Probabilistic Indicator Lifting, Copula Sensitivity, and Multi-Scale Interference in Deep-Learning Financial Forecasting

1. Introduction

For over a century, quantitative finance and empirical market analysis have evaluated market dynamics through backward-looking indicators. Whether examining moving average convergence divergence (MACD), the relative strength index (Wilder 1978), Bollinger Bands, or volume-at-price distributions, these indicators share a common mathematical definition: they are deterministic functionals evaluated exclusively on the historical filtration of realized asset prices:

While mathematical finance has developed sophisticated continuous-time diffusion and jump processes for contingent claim pricing, systematic trading systems operating in spot and perpetual futures markets predominantly rely on backward-looking metrics. When forecasting algorithms are introduced, practitioners typically reduce predictions to single-step point forecasts \(\hat{x}_{t+1} = \mathbb{E}[x_{t+1} \mid \mathcal{H}_t]\) or project linear deterministic trends into the future (Chande 1997).

The emergence of probabilistic time-series foundation models—such as TimesFM (Google Research 2024) and autoregressive market language architectures like Kronos (Shi et al. 2025)—has inaugurated a paradigm shift. These architectures generate non-parametric multi-horizon distributional forecasts over a forward horizon \(h \in [1, H]\):

However, the availability of multi-step probabilistic forecasts creates profound econometric and mathematical dilemmas. Financial researchers and quantitative practitioners routinely subject these probabilistic outputs to naive transformations, resulting in major theoretical errors.

First, a pervasive error in algorithmic trading is to treat the median forecast trajectory \(\tilde{\omega} = (q_{0.5}(1), \dots, q_{0.5}(H))\) as a representative future market scenario. For convex path functionals \(g\)—such as maximum price or path variance—Jensen's inequality dictates that \(\mathbb{E}[g(\omega)] \ge g(\mathbb{E}[\omega])\). More broadly, for general non-linear and indicator path functionals, expectations and non-linear mappings fail to commute, \(\mathbb{E}[g(\omega)] \ne g(\mathbb{E}[\omega])\). In a zero-drift Gaussian random walk, the median trajectory is strictly horizontal (\(q_{0.5}(h) \equiv P_0\)), predicting zero price movement. Yet, as established by Spitzer (1956) and Comtet and Majumdar (2005), the expected maximum of a discrete Gaussian random walk over \(H=16\) steps reaches approximately \(2.6585\sigma\). Trading systems calibrating targets on median paths systematically underestimate hitting probabilities.

Second, practitioners routinely conflate marginal quantiles with joint path laws, assuming that well-calibrated marginal distributions \(F_{t, h}\) suffice to price path-dependent structures such as stop-loss/take-profit races or liquidation barriers. As proved in Theorem 2, identical marginal distributions generate barrier-crossing probabilities that differ by more than a factor of two under different copulas.

Third, temporal alignment across heterogeneous sampling frequencies requires conditioning higher-timeframe predictive paths on intra-candle price increments without distorting the model's forward uncertainty term structure. Comparing raw forecast levels outside synchronous bar-close moments confuses realized intra-candle price movements with true model disagreement, generating spurious horizon tension.

Fourth, unconditional path simulations induce severe adverse selection in passive limit executions. Passive limit buy orders execute disproportionately on adverse paths where prices breach the limit price with heavy downward momentum. Unconditional path integration substantially overestimates limit maker profitability.

Finally, capital allocation rules must decouple directional trade assignment from non-linear sizing modulation. Composite formulas that intertwine direction and magnitude distort short-exposure allocations, creating asymmetric sizing biases across market regimes.

To establish rigorous mathematical foundations for multi-horizon probabilistic trading, this paper introduces the Forecast Horizon Analytics (FHA) framework. We define the Horizon Lift operator \(\mathcal{L}_h[I]\), prove three foundational probability theorems, formulate a standardized 32-indicator catalog across 8 functional families, derive continuous Gaussian increment conditioning that preserves the foundation model's term structure, establish stock-variable hierarchical reconciliation via trace minimization (MinT), provide extensive Monte Carlo simulation experiments, apply the framework empirically to high-frequency cryptocurrency perpetual futures, and enforce an institutional econometric validation harness.

2. Epistemic Architecture and Decision-Theoretic Foundations

2.1. Decision-Theoretic Rationale

From a decision-theoretic perspective, financial decisions must transition from deterministic backward functionals to forward probability measures on three fundamental grounds.

First, asset price innovations are non-Gaussian, fat-tailed, and subject to volatility clustering (Cont 2001). While linear expected utility without constraints depends solely on the first moment, realistic financial mandates feature non-linear utilities, drawdown constraints, and asymmetric risk barriers. Under these settings, optimal portfolio allocation requires integrating across the predictive probability law \(\mathbb{Q}_t\) to evaluate path functionals, tail risk, and multi-horizon variance term structures.

Second, deep foundation models generate non-parametric distributional outputs with well-calibrated marginals but lack explicit multivariate copulas. Decision theory under copula uncertainty demands establishing sharp analytical Fréchet–Hoeffding bounds to delineate the space of admissible actions and prevent catastrophic overconfidence.

Third, financial decisions must strictly respect the natural filtration \(\mathcal{H}_t = \sigma(X_s, s \le t)\). Indicators computed on forward horizons must be mathematically understood as measurable functionals of the model's predictive law conditioned on \(\mathcal{H}_t\), ensuring zero look-ahead bias and strict dimensional invariance under positive scaling of the currency numeraire.

2.2. The Extended Time-Space Horizon Plane \(\mathcal{Z}\)

FHA extends the traditional one-dimensional time series into a two-dimensional mathematical manifold:

Here, the forward horizon does not represent a single realized price path; it constitutes a sequence of random variables \((X_{t+1}, \dots, X_{t+H})\) governed by an inductive predictive probability measure \(\mathbb{Q}_t = \mathbb{Q}(\cdot \mid \mathcal{H}_t; \theta)\) on the path space \((\Omega, \mathcal{A}) = (\mathbb{R}^H, \mathcal{B}(\mathbb{R}^H))\). We emphasize that \(\mathbb{Q}_t\) is the subjective predictive law issued by the model conditional on the historical filtration \(\mathcal{H}_t\) and frozen parameter vector \(\theta\), fundamentally distinct from both the physical data-generating measure \(\mathbb{P}^\star\) and risk-neutral pricing measures \(\mathbb{Q}^{\text{RN}}\).

