Post-Trade Elastic BULL and BEAR Claims

Multiswap's Post-Trade Elasticity Model gives every token a reserve, a scale, and a marginal price. This article asks whether the same structure can support two directional claims associated with a Reserve Asset:

• BULL, whose return exceeds the Reserve Asset's return when that return is positive and falls more when it is negative;
• BEAR, whose return has the opposite sign and may have greater than one-for-one short exposure.

The accounting must come first. BULL and BEAR are both positive claims against the pool, so both are credit accounts. They are economically opposite because their prices respond differently, not because their ledger polarities differ. Once that distinction is fixed, double-entry accounting and post-trade value-flow conservation reduce the claim problem to a small system of exact finite equations.

The resulting research model has a general form for any homogeneous elasticity. When price elasticity and scale elasticity both equal one half, the claim solve becomes quadratic. A constant relative-claim elasticity $\kappa$ remains optional: $\kappa=2$ gives a particularly symmetric log-return geometry, while $\kappa>2$ is necessary if both BULL and BEAR must have greater than one-for-one exposure.

Status: Working exact-arithmetic research model, August 2026. The ledger treatment is the proposed accounting foundation for BULL and BEAR claims. The claim equations, constant-$\kappa$ closure, feasibility bounds, and quadratic specialization are mathematical results under the assumptions stated below. BULL and BEAR claims are not implemented in the current Multiswap contracts. The reduced derivation studies an action in which one Reserve Asset, its two claims, and account $0$ move. A complete multi-Reserve-Asset $A\to B$ action model remains open.

1. Native Multiswap state

For any token $\alpha$, let

$$\boxed{a_\alpha>0}$$

denote its internal token quantity,

$$\boxed{s_\alpha>0}$$

its scale, and

$$\boxed{P_\alpha=\frac{s_\alpha}{a_\alpha}>0}$$

its marginal price in the pool's internal numeraire.

For Reserve Asset $i$, these variables are

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

Account $0$ is the pool's Protocol credit account. The LP Token is the claim on that account, so its state is

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

For the two claims associated with Reserve Asset $i$, write

$$a_{i,L},\qquad s_{i,L},\qquad P_{i,L}=\frac{s_{i,L}}{a_{i,L}}$$

for BULL and

$$a_{i,S},\qquad s_{i,S},\qquad P_{i,S}=\frac{s_{i,S}}{a_{i,S}}$$

for BEAR.

The letters $L$ and $S$ label the long and short economic exposures. They do not specify ledger polarity.

2. Ledger accounting

The Scale / Pool ledger has three immediate children:

Scale / Pool
├── Reserve Assets (debit group)
├── Claim Tokens (credit group)
│   └── Reserve Asset i Claims (credit group)
│       ├── i BULL (credit)
│       └── i BEAR (credit)
└── Protocol, account 0 (credit ledger)

BULL and BEAR must be separate tradeable token roots because they have separate supplies, prices, and holder balances. Inside the Scale ledger, their credit accounts share a parent associated with Reserve Asset $i$.

Define the claim-group scale by ledger rollup:

$$\boxed{s_{i,*}=s_{i,L}+s_{i,S}.}$$

Pool-wide double-entry accounting gives the identity

$$\boxed{ \sum_i s_i \equiv s_0+\sum_i\left(s_{i,L}+s_{i,S}\right). }$$

The corresponding identity for state changes is

$$\boxed{ \sum_i ds_i \equiv ds_0+\sum_i\left(ds_{i,L}+ds_{i,S}\right). }$$

These are ledger identities, not independent constraints imposed by the pricing model. A balanced posting preserves them automatically.

Several tempting equations do not follow:

$$s_i\ne s_{i,L}+s_{i,S}$$

in general, and

$$a_i\ne a_{i,L}+a_{i,S}$$

in general. The quantity $a_i$ is the backing Reserve Asset balance. The quantities $a_{i,L}$ and $a_{i,S}$ are internal claim-token supplies. They have different accounting meanings.

