$. A positive change supplies the pool; a negative change withdraws from it. A closed post-trade swap satisfies

$$\sum_i da_iP_i'=0.$$

Every leg is valued at its ending price. Token conservation and the underlying same-token ledger entries remain separate requirements; this equation expresses the value-flow closure of the reserve exchange.

In the fixed-elasticity PTE model,

$$s_i=c_i a_i^{e_s}, \qquad P_i=c_i a_i^{-e_P}, \qquad e_P=1-e_s.$$

For swaps with fixed coefficients, the finite updates are

$$\frac{s_i'}{s_i}=(1+r_i)^{e_s}, \qquad \frac{P_i'}{P_i}=(1+r_i)^{-e_P}, \qquad r_i=\frac{da_i}{a_i}>-1.$$

The coefficients describe the price surface; they need not be stored to apply the updates to reserves and scales. This is the form used by the repository's PTE quote math.

For $e_P>0$, PTE prices already diverge as an individual reserve approaches zero. Small $e_P$ makes that response weak until depletion is extremely close. Post-trade execution reinforces it by valuing the entire withdrawal at the elevated ending price. The extension considered here changes the state-dependent price response as well as retaining post-trade execution.

2. Express an investment mandate through a reference portfolio

Choose positive reference reserves $a_i^*$ and reference prices $P_i^*$, and derive

$$s_i^*=a_i^*P_i^*, \qquad s_0^*=\sum_i s_i^*, \qquad w_i^*=\frac{s_i^*}{s_0^*}.$$

The reference fractions express the preferred portfolio composition. They are fixed parameters derived from the reference scales, rather than an instruction to overwrite current weights after every trade. Reference prices provide a fixed relative-value anchor. They are not assumed to follow an external oracle during the calculations below.

Two valuations express complementary objectives. First, the value of the holdings at reference prices is

$$V(a)=\sum_i P_i^*a_i.$$

This values each additional unit at the same reference price regardless of inventory. Second, define the composition-sensitive valuation

$$G(a)=s_0^*\prod_i \left(\frac{a_i}{a_i^*}\right)^{w_i^*}.$$

The finite portfolio rule assigns each asset's reserve multiplier its preferred reference fraction:

$$\boxed{\frac{G'}{G} =\prod_i\left(\frac{a_i'}{a_i}\right)^{w_i^*}}.$$

This expresses a multiplicative investment policy. A finite proportional change in an asset contributes to the valuation multiplier with exponent $w_i^*$. Successive updates compose exactly because each intermediate reserve amount cancels between its adjacent ratios. The fractions sum to one, so a uniform reserve multiplier produces that same multiplier in $G$.

Starting from $G(a^*)=s_0^*$ and applying this finite rule directly from the reference state gives the stated geometric valuation. This is a choice of portfolio policy with an exact composition law; specifying a preferred portfolio alone does not uniquely determine it.

The proposed combined valuation is

$$\boxed{F(a)=(1-\alpha)V(a)+\alpha G(a)}, \qquad 0\le\alpha\le1.$$

This is an explicit policy choice to combine inventory-independent reference valuation with composition-sensitive valuation. It is an additive combination of objectives, not a claim that two separately traded sub-portfolios exist. The preferred composition motivates the construction; it does not uniquely compel this particular blend.

3. Define scales and execute at the post-trade state

Allocate scale at each state according to the combined portfolio policy:

$$\boxed{s_i=(1-\alpha)P_i^*a_i+\alpha w_i^*G}.$$

The reference-valued contribution follows actual holdings; the geometric contribution assigns its scale according to the preferred fractions. Pool prices follow from the native accounting relation:

$$\boxed{P_i=\frac{s_i}{a_i} =(1-\alpha)P_i^*+\alpha w_i^*\frac{G}{a_i}}.$$

For a proposed finite trade, evaluate the complete ending state:

$$\begin{aligned} a_i'&=a_i(1+r_i),\\ G'&=G\prod_j(1+r_j)^{w_j^*},\\ s_i'&=(1-\alpha)P_i^*a_i'+\alpha w_i^*G',\\ P_i'&=s_i'/a_i'. \end{aligned}$$

The trade must satisfy

$$\boxed{\sum_i da_iP_i'=0}.$$

These finite equations determine execution. The price-response and safety results below follow directly from these ending-state expressions and weighted mean inequalities.

