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REVIEW 5 major objections 5 minor 110 references

Bank Run Exposure in a Paycheck-to-Paycheck Economy with Loss-Averse Depositors

T0 review · 5 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper claims that bank runs become possible exactly when depositor loss aversion exceeds a belief-weighted threshold set by the odds of a bad paycheck state, and identifies that threshold in closed form.

desk verdict A plausible behavioral trigger (Lemma 2) is buried under an unsupported state-space construction that conflates the loss-aversion parameter with a marginal-utility ratio, so the paper's central formal contribution needs substantial rework before it is credible. read the letter →

arxiv 2608.11266 v1 pith:D2PUYF67 submitted 2026-08-10 econ.TH q-fin.RM

G. Charles-Cadogan This is my paper · ORCID
classification econ.THq-fin.RM
keywords bankrunslossaversionliquidityriskconsumptionratchetingsuspensionofconvertibilitypaycheck-to-paycheckeconomyRunExposureStateSpacestochastic
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper tries to establish that bank run exposure is not only a balance-sheet object but an endogenous behavioral outcome: in a paycheck-to-paycheck economy, depositors who are loss averse over declines in income and consumption will suddenly demand more liquidity when they attach sufficiently high probability to a bad income state. The paper pins down a precise trigger — the loss-aversion index exceeds a probability-weighted ratio of marginal gains to marginal losses — and shows that the states satisfying this condition form a Bank Run Exposure State Space. If true, regulatory run-off factors of the LCR type systematically understate reserve needs in stress states, because they treat withdrawals as exogenous averages rather than as a function of depositor fear. The empirical proof of concept on quarterly Call Report data is deliberately modest: a composite balance-sheet proxy improves fit slightly over retail share, and the theory's real test awaits account-level pay-cycle data.

What carries the argument

The load-bearing object is the loss-aversion-adjusted marginal propensity to consume, $\tilde{k}(\rho,d;\lambda)=\tilde{k}(\rho,d)\lambda$, coupled with the consumption ratchet: the one-period ratchet increment is nonnegative exactly when $\tilde{k}\ge k$ (Theorem 2.1). This linear separability converts loss aversion into a threshold on the marginal propensity to consume and into the admissible-state inequality of Lemma 2. Around that threshold the paper builds the Bank Run Exposure State Space, the stopping time at which cumulative withdrawals exhaust reserves, and the half-Cauchy stochastic process $\bar{\lambda}(t)=m_\lambda\tan(\pi\Phi(Z_t)/2)$ that turns the static liquidity-risk slope into a dynamic exposure coefficient.

What would settle it

Measure a depositor population's subjective bad-state probability, loss-aversion index, and marginal value-function slopes, then check whether suspension of convertibility actually occurs in the states satisfying the Lemma 2 inequality; if reserve shortfalls arise outside those states, or fail to arise inside them, the central claim is falsified.

Watch

Extended reading notes

Core claim

The paper's central claim is that a bank run is triggered when a loss-averse depositor's belief in a bad income state is strong enough that the state-dependent loss aversion $\lambda(\omega')$ exceeds $\pi^+_{\omega'} v'_g(c_{1\omega'}) / (\pi^-_{\omega'} v'_\ell(-c_{1\omega'}))$ (Lemma 2). Because $\lambda$ is itself endogenous and random, the collection of admissible and taboo states satisfying this inequality defines a Bank Run Exposure State Space, where the subjective bad-state probability $\pi^-_{\omega'}$ is large relative to $\pi^+_{\omega'}$. The model converts this state-space description into quantitative run probabilities through a stopped withdrawal process: suspension occurs when cumulative withdrawal demand first exhausts cash reserves, giving a negative-binomial count with a Poisson approximation. It then represents the exposure coefficient as a martingale plus a liquidity-risk stochastic integral, with loss aversion evolving as a positive half-Cauchy process around a median near 2.25. The proof-of-concept Call Report exercise finds that a regression-weighted composite of transaction-deposit, core-deposit, and consumer-loan shares modestly improves fit relative to retail share alone, with suggestive but proxy-sensitive amplification for small banks and the post-SVB window.