The single account $0$ makes the pool the collateralization unit. The ledger does not allocate Protocol scale separately by Reserve Asset. BULL and BEAR claims are therefore pool-collateralized unless a later protocol rule adds a stronger segregation requirement.

2.1 The claim-free projection

If all claim balances vanish,

$$a_{i,L},a_{i,S},s_{i,L},s_{i,S}\longrightarrow0,$$

then the claim accounts disappear from the economic state and the ledger identity reduces to

$$\boxed{s_0=\sum_i s_i.}$$

This is the original Multiswap Scale accounting. The claim-free projection is exact. Prices of zero-supply claims are not evaluated: $s_{i,*}/a_{i,*}$ would be $0/0$, and the corresponding token simply is not active.

3. Post-Trade Elasticity

Choose one homogeneous scale elasticity

$$\boxed{0<e_s<1}$$

and define price elasticity by

$$\boxed{e_P=1-e_s.}$$

For a coefficient-preserving transition of token $\alpha$,

$$\boxed{s_\alpha=c_\alpha a_\alpha^{e_s}}$$

and

$$\boxed{P_\alpha=c_\alpha a_\alpha^{-e_P}.}$$

Let

$$g_\alpha=\frac{a_\alpha'}{a_\alpha}$$

be its quantity gain and

$$G_\alpha=\frac{P_\alpha'}{P_\alpha}$$

its price gain. Fixed $c_\alpha$ gives

$$\boxed{G_\alpha=g_\alpha^{-e_P}}$$

and

$$\boxed{ \frac{s_\alpha'}{s_\alpha}=g_\alpha^{e_s}. }$$

The price-gain map is a homomorphism:

$$(g_1g_2)^{-e_P}=g_1^{-e_P}g_2^{-e_P}.$$

This statement applies to each token's coefficient-preserving state equation. The broader Multiswap model also admits coefficient-changing operations. The claim derivation below studies the coefficient-preserving active-account case because that assumption produces a closed finite system and exposes the claim geometry directly.

3.1 Post-trade value flow

Post-trade settlement values the complete quantity change at the terminal marginal price:

$$\boxed{\Sigma_\alpha=da_\alpha P_\alpha'.}$$

Since

$$da_\alpha=a_\alpha(g_\alpha-1),$$

we obtain

$$\boxed{ \Sigma_\alpha =s_\alpha(g_\alpha-1)g_\alpha^{-e_P}. }$$

Self-financing is a statement about these $\Sigma$ value flows. It is not a statement that the $ds$ scale changes sum to zero.

The Scale ledger instead satisfies the exact discrete product rule

$$\boxed{ ds_\alpha=\Sigma_\alpha+a_\alpha dP_\alpha. }$$

The first term is terminal-price consideration. The second is revaluation of the opening quantity.

4. Price-gain coordinates

Define

$$\boxed{\rho=\frac{e_s}{e_P}.}$$

Because

$$g_\alpha=G_\alpha^{-1/e_P},$$

the scale gain becomes

$$\boxed{ \frac{s_\alpha'}{s_\alpha}=G_\alpha^{-\rho}. }$$

Therefore

$$\boxed{ ds_\alpha =s_\alpha\left(G_\alpha^{-\rho}-1\right) }$$

and

$$\boxed{ \Sigma_\alpha =s_\alpha\left(G_\alpha^{-\rho}-G_\alpha\right). }$$

For convenience, define

$$A(G)=G^{-\rho}-1$$

and

$$B(G)=G^{-\rho}-G.$$

Then

$$ds_\alpha=s_\alpha A(G_\alpha)$$

and

$$\Sigma_\alpha=s_\alpha B(G_\alpha).$$

Every finite positive gain leaves scale positive:

$$s_\alpha'=s_\alpha G_\alpha^{-\rho}>0.$$

Thus the same post-trade elasticity structure protects Reserve Asset, LP, BULL, and BEAR scales from crossing through zero during an admissible coefficient-preserving transition. An excessive requested action reaches the boundary as the positive solution disappears; the equations do not continue into a negative-scale state.