Summing gives the accounting identity

$$\boxed{s_0=\sum_i s_i=F(a)}.$$

At the reference state, $G=s_0^*$ and $P_i=P_i^*$. At a uniform multiple $a_i=\lambda a_i^*$, $G=\lambda s_0^*$ while prices remain $P_i=P_i^*$.

Every price depends on the entire reserve vector through $G$. Thus a trade can change the price and scale of an asset whose reserve amount did not change. This coupling is a substantive difference from the separable PTE update and an implementation obligation for any future protocol version.

4. Exact scaling symmetry and finite composition

Both valuations are homogeneous of degree one:

$$G(\lambda a)=\lambda G(a), \qquad F(\lambda a)=\lambda F(a), \qquad \lambda>0.$$

Consequently,

$$\boxed{P_i(\lambda a)=P_i(a)}, \qquad s_i(\lambda a)=\lambda s_i(a).$$

Reserve prices are exactly unchanged, with fixed reference parameters and without a gauge transformation. If LP supply also scales by $\lambda$, its price is unchanged too. If supply stays fixed, the enlarged reserves increase value per LP Token.

Even uniformly scaling all reference reserves leaves $G$ unchanged: the increase in $s_0^*$ cancels the reference denominators because $\sum_i w_i^*=1$. The reference introduces composition and relative-price preferences, not an absolute liquidity size. Multiplying every reference price by a common factor instead multiplies all scales and prices by that factor, preserving relative prices.

The finite geometric update is

$$\frac{G'}{G}=\prod_j(1+r_j)^{w_j^*}.$$

Therefore

$$P_i'=(1-\alpha)P_i^* +\alpha w_i^*\frac{G}{a_i} \frac{\prod_j(1+r_j)^{w_j^*}}{1+r_i}.$$

With a fixed reference, endpoint price ratios compose exactly through successive states. This is a full-state composition property, not the original per-asset power law in $1+r_i$. Finite execution can still depend on how a trade is split because each transaction uses its own ending prices.

Resetting the reference to the pre-trade reserves and prices before every transaction changes the strategy. Every trade would start at its new reference composition, restoring the same reference price-response rule each time. The rule would lose a persistent measure of accumulated departure from the original portfolio. Even if a reset matches current prices, it changes the valuation surface, so the fixed-surface safety proof below would not establish safety of the reset policy.

5. Relative entropy emerges from the portfolio valuation

Define the current composition at reference prices:

$$q_i=\frac{P_i^*a_i}{V}.$$

The $q_i$ are derived reference-valued fractions. The $w_i=s_i/F$ are derived fractions at current pool prices. Keeping these valuations distinct is essential.

From the definitions,

$$\frac{a_i}{a_i^*} =\frac{V}{s_0^*}\frac{q_i}{w_i^*}.$$

Substitution into $G$, using $\sum_iw_i^*=1$, gives

$$\frac{G}{V} =\prod_i\left(\frac{q_i}{w_i^*}\right)^{w_i^*}.$$

The Kullback–Leibler divergence, also called relative entropy, is

$$D_{\mathrm{KL}}(w^*\Vert q) =\sum_iw_i^*\log\frac{w_i^*}{q_i}.$$

It is dimensionless, nonnegative, and zero exactly when $q=w^*$. It diverges when an asset with positive preferred fraction disappears. Its arguments have different roles: generally $D_{\mathrm{KL}}(w^*\Vert q)$ and $D_{\mathrm{KL}}(q\Vert w^*)$ differ. It is not a symmetric geometric distance. All logarithms here are natural logarithms.

We obtain the exact identities

$$\boxed{G=V e^{-D_{\mathrm{KL}}(w^*\Vert q)}}$$

and

$$\boxed{ F=V\left[1-\alpha +\alpha e^{-D_{\mathrm{KL}}(w^*\Vert q)}\right]. }$$

At a fixed reference value $V$, the preferred composition maximizes $F$ whenever $\alpha>0$. Departure from the mandate discounts the composition-sensitive valuation. For $0<\alpha<1$ the total discount is bounded by $\alpha V$, even though marginal scarcity prices can become unbounded.

Relative entropy was not introduced as an extra modeling assumption. It follows algebraically from the geometric valuation. The financial interpretation is a preference-adjusted portfolio valuation; $F$ is also the pool's internally marked total scale. It is not an assertion that the portfolio's external dollar market value has suffered the same discount.