Load-bearing premise

The load-bearing premise is Assumption 8: a depositor's change in consumption is exactly proportional to her change in income, with loss aversion multiplying a fixed marginal propensity to consume; if that linear separability fails, the ratchet threshold and the run-state inequalities built on it do not follow.

Editorial extensions

If this is right

  • Liquidity regulation that treats run-off as an exogenous stress parameter will understate reserve needs: in the paper's simulation, omitting loss aversion cuts fitted reserve-need volatility by roughly 87 percent, making cash demand look smooth just before suspension.
  • Banks facing loss-averse clienteles will optimally hold higher cash reserves, and those reserves crowd out positive-net-present-value lending, a liquidity externality the model formalizes.
  • Run probabilities become computable objects: given the withdrawal order, reserves, and depositors' subjective bad-state probability, suspension risk follows a negative-binomial count that can be approximated by a Poisson law.
  • Bank-run exposure is better treated as a dynamic, clientele-specific monitoring statistic than as a fixed balance-sheet ratio, because the exposure coefficient inherits the volatility of liquidity demand and of loss aversion.
  • Quarterly Call Report data can implement the exposure regression, and a composite proxy spanning transaction, core, and consumer-loan shares yields modestly better fit than retail share alone, with small-bank and post-SVB amplification appearing only in some specifications.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Beyond the paper: if the Lemma 2 inequality is right, supervisors could measure depositor subjective bad-state probabilities directly — for instance through survey expectations — and use the threshold as an early-warning screen for run-prone clienteles before balance-sheet deterioration appears.
  • Beyond the paper: a testable extension would compare two banks with identical balance sheets but different depositor paycheck volatility; the model predicts the high-volatility clientele shows larger reserve shortfalls exactly in low-income states.
  • Beyond the paper: the public-data proxy cannot identify the account-level mechanism, so an out-of-sample prediction is that with payroll-cycle data the interaction coefficient $\beta_2$ should spike precisely in the states characterized by Lemma 2.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

5 major / 5 minor

Summary. The paper develops a behavioral model of bank run exposure in which paycheck-to-paycheck depositors are loss averse over changes in income and consumption. The central mechanism is a sufficient condition (Lemma 2) that a run is triggered when the loss-aversion index exceeds a probability-weighted ratio of marginal value in gains versus losses. On this basis the paper constructs a 'Bank Run Exposure State Space' (Section 3), derives probability estimates for suspension of convertibility via a stopped process (Section 4), and claims a martingale representation for the exposure process (Section 5). A proof-of-concept empirical section uses quarterly Call Report data for 232 banks to compare a retail-share proxy with a composite balance-sheet proxy, finding modest R² improvements and proxy-sensitive interaction terms. The paper is candid that the empirical exercise is not a definitive test of the behavioral mechanism. However, several load-bearing formal steps, especially the definition of the taboo state space, the derivation of Lemma 2, and the martingale representation, are either incorrect or unsupported as written.

Significance. The topic is timely, particularly after the 2023 SVB episode, and the idea of endogenizing liquidity demand through loss aversion is a useful complement to the exogenous stress parameters in Basel III-type regulation. If the formal results were established, the paper would contribute a behaviorally grounded characterization of run-prone states and a dynamic exposure measure. The empirical implementation is transparently labeled as a proof of concept and provides a detailed data-cleaning discipline, which is commendable. The paper also makes falsifiable predictions about the role of subjective bad-state probabilities. However, the significance is undercut by the fact that the central state-space theorem and the martingale representation are not validly derived as they stand, so the theoretical contribution cannot currently be taken as established.

