# WP19 — Results audit before drafting (Phase 6)

Date: 2026-10-20. Every claim intended for the abstract, results and conclusion is mapped to (a) a proposition in `proofs.md`,
(b) an exact result file / figure / table, or (c) a verified citation (`literature_matrix.csv`). Classification: **P** = proved,
**N** = numerically illustrated (baseline parameterisation, illustrative), **E** = empirically documented in the literature
(cited, not estimated here), **S** = speculative (must be labelled as such or dropped). Numbers quoted in the manuscript are injected
through the macros of `paper/tables/numbers.tex`, generated by `src/tables.py` from `results/processed/*.csv`; the values below were
read from the same files on 2026-10-20.

## A. Claims and their support

| # | Claim | Class | Support |
|---|---|---|---|
| A1 | For given $U$ and $\Lambda_s$, the seller value has a unique solution, $C^1$ in parameters | P | Lemma 2 |
| A2 | Reservation price and expected transaction price fall with carrying cost, the sale hazard rises; price falls less than the reservation price under IFR; participation threshold $\bar c=\Lambda_s\beta H(w_0+k)-rw_0$ | P + N | Prop. 1; `exp02_urgency_gradient.csv`, Fig. 2–3, Table 2b; test `test_prop1_monotonicity` |
| A3 | Baseline: $\theta^*=4.39$, TOM of sales 2.23 months, buyer duration 10.8 months, price s.d. 3.47\% with between-type share 13.1\%, urgency discount 5.59\% (most urgent) / 1.98\% (weighted), 90.7\% of listings sold | N | `exp01_baseline_moments.csv`, Table 2 |
| A4 | Population exit hazard decreases with duration although individual hazards are constant | P + N | Lemma 3; `exp01_baseline_survival.csv`, Fig. 7; analytic decline 5.42→5.30 over 0.6 y; MC inconclusive (validation.md §3) |
| A5 | Lower effective arrival rate lowers $V$ and $\bar p$ for every type; under IFR it lowers the hazard and widens the reservation gap | P + N | Prop. 2(i)–(iii); `exp02_urgency_gradient.csv` (five $A$ multiples), `exp03_arrival_shock.csv`, Fig. 2–4 |
| A6 | Pareto counterexample: hazard decreasing in $\Lambda_s$ iff $c>\delta w_0+(r+\delta)U$ | P + N | Prop. 2(ii); test `test_prop2_pareto_counterexample` (both signs verified) |
| A7 | Market liquidity gap $\Gamma$ relative to the Walrasian free-entry price changes sign: +43\% at $\Lambda_s=0.05$, zero near $\Lambda_s\approx135$, negative beyond | P (limits) + N | Prop. 2(iv); `exp03_gamma_sign_change.csv`, Fig. 4c; baseline $\Gamma=16.6\%$ with $p^*=1.19$ |
| A8 | Homogeneous sellers: unique $\theta^*$, $d\theta/d\phi>0$, $d\theta/dA>0$, $A$ and $\phi$ observationally equivalent, amplification elasticity $d\ln\Lambda_s/d\ln\phi=1.10>1$ | P + N | Prop. 3(a); `exp07_amplification.csv` (GE 1.10 vs PE 1.00), test `test_prop3_*` |
| A9 | Heterogeneous sellers: equilibrium exists; uniqueness verified numerically (single root in baseline, 60 random draws, 150 global draws) | P (existence) + N (uniqueness) | Prop. 3(b); `exp07_uniqueness_draws.csv`, `exp12_global_sensitivity.csv`, `exp01_free_entry_curve.csv`, Fig. 1a |
| A10 | A 50\% fall in $A$ lowers the mean price by 4.8\% (GE) vs 4.3\% (PE), raises TOM from 2.2 to 3.4 months, inventory from 0.186 to 0.284, $\theta$ from 4.39 to 3.73, urgency discount from 5.6\% to 7.6\% | N | `exp03_arrival_shock.csv`, Table 3, Fig. 4, 6 |
| A11 | Steady-state sales volume falls with liquidity iff $\delta>0$; with $\delta=0$ volume equals inflow and composition equals $f$ | P + N | Prop. 4(iv), Prop. 6(i); test `test_prop4_liquidity_shock` |
| A12 | Between-type variance is positive iff the transaction composition is non-degenerate; $\beta=0$ bound; range bound | P + N | Prop. 5; `exp08_dispersion_vs_beta.csv`, `exp08_heterogeneity_spread.csv`, Table 5, 5b; between share rises from 13.1\% ($c_H=0.3$) to 26.5\% ($c_H=0.6$); and from 13.1\% to 14.0\% when $A\times0.7$ |
