Appendix H. Formal Model of Market Realization, Wrapper Flows, and Price Capture

Jason St George. "Appendix H. Formal Model of Market Realization, Wrapper Flows, and Price Capture" in Next Generation Stores of Value: Privacy, Proofs, Compute. Version v2.2. /v/2.2/read/appendix/h-market-realization/

Formal Model of Market Realization, Wrapper Flows, and Price Capture

This appendix develops the flow machinery behind the Market Realization Plane (§10: Work Credits: Energy-Anchored Claims) and VerifyFlow (§23: Extended Telemetry). It is deliberately quarantined here: the main body needs the distinction between value capture and price capture, not the derivations.

Attribution.

The flow machinery below—flow elasticity, holder recycling, leveraged-rebalancing mechanics, and the return-coupled versus return-decoupled distinction—is adapted from Michael W. Green’s work on how product mechanics and holder behavior set prices, in particular A Semi-Theory of Almost Everything (Tier1 Alpha, July 2026). The market-impact form is the square-root “law” of execution cost from the market-microstructure literature, in which impact scales roughly with the square root of order size relative to available liquidity; it is not Gabaix and Koijen’s aggregate demand-elasticity multiplier, which does not appear in this model. Gabaix and Koijen (2021) is relied on only for the broader inelastic-markets premise—that flows move prices at all, because aggregate demand for financial assets is far less elastic than textbook treatments assume—which is why a flow-based telemetry layer is worth publishing. What is ours is the generalization from equity market structure to monetary telemetry: the claim that these quantities must be published as a distinct verification family (VerifyFlow) precisely so that price movement is not mistaken for monetary adoption. Any errors in that generalization are ours, not theirs.

A caution before the mathematics. These equations describe transmission and magnitude. They do not explain why a narrative ignites a behavioral shift in the first place. Flow elasticity in particular is measured, not explained. Treating any parameter below as a structural constant is a modeling error.

Flow Elasticity

For wrapper $i$, estimate the responsiveness of shares outstanding to changes in value per share:

where $S_{i,t}$ is shares outstanding, $\mathrm{NAV}_{i,t}$ is value per share, and $\varepsilon_{i,t}$ is flow elasticity. Interpretation: $\varepsilon \approx 0$ means holders largely sit still; $\varepsilon \approx -1$ means holders redeem roughly enough after gains to maintain a constant dollar position.

The regression has three parts and each is allocated to a different place downstream, so the decomposition must be stated rather than assumed. The $\varepsilon$-driven component is return-coupled and belongs in $Q^{RC}$ below. The intercept $\alpha_i$ captures baseline creation or redemption that occurs regardless of the day’s return — distribution growth, model-portfolio adoption, scheduled contributions — and therefore belongs in the return-decoupled term $Q^{RD}$, not in $Q^{RC}$. The residual $u_{i,t}$ stays unexplained and must be published rather than absorbed into either. Assigning $\alpha_i$ to $Q^{RC}$ would overstate momentum amplification; discarding it would understate the structural bid.

Identification: the regression is simultaneous, and the bias is not benign.

$\Delta \ln \mathrm{NAV}_{i,t}$ is not exogenous to $\Delta \ln S_{i,t}$. Creations and redemptions force dealer hedging, hedging passes through the impact function of Appendix H: Formal Model of Market Realization, Wrapper Flows, and Price Capture, and impact moves the underlying — which moves the NAV being used as the regressor. The OLS estimate of $\varepsilon$ therefore suffers simultaneity bias, and its direction is the dangerous one: flows that amplify the move induce a correlation between share changes and NAV changes that OLS books as elasticity, so naively fitted $\hat\varepsilon$ overstates amplification precisely when amplification is present, and the fitted values then feed $\kappa$ and WRR downstream. Three disciplines follow. First, prefer instruments for $\Delta \ln \mathrm{NAV}$ that are exogenous to wrapper flow — underlying-index constituent news, macro release days, cross-venue price dislocation — over the raw regression. Second, where instruments are unavailable, publish the OLS estimate as an upper bound on amplification rather than as a point estimate. Third, never recycle a fitted $\hat\varepsilon$ back into the stress harness of Appendix H: Formal Model of Market Realization, Wrapper Flows, and Price Capture without reporting both the instrumented and OLS values; a harness whose elasticity was fitted on the same amplification it simulates will find it.

The second bias runs the other way, and it decides when the upper bound is safe to publish.

