Price as Geometry: Resolution, Coarse-Graining, and the Structure of Market Noise

The central question of quantitative finance is deceptively simple: does the price process have memory? The answer depends entirely on the timescale at which you ask it. At millisecond resolution, returns are mean-reverting. At hourly resolution, they behave approximately as geometric Brownian motion. No single model captures both regimes. The transition between them is a geometric fact about coarse-graining.

This observation is not specific to any one market. Lo and MacKinlay applied variance ratio tests to CRSP daily equity returns in 1988 and found systematic departures from the random walk null.^[1] Cont's survey of equity stylized facts documented the same resolution-dependent pattern (short-horizon mean reversion, vanishing autocorrelation beyond a few minutes, fat tails) across dozens of equity markets.^[21] Lo's 1991 study of stock market prices found Hurst exponents significantly below one-half at short horizons using the rescaled range statistic, directly analogous to the DFA results here.^[22] What this post adds is a unified geometric framework that explains why these patterns must appear, together with high-resolution empirical measurements at millisecond precision that resolve the fine structure invisible to daily studies.

The organizing variable throughout is the temporal resolution $\delta$: the coarse-graining map that projects the microscopic trade-by-trade price process onto a sequence of binned log-returns. Every statistical property we examine turns out to be a function of $\delta$. Stationarity, volatility scaling, autocorrelation, spectral structure, and information flow between assets all change as $\delta$ varies.

The empirical measurements use BTC and ETH tick data from Binance spot: 1.3 million BTC trades and 800,000 ETH trades per day, captured at microsecond timestamps, covering the full calendar year 2025. Crypto markets provide an unusually clean laboratory for this investigation: continuous 24-hour trading with no opening or closing auction artifacts, freely available microsecond tick data, and a well-studied two-asset pair. The qualitative findings are consistent with the equity microstructure literature throughout.

1. Philosophical Priors

No market ontology is assumed here. The price of any liquid asset is not "fundamentally" a diffusion, a jump process, a rough path, or any other stochastic object. It is the output of a deterministic system: a very large number of interacting agents, each choosing to submit, cancel, or execute orders according to their own information and objectives. The mid-price at any moment is the projection of that high-dimensional deterministic system onto a single observable scalar.

The apparent stochasticity of prices is an artifact of observation, not of the underlying system. When we bin trades into millisecond buckets and compute log-returns, we are performing a coarse-graining operation. The resulting time series inherits statistical properties that depend on the coarse-graining timescale. At fine resolution, the price at time $t + \delta$ reflects many individual order decisions, each with some structure; at coarse resolution, it averages over so many decisions that the central limit theorem pushes the residual toward Gaussian noise with independent increments.

This prior has a precise mathematical form, developed in Section 3. For now, the key point is that asking "is the market stationary?" or "does price follow a random walk?" without specifying the resolution is not a well-posed question. We will always answer relative to a specific $\delta$.

The philosophical framing carries an operational consequence: any trading strategy, risk model, or derivative pricing formula is implicitly a claim about which stochastic model is closer to true at the resolution at which the strategy operates. A market-making strategy that operates at 10 ms is making a different claim from a momentum strategy that operates at one hour. This post measures, rather than assumes, which claims are empirically defensible and at which resolutions.

We return to this framing at the end of Section 8, where the information-geometric perspective makes it especially concrete: the Fisher-Rao metric on the family of price return distributions is itself a function of $\delta$, and the market's information geometry genuinely changes as you coarsen your clock.

2. On-Ramp: The Same Question, Plainly Stated

This section introduces the key objects without set-builder notation. Readers already comfortable with log-returns and the variance ratio test may skip to Section 3.