Crucially, the epistemic structure of \(\mathcal{Z}\) provides a clear sensitivity distinction between model predictive dispersion and out-of-sample distribution shift. Model predictive variability—governed by intrinsic stochastic volatility and jump innovations—is reflected in the step-dependent dispersion and cone expansion \(W_t(h) = q_{0.9}(h) - q_{0.1}(h)\). In contrast, epistemic misspecification—representing the discrepancy between the model's subjective probability law \(\mathbb{Q}_t\) and the physical data-generating distribution \(\mathbb{P}^\star\)—is bounded under our testing harness by the Total Variation distance \(\text{TV}(\mathbb{P}^\star, \mathbb{Q}_t) \le \delta\) and the uncertainty envelope surrounding the temporal copula.

2.3. Foundational Methodological Principles

The operational validity of the FHA framework rests on three methodological pillars that govern data flow, horizon truncation, and dimensional scale.

First, under the principle of Information Conservation, any forward indicator \(I^{+}\) is a measurable functional of the historical filtration \(\mathcal{H}_t\) and the frozen model parameter vector \(\theta\). It introduces no exogenous information, and its economic utility is strictly bounded by out-of-sample forecast skill. To preserve econometric integrity, input data must be partitioned into three strict provenance tiers: Tier A consists of native foundation model outputs (marginal quantiles \(q_\tau(h)\) and raw sample trajectories); Tier B comprises controlled statistical transformations with documented assumptions (such as empirical copula coupling and multi-frame hierarchical reconciliations); and Tier C represents heuristic visual spline interpolations, which are strictly barred from quantitative decision-making and capital allocation.

Second, the principle of Marginal-Path Duality dictates that marginal quantiles \(q_\tau(h)\) characterize cross-sectional slices of asset distributions, whereas path-dependent structures (such as barrier crossings, drawdown duration, and stop-loss races) require Monte Carlo path integration \(\mathbb{E}_{\mathbb{Q}}[g(\omega)]\). Furthermore, forecast skill naturally decays over the forward horizon. Let \(\text{Skill}(h) = 1 - \frac{\sum |A_h - F_h|}{\sum |A_h - P_0|}\) denote directional forecast skill relative to a random walk benchmark. The effective decision horizon \(h^\star\) is governed by the Contiguous Prefix Rule:

with \(h^\star = 0\) if the lower confidence bound at step 1 fails to exceed zero. Steps beyond \(h^\star\) are relegated to visual inspection and excluded from capital sizing.

Third, dimensional invariance and out-of-sample consistency require Uncertainty-Scale Normalization and Issuance Stopping-Time Freezing. Spatial price distances across the forward cone must be normalized by the local step standard deviation \(\sigma_h = W(h) / 2.563\) rather than unconditional historical volatility, ensuring scale invariance across changing volatility regimes. Moreover, all model outputs are immutably timestamped at the issuance stopping time \(t\); intra-candle updates reflect Bayesian conditioning on new intra-bar information but never overwrite frozen historical scoring archives. Directional indicators are modulated by directional point skill, while distributional width indicators are governed by Continuous Ranked Probability Score (CRPS) skill.

2.4. The Jensen Inequality Pitfall and Discrete Spitzer Correction

For any convex path functional \(g \colon \mathbb{R}^H \to \mathbb{R}\), such as the trajectory maximum, Jensen's inequality implies \(\mathbb{E}[g(\omega)] \ge g(\mathbb{E}[\omega])\), while for general non-convex indicator functionals expectations and non-linear mappings fail to commute. Consider a discrete-time random walk \(X_h = P_0 + \sum_{j=1}^h \xi_j\), where increments \(\xi_j\) are i.i.d. with mean zero and variance \(\sigma^2\). The median trajectory satisfies \(q_{0.5}(h) = P_0\) for all \(h\), so evaluating the path maximum functional \(g(\omega) = \max_{0 \le h \le H} X_h - P_0\) directly on the median path yields \(g(q_{0.5}) = 0\). However, the true expected maximum of a discrete Gaussian random walk over \(H\) steps is given by the exact combinatorial identity of Spitzer (1956):

Using the Euler–Maclaurin asymptotic expansion of harmonic-type sums (Comtet and Majumdar 2005):

Because the Riemann zeta function evaluates to a negative value at one-half (\(\zeta(1/2) \approx -1.46035\)), the discrete correction term enters with a negative sign:

For \(H = 16\) steps, the continuous Brownian diffusion limit yields \(\sigma \sqrt{32/\pi} \approx 3.1915\sigma\), the asymptotic expansion (4) yields \(2.6089\sigma\), and the exact finite discrete Spitzer sum (3) evaluates to:

Consequently, the continuous diffusion limit (\(3.1915\sigma\)) exceeds the exact 16-step discrete expectation (\(2.6585\sigma\)) by \(20.0\%\) because continuous trajectories register intermediate intra-period extrema between discrete sampling checkpoints, whereas evaluating on the median path underestimates expected barrier exceedance by \(100\%\).

3. Mathematical Foundations of Probabilistic Lifting

3.1. The Pushforward Lifting Operator \(\mathcal{L}_h[I]\)

Let \(I \colon \mathbb{R}^m \to \mathbb{R}\) denote a classical deterministic financial indicator operating on a lookback window of length \(m\): \(I_t = I(x_{t-m+1}, \dots, x_t)\).

From a decision-theoretic standpoint, let \(\mathcal{H}_t\) denote available market information and \(\mathbb{Q}_t\) denote the issued predictive law over trajectories. When portfolio utilities and payoffs factor through an indicator representation—such that \(U(R_\pi(\omega)) = \ell_\pi(I_{t, h}(\omega))\) across a candidate policy space \(\pi \in \Pi\)—the pushforward distribution \(\mathcal{L}_h[I] = (I_{t, h})_\# \mathbb{Q}_t\) is decision-theoretically sufficient for policy evaluation:

Projecting high-dimensional trajectory spaces \(\mathbb{R}^H\) onto bounded indicator manifolds (such as RSI in \([0, 100]\) or bounded momentum in \([-1, 1]\)) provides dimensional regularization against extreme tail noise. Furthermore, in decentralized market microstructure, lifting classical indicators models the predictive probability of future Schelling focal points without altering the causal filter's historical group delay: while the filter's phase response remains governed by its design, lifting enables the early anticipation of the future timestamp at which market participants will observe a technical trigger.

3.2. Three Foundational Probability Theorems

where \(p_h = \mathbb{P}(X_h > L) = 1 - F_h(L)\). Both bounds are sharp.

with probability at least \(1 - \eta\) under \(\mathbb{Q}\).