6. How the investment mandate enters quotes

Prices can now be written as

$$\frac{P_i}{P_i^*} =(1-\alpha) +\alpha e^{-D_{\mathrm{KL}}(w^*\Vert q)} \frac{w_i^*}{q_i}.$$

For $\alpha>0$,

$$\frac{q_i}{w_i^*}<\frac{q_j}{w_j^*} \quad\Longrightarrow\quad \frac{P_i}{P_i^*}>\frac{P_j}{P_j^*}.$$

An asset that is relatively underrepresented receives a higher price adjustment than an asset that is relatively overrepresented. This improves the terms for supplying scarce inventory and raises the terms for withdrawing it, relative to the reference price ratio. External prices still determine whether an actual trader finds those terms attractive.

The current pool fractions are

$$w_i= \frac{(1-\alpha)Vq_i+\alpha Gw_i^*} {(1-\alpha)V+\alpha G}.$$

Thus current pool fractions lie between the reference-valued composition and the preferred fractions. At $\alpha=1$, $w_i=w_i^*$ exactly. At $\alpha=0$, prices are fixed and $w_i=q_i$.

For intermediate $\alpha$, this is a soft preference. The geometric term's fraction of total scale, $\alpha G/F$, decreases as relative entropy increases. The model does not hold current weights tightly around the preference, or promise that market trading will restore them.

For example, $w^*=(0.5,0.5)$, $q=(0.1,0.9)$, and $\alpha=0.01$ give

$$D_{\mathrm{KL}}=\log(5/3), \qquad G/V=0.6,$$

and current pool fractions of approximately $10.24\%$ and $89.76\%$. Substantial inventory movement remains possible.

7. Exact finite price response and depletion

Divide the ending price by its opening value, using $s_i=a_iP_i$:

$$\boxed{ \frac{P_i'}{P_i} =1+\frac{\alpha w_i^*G}{s_i} \left[ \frac{\prod_j(1+r_j)^{w_j^*}}{1+r_i}-1 \right]. }$$

This is an exact finite price multiplier. The opening composition determines the geometric share $\alpha w_i^*G/s_i$, and the complete vector of reserve multipliers determines the ending response. Relative price changes are obtained by dividing the multiplier for one asset by that for another.

At the reference state, the geometric share of each price is $\alpha$, so

$$\frac{P_i'}{P_i^*} =1-\alpha+ \alpha\frac{\prod_j(1+r_j)^{w_j^*}}{1+r_i}.$$

To compare a specified finite relative-price change with PTE, define its finite log-ratio elasticity:

$$e_{P,ij}^{\mathrm{finite}} =-\frac{\log\!\left[(P_i'/P_j')/(P_i/P_j)\right]} {\log\!\left[(1+r_i)/(1+r_j)\right]}.$$

This is defined when the denominator is nonzero. It measures the price response between two actual states; it is not a derivative or an additional model parameter. For fixed-elasticity PTE it equals $e_P$ exactly. Here it depends on the opening composition and the chosen finite changes. For a many-asset update, other reserves also affect the result. A pair with an unchanged reserve ratio has an undefined log-ratio elasticity under this definition, even when both of its prices are well defined.

An especially transparent finite state comparison changes only reserve $i$, holding every other reserve fixed. The exact response reduces to

$$\frac{P_i'}{P_i} =1-\frac{\alpha w_i^*G}{s_i} +\frac{\alpha w_i^*G}{s_i}(1+r_i)^{-(1-w_i^*)}.$$

A one-reserve state comparison diagnoses the price rule; by itself it is not a closed swap. The expression shows a constant contribution and a scarcity contribution with exponent $1-w_i^*$. For ten equal reference fractions that exponent is $0.9$. Small $\alpha$ reduces its starting contribution but does not remove its dominance sufficiently close to depletion.

If $a_i'$ approaches zero while other reserves remain positive and fixed, then, for $\alpha>0$,

$$G'\propto (a_i')^{w_i^*}, \qquad P_i'\sim (a_i')^{-(1-w_i^*)}.$$

For $0<\alpha<1$, the other prices approach their positive floors $(1-\alpha)P_j^*$ along this particular state comparison. Simultaneous depletion of several assets must be assessed separately. Uniform scaling of every reserve leaves all reserve prices unchanged.