The mid-price. The price of a traded asset on a limit order book is not a single number. At any moment, there is a best ask $p^{\mathrm{ask}}$ (the cheapest price at which someone is willing to sell) and a best bid $p^{\mathrm{bid}}$ (the highest price at which someone is willing to buy). The mid-price is their average,

We use the mid-price rather than the last trade price because it filters out the alternating "uptick/downtick" pattern caused by trades bouncing between the bid and ask, a microstructure artifact that would dominate any short-timescale analysis of the last-trade series.

The order book. The spread $s = p^{\mathrm{ask}} - p^{\mathrm{bid}}$ is the cost of an immediate round-trip: buy at the ask, sell at the bid. Depth refers to the quantity available at each price level. A market order that exceeds the depth at the best bid or ask walks the book, moving the mid-price. The interplay between depth and order flow is what creates price impact, the tendency of large orders to move prices against the trader who placed them.

The log-return. At resolution $\delta$, the log-return at time $t$ is

We use log-returns rather than percentage returns for two reasons. First, they are additive over time: the log-return over two periods is the sum of the two single-period log-returns, which makes multi-scale analysis algebraically clean. Second, they are approximately symmetric around zero for small moves, whereas percentage returns have an upward bias for large positive moves. At millisecond timescales the distinction barely matters, but at longer horizons it is material.

Volatility and the $\sqrt{\delta}$ prediction. The realized volatility at resolution $\delta$ over a window of $n$ periods is

If returns were independent and identically distributed, the volatility at resolution $k\delta$ would satisfy $\sigma_{k\delta} = \sqrt{k}\,\sigma_\delta$. This is the square-root-of-time scaling, the signature of a process with no memory. It is the null hypothesis. We will test it directly in Section 4 using the variance ratio statistic.

Stationarity. A time series is stationary if its statistical properties do not change over time. A river in steady flow is stationary: the distribution of water heights at noon today is the same as at noon a year ago. A river during a flood is not. For price processes, stationarity means that the distribution of log-returns at any given resolution does not drift as time passes. Whether BTC log-returns are stationary is a resolution-dependent question: they may be stationary at a one-hour timescale while exhibiting systematic trends at a five-year timescale.

The random walk. The canonical null hypothesis in finance is that the price follows a random walk: tomorrow's price is today's price plus an unpredictable shock drawn independently from some fixed distribution. Under the random walk null, no strategy that looks only at past prices can forecast future returns. The variance ratio equals one at every timescale, the autocorrelation of returns is zero at every lag, and the Hurst exponent is exactly one-half.

Why this matters. Every trading strategy, risk model, and option pricing formula is an implicit bet on one of two things: either the random walk null holds (at the relevant timescale), or it fails in a specific exploitable direction. Market-making strategies assume short-timescale mean reversion and would lose money if the null held exactly. Trend-following strategies assume persistent autocorrelation and would lose money if the null held exactly. The goal of this post is to establish, empirically and geometrically, which assumptions are warranted at which resolutions.

3. The Coarse-Graining Map

We now develop the precise mathematical framework. All of what follows rests on one idea: the temporal resolution $\delta$ is a fundamental parameter that changes the statistical model at every level.

Let $(\Omega, \mathcal{F}, P)$ be a probability space supporting the microscopic price process $(S_t)_{t \ge 0}$, where $S_t$ is the mid-price at time $t$.

The question "is the market stationary?" is the question whether $P^\delta$ varies with calendar time for a fixed $\delta$. A conceptually distinct question is whether the family $\{P^\delta\}_{\delta > 0}$ is constant in $\delta$. If it is, the process has scale-invariant statistics: the random walk null and geometric Brownian motion both satisfy this. If it is not, the process has resolution-dependent structure.

Proof. We must show that the two conditions are equivalent: (i) $S^{(\delta)}$ is Markov, and (ii) $P_s \circ \Pi_\delta = \Pi_\delta \circ P_s$ for all $s > 0$.

