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\section{Cost-approach theory and a formal extraction-to-valuation framework}
\label{sec:framework}
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\subsection{The cost identity and what each term may legitimately contain}
\label{sec:identity}

The cost approach produces an \emph{indication of value} for an improved property by adding the value of the land, assumed vacant and available for its highest and best use, to the cost of producing the improvements new, less the loss in value that the existing improvements have suffered relative to a new equivalent. For property $i$ we write
\begin{equation}
\widehat V_i^{\mathrm{cost}} \;=\; \widehat L_i \;+\; \bigl(\widehat{RCN}_i + \widehat S_i\bigr)\,(1-\widehat D_i),
\label{eq:identity}
\end{equation}
where $\widehat L_i$ is the land value, $\widehat{RCN}_i$ the replacement cost new of the principal improvement, $\widehat S_i$ the cost new of site improvements that are valued separately from the building (driveway, landscaping, pool), and $\widehat D_i\in[0,1)$ the accrued depreciation rate applied to the improvements. Equation~\eqref{eq:identity} differs from the familiar additive display $L + RCN - D + S$ of the author's specification only in that depreciation is written as a rate applied to all improvements, which is how it is implemented and which makes the no-double-counting requirement explicit: physical deterioration, functional obsolescence and external obsolescence are combined multiplicatively, $1-D=(1-D^{\mathrm{phys}})(1-D^{\mathrm{func}})(1-D^{\mathrm{ext}})$, so that no component is charged twice. Three scope statements govern the identity throughout the paper.

First, $RCN$ is \emph{replacement} cost, the cost of a building of equivalent utility built with current materials and methods, not \emph{reproduction} cost, the cost of an exact replica. Replacement cost is the appropriate base for a mass-appraisal setting in which the subject's original materials are unknown, and it is the concept that the published per-area schedules we use describe. The paper never attempts to reproduce historical construction details from photographs.

Second, market price is not construction cost. The identity is an indication of value under the economic principle of substitution; the relation between $\widehat V_i^{\mathrm{cost}}$ and an observed asking or transaction price $P_i$ is an empirical question that depends on supply conditions, land scarcity and the depreciation actually perceived by buyers \citep{glaeser2005why,davis2007price}. We therefore call comparisons with prices \emph{market alignment} and never accuracy.

Third, the assessed values of the Québec roll are themselves indications produced under a cost-based mass-appraisal manual (Section~\ref{sec:setting}); the roll's building value $B_i^{\mathrm{roll}}$ is used as an \emph{assessor benchmark} after temporal indexing, and its land value $L_i^{\mathrm{roll}}$ as the independent evidence for $\widehat L_i$. Both are judgments, with their own errors \citep{mcmillen2020assessment}, not ground truth.

\subsection{Observations, evidence classes and the attribute vector}
\label{sec:evidence}

A listing $i$ consists of a set of photographs $\{x_{ik}\}_{k=1}^{K_i}$ in the order chosen by the seller, a free-text description $t_i$, a set of broker-entered fields $b_i$, and, through a spatial match, the administrative record $a_i$ of the assessment-roll unit. The appraisal-relevant characteristics of the property are collected in an attribute vector $\theta_i=(\theta_{i1},\dots,\theta_{iJ})$ whose components belong to one of five \emph{evidence classes} (Table~\ref{tab:ontology}):
\begin{itemize}
  \item \textsc{admin}: recorded by the assessor (year built, floor area, lot area, storeys, assessed land and building values);
  \item \textsc{visual}: directly observable in a photograph (cladding material, roof covering, garage, kitchen counters, basement finish);
  \item \textsc{proxy}: visual constructs that stand in for a latent cost driver (finish grade, apparent condition, renovation evidence);
  \item \textsc{text}: claimed by the seller or broker (renovations and their years, condition statements, materials listed in structured fields);
  \item \textsc{latent}: not inferable from listing media (foundation integrity, hidden moisture, structural soundness, permits, tenure, exact area).
\end{itemize}
The pipeline never estimates a \textsc{latent} attribute; it treats it as unknown and, where it matters for value, as a scenario. The distinction between \textsc{visual} and \textsc{proxy} is the distinction between a fact that two careful observers would agree on from the same image and a graded judgment on which they could reasonably differ; the evaluation design (Section~\ref{sec:methodology}) tests the former against reference fields and the latter against cross-extractor agreement.