major comments (5)
  1. [Section 3, Eqs. (3.3)-(3.8)] The definition λ(c1ω') = v'_g(c1ω')/v'_ℓ(-c1ω') conflates the Kőbberling-Wakker loss-aversion index, which is a limit of this ratio at the reference point c=0, with the marginal-value ratio at an arbitrary consumption change c1. Under the paper's own Tversky-Kahneman calibration with symmetric power exponents, v'_g(c)/v'_ℓ(-c) equals 1 for all c, not the calibrated λ≈2.25. Moreover, Theorem 2.1 provides only the lower bound λ(ω') ≥ k/k-tilde; the upper bound λ(c1) < 1/k-tilde in (3.4) appears without any derivation. Consequently, the taboo set Ω_taboo in (3.8) and (3.11), and the pullback-topology construction in Theorem 3.1, are not supported by the preceding results. The central state-space object of the paper therefore lacks a valid derivation and must be re-derived or replaced.
  2. [Section 8.3, proof of Lemma 2] The derivation of the trigger condition starts from inequality (8.32), which includes the indicator I{c1ω'<0}. The subsequent text cancels this indicator because 'π−ω' > 0 in a loss state', but this cancellation is only valid when c1ω'<0, i.e., when the state is actually a loss state. Lemma 2 as stated in (2.27) does not impose this sign restriction on c1ω'. If c1ω'≥0, the left side of (8.32) is zero, so (2.27) is not a sufficient condition for a run trigger. The lemma and its proof need to either restrict ω' to states with c1ω'<0 or demonstrate that the condition in (2.27) is sufficient without the indicator, which the current argument does not do.
  3. [Section 5, Theorem 5.1] The martingale representation (5.3) with the specific integrand b(u)=σ_x,N(u)/∫₀^T σ²_x,N(v)dv is asserted without derivation. The paper's appeal to the martingale-difference property of correctly specified regression residuals does not imply that the least-squares exposure coefficient admits a representation of this form; one would need a genuine martingale representation theorem applied to the projection coefficient as a stochastic process, together with verification of its hypotheses. As it stands, the main dynamic result of Section 5 is unsupported. The authors should either provide a rigorous derivation of the representation or state the precise theorem and conditions under which it holds.
  4. [Section 4, Proposition 2] The negative binomial formula in (4.3) uses τ_n(ω), which is defined in (4.1) as a random stopping time, as the fixed parameter of a negative binomial distribution. In a standard negative binomial, the number of 'successes' (or failures) before the stopping event is fixed; substituting a random stopping time changes the distribution into a mixture over the law of τ_n, so the formula (4.3) is not a valid probability mass function as written. The Poisson approximation in (4.4) inherits the same problem because its mean parameter τ_n(ω)(1-π−ω)/π−ω is random. The authors should condition on τ_n and integrate over its distribution, or redefine τ_n as a fixed threshold.
  5. [Section 6, Tables 3-5] The regression-weighted composite proxy is constructed in-sample: the weights in (6.8) are estimated from the interaction coefficients in the same regressions that are then used to evaluate the composite's fit. The reported improvement in R² (0.043 to 0.049 in Table 3) is therefore a within-sample comparison and does not establish out-of-sample content. Moreover, the within R² is reported as 0.000 in all specifications, meaning that after bank and quarter fixed effects the regressors have no explanatory power within banks. The text should present the result as purely illustrative, and the claim of a 'modest improvement' should be qualified by the in-sample weighting scheme and the zero within fit.
minor comments (5)
  1. [Section 2.2.2, Eqs. (2.18)-(2.19)] The symbol σ is used both for the elasticity of intertemporal substitution in (2.18)-(2.19) and for liquidity risk σx in (2.7) and elsewhere; this dual usage is confusing and should be disambiguated.
  2. [Section 8.4, proof of Theorem 3.1] The proof of Theorem 3.1 is a single sentence that refers to the Internet Appendix for the formal topology, measurability, and random-field details, but the Internet Appendix is not included with the manuscript. The formal construction therefore cannot be checked by a reader or referee.
  3. [Section 6.2, Eq. (6.2)] The phrase 'constructed defensively' is vague; please specify the exact rule for when RCFD2200 is used versus the fallback construction, and why the fallback is preferred for all observations in the cleaned panel.
  4. [Section 2.2.2, Eq. (2.18)] The constant a0 in (2.18) is a free proportionality constant, and the calibration to λ ≈ e is not derived from the model; the text should more clearly state that a0 is a normalization rather than a model output.
  5. [Section 6.2, variable construction] The loss-aversion proxy LAbt is defined as the retail-funding share, which is admittedly a coarse balance-sheet proxy rather than a measure of loss aversion; the text acknowledges this, but readers would benefit from an explicit statement about the direction of the measurement error and how it would bias β2 in (6.13).