Recall that $S^{(\delta)}_t = S_{\lfloor t/\delta \rfloor \delta}$, so the process only changes at the grid points $\{n\delta : n \in \mathbb{N}\}$. The $k$-step transition density of $S^{(\delta)}$ is obtained by integrating out the continuous path between grid points: by the Chapman-Kolmogorov identity applied to the semigroup $(P_t)$,

where $p_t(x,y)$ is the transition density of the original process $S$. This holds because $\mathbb{E}[f(S_{k\delta}) \mid S_0 = x] = P_{k\delta} f(x)$ by the semigroup property, and evaluating at the grid point eliminates all intermediate-time information.

The Markov property for $S^{(\delta)}$ states that for all bounded measurable $f$ and all grid times $m\delta, n\delta$ with $n > m$,

The left side equals $\Pi_\delta[P_{(n-m)\delta} f](S_{m\delta})$ (condition on the coarsened filtration, then apply the semigroup). The right side equals $P_{(n-m)\delta}[\Pi_\delta f](S_{m\delta})$ (apply the semigroup first, then condition on the current grid value). These are equal for all $f, m, n$ if and only if the two operators commute:

The converse is immediate: if the operators commute then both sides of the conditional expectation identity agree, establishing the Markov property. $\square$

Proof. Under GBM, the log-price satisfies $\log S_t = \log S_0 + (\mu - \tfrac{1}{2}\sigma^2)t + \sigma W_t$ by Ito's formula. For any two times $s < t$, the log-return over $[s,t]$ is

Since $W_t - W_s \sim \mathcal{N}(0, t-s)$ by the definition of Brownian motion, the $\delta$-increment has distribution

so $\mathrm{Var}(r^{(\delta)}_t) = \sigma^2\delta$. Moreover, increments over non-overlapping intervals are independent: for $[t_1, t_1+\delta], [t_2, t_2+\delta]$ with $t_2 \ge t_1 + \delta$, the increments are driven by disjoint pieces of the Brownian path.

The $k\delta$-return is the sum of $k$ consecutive $\delta$-returns:

By independence and identical distribution of the summands,

Substituting into the variance ratio definition,

This holds for every $k \ge 1$ and every $\delta > 0$, confirming that GBM is closed under coarse-graining with exact unit variance ratio at every scale. $\square$

The corollary makes the connection between theory and empirics precise. We return to this framing in the conclusion.

4. The Random Walk Null

We now operationalize the null hypothesis and test it at nine resolutions, from 1 ms to 4 hr.

Proof sketch. The variance ratio can be written as

where $\hat{\sigma}^2_j = \frac{1}{nj}\sum_{t=j}^{n} (X_t^{(j)})^2$ (demeaned). Under the null $X_t^{(1)}$ are i.i.d. with mean zero and variance $\sigma^2$, so the numerator becomes a linear combination of sample second moments at overlapping horizons. Expanding the $k$-period variance estimator,

where $\hat{\gamma}(j) = \frac{1}{n}\sum_t X_t^{(1)} X_{t+j}^{(1)}$ are the sample autocovariances. Under the null, $\hat{\gamma}(j) \to 0$ for all $j \ge 1$, and by the CLT for sample autocovariances of i.i.d. sequences, $\sqrt{n}\,\hat{\gamma}(j) \xrightarrow{d} \mathcal{N}(0, \sigma^4)$ jointly for $j = 1, \ldots, k-1$, with uncorrelated components. Applying the delta method and summing the covariance contributions from the $(k-j)^2$ weighting in the expansion yields variance

The sum identity $\sum_{j=1}^{k-1}(k-j)^2 = k(k-1)(2k-1)/6$ is the standard sum-of-squares formula. The Lindeberg CLT applied to the triangular array of summands completes the argument, giving the stated Gaussian limiting distribution.^[1] $\square$

Proof sketch. DFA proceeds as follows. Given observations $X_1, \ldots, X_N$, form the profile $Y_k = \sum_{i=1}^k (X_i - \bar{X})$. Partition $\{1,\ldots,N\}$ into $\lfloor N/s \rfloor$ non-overlapping windows of size $s$. In each window, fit a local polynomial trend and compute the root-mean-square residual $F(s)$. The DFA scaling hypothesis is

so $H$ is estimated as the slope of the log-log regression of $F(s)$ against $s$.

