The Extra-Geeky Math

Base Model 1: Markov-Switching VAR with Trust-Threshold Transition

Core bet: Civic stress is governed by three discrete latent regimes (quiescent, primed-mobilizable, unity-inverted). The same indicator carries different coefficients across regimes. Getting the regime right matters more than getting any single weight right.

Empirical basis: E1, E2, E7, E12, E3, E13 — full range from paradoxically-low unity (E7) to near-maximum multi-channel (E13), plus high-unrest-at-low-economic-stress cases (E1, E2).

Base formula:

$$CSI_t = \sum_{s \in {Q,P,U}} \Pr(S_t = s | I_{1:t}) \cdot \mu_s(X_t)$$

where

$$\mu_s(X_t) = c_s + a_s^{\text{Trust}} \cdot \text{Trust}_t + a_s^{\text{Griev}} \cdot G_t + a_s^{\text{Econ}} \cdot E_t^{\text{signed}} + a_s^{\text{Pol}} \cdot P_t$$

Regime-specific calibration:

• QUIESCENT ($Q$): $c_Q = 22$, $a_{\text{Econ}} = 0$ (gate closed), $a_{\text{Griev}} = 0.15$, $a_{\text{Trust}} = 0.10$
• PRIMED ($P$): $c_P = 45$, $a_{\text{Econ}} = 0.4$, $a_{\text{Griev}} = 0.6$, $a_{\text{Trust}} = 0.5$
• UNITY-INVERT ($U$): $c_U = 12$, $a_{\text{Econ}} = -0.1$, $a_{\text{Griev}} = -0.15$, $a_{\text{Trust}} = -0.2$

Transition gates: $$\Pr(S_{t+1} = U | S_t) = \text{logit}^{-1}(-3 + 8 \cdot \text{ExternalAttrib}t)$$ $$\Pr(S{t+1} = P | S_t) = \text{logit}^{-1}(-2 + 3 \cdot \text{TrustDeficit}_t)$$

BM1-T1: Citation-Locked 3-Regime Threshold Filter

Verdict: ⚠️ FLAG

Issue: External-attribution definition ambiguous; design is salvageable with definition clarification.

BM1-T2: 3-Regime MS-VAR, Validated-Central

Verdict: ✗ FAIL

Issue: $\text{CSI}_{\text{obs},t}$ (ground-truth observation target for Hamilton filter) is undefined. The formula claims to estimate via EM, but there is no dependent variable to condition the filter on.

BM1-T3: Recency-Dominated 3-Regime MS-VAR (EWMA Trajectory)

Verdict: ⚠️ FLAG

Issue: PriorUnrestStock is undefined; two of three core data sources (event-count trajectory, institutional-trust survey) are missing for pre-1990 episodes.

BM1-T4: Structural-Floor MS-VECM

Verdict: ⚠️ FLAG

Issue: Sub-model relationships (cointegration rank, adjustment speeds) are under-specified; the error-correction matrix is stated but not operationalized.

BM1-T5: Operational-Saturated 3-Regime MS-VAR

Verdict: ⚠️ FLAG

Issue: The OPS/FUEL weight creates a falsifiable contradiction: formula predicts CSI=35 at 2020 when the regime switch to Primed should yield CSI≥55.

BM1-T6: SETAR Self-Amplification

Verdict: ✗ FAIL

Issue: This is a template, not a runnable formula. No threshold variable $z_t$ is specified; episode date ranges needed for the autoregressive history are missing.

BM1-T7: Reduced-Form VAR with Granger-Ordered Indicators

Verdict: ⚠️ FLAG

Issue: A release-valve term (channeling of grievance into institutions) is described in the rationale but not implemented in the matrix specification.

BM1-T8: Stratified-Cox Trust-Gated Hazard Survival

Verdict: ✗ FAIL

Issue: Episode chronology is absent — Cox model requires entry and exit times; ExternalAttributedShock is undefined operationally.

BM1-T9: Kalman Latent-Stress State-Space

Verdict: ✗ FAIL

Issue: The nonlinearity $H(p_t)$ (a logistic transformation of polarization) is incompatible with the linear Kalman filter. A particle filter or EKF would be required but is not specified.

BM1-T10: Orthogonal-Factor MS-VAR (Factor-PCA)

Verdict: ⚠️ FLAG

Issue: Annual survey data for the latent stress factors are sparse; DW-NOMINATE congressional polarization has gaps; specification is incomplete for monthly/quarterly output.

BM1-T11: Asymmetric-Loss MS-VAR (Reference-Dependent)

Verdict: ⚠️ FLAG

Issue: Technically salvageable; the ratchet reference (expectations rise with prior peaks) is a genuine behavioral mechanism supported by behavioral-economics literature.