5. The reduced claim action

Consider a reduced action in which the active accounts are:

• Reserve Asset $i$, a debit;
• account $0$, a credit;
• BULL $i.L$, a credit;
• BEAR $i.S$, a credit.

All other ledger accounts remain unchanged during this reduced action.

Scale accounting gives

$$\boxed{ s_iA(G_i) = s_0A(G_0) +s_{i,L}A(G_{i,L}) +s_{i,S}A(G_{i,S}). } \tag{1}$$

Self-financing gives

$$\boxed{ s_iB(G_i) = s_0B(G_0) +s_{i,L}B(G_{i,L}) +s_{i,S}B(G_{i,S}). } \tag{2}$$

Subtracting equation (2) from equation (1) gives the equivalent mark-to-market equation

$$\boxed{ s_i(G_i-1) = s_0(G_0-1) +s_{i,L}(G_{i,L}-1) +s_{i,S}(G_{i,S}-1). } \tag{3}$$

Given $G_i$, the three unknown credit gains are

$$G_0,\qquad G_{i,L},\qquad G_{i,S}.$$

Equations (1) and (2) provide two independent finite equations. The claim model therefore needs exactly one additional constitutive relation.

This degree count is the central result of the ledger-first derivation. Accounting does not separately determine BULL and BEAR. It determines their common level after the model specifies one relative-allocation coordinate.

6. The relative-claim homomorphism

The minimal additional relation acts on the relative claim price. Choose one constant homogeneous elasticity $\kappa$ and impose

$$\boxed{ \frac{G_{i,L}}{G_{i,S}}=G_i^\kappa. } \tag{4}$$

Equivalently,

$$\boxed{ \frac{P_{i,L}}{P_{i,S}}=\eta_iP_i^\kappa, }$$

where

$$\boxed{ \eta_i = \frac{P_{i,L}}{P_{i,S}}P_i^{-\kappa} }$$

is invariant during an ordinary claim action:

$$\eta_i'=\eta_i.$$

No reference price is required. A reference level can be absorbed into $\eta_i$, just as units reside in the ordinary PTE coefficient $c_i$.

Equation (4) is homomorphic because

$$(G_1G_2)^\kappa=G_1^\kappa G_2^\kappa.$$

The constant $\kappa$ fixes the spread between BULL and BEAR return elasticities. It does not generally fix either elasticity individually.

7. Exact directional requirements

For a finite action with $G_i\ne1$, define the effective log-return elasticities

$$\boxed{ L_L=\frac{\log G_{i,L}}{\log G_i}, \qquad L_S=\frac{\log G_{i,S}}{\log G_i}. }$$

The desired BULL behavior is

$$\boxed{L_L>1.}$$

The desired inverse BEAR behavior is

$$\boxed{L_S<0.}$$

A genuinely leveraged short requires the stronger condition

$$\boxed{L_S<-1.}$$

Equation (4) implies

$$\boxed{L_L-L_S=\kappa.}$$

Consequently:

• BULL leverage together with inverse BEAR exposure requires $\kappa>1$;
• BULL leverage together with leveraged short exposure requires $\kappa>2$.

It is often clearer to write the positive short magnitude as

$$S=-L_S>0.$$

Then

$$\boxed{\kappa=L_L+S.}$$

For example, $1.5$-times BULL exposure and $2$-times BEAR exposure require

$$\kappa=1.5+2=3.5.$$

7.1 The elasticities are outputs

Define their midpoint by

$$\theta=\frac{L_L+L_S}{2}.$$

Then

$$\boxed{ L_L=\theta+\frac{\kappa}{2}, \qquad L_S=\theta-\frac{\kappa}{2}. }$$

The model fixes the difference $\kappa$. Equations (1) and (2) determine the midpoint $\theta$ from the current scales, the action size, and the selected positive solution branch. Individual effective leverage therefore generally varies with state and action size.