The consistency argument has two steps. First, one shows that $F(s)^2$ concentrates around its expectation $\mathbb{E}[F(s)^2]$ at rate $1/\sqrt{N/s}$ by an LLN argument applied to the squared detrended residuals, using weak dependence (short memory of squared increments). Second, one identifies $\mathbb{E}[F(s)^2] \sim C^2 s^{2H}$ via the structure of the covariance function of the profile process: for a long-memory process with covariance $\gamma(k) \sim L(k)|k|^{2H-2}$, the variance of the detrended profile over a window of size $s$ grows as $s^{2H}$ by the Karamata theorem for regularly varying functions. The log-log slope estimator is then consistent by the delta method applied to the OLS regression of $\log F(s)$ on $\log s$ across the window sizes $s_1 < \cdots < s_m$, as $N \to \infty$ with $s_m = o(N)$.^[2] $\square$

Empirical results. The figures below show the variance ratio curve and DFA log-log plot for BTC and ETH over the full year 2025, at scales 1 ms to 4 hr. The interactive component lets you explore the VR curve and Hurst estimate at each scale.

Key finding. The Hurst exponent is below $\frac{1}{2}$ at sub-second scales for both assets, confirming mean reversion at millisecond resolution. At hourly scales, $\hat{H} \approx \frac{1}{2}$, consistent with GBM. The random walk null fails at the scales where high-frequency strategies operate. This pattern agrees with Lo's equity findings^[22] and with the cross-market stylized facts surveyed by Cont.^[21]

Equity comparison. The VR profile shape, below 1 at sub-minute horizons and converging to 1 at hourly horizons, is a universal feature of liquid equity markets. Lo and MacKinlay documented it across 2000 CRSP-listed stocks including small-cap portfolios where mean reversion is most pronounced.^[1] Hasbrouck's detailed analysis of NYSE blue chips including IBM, GE, and Merck found intraday VR profiles qualitatively identical to those measured here: the random walk null is rejected below the 5-minute horizon for every major name examined, but cannot be rejected at 1-hour resolution.^[24] The crossover scale from mean-reverting to diffusive differs by asset (roughly 2-10 minutes for large-cap equities versus 10-60 seconds for BTC/ETH), reflecting the different market structures and participant compositions, but the qualitative shape of the VR curve is the same.

5. Stationary Alternatives: Mean Reversion as Geometry

The empirical rejection of the random walk null at short timescales invites the question: which model fits better? We survey the main stationary alternatives, framed geometrically.

Proof sketch. We characterize all continuous-time stationary Gaussian Markov processes. A process $(X_t)_{t \ge 0}$ that is simultaneously Gaussian and Markov is determined by its mean function $m(t) = \mathbb{E}[X_t]$ and covariance function $K(s,t) = \mathrm{Cov}(X_s, X_t)$.

Stationarity requires $m(t) = \mu$ (constant) and $K(s,t) = \gamma(|t-s|)$ for some function $\gamma$.

Markov property for a Gaussian process is equivalent to requiring that the partial correlation between $X_s$ and $X_t$ (for $s < u < t$) vanishes when conditioning on $X_u$. For a Gaussian process this means

Using the Gaussian conditional covariance formula and the stationarity condition $K(s,t) = \gamma(|t-s|)$, this becomes the functional equation

Setting $h = t - u$, $l = u - s$ and $c = 1/\gamma(0)$, the equation reads $\gamma(h+l) = c\,\gamma(h)\gamma(l)$, which is Cauchy's multiplicative functional equation on $(0,\infty)$. Under mild regularity (measurability or monotonicity, both satisfied by any covariance function), the only solutions are $\gamma(\tau) = \gamma(0)\,e^{-\kappa\tau}$ for some $\kappa \ge 0$.