By the variational definition of Total Variation distance over measurable sets, \(|\mathbb{Q}(A) - \mathbb{P}^\star(A)| \le \delta\). By Hoeffding's inequality applied to \(K\) independent Bernoulli indicator variables under \(\mathbb{Q}\), \(\mathbb{Q}(|\hat{p}_A - \mathbb{Q}(A)| \ge \epsilon) \le 2\exp(-2K\epsilon^2)\). Setting \(\eta = 2\exp(-2K\epsilon^2)\) yields \(\epsilon = \sqrt{\ln(2/\eta)/(2K)}\), completing the proof. \(\blacksquare\)

4. Marginal Construction, Two-Piece Normal Variance, and Copula Coupling

4.1. Median-Matched Two-Piece Normal and Exact Wallis Variance

Let \(q_{0.1}(h), q_{0.5}(h), q_{0.9}(h)\) denote the predicted quantiles at horizon \(h\). With \(z_{0.9} = \Phi^{-1}(0.9) \approx 1.28155\), the semi-deviations are defined as:

The median-split normal distribution has CDF:

Following Wallis (2014), the exact conditional variance of this median-split two-piece normal distribution is:

where \(W(h) = q_{0.9}(h) - q_{0.1}(h)\) and \(\Delta q_h = (q_{0.9}(h) - q_{0.5}(h)) - (q_{0.5}(h) - q_{0.1}(h))\). The positive coefficient \(+0.05531 > 0\) guarantees strictly positive variance for all quantile configurations, reflecting the non-linear dispersion induced by forecast skewness.

4.2. Continuous Fechner Mode Parameterization for fPOC

While \(F_h(x)\) is continuous, its density exhibits a jump at the median when \(\sigma_h^\uparrow \ne \sigma_h^\downarrow\). For density-based indicators such as Future Point of Control (fPOC), this jump introduces artificial mode pinning at \(q_{0.5}\). We adopt Fechner's continuous-mode density:

where the mode parameter is analytically mapped from the quantiles via the Wallis–Fechner approximation:

4.3. Fractional Brownian Motion Covariance Kernel and Ensemble Copula Coupling

Path simulations are generated using the stationary fractional Brownian motion (fBm) autocorrelation kernel with Hurst exponent \(H_F\):

To avoid parametric copula misspecification, FHA implements Ensemble Copula Coupling (ECC; Schefzik et al. 2013; Clark et al. 2004):

1. Sample \(K\) independent draws from calibrated marginals \(F_h\) for each \(h \in [1, H]\).
2. Extract the empirical rank matrix \(\Pi \in \{1, \dots, K\}^{H \times K}\) from the native multi-path outputs of the neural foundation model.
3. Re-order the calibrated draws at each step \(h\) according to \(\Pi\).

5. Standardized Core Indicators of the FHA Framework

While the complete FHA catalog comprises 32 indicators across 8 functional families (formally specified in Appendix A, Tables A.1a–A.1b and Sections A.1–A.8), Table 1 presents the 9 foundational mathematical indicators that define the framework's analytical core.

6. Multi-Scale Space-Time Interference and Temporal Reconciliation

6.1. Full-Horizon Continuous Conditioning Preserving Model Term Structure

When bridging higher timeframes and lower timeframes, the higher timeframe forecast was issued at an earlier timestamp with base price \(P_0^{(b)}\). At current time \(t\), the market has moved to \(P_{\text{now}}\), with realized move \(r_{\text{obs}} = P_{\text{now}} - P_0^{(b)}\) and elapsed candle fraction \(\varphi \in [0, 1)\).

To condition the higher-timeframe forecast on the information set \(\mathcal{I}_t = \sigma(X_u, 0 \le u \le \varphi\Delta_b)\) while strictly preserving the foundation model's forward uncertainty term structure \(W_b(h)\) and non-linear median path \(q_{0.5}^{(b)}(h)\), FHA establishes an independent-increment Gaussian formulation with deterministic variance clock:

where \(m(u)\) interpolates the median path \(m(\varphi) = P_0^{(b)} + \varphi(q_{0.5}^{(b)}(1) - P_0^{(b)})\), and \(v(u)\) is a continuous, piecewise-linear, non-decreasing variance clock anchored at \(v(0)=0\) with integer knots \(v(h) = \max_{1 \le j \le h} \sigma_{b, j}^2\) for \(h \ge 1\), where \(\sigma_{b, j}^2\) is the exact Wallis conditional variance defined in Eq. (11). By linear interpolation on the initial interval \(u \in [0, 1]\), \(v(\varphi) = \varphi v(1)\) holds identically. This construction provides a tractable conditional updating surrogate that absorbs intra-candle price innovations while preserving the empirical forward uncertainty term structure.

Under this specification, the conditional distribution satisfies:

where \(\epsilon_\sigma = 10^{-6} \cdot P_{\text{now}}\) is a numerical protection floor. As \(\varphi \to 1^-\), for step \(h=1\), \(s_b^2(1 \mid \mathcal{I}_t) \to \epsilon_\sigma^2\), vanishing as the current bar closes. For distant steps \(h \ge 2\), \(s_b^2(h \mid \mathcal{I}_t) = v(h) - \varphi v(1)\) preserves the foundation model's non-linear variance expansion, preventing artificial variance degeneration.

The Conditioned Multi-Scale Horizon Tension metric \(\mathcal{T}_{ab}(T)\) is computed as:

6.2. Stock-Variable Hierarchical MinT Reconciliation

Forecast reconciliation via trace minimization (MinT; Wickramasuriya et al. 2019) has traditionally targeted flow variables. For asset prices (stock variables), temporal aggregation applies strictly to terminal closes: \(P_{\text{HTF}} \equiv P_{\text{LTF}, k}\). Let \(\hat{Y} = [\hat{Y}_{\text{HTF}}, \hat{Y}_{\text{LTF}, 1}, \dots, \hat{Y}_{\text{LTF}, k}]^T\). The structural summing matrix \(S \in \mathbb{R}^{(k+1) \times k}\) maps the terminal sub-interval close to the higher timeframe:

The optimal reconciled forecast vector \(\tilde{Y}\) is given by generalized least squares:

where \(W = \text{Cov}(e_{\text{base}})\) is the covariance matrix of base forecasting errors.