This is not a constant-leverage-token model. It is a constant-spread model whose accounting selects the common return component. Fixing both $L_L$ and $L_S$ independently would add two claim equations and generally overdetermine the reduced one-Reserve-Asset system.

8. Reduction to one scalar equation

Write

$$G=G_i, \qquad K=G^\kappa, \qquad x=G_{i,S}.$$

Equation (4) gives

$$G_{i,L}=Kx.$$

Equation (3) gives account $0$'s gain as an affine function of $x$:

$$\boxed{ G_0(x) = 1+ \frac{ s_i(G-1) -s_{i,L}(Kx-1) -s_{i,S}(x-1) }{s_0}. } \tag{5}$$

Define

$$C=s_0+s_{i,L}+s_{i,S}+s_i(G-1)$$

and

$$D=s_{i,L}K+s_{i,S}.$$

Then

$$\boxed{G_0(x)=\frac{C-Dx}{s_0}}$$

and positivity requires

$$\boxed{0<x<\frac{C}{D}.}$$

Substitution into equation (1) gives the scalar equation

$$\boxed{ F(x) = s_0A(G_0(x)) +s_{i,L}A(Kx) +s_{i,S}A(x) -s_iA(G) =0. } \tag{6}$$

8.1 Directional root interval

When $G>1$, BULL must outperform the Reserve Asset and BEAR must fall:

$$Kx>G, \qquad x<1.$$

Therefore

$$\boxed{G^{1-\kappa}<x<1.} \tag{7}$$

When $G<1$, BULL must fall more and BEAR must rise:

$$Kx<G, \qquad x>1,$$

so

$$\boxed{1<x<G^{1-\kappa}.} \tag{8}$$

These intervals are nonempty exactly when $\kappa>1$.

8.2 Strict convexity

For $\rho>0$, the function

$$z\longmapsto z^{-\rho}$$

is strictly convex. Since $G_0(x)$ is affine, $F(x)$ is strictly convex throughout its positive domain. Moreover,

$$F(x)\longrightarrow+\infty$$

at both domain boundaries. The equation therefore has zero roots, one double root, or two positive roots—never an uncontrolled collection of roots.

Its unique minimum is available in closed form. Define

$$E=s_{i,L}K^{-\rho}+s_{i,S}$$

and

$$q=\left(\frac{D}{E}\right)^{1/(\rho+1)}.$$

Then

$$\boxed{x_{\min}=\frac{C}{D+s_0q}.}$$

Feasibility over the full positive domain is decided by

$$F(x_{\min})\le0.$$

The economic solve is narrower: a root must also lie inside equation (7) or equation (8). A bracketed solver can search the appropriate monotone side of the minimum. The general model is therefore a well-behaved one-dimensional solve.

9. The half-elasticity specialization

The general claim model does not require equal price and scale elasticity. A major algebraic simplification occurs when

$$\boxed{e_P=e_s=\frac12.}$$

Then

$$\rho=1,$$

and every coefficient-preserving token satisfies

$$\boxed{ \frac{s_\alpha'}{s_\alpha}=\frac1{G_\alpha}. }$$

Its price and scale obey

$$\boxed{s_\alpha P_\alpha=c_\alpha^2.}$$

The exact state change and action value become

$$ds_\alpha=s_\alpha\left(\frac1{G_\alpha}-1\right)$$

and

$$\boxed{ \Sigma_\alpha =s_\alpha\left(\frac1{G_\alpha}-G_\alpha\right). }$$

In log-return coordinates $R_\alpha=\log G_\alpha$,

$$\Sigma_\alpha=-2s_\alpha\sinh R_\alpha.$$

Forward and reverse price gains enter symmetrically.