Circularity Check

3 steps flagged · score 6.0 of 10

The Bank Run Exposure State Space is defined as the set obeying Lemma 2's inequality, so Theorem 3.1 restates its own trigger; the taboo refinement relies on replacing the Kőbberling–Wakker limit by a ratio at arbitrary c1, and the empirical composite is fit and evaluated on the same sample.

  1. self definitional [Section 3, Lemma 3 (eq. 3.9), Lemma 4 (eq. 3.12), Theorem 3.1]
    "Let Ωadmiss be the set of admissible stress states for which the inequality in Lemma 2 holds. ... Then the set of admissible states for bank runs is given by Ωadmiss = {ω′ ∈ Ω : λ(ω′) > π+ω′v′g(c1ω′)/π−ω′v′ℓ(−c1ω′)}. ... Thus, the set of states that support a bank run is given by Ωrun = Ωadmiss ∩ Ωtaboo. ... Theorem 3.1 ... The Bank Run Exposure State Space (Ωrun, T run) is induced by loss aversion and uncertainty over comparatively large probabilities of bad shocks π−ω′ ≫ π+ω′ to fluctuations in income (x) and consumption (c)."

    By the paper's own sentence, Ωadmiss is exactly the set of states satisfying Lemma 2's inequality. Lemma 2 already asserts that this inequality is a sufficient stress-state condition for a bank run. Defining Ωrun as Ωadmiss ∩ Ωtaboo and then announcing that this space is induced by loss aversion and large bad-state probabilities restates the trigger condition under a new name. The Ωtaboo intersection cannot add independent content because its bounds (3.4) are asserted from Theorem 2.1, but Theorem 2.1 with separability gives only the lower bound λ ≥ k/k̃ (from c1ω ≥ 0); no upper bound λ < 1/k̃ follows. The state-space theorem is therefore the definition of the run-eligible set plus an unproved interval, not an implication.

  2. ansatz smuggled in via citation [Section 3, eqs. (3.2)-(3.4)]
    "Köbberling and Wakker (2005, pg. 121) used the ratio of left and right derivatives at the reference point c = 0 to define the loss aversion parameter with a L'Hospital type rule as λ = lim c↓0 v′g(c)/v′ℓ(−c). In the context of (3.1) this implies that λ(c1ω′) = v′g(c1ω′)/v′ℓ(−c1ω′)."

    The cited KW index is the limit of the marginal-value ratio as the gain/loss magnitude goes to zero; replacing it by the ratio at an arbitrary c1 is not an implication of that definition. It is an identification chosen precisely because the same ratio already appears inside Lemma 2's trigger inequality. Under the paper's own Tversky-Kahneman power calibration with equal exponents, v′g(c1)/v′ℓ(−c1) equals 1 for every c1, not the calibrated λ ≈ 2.25, so the two objects are not interchangeable. The bounds (3.4), k/k̃ ≤ λ(c1) < 1/k̃, are likewise not consequences of Theorem 2.1, which supplies only the lower bound.

1 more flagged steps
  1. fitted input called prediction [Section 6.2, eqs. (6.8)-(6.9); Section 6.4 and Table 3]
    "The regression-weighted index is constructed in two steps. First, each standardized component is interacted separately with liquidity volatility in a bank and quarter fixed-effect regression. Second, the absolute values of the three interaction coefficients are normalized to sum to one: wk = |β̂2k|/Σj|β̂2j|. ... In the present sample, the normalized weights are 0.169 for transaction share, 0.537 for core share, and 0.294 for consumer-loan share. ..."