| A13 | Impact decomposition of the index into within-type and composition terms; sign of the composition term = sign of $\mathrm{Cov}_\omega(\bar p,\epsilon)\,d\ln\Lambda_s$; exponential/Pareto amplify, light-tailed (truncated normal, Weibull shape 2) dampen | P + N | Prop. 6(ii)–(iii); `exp10_selection_decomposition.csv`, `exp10_hazard_times_mrl.csv`, Table 6, Fig. 9; test `test_prop6_impact_composition` |
| A14 | Composition term is quantitatively small: at most 0.15 log-\% (Pareto, $c_H=0.6$, $A\times0.7$) against within-type changes of 2–10\%; amplification ratio between 0.979 and 1.023 | N | `exp10_selection_decomposition.csv`, Table 6 |
| A15 | Transition after a permanent 20\% fall in $A$: price index jumps by −1.46\% on impact and stays; sales fall 13.1\% on impact and recover to −1.4\%; inventory rises 13.2\% with a half-life of 1.9 months; composition term ≤ 1 bp; temporary shock produces a 27.9\% rebound of sales when liquidity returns | N | `exp09_transition_*.csv`, Table 6b, Fig. 10 |
| A16 | Welfare: steady-state flow welfare maximised at $\beta=0.55$ for both homogeneous and heterogeneous sellers (Hosios value 0.50); loss at $\beta=0.5$ is 0.1\%; loss at $\beta=0.2$ is 14.1\%; planner's flow optimum exceeds the decentralised one by 0.30\% (het.) / 0.24\% (hom.) with lower tightness (3.80 vs 4.39) | N (not a theorem; flow criterion, see limitations) | `exp05_bargaining_power.csv`, `exp05_welfare_surface.csv`, `exp05_planner_benchmark.csv`, Table 8, 8b, Fig. 11; cites hosios1990efficiency, gabrovski2025note |
| A17 | Financing extension: mortgage–cash gap = delay-bargaining + outside-option + selection; $\Delta_b$ increasing in $c$; selection term has the sign of $v^*_m-v^*_c$ | P | Prop. 7; test `test_financing_extension` (closed form of $\Delta_b$ verified) |
| A18 | Baseline extension: weighted gap 1.73\% (patient 1.12\%, urgent 2.96\%) = $\Delta_b$ 0.49\% + outside option 1.61\% + selection −0.37\%; gap rises to 2.42\% at $\phi_m=0.70$ and to 6.0\% with a four-month approval delay; $U_c=0.0426>U_m=0.0103$ | N | `exp04_financing_phi.csv`, `exp04_financing_nu.csv`, Table 4b–4c, Fig. 5 |
| A19 | With $\phi_m\le0.60$ mortgage buyers' value is negative (participation violated); results reported only for $\phi_m\ge0.70$ | N (failure region) | Table 4b column $U_m\ge0$ |
| A20 | Matching-efficiency vs approval shocks are distinct in the extension: a 30\% fall in $A$ lowers prices by 2.34\% and raises time to closing from 2.97 to 3.44 months; a fall of $\phi_m$ to 0.70 lowers prices by 0.94\%, raises time to closing to 3.31 months and widens the gap to 2.41\% (vs 1.81\%) | N | `exp06_matching_vs_approval.csv`, Table 4e |
| A21 | Posted prices: pickier sellers (TOM 4.38 vs 2.23 months), higher prices conditional on sale (1.0195 vs 0.996), lower tightness (2.91), all dispersion between-type by construction (within-type s.d. zero; total s.d. 2.80\%), welfare 11.2\% lower, urgency gradient preserved | P (Prop. 8) + N | `exp11_protocol_comparison.csv`, `exp11_types_*.csv`, Table 9, Fig. 14 |
| A22 | Signs robust: in 150 random parameter draws, $D_{urg}>0$ in 100\%, $\bar p$ decreasing in 100\%, $\eta$ increasing in 98.7\% (the exceptions are flat-hazard draws at the discretisation noise level, not reversals); TOM ranges 0.6–10.7 months, $D_{urg}$ 2.9–20.3\% | N | `exp12_global_sensitivity.csv`, `exp12_sensitivity_oat.csv`, Table 7, Fig. 12 |
| A23 | Monte Carlo listing simulation reproduces analytic moments within 0.39\% (100,000 listings) | N (validation) | `val_montecarlo.csv`, `validation.json` |
| A24 | Empirical magnitudes used for comparison (forced-sale discount ≈27\%; cash discount 2–3.9\%; mortgage premium 11\%; contact-hazard elasticity 0.84) | E | campbell2011forced, han2024cash, reher2024mortgage, genesove2012search |