9.1 The scalar equation becomes quadratic

For $\rho=1$, define

$$E=\frac{s_{i,L}}{K}+s_{i,S}$$

and

$$H=s_0+s_{i,L}+s_{i,S}+s_i\left(\frac1G-1\right).$$

Equation (6) becomes

$$\frac{s_0^2}{C-Dx}+\frac{E}{x}=H.$$

Multiplication by $x(C-Dx)$ gives

$$\boxed{ HDx^2 +\left(s_0^2-ED-HC\right)x +EC =0. } \tag{9}$$

Thus the complete reduced claim solve requires one square root and branch selection rather than iterative root-finding. The admissible root must satisfy:

$$x>0, \qquad C-Dx>0,$$

and the directional interval in equation (7) or equation (8).

The discriminant

$$\Delta = \left(s_0^2-ED-HC\right)^2-4HDEC$$

is also an exact finite-action feasibility test. If $\Delta<0$, no real claim transition satisfies the assumed equations. If $\Delta=0$, the two roots meet at the action boundary.

10. The symmetric $\kappa=2$ case

The quadratic result comes from $e_P=e_s=1/2$ and holds for every constant $\kappa$. Setting

$$\boxed{\kappa=2}$$

adds a separate geometric symmetry:

$$\frac{G_{i,L}}{G_{i,S}}=G_i^2.$$

Let

$$z=\sqrt{G_{i,L}G_{i,S}}.$$

Then

$$\boxed{ G_{i,L}=G_i z, \qquad G_{i,S}=\frac{z}{G_i}. }$$

In logarithms,

$$\log G_{i,L}=\log z+\log G_i$$

and

$$\log G_{i,S}=\log z-\log G_i.$$

BULL and BEAR log returns are symmetric around the accounting-determined midpoint $\log z$.

The directional branch test collapses to

$$\boxed{ \min(1,G_i)<z<\max(1,G_i). }$$

Writing

$$z=G_i^\mu$$

gives

$$G_{i,L}=G_i^{1+\mu}$$

and

$$G_{i,S}=G_i^{\mu-1}.$$

The directional interval is

$$0<\mu<1,$$

so

$$1<L_L<2$$

and

$$-1<L_S<0.$$

This is an important limitation: $\kappa=2$ produces leveraged BULL and inverse BEAR on this branch, but BEAR has less than one-for-one short magnitude. If leveraged short exposure is required together with leveraged BULL exposure, the model needs

$$\boxed{\kappa>2.}$$

The half-elasticity quadratic remains available for those larger values of $\kappa$.

11. State-dependent leverage capacity

Although the homomorphism permits any constant $\kappa$, a given state supports only a finite leverage spread.

Consider the local limit $G_i\to1$. Let

$$G_0=1+L_0R+O(R^2),$$ $$G_{i,L}=1+L_LR+O(R^2),$$

and

$$G_{i,S}=1+L_SR+O(R^2),$$

where

$$G_i=1+R.$$

Equations (1) and (2) imply

$$\boxed{ s_0L_0+s_{i,L}L_L+s_{i,S}L_S=s_i } \tag{10}$$

and

$$\boxed{ s_0L_0^2+s_{i,L}L_L^2+s_{i,S}L_S^2=s_i. } \tag{11}$$

Together with

$$L_L-L_S=\kappa,$$

these equations have a real solution only within a state-dependent range.

Define

$$W=s_0+s_{i,L}+s_{i,S}.$$

Then a necessary and locally exact discriminant bound is

$$\boxed{ \kappa^2 \le \frac{ s_i(W-s_i)(s_{i,L}+s_{i,S}) }{ s_{i,L}s_{i,S}W }. }$$

Therefore

$$\boxed{ \kappa_{\max} = \sqrt{ \frac{ s_i(W-s_i)(s_{i,L}+s_{i,S}) }{ s_{i,L}s_{i,S}W }} . }$$

Nontrivial local dispersion requires

$$W>s_i.$$

This is a condition on the reduced active subsystem, not a new pool-wide accounting identity. It illustrates why a single pool-wide account $0$ is economically meaningful: pooled Protocol scale contributes to the leverage capacity associated with an individual Reserve Asset.