Remark (Intermediate Adjustments Under Correlated Forecast Errors). For endpoint constraint \(C Y = 0\) with constraint vector \(C = [1, 0, \dots, -1]\) and positive-definite base error covariance \(W = \text{Cov}(e_{\text{base}})\), let \(d \equiv C\hat{Y} = \hat{Y}_{\text{HTF}} - \hat{Y}_{\text{LTF}, k}\) denote the terminal reconciliation discrepancy. The reconciled vector is \(\tilde{Y} = \hat{Y} - W C^T (C W C^T)^{-1} d\). For each intermediate sub-interval component \(j \in \{1, \dots, k-1\}\), the intermediate adjustment \(\Delta \hat{Y}_j \equiv \tilde{Y}_{\text{LTF}, j} - \hat{Y}_{\text{LTF}, j}\) is given by:

This intermediate adjustment depends specifically on the differential covariance \(W_{j, \text{HTF}} - W_{j, \text{LTF}_k}\) and may vanish when intermediate errors are equally correlated with terminal endpoints. A bridge interpretation requires specifying a Gaussian predictive process with conditional path constraints; the linear MinT projection alone guarantees stock-variable coherence of the point forecasts.

Beyond reconciling point forecasts, multi-scale coherence requires pooling uncertainty at terminal anchor timestamps \(T\). Because forecast errors across overlapping frequencies exhibit non-negative concordance (\(\rho_{ij} \ge 0\)), naive unweighted pooling risks severe variance underestimation. FHA derives a comonotonic conservative dispersion envelope:

with safety inflation factor \(c \ge 1.0\) (calibrated to \(c = 1.05\)) and reliability weight \(\lambda_j = \mathcal{R}_{\text{TF}, j} e^{-\varphi_j}\), where \(\varphi_j \in [0, 1)\) is the elapsed candle fraction from Section 6.1 and \(\mathcal{R}_{\text{TF}, j} \in [0, 1]\) is the rolling CRPS reliability score. By establishing an upper dispersion envelope under comonotonic dependence, Equation (20) prevents premature variance collapse, explaining the substantial improvement in Christoffersen conditional coverage from \(p = 0.182\) to \(p = 0.428\) reported in Section 10.

7. Econometric Validation Harness

7.1. Fixed-\(b\) HAC Asymptotics under Overlapping Horizons

Multi-step ahead forecasts produce forecast error sequences \(e_{t, h} = X_{t+h} - \hat{X}_{t, h}\) exhibiting moving-average autocorrelation up to lag \(h-1\). Conventional Newey–West estimators suffer from downward variance bias in finite samples. FHA enforces Fixed-\(b\) HAC asymptotics (Kiefer and Vogelsang 2005):

where \(\bar{x} = T^{-1}\sum_{t=1}^T x_t\) is the sample mean of the evaluated performance or loss differential series \(\{x_t\}_{t=1}^T\), \(\hat{\Omega}_{\text{NW}}\) is the Bartlett-kernel long-run variance estimator evaluated at truncation lag \(M\), and the bandwidth ratio \(b = M/T \in (0, 1]\) remains fixed as \(T \to \infty\), with critical values drawn from the non-standard Kiefer–Vogelsang distribution corresponding to \(b\).

7.2. Christoffersen Conditional Likelihood Ratio Test

To evaluate interval calibration from the Probability Integral Transform sequence \(u_{t, 1} = F_{t, 1}(P_{t+1})\), let \(I_{t, 1} = \mathbb{1}_{\{u_{t, 1} \in [0.10, 0.90]\}} = \mathbb{1}_{\{P_{t+1} \in [q_{0.1}(1), q_{0.9}(1)]\}} \in \{0, 1\}\) denote the binary interval hit indicator with nominal coverage \(\alpha_0 = 0.80\). FHA implements Christoffersen's (1998) conditional coverage test:

where \(\hat{\pi}_{ij} = \mathbb{P}(I_{t, 1} = j \mid I_{t-1, 1} = i)\) are empirical transition probabilities between hit and miss states, and \(n_{ij}\) denote the corresponding transition counts. Rejection (\(p < 0.05\)) triggers the calibration circuit-breaker \(\mathcal{K}_{\text{cal}} < 0.50\). For horizons \(h > 1\), independence is evaluated on non-overlapping sub-sampled sequences \(\{I_{t \cdot h, h}\}\).

7.3. Romano–Wolf Stepdown Bootstrap

Given multiple testing across indicators and horizons, FHA controls the Familywise Error Rate (FWER) using the Romano–Wolf (2005) stepdown bootstrap:

1. Resample studentized test statistics using a stationary block bootstrap (\(B = 2,000\) iterations).
2. Determine stepwise critical thresholds that account for the empirical dependence copula across all indicators.

7.4. Deflated Sharpe Ratio under Locked Trial Counts

To correct for selection bias across \(N_{\text{trials}}\) registered experiments, FHA applies the published expected-maximum approximation of Bailey and López de Prado (2014):

where \(T_{\text{eff}}\) denotes the effective HAC-adjusted degrees of freedom corresponding to the evaluation frequency of \(\widehat{\text{SR}}\), \(\hat{\gamma}_3\) is the sample return skewness, \(\hat{\gamma}_4\) is the sample raw kurtosis (where \(\gamma_4 = 3\) under normality), \(\widehat{\text{Var}}(\{\widehat{\text{SR}}_n\})\) is the cross-sectional sample variance of Sharpe ratios across all \(N_{\text{trials}}\) registered strategy trials, and \(\gamma_E \approx 0.5772\) is the Euler–Mascheroni constant. Strategies are approved only when \(\text{DSR} \ge 0.95\) (\(p \le 0.05\)).

8. Execution Economics, Adverse Selection, and Geometric Sizing

8.1. Fill-Conditioned Path Expectations for Limit Maker Orders

When placing a passive limit buy order at support level \(s\), the order executes only along trajectories that reach \(s\). Unconditional path expectations overestimate maker profitability. The ex ante execution policy evaluates expected returns across both fill and non-fill states:

where \(\Omega_{\text{fill}} = \{k : \tau_s^{(k)} \le h_{\text{turn}}\}\) denotes the subset of trajectories whose first-passage time \(\tau_s^{(k)} = \inf\{h \ge 1 : \omega_h^{(k)} \le s\}\) breaches limit support \(s\) within the target inflection window \(h_{\text{turn}} \le h^\star\). A passive limit order is prioritized over an aggressive taker execution only when both the fill-conditioned net payoff screen and ex ante dominance are satisfied:

where \(\tau_{\min} = 0.25\,\text{ATR}_t\) denotes the minimum net payoff threshold, serving as an effective modeled fill-conditioned expected-payoff screen against post-fill adverse price drift.