## B. Claims removed or weakened after the audit

- "Selection bias amplifies measured price responses" — **weakened**: the sign depends on $G$ (Prop. 6(iii)); for the light-tailed
  baseline it dampens; magnitudes are second-order (A14). The manuscript states the sign condition and the magnitudes.
- "Mortgage deals require better matches" — **weakened** to the conditional statement of Prop. 7(ii); in the baseline the selection
  term is negative for patient sellers because cash buyers' outside option is better.
- "Hosios efficiency with heterogeneous sellers" — **not claimed**; the welfare criterion is the steady-state flow surplus, the optimum
  is found numerically at $\beta=0.55$ for both economies, and the heterogeneity wedge is negligible (0.01 pp). Presented as a
  numerical result with the caveat on the flow criterion.
- "Prices and volume co-move" — stated with the $\delta>0$ qualification (A11).
- Any statement about Québec or Canada — **removed**; no data.
- "Liquidity premium $D=(p^*-p)/p^*$ is positive" — **replaced** by the two-object convention (urgency discount $D(c)\ge0$; market gap
  $\Gamma$ of either sign, A7).

## C. Failed experiments and regions without equilibrium

- Financing extension, first solver (damped fixed point): divergent; replaced (validation.md §1).
- Financing extension with $\phi_m\le0.60$: equilibrium exists but violates buyer participation ($U_m<0$); excluded from the text.
- No parameter draw without an equilibrium; no multiple roots detected.
- Heavy-tailed match values: equilibria exist but are empirically implausible (prices above the mean valuation); used only as
  theoretical illustrations.

## D. Sign conventions, units, sample selection — checked

- All prices and values in units of the mean buyer valuation $q=1$; time in years; hazards per year; months = 12 × years.
- $D(c)=1-\bar p(c)/\bar p(c_L)\ge0$; $\Gamma=1-\bar p_T/p^*$ of either sign; composition term $\ln(P^{obs}/P^{fix})$ relative to the
  pre-shock transaction weights.
- Transaction-weighted moments use $\omega(c)=\eta(c)n(c)/Q$; listing-cohort moments use $f(c)$; the stock uses $N(c)$. Each table
  states which weighting is used.
- JEL: R31, R21, D83, C63, G51 (G51 retained because of the financing extension; G12 dropped: no asset-pricing content).