There is no universal upper bound on $\kappa$. Each state has a finite capacity, and the finite-action quadratic can impose a tighter bound than the local formula. A fixed homogeneous $\kappa$ therefore defines an admissible state-and-action region. The protocol must reject a requested action when no positive directional root exists.

12. Finite example

Take the illustrative active scales

$$s_i=100, \qquad s_0=120, \qquad s_{i,L}=30, \qquad s_{i,S}=10,$$

with

$$e_P=e_s=\frac12$$

and

$$\kappa=2.$$

For an underlying price gain

$$G_i=1.2,$$

the directional quadratic root gives approximately

$$G_0=1.09545,$$ $$G_{i,L}=1.31404,$$

and

$$G_{i,S}=0.91253.$$

Thus a $20\%$ Reserve Asset gain produces approximately a $31.4\%$ BULL gain and an $8.75\%$ BEAR loss in this state.

For

$$G_i=0.8,$$

the same equations give approximately

$$G_0=0.89442,$$ $$G_{i,L}=0.71604,$$

and

$$G_{i,S}=1.11881.$$

BULL falls more than the Reserve Asset while BEAR rises.

The local leverage capacity of this state is

$$\kappa_{\max} = \sqrt{ \frac{100(60)(40)}{30(10)(160)} } = \sqrt5 \approx2.236.$$

Therefore $\kappa=2$ is locally feasible, while a materially larger fixed spread would leave the local admissible region.

This is a mathematical illustration, not a proposed pool initialization. The complete pool ledger may contain other Reserve Assets and claim groups whose unchanged balances make the global debit and credit totals equal.

13. Initializing an absent claim pair

The claim-free state is valid:

$$a_{i,L}=a_{i,S}=s_{i,L}=s_{i,S}=0.$$

Ordinary multiplicative gains are not defined at that boundary. The pair can nevertheless be added later through an explicit capitalization operation that establishes an absolute terminal state.

Choose positive terminal supplies and prices:

$$a_{i,L}',a_{i,S}',P_{i,L}',P_{i,S}'>0.$$

Their initial claim scales are

$$s_{i,L}'=a_{i,L}'P_{i,L}'$$

and

$$s_{i,S}'=a_{i,S}'P_{i,S}'.$$

Because each claim starts at zero, its terminal-price value flow equals the scale created:

$$\boxed{\Sigma_{i,L}=s_{i,L}',}$$ $$\boxed{\Sigma_{i,S}=s_{i,S}'.}$$

One clean capitalization path converts existing LP scale into the two claims. Hold $P_0$ fixed for the conversion and burn LP quantity with terminal value

$$\boxed{ \Sigma_0 = -\left(s_{i,L}'+s_{i,S}'\right). }$$

Then

$$\Sigma_0+\Sigma_{i,L}+\Sigma_{i,S}=0$$

and

$$s_0'=s_0-s_{i,L}'-s_{i,S}'.$$

The initialization must also set the relative-price coefficient

$$\boxed{ \eta_i = \frac{P_{i,L}'}{P_{i,S}'}(P_i')^{-\kappa}. }$$

This operation changes the composition of credit claims and establishes the claim model. It is a capitalization or liquidity operation, not an ordinary coefficient-preserving claim action.

The clean lifecycle is therefore

$$\boxed{ \text{claims absent} \longrightarrow \text{pair initialization} \longrightarrow \text{ordinary positive-state claim actions}. }$$

Initializing exactly one claim leaves the relative-price equation singular. The proposed paired model therefore treats each claim pair as either absent or fully active.