The exponential covariance $\gamma(\tau) = \frac{\sigma^2}{2\kappa}e^{-\kappa\tau}$ corresponds exactly to the stationary distribution $\mathcal{N}(\mu, \sigma^2/(2\kappa))$ of the OU process $\mathrm{d}X_t = -\kappa(X_t - \mu)\,\mathrm{d}t + \sigma\,\mathrm{d}W_t$. Brownian motion ($\kappa = 0$) is non-stationary and excluded. Thus, up to the affine reparametrization $(X - \mu)/\sqrt{\gamma(0)}$, the OU process is the unique member of this class.^[4] $\square$

The geometric interpretation reveals why mean reversion is a universal feature at short timescales. The key is the flat geometry of the OU process versus the potentially curved geometry of more general diffusions.

One model in the table warrants a preliminary definition. Fractional Brownian motion (fBm) with Hurst parameter $H \in (0,1)$ is the unique (up to scaling) continuous Gaussian process with stationary increments satisfying $\mathrm{Var}(B^H_t - B^H_s) = |t - s|^{2H}$. Standard Brownian motion is the special case $H = \frac{1}{2}$, where increments are independent. For $H \neq \frac{1}{2}$ the increments are correlated across all lags: positively for $H > \frac{1}{2}$ (long-range dependence) and negatively for $H < \frac{1}{2}$ (anti-persistence). The covariance matrix of fBm increments $(\Delta B^H_0, \Delta B^H_1, \ldots, \Delta B^H_{n-1})$ is Toeplitz with entries $\gamma_H(k) = \frac{1}{2}\bigl(|k+1|^{2H} - 2|k|^{2H} + |k-1|^{2H}\bigr)$; for $n = 3$ this is:

At $H = \frac{1}{2}$ we have $\gamma_H(k) = 0$ for all $k \ge 1$, recovering the identity covariance of standard Brownian increments. At $H = 0.1$ the off-diagonal entries are negative, encoding the anti-persistent structure of rough volatility. The rough volatility model of Gatheral, Jaisson, and Rosenbaum calibrated the log-volatility process of equity and index options and found $H \approx 0.1$, far below the Brownian value.^[7] At $H = 0.1$ the sample paths of fBm are far rougher than Brownian motion: the classical Ito calculus does not apply, and a theory of integration against such paths requires the rough paths framework.^[8]

The table below summarizes the geometric interpretation of the standard models.

The Avellaneda-Stoikov market-making model^[6] is worth examining in this context. It assumes a flat GBM mid-price and derives the optimal bid and ask quotes by solving a Hamilton-Jacobi-Bellman equation. The reservation price is

a linear (affine) correction for inventory $q$ with risk aversion $\gamma$. The optimal spread is

The linearity of the inventory correction and the symmetry of the optimal quotes are both consequences of the flat geometry. On a curved state space, the Hamilton-Jacobi-Bellman equation acquires curvature corrections, the inventory correction becomes nonlinear, and the optimal bid and ask are no longer symmetric around the mid-price. Whether these corrections are material at millisecond scales is an open question.

6. Autocorrelation Structure Across Scales

The variance ratio test detects memory in aggregate. The autocorrelation function resolves it by lag.

Empirical results. We compute the signed-volume autocorrelation at sub-second lags (1 ms to 1 s) and the log-return autocorrelation at super-second lags (1 s to 1 hr) for both assets. The heatmap below shows both dimensions simultaneously.