\subsection{Per-image observation and cross-image aggregation}
\label{sec:aggregation}

For each image $x_{ik}$ the extractor $m$ returns a scene label $s_{ik}^m\in\mathcal S$, a vector of attribute observations $\hat\theta_{ik}^m=(\hat\theta_{ikj}^m)_j$ whose components may take the value \texttt{not\_visible} (the attribute is relevant to the scene but cannot be seen) or \texttt{not\_applicable} (the attribute does not pertain to the scene), and a stated confidence $c_{ik}^m\in[0,1]$. Let $\mathcal K_{ij}$ denote the set of images whose scene is relevant to attribute $j$ (exterior scenes for cladding, kitchen scenes for counters, and so on) and $\mathcal I_{ij}\subseteq\mathcal K_{ij}$ the subset in which the observation is informative. The property-level estimate is a deterministic function of the per-image observations: for a nominal attribute, the confidence-weighted plurality
\begin{align}
w_{ij}^m(v)&=\sum_{k\in\mathcal I_{ij}}\max(c_{ik}^m,\underline c)\,\mathbf 1\{\hat\theta_{ikj}^m=v\},\qquad
\hat\theta_{ij}^m=\arg\max_{v}\,w_{ij}^m(v),
\label{eq:vote}\\
\pi_{ij}^m&=\frac{w_{ij}^m(\hat\theta_{ij}^m)}{\sum_{v}w_{ij}^m(v)},
\label{eq:voteshare}
\end{align}
with $\underline c=0.05$ a floor that keeps zero-confidence votes from vanishing and $\pi_{ij}^m$ the vote share, which we use as an internal consistency measure; for an ordinal grade, the weighted median of the ordinal codes; for a condition score, the weighted mean over informative images. When $\mathcal I_{ij}=\emptyset$ the attribute is \texttt{not\_visible} at the property level and the cost model falls back to a declared default. Truncating the image set to the first $K$ photographs in seller order, $\mathcal K_{ij}(K)$, and recomputing \eqref{eq:vote} gives the image-count sensitivity of Section~\ref{sec:results_extraction} without any additional model call.

The composite condition $\bar c_i^m$ is the mean of the exterior and interior condition scores when both exist, and the composite finish grade $g_i^m\in\{\text{basic},\text{standard},\text{upgraded},\text{luxury}\}$ is the rounded weighted average of the kitchen grade (weight~0.5), the bathroom grade (0.3) and the interior finish-quality score rescaled to the grade range (0.2), each term entering only when observed. These weights are declared, not fitted; Section~\ref{sec:robustness} varies them.

\subsection{Text--image--record reconciliation}
\label{sec:reconciliation}

Let $\tilde\theta_{ij}$ be the text-claimed value of attribute $j$ (from the broker fields and description, with prices and assessments masked) and $a_{ij}$ the administrative value when the attribute is in the roll. The reconciled value and its status are
\begin{equation}
\theta_{ij}^{\star}=
\begin{cases}
a_{ij} & j\in\textsc{admin},\\
\hat\theta_{ij}^m & \hat\theta_{ij}^m \text{ informative},\\
\tilde\theta_{ij} & \text{otherwise, if informative},\\
\text{default}_j & \text{otherwise},
\end{cases}
\label{eq:reconcile}
\end{equation}
with a status $\sigma_{ij}\in\{\text{agree},\text{contradict},\text{visual only},\text{text only},\text{unknown}\}$ that compares the visual and text channels when both are informative. The ordering in \eqref{eq:reconcile} gives precedence to the record, then to what is seen, then to what is claimed; it is a design choice motivated by the incentive structure of listings \citep{levitt2008market,nowak2017textual} and we report how often it is exercised. Contradictions are not resolved silently: they widen the uncertainty interval of Section~\ref{sec:uncertainty}.

\subsection{Mapping attributes to the cost schedule}
\label{sec:schedule}

The replacement cost new is the product of a quantity, which is administrative, and a unit cost, which is a function of attributes:
\begin{equation}
\widehat{RCN}_i = A_i\;\kappa\bigl(g_i\bigr)\,\mu_{\mathrm{st}}(n_i)\,\mu_{\mathrm{att}}(\tau_i)\,\phi_{r(i)}\,\psi_{t}
\;+\;\beta_{\mathrm{bsmt}}(f_i)\,\frac{A_i}{n_i}\;+\;\gamma_{\mathrm{gar}}(h_i),
\label{eq:rcn}
\end{equation}
where $A_i$ is the roll floor area (sum of above-grade floor areas), $\kappa(g)$ the per-area hard construction cost for finish grade $g$ taken from a published schedule, $\mu_{\mathrm{st}}(n)$ a storey multiplier that spreads foundation and roof costs over the floor area, $\mu_{\mathrm{att}}(\tau)$ a party-wall adjustment for attached houses, $\phi_r$ a regional level factor, $\psi_t$ the temporal update of the schedule to the valuation quarter by the Statistics Canada building construction price index, $\beta_{\mathrm{bsmt}}(f)$ the basement-finish add-on per unit of footprint and $\gamma_{\mathrm{gar}}(h)$ the garage add-on. Indirect costs and entrepreneurial incentive are applied as proportional mark-ups, $RCN_i=\widehat{RCN}_i^{\mathrm{hard}}(1+s)(1+e)$. Every parameter is listed with its source or its status as an assumption in Table~\ref{tab:schedule_parameters}; the two published anchors for $\kappa(\cdot)$ are compared in Section~\ref{sec:results_cost}.