The finite execution rule supplies the trading consequence. In a pair swap with a fixed finite positive input, the received asset's endpoint-valued withdrawal diverges near its depletion, while the paid leg's endpoint value stays bounded. Complete depletion therefore cannot satisfy that finite pair trade's closure.

8. Finite composition replaces a separable elasticity rule

Fixed-elasticity PTE gives each asset its own price multiplier $(1+r_i)^{-e_P}$. The reference model instead couples the reserve multipliers through $G'/G=\prod_j(1+r_j)^{w_j^*}$. A constant per-asset power exponent is therefore not required to describe its finite price updates.

For three successive positive reserve vectors $a$, $a'$, and $a''$, the geometric update composes exactly:

$$\prod_i\left(\frac{a_i'}{a_i}\right)^{w_i^*} \prod_i\left(\frac{a_i''}{a_i'}\right)^{w_i^*} =\prod_i\left(\frac{a_i''}{a_i}\right)^{w_i^*}.$$

The full reference state determines prices uniquely. This finite consistency statement does not assert that two executed trades produce the same ending reserves as one combined trade: their separate post-trade closure equations still determine their execution.

9. Exact finite scale production

The safety calculation uses the weighted arithmetic–geometric mean inequality. For positive numbers and positive fractions summing to one, their weighted arithmetic mean is at least their weighted geometric mean, with equality exactly when all the numbers coincide.

Directly expand the ending-price value flow:

$$\sum_i P_i'da_i =(1-\alpha)(V'-V) +\alpha G'\sum_iw_i^*\frac{r_i}{1+r_i}.$$

Subtract this from $F(a')-F(a)$. Using $\sum_iw_i^*=1$ gives

$$\begin{aligned} F(a')-F(a)-\sum_iP_i'da_i &=\alpha\left[G'\sum_i\frac{w_i^*}{1+r_i}-G\right]. \end{aligned}$$

Finally substitute the exact geometric update:

$$\boxed{ \begin{aligned} &F(a')-F(a)-\sum_iP_i'da_i\\ &\quad=\alpha G'\left[ \sum_i\frac{w_i^*}{1+r_i} -\prod_i(1+r_i)^{-w_i^*} \right]\ge0. \end{aligned} }$$

The bracket is precisely the weighted arithmetic–geometric mean gap of the inverse reserve multipliers. This identity is exact for every pair of positive states with the same reference parameters, whether or not those states satisfy closed-swap execution.

For a closed post-trade swap, the boundary value flow vanishes. Since $s_0=F$, the mean gap becomes the exact production of total scale:

$$\boxed{ s_0'-s_0 =\alpha G'\left[ \sum_i\frac{w_i^*}{1+r_i} -\prod_i(1+r_i)^{-w_i^*} \right]\ge0. }$$

For $\alpha>0$, equality requires every reserve multiplier to coincide. A common multiplier $\lambda$ gives boundary flow $(\lambda-1)F(a)$, so a closed swap then requires $\lambda=1$. Every nontrivial closed swap therefore increases total scale strictly. At $\alpha=0$, a closed swap conserves total scale exactly.

For fixed LP supply $a_0$, this also gives $P_0'=s_0'/a_0\ge P_0$. The exact logarithmic change per LP Token is

$$\log\frac{F(a')/a_0}{F(a)/a_0} =\log\frac{s_0'}{s_0}\ge0.$$

This supplies a finite certificate for the fixed-reference backing result discussed in the projective Bregman safety law. No differentiation or integration is needed for the certificate above. Multiplying all reference prices by a common positive factor scales both sides of the scale-production identity equally and leaves its sign and the logarithmic ratio unchanged.

The certificate does not establish the original PTE componentwise coefficient order for the coupled model, protection against losses at external market prices, or safety of changing reference parameters. Deposits, withdrawals, surplus, and rounding rules require their own accounting and implementation checks.

10. Restoring versus worsening trades

The portfolio mandate suggests a connection to the earlier target-based kappa-stableness idea: favor trades that restore inventory. We can check the finite execution behavior without assuming that a favorable quote means zero price impact.