For a continuously traded underlying, the wrapper’s official NAV is computed on a stale or synchronized print, not on the price at which flows actually cleared. The regressor is then a noisy measurement of the true underlying return, and classical errors-in-variables attenuation pulls $\hat\varepsilon$ toward zero — understating amplification. The two biases are opposite in sign: simultaneity inflates $\hat\varepsilon$, staleness attenuates it, and which dominates is not a property of the estimator but of the underlying’s liquidity and the wrapper’s pricing convention. The upper-bound publishing rule above is therefore conditional, not absolute. A fourth discipline: publish alongside $\hat\varepsilon$ a staleness diagnostic — the divergence between the wrapper’s pricing print and the underlying’s contemporaneous venue price — and treat the OLS value as an upper bound only where that divergence is small. Where staleness is material, the OLS estimate bounds nothing in either direction, and the reading must be published as unidentified rather than as a bound.

Because a daily-reset leveraged wrapper’s NAV return is approximately $L_i r_t$ for underlying return $r_t$:

Leveraged Rebalancing and Flow-Induced Exposure

A daily-reset fund must trade to restore its target exposure. The gross required rebalance is

where $A_{i,t}$ is fund assets. Note that $L(L-1) > 0$ both for $L > 1$ and for negative leverage such as $-2$ or $-3$: long and inverse leveraged funds both mechanically chase the underlying move. The inverse product is not a stabilizer.

Separately, creations and redemptions carry exposure equal to leverage times the dollar flow:

Combining the two gives total return-coupled mechanical exposure:

Net Mechanical Gain

Define the net mechanical gain coefficient

so that $Q^{RC}_{i,t} = \kappa_{i,t} A_{i,t} r_t$. The sign and magnitude of $\kappa$ determine whether a wrapper amplifies or absorbs the move it sits on.

The Recycling Boundary

A wrapper fully offsets its own gross mechanical trade when $\kappa_i = 0$, which implies a critical elasticity

Illustrative recycling boundaries. Holder behavior stops being a vague qualitative concern and becomes an observable regime boundary.

This is the single most useful conversion in the appendix: it turns “holder behavior matters” into a number that can be measured, plotted, and breached. Empirically, a $3\times$ product whose elasticity sits near $-0.68$ is close to self-neutralizing; if that elasticity weakens toward $-0.40$, the same product becomes a material momentum amplifier, and the underlying’s behavior can shift from mean-reverting toward higher volatility with greater trend persistence.

Wrapper Recycling Ratio

A more legible operator metric normalizes elasticity by its own boundary:

where $\mathrm{WRR} = 1$ means holder flows fully offset gross rebalancing; $\mathrm{WRR} < 1$ means the wrapper amplifies the underlying move; and $\mathrm{WRR} > 1$ means holder flows more than offset it and become countercyclical. This belongs on the public Market Realization & Wrapper Board.

The ratio and the gain coefficient are one object, and stating the identity prevents misreadings of both:

so the unnormalized quantity and its normalized image always move together, and any dashboard showing one can be checked against the other.

The unlevered case is the dominant one, and WRR is undefined there.

$\mathrm{WRR}$ divides by $L_i - 1$, so it does not exist for an unlevered wrapper ($L_i = 1$) — and the spot ETF, the instrument most likely to become this asset’s largest wrapper, is exactly that. The degeneracy is not a defect in the metric; it is the metric correctly reporting that an unlevered wrapper has no rebalancing obligation and therefore nothing to recycle. What remains meaningful at $L_i = 1$ is the elasticity itself: holder flows into a spot wrapper scale with $\varepsilon_{i,t} A_{i,t} r_t$ through the $Q^{RC}$ term, with no rebalancing component to offset. The publication rule is therefore: leveraged and inverse wrappers publish $\kappa$ and WRR; unlevered wrappers publish $\varepsilon$ and $\kappa = \varepsilon A r$; and the board never carries a WRR column for $L = 1$. A reader who wants a single cross-class reading should use $\kappa_{i,t} A_{i,t} r_t$ directly, which is defined everywhere.

Wrapper-Class Applicability

The machinery above was derived for fund-like wrappers with shares outstanding, NAV, and creation/redemption mechanics — the equity-ETF structure it was adapted from. Not every wrapper class that matters for a bearer asset has those observables, and applying an equation to a class that lacks its inputs does not produce a number; it produces a fabricated one. The publication contract is therefore class-conditional:

Wrapper-class applicability of the VerifyFlow apparatus. A class without the observables an equation consumes does not get a fabricated reading; it gets the substitute construction named in the last column, or nothing.