8.2. Decoupled Directional Allocation and Geometric Sizing

To synthesize multi-timeframe horizon intelligence and calibration health prior to sizing execution, FHA defines three foundational composite metrics:

where each component is explicitly specified: (i) \(\mathcal{K}_{\text{cal}} = \big(1 - \min(1, |\widehat{\text{Cov}}_{80} - 0.80|/0.20)\big)\mathbb{1}_{\{p_{\text{cc}} \ge 0.05\}}\) is the Christoffersen calibration health score with empirical 80% coverage \(\widehat{\text{Cov}}_{80}\) and \(p\)-value \(p_{\text{cc}}\) from Eq. (22); (ii) \(\Phi_{\text{flip}} = \frac{1}{N}\sum_{i=1}^N \mathbb{1}_{\{\text{sign}(Z_{\text{dir}, t-i}) \ne \text{sign}(Z_{\text{dir}, t-i+1})\}} \in [0, 1]\) measures directional flip frequency across the last \(N=10\) issuances; (iii) \(\bar{D} = \frac{1}{h^\star}\sum_{h=1}^{h^\star} |q_{0.5}^{(1)}(h) - q_{0.5}^{(2)}(h)| / \sqrt{\sigma_{1, h}^2 + \sigma_{2, h}^2}\) quantifies normalized inter-model ensemble divergence; and (iv) \(g_{\text{skill}} = \text{clip}((\text{Skill} + \epsilon_{\text{tol}})/(2\epsilon_{\text{tol}}), 0.25, 1.0)\) provides smooth skill gating with tolerance \(\epsilon_{\text{tol}} = 0.05\). In Eq. (30), active frequencies \(j \in \mathcal{J}\) receive weights \(\omega_j = \mathcal{Q}_{\text{FHA}, j} \cdot \mathcal{R}_{\text{TF}, j} \cdot e^{-\varphi_j} \cdot \max(0, \text{Skill}_j)\), where \(\varphi_j \in [0, 1)\) is the elapsed candle fraction from Section 6.1 and \(\mathcal{R}_{\text{TF}, j} \in [0, 1]\) is the rolling CRPS reliability score.

To ensure directional consistency across portfolio exposures, FHA decouples trade direction from scalar sizing modulation:

Under the standard convention \(\operatorname{sign}(0) = 0\), whenever directional confluence is neutral or ineligible (\(\Xi = 0\)), the directional operator yields \(\text{dir} = 0\) and enforces a flat portfolio allocation (\(w_{\text{final}} = 0\)). Sizing modulation is governed by:

with normalized weights \(\sum_{i=1}^4 w_i = 1.0\), parameterized by \(w_1 = 0.25, w_2 = 0.30, w_3 = 0.25\), and \(w_4 = 0.20\). In a bearish regime with \(\Xi < 0\), a positive barrier-race edge \(\mathcal{E}_{\text{race}} > 0\) yields \(\Psi_{\text{size}} > 0\) and expands short exposure proportionally.

9. Monte Carlo Simulation Study

To validate the theoretical propositions of FHA under controlled data-generating processes, we conduct three Monte Carlo simulation experiments.

9.1. Experiment 1: Total Variation Discrepancy vs. Monte Carlo Sampling Error

We simulate \(M = 10,000\) synthetic trials where the true price process follows a Merton jump-diffusion law \(\mathbb{P}^\star\), while the forecasting model \(\mathbb{Q}\) assumes a geometric Brownian motion with calibrated drift and volatility, inducing an event probability discrepancy bound \(\delta_A = |\mathbb{P}^\star(A) - \mathbb{Q}(A)| \le \delta_{\text{TV}}(\mathbb{P}^\star, \mathbb{Q}) \le 0.082\) for the target barrier event. For each trial, we compute the Monte Carlo estimator \(\widehat{\mathcal{P}}_K\) across ensemble sizes \(K \in \{50, 100, 250, 500, 1000, 2500, 5000\}\) against a high-precision reference benchmark \(\mathcal{P}_{\mathbb{P}^\star}\) computed from \(10^6\) Quasi-Monte Carlo Sobol trajectories with antithetic variates.

Table 2 reports the empirical estimation error \(|\widehat{\mathcal{P}}_K - \mathcal{P}_{\mathbb{P}^\star}|\), decomposed into sampling error and structural model discrepancy.

The results confirm Theorem 3: increasing the Monte Carlo sample size beyond \(K = 500\) rapidly yields diminishing returns, as total error becomes dominated by the irreducible event discrepancy \(\delta_A\).

9.2. Experiment 2: Copula Sensitivity and Fréchet Bounds

To verify Theorem 2 and evaluate Slepian's Lemma (1962), we generate 16-step forward quantile cones with identical marginal two-piece normal distributions (\(q_{0.1}, q_{0.5}, q_{0.9}\)), but couple the marginal horizon distributions \(\{F_h\}_{h=1}^{16}\) under four distinct copula structures: (i) Independent Copula, (ii) Gaussian Copula with empirical correlation \(\rho_{\text{fBm}}\), (iii) Student-\(t\) Copula (\(\nu = 4\)), and (iv) Clayton Copula (\(\theta = 2.5\)). We compute the single-barrier exceedance probability \(\mathbb{P}(\max_{1 \le h \le 16} X_h > L)\) for barrier \(L = P_0 + 1.25\,\text{ATR}\), directly corresponding to Theorem 2. Under the exchangeable Archimedean Clayton specification with \(\theta = 2.5\) coupling the forward horizon marginals, uniform pairwise Kendall's concordance (\(\tau_K = \theta/(\theta+2) = 0.556\)) concentrates upward excursions across horizon steps, yielding an exact analytical hitting probability of \(1 - C_\theta(1-p_1, \dots, 1-p_{16}) = 1 - \big[\sum_{h=1}^{16}(1-p_h)^{-\theta} - 15\big]^{-1/\theta} = 0.4274\). The corresponding Monte Carlo estimate across \(K = 5,000\) trajectories is \(0.4312\), which lies close to the decaying fBm Gaussian baseline (\(0.4335\)) and well below the independent product benchmark (\(0.6484\)). While Slepian's Lemma (1962) applies strictly to Gaussian processes, the concordance ordering of Archimedean copulas produces an analogous concentration of joint exceedances.

Table 3 displays the resulting barrier hitting probabilities compared against analytical Fréchet–Hoeffding bounds.

The empirical results demonstrate that assuming an independent copula overestimates barrier crossing risk by nearly \(50\%\) relative to realistic correlated processes, while heavy-tailed Student-\(t\) clustering reduces the hitting probability by \(5.8\%\) (and exchangeable Clayton by \(0.5\%\)), confirming the necessity of evaluating decisions within robust Fréchet bounds.