Key finding. The signed-volume autocorrelation at lag 1 ms is negative for both assets, with the lag-1 return autocorrelation at $\delta t = 1\,\mathrm{s}$ agreeing with the PolyBackTest finding of $-0.048$ at $p < 0.001$. The negative autocorrelation decays rapidly: by lag 10 ms it is near zero, and by 1 s it is indistinguishable from zero at the 95% level. Short-horizon negative autocorrelation followed by rapid decay to zero is a universal stylized fact across equity markets.^[21]

Equity comparison. Roll derived the mechanism from first principles: in any two-sided market with a positive bid-ask spread $s$, the serial covariance of transaction-price changes satisfies $\mathrm{Cov}(\Delta p_t, \Delta p_{t-1}) = -s^2/4$, generating a mechanical negative lag-1 autocorrelation regardless of the underlying fundamental process.^[23] For NYSE-listed equities including AAPL, GE, IBM, and MSFT, this bid-ask bounce accounts for the majority of the negative serial correlation at tick-by-tick resolution, with the residual attributable to order-book replenishment dynamics analogous to the microstructure mean reversion measured here. Roll's formula implies that the lag-1 autocorrelation should be more negative for illiquid securities (wider spread) and less negative for highly liquid ones. BTC and ETH on Binance spot have spreads of roughly 0.01-0.02% at millisecond resolution, placing them in the same range as large-cap equities like AAPL and SPY, and the observed autocorrelation magnitudes are consistent with this prediction.

7. Spectral Decomposition: Wavelets and Geometric Harmonics

The Wiener-Khinchin theorem decomposes variance by frequency. Wavelets refine this by isolating variance at specific timescales without losing time localization.

Proof sketch. The MRA orthogonal decomposition $L^2(\mathbb{R}) = \bigoplus_{j=1}^{\infty} W_j$ (where $W_j = V_{j+1} \ominus V_j$) gives an orthogonal direct sum. Any $X \in L^2(\Omega)$ can be written as

where the scaling function contribution vanishes in the $L^2$ limit because the approximation spaces $V_j$ capture only coarser-and-coarser structure as $j \to -\infty$ (the completeness condition of the MRA). The orthogonality of the $W_j$ subspaces gives

For a stationary process, $\mathbb{E}[X_t] = \mu$ is constant, so $\mathrm{Var}(X) = \|X - \mu\|^2_{L^2(\Omega)}$. The wavelet filter $\psi_j(t) = 2^{-j/2}\psi(2^{-j}t)$ is a bandpass filter that extracts the variance in the octave band $[2^{-(j+1)}, 2^{-j}]$ of the power spectrum. The wavelet variance at scale $2^j$ is therefore

where $S_X$ is the power spectral density. Summing over all scales and using the Parseval-Wiener relation $\mathrm{Var}(X) = \int_{-\infty}^{\infty} S_X(f)\,\mathrm{d}f$ together with the fact that the wavelet filter banks partition the frequency axis, one recovers $\mathrm{Var}(X) = \sum_{j=1}^{\infty} \nu^2_j$.^[10] $\square$

The wavelet variance profile $j \mapsto \nu^2_j$ is the time-localized analogue of the power spectrum: it shows which scales carry the variance, without the stationarity assumption required by the Fourier transform.

Proof sketch. Let $W = D^{-1/2}(I - D^{-1}W_\varepsilon)D^{1/2}$ denote the symmetrized graph Laplacian (here temporarily overloading $W_\varepsilon$ for the weight matrix). Its eigendecomposition gives orthonormal eigenvectors $\phi_k$ with eigenvalues $1 \ge \lambda_1 \ge \lambda_2 \ge \cdots \ge 0$. The diffusion map embeds each point $x_i$ into $\mathbb{R}^m$ via

where $\psi_k = D^{-1/2}\phi_k$ are the (right) eigenvectors of the row-normalized Markov matrix $M = D^{-1}W_\varepsilon$. The squared Euclidean distance in this embedding is

On the other hand, the $t$-step Markov transition kernel satisfies $p_t(x, \cdot) = \sum_k \lambda_k^t \psi_k(x)\psi_k(\cdot)$ by the spectral expansion. Hence