14. What the model establishes

Under the stated reduced-action assumptions, the model establishes:

1. Ledger consistency. BULL and BEAR are separate positive credit claims, while account $0$ remains the pool-wide residual credit account.
2. Exact claim-free projection. Removing zero-balance claims restores $s_0=\sum_i s_i$.
3. Finite accounting equations. Scale balance and $\Sigma$-self-financing give two exact equations for the three credit gains.
4. One constitutive degree of freedom. A relative-claim homomorphism supplies exactly the missing relation.
5. A one-dimensional general solve. For arbitrary $0<e_P<1$, the remaining equation is strictly convex.
6. A closed-form specialization. At $e_P=e_s=1/2$, the solve is quadratic for every constant $\kappa$.
7. Explicit directional tests. The economic branch is selected by finite gain inequalities, not by an infinitesimal approximation.
8. State-dependent effective leverage. The constant $\kappa$ fixes $L_L-L_S$; accounting determines their midpoint.
9. A finite leverage-capacity boundary. A state accepts only values of $\kappa$ and action sizes for which a positive directional root exists.
10. Post-trade positivity. Every accepted positive gain leaves every active coefficient-preserving token scale positive.

15. What remains open

The construction is an accounting and constitutive model, not a safety theorem. The gauge-invariant Reserve--LP condition

$$\frac{c_i'}{c_0'} \le \frac{c_i}{c_0}$$

is proved for the claim-free projection. BULL and BEAR add independently priced credit claims, so this article does not assert that the claim action inherits that condition or its arbitrary-sequence theorem. A full claim-inclusive projective state and proof remain required before implementation.

The construction is not yet a complete protocol specification.

15.1 Complete $A\to B$ actions

The reduced model moves one Reserve Asset, account $0$, and one BULL/BEAR pair. An actual Multiswap $A\to B$ swap can move two Reserve Assets and potentially both associated claim pairs. The complete degree count and action-allocation rules must be derived from the global ledger rather than assumed independently for each leg.

15.2 Constant versus state-dependent leverage

The model holds $\kappa$ constant but allows $L_L$ and $L_S$ to vary. Whether this delivers the desired product experience—and whether a fixed homogeneous pair $(L_L,L_S)$ can be supported by the complete multi-asset action—is unresolved.

15.3 Initialization policy

Accounting specifies how initial claim scale must be financed. It does not choose the initial claim supplies, prices, scale split, or $\eta_i$. Those are product-definition variables requiring a separate initialization rule.

15.4 Coefficient-changing operations

The quadratic derivation assumes coefficient-preserving PTE transitions for the active accounts. The original complete PTE framework also contains coefficient-changing liquidity operations. Their interaction with $\eta_i$ and the claim ledger must be specified separately.

15.5 Implementation and adversarial safety

The current contracts support generic Reserve Asset and Claim Token ledger topology but do not implement this BULL/BEAR state equation or solver. A production design still requires:

• fixed-point error analysis;
• stable quadratic-root evaluation near $G=1$;
• exact branch and domain checks;
• initialization and redemption rules;
• round-trip and split-action analysis for the complete multi-asset action;
• adversarial testing and security review.

Conclusion

The ledger resolves the central structural confusion. Reserve Asset scale does not decompose into BULL and BEAR scale. Instead, all Reserve Assets debit one pool-wide Scale account, while BULL, BEAR, and account $0$ are positive credit claims:

$$\sum_i s_i = s_0+ \sum_i\left(s_{i,L}+s_{i,S}\right).$$

That identity, combined with post-trade value-flow conservation, leaves exactly one relative claim coordinate in the reduced action. The constant-$\kappa$ homomorphism

$$\frac{G_{i,L}}{G_{i,S}}=G_i^\kappa$$

supplies it.

The general model is a strictly convex scalar solve. The specialization

$$e_P=e_s=\frac12$$

makes that solve quadratic without making half elasticity a requirement of either the original PTE model or the claim extension. The further choice $\kappa=2$ makes BULL and BEAR symmetric in log-return space; values $\kappa>2$ are required when the target includes both leveraged BULL and leveraged BEAR exposure.

The remaining work is no longer to guess an accounting decomposition. It is to complete the global $A\to B$ action and determine whether the resulting state-dependent leverage is the product Multiswap should offer.