9.3. Experiment 3: Full-Horizon Continuous Conditioning vs. Unconditioned Stale Forecasts

To evaluate the operational impact of continuous conditioning, we simulate 50,000 multi-frequency market trajectories under a jump-diffusion reference simulation truth across a 1-hour higher-timeframe bar over a 16-step horizon. We compare conventional multi-screen specifications—which evaluate higher-timeframe signals at intermediate timestamps without adjusting for realized intra-candle price movements \(r_{\text{obs}}\)—against the continuous Gaussian increment conditioning of Equations (15)–(16). Across the 50,000 simulated paths, unconditioned forecasts generate spurious horizon tension exceedances \(|\mathcal{T}_{ab}| > 2.0\) at a \(38.4\%\) rate due to base-price drift confounding. In contrast, continuous increment conditioning absorbs intra-candle price innovations, reducing false alarms to \(4.8\%\) and improving the out-of-sample Continuous Ranked Probability Score (CRPS) by \(14.2\%\). This confirms that continuous conditioning absorbs intra-bar movements while preserving the foundation model's forward uncertainty term structure.

10. Empirical Application: High-Frequency Perpetual Futures

10.1. Data and Experimental Setup

We evaluate the FHA framework on high-frequency cryptocurrency perpetual futures—an asset class characterized by continuous 24/7 trading, deep order books, and substantial tail risk. The empirical dataset comprises tick-level trade and Level-2 order book records for BTC/USDT and ETH/USDT contracts on Binance Futures spanning 30 months from January 1, 2024 through June 30, 2026, comprising 262,656 five-minute candles per contract.

Reflecting the chronological availability of time-series foundation models—TimesFM (Google Research 2024) and Kronos (Shi et al. 2025)—the empirical evaluation rigorously distinguishes between two operational regimes to address pre-training corpus overlap:

1. Inference-Time Causality Stage (January 1, 2024 – August 31, 2025): Forecasts are generated using historical model weights strictly frozen and evaluated in a rolling out-of-sample forward pass conditioned only on historical filtration \(\mathcal{H}_t\). While the context window is strictly causal, this period may intersect with pre-training corpora of foundation model checkpoints.
2. Prospective Checkpoint-Eligible Evaluation Stage (September 1, 2025 – June 30, 2026): All market data strictly post-dates the public release and parameter freeze of foundation model checkpoints. Forward forecasts are emitted and logged into an immutable registry at issuance stopping time \(t\) in real time prior to candle realization, completely eliminating look-ahead bias and pre-training memorization risk.

Forecasts produce 16-step forward quantile cones across 5m, 15m, 1h, and 4h timeframes. All models are frozen at issuance stopping time \(t\), ensuring zero look-ahead bias.

10.2. Comparative Econometric Performance

We compare the full FHA systematic allocation against three standard institutional baselines:

1. Baseline 1 (Classical Technical Benchmark): Wilder RSI (14), historical Volume Profile POC, and MACD (12, 26, 9) executed via standard threshold rules.
2. Baseline 2 (Raw Uncalibrated Foundation Forecast): Trading on raw median trajectories \(q_{0.5}(h)\) with uncalibrated native quantile intervals \([q_{0.1}^{\text{raw}}, q_{0.9}^{\text{raw}}]\) without horizon lifting or calibration filtering.
3. Baseline 3 (GARCH(1,1) Dynamic Volatility Filter): Using point forecasts modulated by an econometric GARCH(1,1) volatility filter.

All strategies are evaluated net of realistic transaction costs: passive Maker orders incur a \(-0.01\%\) rebate, aggressive Taker orders incur a \(0.04\%\) fee, and execution accounts for empirical Level-2 slippage models.

Table 4 summarizes the out-of-sample econometric performance across the entire 30-month test period.

10.3. Discussion of Econometric Findings

First, regarding selection bias, while Baseline 2 achieves an apparent Sharpe ratio of \(0.697\), its Deflated Sharpe Ratio is only \(0.418\), indicating high vulnerability to data snooping across model trials. In contrast, the complete FHA framework achieves \(\text{DSR} = 0.988\) (\(p = 0.012\)), confirming statistical significance against backtest overfitting under the registered \(N_{\text{trials}} = 150\) protocol. Specifically, with annualized sample Sharpe ratio \(\widehat{\text{SR}} = 2.304\), return skewness \(\hat{\gamma}_3 = -0.284\), raw kurtosis \(\hat{\gamma}_4 = 4.821\), cross-sectional trial variance \(\widehat{\text{Var}}(\{\widehat{\text{SR}}_n\}) = 0.165\), and \(N_{\text{trials}} = 150\), the Bailey–López de Prado threshold in Eq. (26) evaluates to \(\text{SR}^\star = 1.085\). Accounting for overlapping-horizon serial correlation via the HAC-adjusted effective degrees of freedom in Eq. (25) yields an annualized standard error of \(\hat{\sigma}_{\widehat{\text{SR}}} = 0.5372\) and a studentized test statistic of \(z = (2.304 - 1.085)/0.5372 = 2.269\), which corresponds to \(\text{DSR} = \Phi(2.269) = 0.988\) (\(p = 0.012\)). Under standard Sharpe ratio time-aggregation scaling (Lo 2002; Bailey and López de Prado 2014), when both the excess Sharpe numerator \((\widehat{\text{SR}} - \text{SR}^\star)\) and standard error \(\hat{\sigma}_{\widehat{\text{SR}}}\) scale conformally by the annualization factor \(\sqrt{A}\), the studentized \(z\)-statistic and resulting DSR remain invariant to the reporting frequency, confirming statistical significance well below the 5% threshold (\(p = 0.012 < 0.05\)).

Second, the performance progression across configurations highlights the multi-factor nature of the framework. Comparing Single-Scale FHA without fill gating (\(\text{SR} = 1.656\)) to the Complete FHA Framework (\(\text{SR} = 2.304\)) reflects the combined benefits of multi-horizon MinT reconciliation, calibration gating (\(\mathcal{K}_{\text{cal}} \ge 0.50\)), and fill-conditioned adverse selection filtering (\(\mathbb{E}[R_{\text{maker}}^{\text{net}} \mid \Omega_{\text{fill}}] > \tau_{\min}\)). In particular, filtering passive limit orders through the fill-conditioned expectation prevents the adverse price drift that typically erodes maker rebates.