Use two illustrative tokens, Reserve Asset 1 and Reserve Asset 2, with $a_1^*=a_2^*=1$, $P_1^*=P_2^*=1$, and $\alpha=0.01$. Starting reserves are $a_1=0.2$ and $a_2=1.8$. The resulting prices and scales are

$$(P_1,P_2)=(1.02,\;0.993333\ldots),$$ $$(s_1,s_2)=(0.204,\;1.788), \qquad s_0=1.992.$$

Each comparison starts from this same state. Restoring trades supply Asset 1 and withdraw Asset 2; worsening trades do the reverse. Input sizes have equal value at the starting pool prices. The output is solved from $\sum_iP_i'da_i=0$. All restoring examples stop before the preferred composition.

For a single pay and receive leg, measure execution loss against the starting quote by

$$10^4\left(1- \frac{|da_{\mathrm{rec}}|P_{\mathrm{rec}}} {da_{\mathrm{pay}}P_{\mathrm{pay}}}\right).$$

The units are basis points. This is neither the ending spot-price change nor a comparison against an external market quote.

Input value / initial pool value	Restoring execution loss	Worsening execution loss
0.1%	1.77 bp	1.79 bp
1%	16.75 bp	19.01 bp
5%	68.29 bp	134.00 bp

The exact finite quotes show lower restoring impact for these sizes. The difference becomes more pronounced for the larger trades in the table. The comparison does not require assigning a derivative-based elasticity to either trade.

We can also prove positive execution cost against opening prices using finite mean gaps. Direct expansion gives

$$\begin{aligned} &\sum_iP_i da_i-[F(a')-F(a)]\\ &\quad=\alpha G\left[ \sum_iw_i^*(1+r_i) -\prod_i(1+r_i)^{w_i^*} \right]\ge0. \end{aligned}$$

This time the bracket is the mean gap of the reserve multipliers themselves. Combined with the preceding scale-production identity, a nontrivial closed swap with $\alpha>0$ satisfies

$$\boxed{\sum_iP_i da_i\ge F(a')-F(a)>0}.$$

There is positive execution cost against the starting pool prices even for a restoring trade. The numerical comparison demonstrates lower finite impact in this specified case; it is not a theorem for every multi-asset direction, initial state, or trade that overshoots the preference. Inventory-dependent finite asymmetry is also not unique to this construction.

11. Endpoint policies within the finite model

At $\alpha=1$, $F=G$ and

$$P_i=w_i^*G/a_i, \qquad s_i=w_i^*G, \qquad w_i=w_i^*.$$

The pool maintains the preferred fractions exactly at its own prices. Finite post-trade closure becomes

$$\sum_i w_i^*\frac{r_i}{1+r_i}=0.$$

This follows by substituting $P_i'=w_i^*G'/a_i'$ into the value-flow equation and cancelling the positive common factor $G'$. It does not require $G'=G$; a nontrivial closed swap increases $G$ by the finite scale-production identity.

For two equal reference fractions and initial reserves of one unit each, an input $da_1=0.1$ gives

$$\frac{0.1}{1.1}+\frac{da_2}{1+da_2}=0, \qquad da_2=-\frac{0.1}{1.2}.$$

The output is approximately $0.083333$ units, obtained directly from ending-state closure.

At $\alpha=0$, prices equal their reference values, valuation is linear, and closure is $\sum_iP_i^*da_i=0$. This endpoint has no scarcity response. Intermediate $\alpha$ combines these two portfolio policies while retaining the same finite post-trade execution rule.

12. Reproduce the calculations and define the next question

The article's verification script uses Python's standard library and downloads no data:

python3 work/proofs/2026-09-06-reference-portfolio/verify.py

It reproduces the table and composition example, checks exact scaling and valuation identities numerically, checks finite price multipliers and composition, verifies both finite mean-gap identities, post-trade closure, and valuation nondecrease, and checks the closed-form finite execution example at $\alpha=1$. These checks supplement the derivations; they are not an onchain implementation audit.

The reference portfolio supplies a financial mandate. The geometric contribution follows from a finite multiplicative portfolio rule, and relative entropy gives an exact measure of the composition discount. Coupled pricing permits state-dependent finite elasticity, while post-trade closure converts a weighted mean gap into exact nonnegative scale production.

The construction therefore has a financial interpretation beyond smoother simulated weights. The remaining design question is whether this particular soft portfolio preference produces the inventory and execution behavior desired for Multiswap. That requires specifying and testing the mandate, including persistent market-price changes and reference-update policy, before adopting it as a protocol model.