    The composite exposure proxy is built from interaction coefficients estimated on the same bank-quarter panel used for the reported R2 comparison. The weights 0.169/0.537/0.294 are data-dependent, so comparing fit on that same sample measures in-sample accommodation of the weight-selection procedure rather than independent explanatory content. The abstract's claim that the composite modestly improves fit is therefore a fitted-input result, not an out-of-sample or cross-validated prediction, and the reported standard errors are not adjusted for the weight-estimation step. The paper's own proof-of-concept caveat lowers the weight of this step but does not remove the in-sample construction.

full rationale

Lemma 2 itself is a genuine KKT-derived inequality and is not circular: the model maximizes expected value subject to c0ω = x0ω and c1ω ≥ x1ω and compares weighted marginal loss against weighted marginal gain. The external calibrations (Tversky-Kahneman λ≈2.25; Abdellaoui et al.; Hall-CRRA) are parameter inputs rather than circular shortcuts, and the half-Cauchy dynamic construction is built up in the text, so the self-citations to Charles-Cadogan (2018, 2026) are not load-bearing for the main trigger condition. The circularity is concentrated in Section 3: Ωadmiss is declared to be the set where Lemma 2's inequality holds, so Theorem 3.1's state-space claim is a relabeling of its own sufficient condition; the taboo refinement (3.4) cannot be obtained from Theorem 2.1 (only λ ≥ k/k̃ follows from separability), and its construction depends on treating the KW limit as a ratio at arbitrary c1, an assumed identification rather than a derived result. The empirical proof-of-concept also builds LAreg from interaction coefficients estimated on the same panel used for the R2 comparison, so the modest fit gain is in-sample; the paper's own caveats (proof of concept, public data too coarse) appropriately limit the weight of this step. Overall, the central trigger condition has independent content, but the central state-space contribution is largely self-definitional, giving a partial-circularity score of 6.

Assumptions & free parameters 4 free parameters · 10 assumptions · 3 invented entities

The model rests on a large set of behavioral assumptions (paycheck dependence, zero wealth, loss aversion, ratcheting), one ad hoc linearity (Assumption 8), and a market-clearing identity that links income fluctuations to reserve fluctuations. The only fitted numbers in the empirical proxy are the composite weights; the loss-aversion median is calibrated from prior literature. No new physical entities are introduced; the state space and k-tilde are mathematical constructs.

free parameters (4)
  • a0 = not estimated; chosen so lambda approx 2.25-2.78
    Proportionality constant in the Hall/CRRA loss-aversion calibration, equations (2.18)-(2.20).
  • m_lambda = 2.25
    Median of the half-Cauchy loss-aversion process in Section 5, taken from Tversky and Kahneman (1992).
  • subjective bad-state probability pi-minus-omega = exogenous input
    Central to Lemma 2 and Proposition 2; assumed high in run states, never estimated.
  • composite proxy weights = 0.169, 0.537, 0.294
    Fitted to Call Report data in Section 6.2, equations (6.8)-(6.9), then used to claim improved fit on the same sample.
assumptions (10)
  • domain assumption Assumption 1: Agents have liquidity preference paycheck to paycheck.
    Basis for the whole two-period model; Section 2.1.
  • domain assumption Assumption 2: Agents are loss averse to fluctuations in wealth and income.
    Brings prospect theory into liquidity demand; Section 2.1.
  • domain assumption Assumption 3: Agents have zero wealth.
    Rules out buffer-stock saving, forcing paycheck dependence; Section 2.2.1.
  • domain assumption Assumption 4: Consumption follows the permanent income hypothesis.
    Justifies the MPC framework and reference consumption; Section 2.2.1.
  • ad hoc to paper Assumption 8: Fluctuations in consumption relate linearly to income fluctuations, ci = k-tilde(rho,d;lambda) xi, with separability in lambda.
    Load-bearing linearity for Theorem 2.1; not derived from deeper axioms; Section 2.2.1.
  • domain assumption Assumption 10: Agents ratchet consumption (consumption never falls).
    Ratcheting embeds reference dependence; Section 2.2.1, Definition 2.
  • domain assumption Market clearing: per-capita cash-reserve fluctuations coincide with per-capita income fluctuations.
    Maps income fluctuations to bank cash reserves; Section 5, paragraph before eq (5.1).
  • standard math The probability weighting measure w(dPhi(z)) is fixed with respect to local changes in (mu_x, sigma_x) and differentiation under the integral is valid.
    Needed for the mean-risk slope derivation (2.4); Section 8.1.
  • standard math Martingale-difference and previsibility conditions for the regression residuals, and existence of a Brownian martingale representation.
    Underpins Theorem 5.1; Section 5, footnote 25.
  • standard math Heat-bath construction: an OU process with stationary N(0,1), probability integral transform, and inverse half-Cauchy quantile.
    Builds the half-Cauchy loss-aversion process; Section 5, eq (5.5)-(5.7).
invented entities (3)
  • Bank Run Exposure State Space (Omega_run, T_run)
    purpose: Formal object collecting stress states in which loss aversion supports a bank run; used to define exposure dynamics.
    Defined in Section 3 from the model's own trigger condition; no falsifiable handle outside the model.
  • Loss-aversion-adjusted marginal propensity to consume k-tilde(rho,d;lambda)
    purpose: Intermediary that maps income fluctuations to consumption fluctuations and determines the ratchet condition.
    Postulated in Assumption 8; not separately measured.
  • Stochastic loss-aversion process lambda-bar(t) with half-Cauchy marginal
    purpose: Allows rare fear spikes around the Hall/CRRA benchmark in dynamic exposure.
    A modeling choice built from a Gaussian OU and quantile transform; no independent data support in this paper.