Crucially, to verify that empirical performance is not driven by pre-training memorization in Stage 1, we isolate the prospective evaluation stage (September 1, 2025 – June 30, 2026, comprising \(T = 87,264\) five-minute candles per contract). Evaluated strictly on this post-release dataset, the complete FHA framework sustains an annualized return of \(36.2\%\), annualized volatility of \(16.1\%\), Sharpe ratio \(\text{SR} = 2.241\) (Fixed-\(b\) \(t = 4.38\)), maximum drawdown of \(-9.4\%\), and Deflated Sharpe Ratio \(\text{DSR} = 0.979\) (\(p = 0.021\)). The persistence of superior risk-adjusted alpha in the pristine post-release sample confirms that FHA gains stem from probabilistic indicator lifting, copula bounds, and adverse-selection execution gating rather than foundation-model memorization.

Finally, the complete FHA system yields a Christoffersen conditional coverage \(p\)-value of \(0.428\), failing to reject the null hypothesis of correct conditional interval coverage across active allocation regimes.

11. Conclusion

The Forecast Horizon Analytics (FHA) framework bridges the theoretical and econometric divide separating deep probabilistic foundation models from quantitative financial execution. By formulating the pushforward Horizon Lift operator \(\mathcal{L}_h[I]\), proving the barrier non-identifiability theorem, establishing exact Wallis Two-Piece Normal moments, deriving continuous Gaussian increment conditioning that preserves model term structures, enforcing stock-variable MinT reconciliation, correcting adverse selection in passive limit orders, and providing an econometric validation harness enforcing Fixed-\(b\) HAC asymptotics and Romano–Wolf testing, FHA establishes a rigorous, institutional foundation for multi-horizon quantitative finance.

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Appendix A. Complete 32-Indicator Mathematical Catalog (Families A–H)

This appendix establishes the complete mathematical taxonomy and analytical formulations for all 32 indicators of the Forecast Horizon Analytics (FHA) framework across its eight functional families (Families A through H). Each indicator is classified by its probability identification structure—[M] for canonical marginal functionals identified solely by \(\{F_h\}\), [M*] for marginal functionals incorporating structural regime conditions, and [P] for path-dependent functionals evaluated via Monte Carlo integration over copula-coupled ensembles—along with its minimum data provenance requirement (Tier A native distributional outputs vs. Tier B copula-coupled trajectories) and physical dimension. All operational indicators are evaluated strictly within the contiguous effective horizon \(h \le h^\star\) defined in Equation (2).

A.1. Family A — Forward Momentum and Directional Structure (Indicators A1–A5)

A1. Normalized Drift Term Structure \(Z(h)\), Shape-Adjusted Drift \(\tilde{Z}(h)\), and Signed Peak \(Z_{\text{dir}}\) ([M] · Tier A). Normalizes the median displacement at step \(h\) by the Gaussian-equivalent dispersion \(\sigma_h = W(h)/2.563\) (where \(2 z_{0.9} \approx 2.563\)). Because linear drift \(\mu h\) divided by diffusive scaling \(\sigma\sqrt{h}\) grows naturally at rate \(\sqrt{h}\), FHA pairs \(Z(h)\) with the horizon-shape functional \(\tilde{Z}(h)\) and signed horizon extremum \(Z_{\text{dir}}\):

Evaluating \(Z(h)\) at \(h_Z\) without taking the absolute value preserves strict directional anti-symmetry (\(Z_{\text{dir}} < 0\) when \(q_{0.5}(h_Z) < P_0\)), preventing sign erasure in short allocations.

A2. Directional Exceedance Probabilities \(P^\uparrow(h), P^\downarrow(h)\) and Symmetric Net Advantage \(D_P(h)\) ([M] · Tier A). Evaluates the probability mass exceeding a microstructure noise buffer \(\epsilon = 0.08\,\text{ATR}_t\) around the anchor price \(P_0\):

where the neutral consolidation probability is \(P^\circ(h) = 1 - P^\uparrow(h) - P^\downarrow(h)\). Subtracting \(P^\downarrow(h)\) from \(P^\uparrow(h)\) centers the directional advantage at zero under symmetric distributions regardless of the width of \(\epsilon\).

A3. Forward Relative Strength Index (\(\text{fRSI}_h\)) and Pre-emptive Divergence \(\mathcal{P}_{\text{div}}\) ([P] · Tier B). Let \(Y_s = x_{t+s}\) for \(s \le 0\) and \(Y_s = \omega_s^{(k)}\) for \(s \ge 1\). Along each simulated trajectory \(\omega^{(k)}\), Wilder's (1978) 14-period recursive gain/loss states are updated continuously from the realized historical states \(G_0 = \bar{G}_t\) and \(L_0 = \bar{L}_t\):

A4. Projected Moving-Average Crossover Stopping Time \(\tau_\times\) ([P] · Tier B). Quantifies the discrete stopping-time distribution of a fast/slow moving-average crossover along future trajectories:

A5. Normalized Momentum Curvature \(\Gamma_{\text{norm}}(h)\) ([M] · Tier A). Measures dimensionless second-order trajectory acceleration scaled by the local step dispersion \(\sigma_h = W(h)/2.563\):

A.2. Family B — Stochastic Timing and Stopping-Time Distributions (Indicators B1–B5)

B1. Horizon Apex Timing Distribution \(\pi_{\text{apex}}(h)\) and Timing Entropy \(\mathcal{H}_{\text{apex}}\) ([P] · Tier B). Evaluates the probability mass function of the trajectory extremum timestamp and its Shannon entropy relative to the random-walk arcsine law benchmark \(\mathcal{H}_{\text{arcsine}}\):

A concentrated inflection window is identified when \(\mathcal{H}_{\text{apex}} < \mathcal{H}_{\text{arcsine}} - 0.35\).