The equality shows that the Euclidean distance in the diffusion map coordinates is exactly the $L^2$ distance between the diffusion kernels $p_t(x,\cdot)$ and $p_t(y,\cdot)$. As $\varepsilon \to 0$ and $n \to \infty$, the Markov matrix $M$ converges to the heat semigroup of the Laplace-Beltrami operator on $M$, and the diffusion distance converges to the geometric diffusion distance on the underlying manifold.^[11] $\square$

The diffusion distance provides a notion of geometric proximity that is intrinsic to the data manifold, rather than imposed by a Euclidean embedding. For price data, it means that two market states are "similar" not because their log-return values happen to be close, but because the conditional distributions of future returns are close in $L^2$.

Empirical results. We apply a Daubechies-4 (D4) wavelet decomposition to BTC and ETH log-returns, computing the wavelet variance at each of eight decomposition levels in a 30-day rolling window stepped daily across 2025.

Key finding. The wavelet variance profile is not flat: the bulk of the variance sits at the finest scales (sub-100 ms), consistent with the mean-reversion and negative autocorrelation evidence. The profile shifts slightly toward coarser scales during periods of high market volatility (Q1 2025 BTC bull run), suggesting that trending behaviour temporarily transfers variance from fine scales to coarse ones.

Equity comparison. Andersen, Bollerslev, Diebold, and Ebens computed realized volatility for all 30 DJIA component stocks at 5-minute resolution and found the same qualitative wavelet structure: variance is concentrated at fine timescales under normal conditions, with spectral mass shifting toward daily and weekly scales during volatility episodes such as the 1998 LTCM crisis.^[25] Percival and Walden's textbook treatment of wavelet variance uses S&P 500 return data as the primary example, demonstrating the bottom-heavy profile for large-cap equity indices as the canonical case.^[10] The BTC/ETH profile here is consistent with what is observed for GE, MSFT, JPMorgan, and the DJIA aggregate, shifted to finer timescales by roughly one order of magnitude owing to the higher trade frequency on cryptocurrency exchanges.

8. Information Geometry and Cross-Asset Flow

The statistical models that describe the price process at different resolutions are themselves elements of a geometric space. This is the domain of information geometry.

Proof (scalar case). Let $T = T(X)$ be an unbiased estimator of $\theta$, so $\mathbb{E}_\theta[T] = \theta$ for all $\theta$. Let $s_\theta = \partial_\theta \log p_\theta$ denote the score function.

Step 1: Score has mean zero. Differentiating the identity $\int p_\theta(x)\,\mathrm{d}x = 1$ with respect to $\theta$ under the integral sign (justified by dominated convergence under regularity conditions),

Step 2: Covariance identity. Differentiating $\mathbb{E}_\theta[T] = \theta$ with respect to $\theta$,

Since $\mathbb{E}[s_\theta] = 0$, this gives $\mathrm{Cov}(T, s_\theta) = \mathbb{E}[T \cdot s_\theta] - \mathbb{E}[T]\mathbb{E}[s_\theta] = 1$.

Step 3: Cauchy-Schwarz. By the Cauchy-Schwarz inequality for covariances,

where $I(\theta) = \mathbb{E}[s_\theta^2]$ is the Fisher information (using Step 1: $\mathrm{Var}(s_\theta) = \mathbb{E}[s_\theta^2]$). Substituting $\mathrm{Cov}(T, s_\theta) = 1$ gives

Equality. The Cauchy-Schwarz inequality is tight if and only if $T - \theta$ is proportional to $s_\theta$ almost surely. This means $T = \theta + c(\theta)\,s_\theta$ for some function $c(\theta)$. For an exponential family $p_\theta(x) = h(x)\exp(\eta(\theta)T(x) - A(\theta))$, the sufficient statistic $T(x)$ satisfies exactly this condition, so the MLE achieves the bound. $\square$