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Pith. "Pith review of Bank Run Exposure in a Paycheck-to-Paycheck Economy with Loss-Averse Depositors." pith.science (2026). https://pith.science/paper/D2PUYF67

@misc{pith2026260811266,
  author       = {Pith},
  title        = {Pith review of: Bank Run Exposure in a Paycheck-to-Paycheck Economy with Loss-Averse Depositors},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/D2PUYF67}},
  note         = {Machine review of arXiv:2608.11266}
}
read the original abstract

We develop a behavioural model of bank run exposure in a paycheck-to-paycheck economy with loss averse depositors. Income is received through demand deposits, and consumption ratcheting embeds reference dependence in a parsimonious asset-pricing framework. We show that sufficiently high subjective bad-state probabilities endogenously increase liquidity demand and generate equilibrium stress states supporting bank runs. These states define a Bank Run Exposure State Space and yield a martingale representation for exposure dynamics. A proof-of-concept empirical implementation using Call Report data constructs bank-level exposure proxies from funding and lending composition. A regression-weighted composite measure modestly improves fit relative to a retail-share benchmark, with stronger amplification among small banks and during the post-Silicon Valley Bank (SVB) collapse period. The framework highlights how behavioural liquidity demand alters equilibrium reserve holdings and can crowd out productive lending.

Figures

Figures reproduced from arXiv: 2608.11266 by the authors.

Figure 1
Figure 1. Calibration of the loss-aversion index 1 2 3 4 5 6 2.2 2.3 2.4 2.5 2.6 2.7 Hall/CRRA calibration curve Hall/CRRA risk parameter σ Loss−aversion index λ lambda(sigma) lambda = 2.25 lambda = e Heavy−tailed λ distribution Loss−aversion index λ (99% cap) Density 0 50 100 150 0.00 0.05 0.10 0.15 0.20 median 5% and 95% 99% plot cap 0 50 100 150 0.00 0.05 0.10 0.15 0.20 0.25 Kernel density of λ Loss−aversion index λ (99% c… view at source ↗
Figure 2
Figure 2. Simulated bank-run exposure regressions (a) Baseline (b) Raised reserve ratio (c) Lower withdrawal intensity (d) High loss aversion and fitted residuals Note: Cornflower-blue observations are ordinary states, transparent black observations are extreme or run states, and red observations are suspension states. The dashed bands are the 95 percent confidence bands around the linear fit. The scenario panels show that th… view at source ↗
Figure 3
Figure 3. Reserve-need stability when latent loss aversion [PITH_FULL_IMAGE:figures/full_fig_p040_3.png] view at source ↗
Figures from the paper (1 more)
Figure 5
Figure 5. Figure 5: Log bank-run exposure diagnostics 0 1 2 3 4 5 0.0 0.5 1.0 1.5 2.0 Median Log Bank−Run Exposure Path Time log(1 + βx(λ(t))) Log simulated paths Median log−exposure path Mean log−exposure path Log scale reveals ordinary−state movements hidden by heavy−tail spikes (a) Med…

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Pith tools

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