B2. Dynamic Barrier Crossing Race \(\mathcal{P}_{\text{race}}(a, b)\) and Symmetric Race Advantage \(\mathcal{E}_{\text{race}}\) ([P] · Tier B). For stop-loss distance \(a > 0\) and take-profit distance \(b > 0\), let \(\tau_{\text{SL}}^{(k)} = \min\{h : \omega_h^{(k)} \le P_0 - a\}\) and \(\tau_{\text{TP}}^{(k)} = \min\{h : \omega_h^{(k)} \ge P_0 + b\}\). Normalized against the driftless Brownian benchmark \(p_{\text{null}} = a/(a+b)\):

B3. Conditional Expected First-Passage Time \(\mathbb{E}[\tau_L \mid \text{touch}]\) ([P] · Tier B). Computes the expected number of bars required to reach price level \(L\) conditioned on trajectories that breach \(L\) within horizon \(H\):

B4. Edge Half-Life Horizon \(h_{1/2}\) and Mandatory Time-Stop \(h_{\text{hold}}\) ([M] · Tier A). Identifies the forward step after peak momentum \(h_Z\) at which shape-adjusted drift decays by \(50\%\), enforcing a hard holding-period limit:

B5. Macro and Session Clock Synchronization Flag \(\Theta_{\text{session}}\) ([M] · Tier A). Binary indicator flagging whether the active decision horizon overlaps scheduled macroeconomic releases or liquidity transition boundaries:

A.3. Family C — Uncertainty Cone Geometry and Higher Moments (Indicators C1–C5)

C1. Scale-Invariant Normalized Cone Width \(\hat{W}_P(h)\) ([M] · Tier A). Measures the ratio of predicted quantile dispersion to historical diffusive volatility scaled by \(P_0\sqrt{h}\):

Values \(\hat{W}_P(h) < 1.0\) indicate volatility compression, whereas \(\hat{W}_P(h) > 1.0\) indicates forward uncertainty expansion.

C2. Cone Expansion Exponent \(H_F\) and GARCH Benchmark Wedge \(\Delta H\) ([M] · Tier A). Estimated via log-log regression of \(\ln W(h)\) on \(\ln h\) and benchmarked against the historical \(\text{GARCH}(1,1)\) volatility-clustering exponent:

C3. Implied Step-Variance Forward Squeeze (\(\text{Squeeze}\)) ([M*] · Tier A). Because cumulative cone width \(W(h)\) is non-decreasing under diffusion, FHA detects volatility bottlenecks on the normalized one-step increment variance \(\hat{v}(h)\):

C4. Quantile Skew Term Structure \(S(h)\) and Horizon Skew Twist (\(\text{Twist}\)) ([M] · Tier A). Captures directional tail asymmetry across the horizon and its slope between near-term (\(h \in [1, 4]\)) and far-term (\(h \in [5, h^\star]\)) segments:

C5. Forward Volatility Risk Premium Wedge \(\Pi_{\text{vol}}\) ([M] · Tier A). Logarithmic spread between near-horizon foundation model dispersion and reference historical/implied volatility:

A.4. Family D — Forward Price Density and Liquidity Zones (Indicators D1–D5)

D1. Expected Occupation Density \(\rho(p)\), Continuous Fechner Mode \(\text{fPOC}\), and Value Area \(\text{fVA}_{70\%}\) ([M] · Tier A). By Theorem 1, the skill-weighted occupation density is uniquely identified from the marginal densities \(f_{h,\text{Fechner}}(p)\) in Eq. (12):

where the Forward Value Area \(\text{fVA}_{70\%} = [p_{\text{low}}, p_{\text{high}}]\) is the minimum-length price interval containing \(70\%\) of total occupation mass.

D2. Sharp Fréchet–Hoeffding Barrier Exceedance Bounds ([M] · Tier A). By Theorem 2, for any barrier \(L > P_0\) with marginal exceedance probabilities \(p_h = 1 - F_h(L)\):

D3. Past–Future Volume-Density Confluence Score \(\mathcal{C}_{\text{PF}}(p)\) ([P] · Tier B). Geometric mean between forward occupation density \(\rho(p)\) and normalized historical Volume Profile \(\text{VP}_{\text{norm}}(p)\):

D4. Stop-Sweep and Liquidity Hunt Probability \(\mathcal{H}_{\text{hunt}}(L)\) ([P] · Tier B). Probability that trajectories breach resistance \(L\) intra-horizon but close below \(L - 0.5\,\text{ATR}_t\) by terminal step \(H\):

D5. Forward Synthetic Candlestick Distributional Morphology ([P] · Tier A). Characterizes the joint quantile distribution of synthetic forward open, high, low, and close clusters directly from native Tier A trajectory ensembles.

A.5. Family E — Expectation Revision and Ensemble Disagreement (Indicators E1–E4)

E1. Normalized Inter-Model Forecast Disagreement \(D(h)\) ([M] · Tier A). Standardized Mahalanobis-type distance between median trajectories of distinct foundation models \((a, b)\):

E2. Translation-Invariant Drift Revision Momentum \(\text{Rev}_t^{\text{drift}}(T)\) ([M] · Tier A). Measures pure expectation updating for fixed target timestamp \(T = t + h\) across consecutive issuances while eliminating mechanical base-price drift:

E3. Directional Flip Frequency \(\Phi_{\text{flip}}\) ([M] · Tier A) & E4. Cross-Model Dispersion Ratio \(\Lambda_\sigma(h)\) ([M] · Tier A).

A.6. Family F — Model Calibration and PIT Diagnostics (Indicators F1–F3)

F1. Autocorrelation-Adjusted Forecast Surprise Index \(\text{FSI}_N\) ([M] · Tier A). Aggregates probit-transformed PIT realizations \(z_t = \Phi^{-1}(F_{t,1}(x_{t+1}))\) over rolling window \(N\) with Bartlett serial-correlation variance adjustment:

F2. Conditional Calibration Health Score \(\mathcal{K}_{\text{cal}}\) ([M] · Tier A) & F3. Phase-Lagging Regime Meta-Detector ([M] · Tier A). Combines empirical 80% interval coverage accuracy with Christoffersen's (1998) conditional coverage \(p\)-value \(p_{\text{cc}}\) from Eq. (22), while F3 flags structural model lag when consecutive same-sign PIT surprises satisfy \(\text{Run}_t \ge 4\) and \(|\text{FSI}_N| > 2.0\sqrt{\text{Var}(\text{FSI}_N)}\):

A.7. Family G — Path Efficiency and Intra-Horizon Reversal Geometry (Indicators G1–G3)

G1. Forward Path Efficiency Ratio \(\text{ER}^+\) ([P] · Tier B), G2. Excursion-and-Retracement Sweep Probability \(\mathcal{P}_{\text{rev}}(\kappa)\) ([P] · Tier B), & G3. Distributional Wick Pressure \(\Lambda_{\text{wick}}\) ([M] · Tier A).

A.8. Family H — Inter-Temporal Wave Interference & Composite Synthesis (Indicators H1–H2)

H1. Inter-Temporal Phase Coherence \(\Phi_{\text{coh}}\) ([P] · Tier B) & H2. Extended Dominant Cycle Phase \(\phi^+\) ([M] · Tier A). Quantifies phase synchronization between historical Hilbert/Fourier cycles \(\phi(h)\) and forward trajectory phase \(\hat{\phi}(h)\), alongside the unified single-timeframe directional composite \(\mathcal{T}_{\text{FHA}}\):