Two consequences. First, the perpetual-futures substitute matters most in practice: for a crypto-native asset, perps are plausibly the largest leveraged wrapper class, and the funding-rate-to-price response is the closest available analog to elasticity — publish it, but do not label it $\varepsilon$, because it is not estimated from the same regression and does not carry the same interpretation. Second, the convenience-yield instruments of §23: Extended Telemetry already read on the derivative complex (funding, basis); the two families meet exactly here, and a reading published by one should reconcile with the other where both touch the same venue.

Return-Decoupled Allocation Flow

Not all flow responds to returns. A thematic or passive wrapper can receive creations largely independent of the current day’s return, arriving on up days and down days alike. For underlying asset $j$:

where $F^{\alpha}_{k,t}$ is the return-decoupled component of net creation into wrapper $k$ and $w_{jk,t}$ is that wrapper’s exposure weight to $j$. For leveraged or derivative wrappers, $w_{jk,t}$ must be a delta-equivalent exposure weight rather than a portfolio accounting weight, or the allocation flow will be understated by roughly the leverage factor.

The input discipline is the decomposition rule of Appendix H: Formal Model of Market Realization, Wrapper Flows, and Price Capture, applied here as a boundary: $F^{\alpha}_{k,t} = \bigl(\alpha_k + u_{k,t}\bigr) A_{k,t}$, the intercept-plus-residual component of the creation regression. The $\varepsilon$-driven component is excluded from this sum because it is already inside $\kappa_i$ and therefore inside $Q^{RC}$; feeding raw net creation into $Q^{RD}$ would count the same dollars twice — once as elasticity-driven exposure and once as allocation — and would propagate the overstatement into MPR, the stress harness, and any red line that reads on them. The two equations of this appendix therefore partition demand by return-dependence: $Q^{RC}$ owns the $\varepsilon$-component; $Q^{RD}$ owns $\alpha$ and $u$. Where a wrapper’s creation series cannot yet be decomposed, publish raw $F_{k,t}$ flagged as undecomposed so the aggregation layer can exclude it from $Q^{RD}$ rather than silently double-count it.

The two flow types do different work. Return-coupled flow governs volatility, momentum, persistence, and reversal severity. Return-decoupled flow governs destination and level: which assets receive the structural bid, how concentrated it becomes, and which prices are accepted without valuation-sensitive selling. For a triad asset, return-decoupled flow could arrive from a spot ETF, a crypto index product, a corporate treasury mandate, an automated wealth-allocation model, a retirement default, an agentic treasury system, or a regulated “digital hard asset” basket — all potentially enormous for price and nearly irrelevant to native triad usage.

Market Impact and Liquidity

Flow becomes price only through a liquidity-dependent impact function. The standard empirical form has impact growing roughly with the square root of trade size relative to available liquidity:

where $Y_j$ is an impact coefficient, $\sigma_{j,t}$ is volatility, $Q_{j,t}$ is exposure demand, and $V_{j,t}$ is executable liquidity over the relevant horizon. Splitting impact into permanent and temporary components with permanence share $\rho_j$:

Fitted permanence is case-specific. Import the structure; estimate $Y_j$, $\rho_j$, and $V_{j,t}$ asset by asset.

Volatility Drag on Leveraged Wrappers

For a daily-reset wrapper with leverage $L$ on an underlying with drift $\mu$ and volatility $\sigma$, the continuous approximation to compound growth is

where $r_f$ is the financing rate on the levered portion and $f$ is the product’s expense ratio. The first two terms are the standard continuous approximation; the last two are the carry the wrapper actually bears and are frequently omitted, which flatters leveraged products in exactly the high-rate environments where they are most costly to hold. Note also that this section’s results, along with the recycling boundary and WRR of Appendix H: Formal Model of Market Realization, Wrapper Flows, and Price Capture, apply to daily-reset products with $L \neq 0, 1$; an unlevered or non-resetting wrapper has no rebalancing obligation and no drag term.

The variance penalty grows with the square of leverage. At sufficiently high volatility a leveraged wrapper destroys capital even when the underlying has a positive average return. This is the formal basis for the classification in §6: The Triad and the Monetary Candidate: a daily-reset leveraged wrapper cannot inherit the underlying asset’s store-of-value status, because its terminal value depends on the path and not merely the endpoints. It is a path-dependent trading instrument that happens to reference a monetary object.