Physical depreciation follows the age--life method with a condition-adjusted effective age,
\begin{equation}
D_i^{\mathrm{phys}}=\min\Bigl\{\frac{\max\{\mathrm{age}_i\,\lambda(\bar c_i)-\rho(\mathrm{ren}_i),\,1\}}{\mathrm{EL}},\;\bar D\Bigr\},
\label{eq:dep}
\end{equation}
where $\lambda(\cdot)$ is a decreasing function of the composite condition equal to one at the average score, $\rho(\cdot)$ removes a fixed number of years when renovation evidence is partial or extensive, $\mathrm{EL}$ is the economic life and $\bar D$ a cap. In the baseline the functional and external components are set to zero, because nothing in a listing supports their measurement; an external-obsolescence scenario is reported.

\subsection{Uncertainty: what propagates and what does not}
\label{sec:uncertainty}

The cost indication inherits three kinds of uncertainty, and we treat them differently. \emph{Attribute uncertainty} arises because $\hat\theta_{ij}^m$ may be wrong or missing; we represent it by a distribution over the finish grade and the condition whose spread increases with $1-c_i^m$, with the absence of an informative observation, and with a recorded contradiction. \emph{Parameter uncertainty} arises because unit costs, economic life, mark-ups, the land index and the regional factor are known only within bands; we represent it by uniform draws over the published or stated bands. \emph{Latent uncertainty}---the possibility that a sound-looking house hides a defective foundation---cannot be propagated because it has no measurable distribution in our data; it is excluded from the intervals, which are therefore explicitly \emph{conditional on the absence of hidden defects}, and it is discussed as the main reason why the cost approach from listing media is a screening tool rather than an appraisal.

Two elementary results frame the empirical analysis. Both follow directly from the identity and from standard measurement-error arguments; proofs are in Appendix~\ref{app:proofs}.

\begin{proposition}[Value of an image observation for the cost indication]
\label{prop:value}
Let the finish grade $g_i$ be a cost driver with unit-cost schedule $\kappa(g)$, and let the appraiser's prior over $g_i$ in the absence of images be the frame distribution $p_g$. Under a quadratic loss on the improvements component, the expected reduction in loss from observing a visual grade $\hat g_i$ with confusion matrix $Q=[q_{g\hat g}]$ equals $A_i^2\mu_i^2\,[\operatorname{Var}_{p}(\kappa(g))-\mathbb E_{\hat g}\operatorname{Var}(\kappa(g)\mid\hat g)]$, where $\mu_i$ collects the multiplicative terms of \eqref{eq:rcn}; it is zero when $Q$ is uninformative (rows equal) and increases with the diagonal dominance of $Q$ and with the spread of the schedule. In particular, an observation that is correct with probability $\pi$ and otherwise uniform over the other grades reduces the loss by a factor that is increasing in $\pi$ and in the between-grade dispersion of $\kappa$, and is scaled by the depreciation factor $(1-D_i)^2$: the value of a visual grade observation is smaller for older houses.
\end{proposition}

\begin{proposition}[Misclassification of condition and bias of depreciation]
\label{prop:condition}
Suppose the true composite condition is $c_i$ and the extractor reports $\hat c_i=c_i+\varepsilon_i$ with $\mathbb E[\varepsilon_i\mid c_i]=\delta(c_i)$. If $\lambda$ is affine on the relevant range, the age--life depreciation computed from $\hat c_i$ has bias $\mathrm{age}_i\,\lambda'\,\delta(c_i)/\mathrm{EL}$ before the cap; a uniform upward bias in reported condition (``everything looks well kept'') therefore understates depreciation by an amount that grows linearly with age and shrinks with the assumed economic life, and a condition-dependent bias ($\delta'<0$, extremes pulled toward the middle) compresses the dispersion of depreciation across properties without changing its mean.
\end{proposition}

Proposition~\ref{prop:value} says what the experiments on grade agreement and on the spread of the cost schedule jointly measure; Proposition~\ref{prop:condition} says why the calibration of condition scores against an external reference, which this study could not perform, matters more for old houses than for new ones. Neither proposition asserts that images improve valuation; that is the empirical question.