Dealer Hedge Decomposition

Product-level exposure demand is not equal to spot orders. Decompose realized hedging by instrument — physical holdings, futures, total-return swaps, options — and track delta- and gamma-adjusted exposure per counterparty, average derivative maturity, collateral and margin requirements, and evidence of counterparty caps. A migration from swaps toward options is a candidate signal that dealer balance-sheet capacity is binding, with the attendant gamma feedback when dealers are short calls. Treat this as a hypothesis to test per asset, not a law.

Measurement Contract

VerifyFlow Measurement Contract

• Use daily or intraday shares outstanding, NAV, price, AUM, holdings, leverage, and derivative disclosures.

• Chain reverse splits and corporate actions before estimating anything.

• Estimate elasticity over multiple rolling windows, never a single fixed window.

• Use structural-break tests rather than assuming fixed holder behavior.

• Address the simultaneity of elasticity estimation explicitly: instrument $\Delta \ln \mathrm{NAV}$ or publish the OLS value as an upper bound on amplification, and only where the staleness diagnostic of Appendix H: Formal Model of Market Realization, Wrapper Flows, and Price Capture shows wrapper pricing tracks the contemporaneous underlying (Appendix H: Formal Model of Market Realization, Wrapper Flows, and Price Capture).

• Report exposure demand separately from realized spot orders.

• Delta-adjust swaps and options.

• Publish top-$N$ counterparty and constituent concentration.

• Separate return-coupled and return-decoupled flows.

• Compare market-exposure growth against native protocol-use growth (WNG).

• Publish confidence intervals and model residuals.

• Report failed hypotheses rather than silently dropping them.

That last clause is not decoration. A flow model that only publishes the specifications that worked is indistinguishable from a narrative, and this thesis has no standing to demand receipts from protocols while exempting its own econometrics.

Reference Stress-Harness Sketch

For each scenario in §23: Extended Telemetry: enumerate wrappers and their $(L_i, A_i, \varepsilon_i)$; compute $\kappa_i$ and $\mathrm{WRR}_i$; simulate a return path; accumulate $Q^{RC}$ and $Q^{RD}$; map aggregate $Q$ through the impact function given $V$; propagate the resulting return back into the next period’s rebalance and creation terms; and record MPR, WNG, and native-use series alongside price. The output of interest is never the simulated price. It is whether the native series moved at all.

Boundaries and Caveats

• This machinery is evidence about market transmission. It is not evidence that Privacy, Proofs, or Compute will earn monetary premium.

• Parameters fitted in one market (equities, semiconductors) do not transfer. Re-estimate everything.

• Unlevered wrappers are not adversaries by construction. The failure mode is the inability to distinguish accessibility, custody concentration, native use, synthetic leverage, and price-insensitive demand.

• Price is never a protocol control target. The purpose of this appendix is to explain price formation, not to defend a price.

• Market realization is orthogonal to the stack. Nothing here becomes a Layer 7.

Buyer Quality and Price Provenance

The vectors above measure product mechanics. They do not measure buyer quality. Green’s later work on the Treasury long end makes the gap precise: a recurring dollar inflow can absorb less duration as prices fall, because market-value weights shrink, and the residual then clears with whoever is left. The same structure can appear in a monetary wrapper whose creations continue while the holders behind those creations are default-enrolled, unlevered, and duration-unaware—or leveraged, short-horizon, and one mark from a forced sale.

VerifyFlow therefore adds a buyer-quality checklist, not a new elasticity (§23: Extended Telemetry). For any material source of demand, record persistence of funding, investment horizon, match to an enduring liability, capacity to bear losses without forced liquidation, and whether incremental risk absorption rises or falls after a drawdown. Do not publish a multiplicative “anchor quality” index. The dimensions are the contribution; collapsing them into a scalar would recreate the scoreboard problem this appendix exists to prevent.

Approximate new DV01 absorption by an index receiving inflow $F_t$ is

where $w_{j,t}$ is the market-value weight of instrument $j$ and $D_{j,t}$ its modified duration. As $w_{j,t}$ falls with price, $A_t$ can decline even as $F_t$ is unchanged. That identity is why “inflows remain healthy” is not evidence that the residual is shrinking.

The measurement contract of this appendix applies here without modification: publish the classification of each buyer, the fields that support it, and the cases in which the classification failed. A flow model that reports only that demand arrived, and not what kind of demand it was, has not instrumented the plane.

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