BM1-T12: Cointegration-ECM with Regime-Dependent Adjustment

Verdict: ✗ FAIL

Issue: Calibration check fails: the formula produces the wrong sign on multi-episode checks. The long-run equilibrium is under-claimed.

BM1-T13: Antagonistic-Suppressor MS-VAR

Verdict: ⚠️ FLAG

Issue: No disqualifying defect. Three orthogonalized suppressors (unity, employment, infrastructure) are theoretically motivated but empirically unverified.

BM1-T14: Hierarchical Bayesian Regime-Switching CSI

Verdict: ⚠️ FLAG

Issue: The observation likelihood $\Pr(X_t^e | S_{t,e}, \theta)$ is not specified. Without a stated likelihood, the Bayesian update is meaningless.

BM1-T15: Wavelet-Multiscale Regime-Gated MS-VAR

Verdict: ✗ FAIL

Issue: The weight vector $w_U$ for the Unity regime sums to 0; the regime is degenerate and contributes nothing to the model.

BM1-T16: Copula-Coupled Regime-Switching (Transfer-Entropy)

Verdict: ✗ FAIL

Issue: Transfer-entropy weights and Gumbel copula tail-lift parameters are unidentified. No calibration procedure is given.

Base Model 2: Self-Exciting Hawkes Intensity with Attribution-Signed Marks

Core bet: Civic conflict is a marked self-exciting point process. Each unrest event transiently raises the probability of subsequent events. The contagion term dominates near-term prediction. Attribution-signed marks allow out-group shocks (E7) to depress rather than excite.

Empirical basis: E1, E10, E13, E14, E2, E5 — episodes with cascade and self-excitation; challenged by economic suppressors (E3, E4, E8, E12).

Base formula:

$$\text{CSI}t = 100 \cdot \Lambda\left(\frac{\mu(t)}{\mu{\max}}\right)$$

where

$$\mu(t) = \beta_0 + \beta_{\text{Trust}} (1 - \text{Trust}t) + \beta{\text{Griev}} \cdot G_t + \beta_{\text{Econ}} \cdot \max(0, \text{NecessityCPIgap}t) - \beta{\text{Unity}} \cdot \text{ExternalUnity}_t$$

Calibration targets: $\mu(2000\text{-}2005) \to \text{CSI} \approx 22$; $\mu(2020) \to \text{CSI} \approx 65$ (slow component only).

Full intensity (including contagion):

$$\lambda(t) = \mu(t) + \sum_{t_i < t} m_i \cdot \alpha \cdot \exp(-\delta \cdot (t - t_i))$$

where $m_i = (+1) \cdot \text{Magnitude}_i \cdot \text{Attribution}_i$ and $\text{Attribution}_i = -1$ if external out-group event, $+1$ otherwise.

$$\text{CSSI}t = 100 \cdot \Lambda\left(\frac{\lambda(t)}{\lambda{\max}}\right)$$

BM2-T1: Hawkes Background-Only: PITF-Hazard Conditional

Verdict: ⚠️ FLAG

Issue: PITF U6 indicator unavailable before 1994; 2 of 15 episodes (E1, E2) are uncomputable.

BM2-T2: Validated-Central Marked Hawkes

Verdict: ✗ FAIL

Issue: E1 (1968) cannot be computed; GDELT begins 1979, RSUI 1985. The dependent-variable baseline is undefined.

BM2-T3: Short-Kernel Marked Hawkes (EWMA Trajectory)

Verdict: ⚠️ FLAG

Issue: GrievanceStock proxy from GDELT v1.0 covers 1979+ only; pre-1979 episodes (E1, E2) are uncomputable.

BM2-T4: Structural-Floor Hawkes: Kalman-Filtered Slow Substrate

Verdict: ✗ FAIL

Issue: U6 data absent before 1994; hyperparameters of the Kalman filter are unspecified.

BM2-T5: Operational-Saturated Hawkes: Realized-Intensity Nowcast

Verdict: ⚠️ FLAG

Issue: Conceptually sound; specification is incomplete for actual execution.

BM2-T6: Threshold-Gated Self-Exciting Hawkes (SETAR Branching)

Verdict: ⚠️ FLAG

Issue: No disqualifying defect; specification gaps exist but are technical rather than conceptual.

BM2-T7: VAR-Granger Reduced-Form Linearization of Hawkes

Verdict: ✓ PASS

Issue: No disqualifying defects. Fully computable pre-2020 using Granger-ordered VAR as a reduced form. Handles the branching ratio through residual covariance.

BM2-T8: Cox-Hawkes Hazard-Survival

Verdict: ⚠️ FLAG

Issue: ExternalUnity indicator is undefined; construction method is missing.

BM2-T9: Self-Exciting Stress as Latent Linear-Gaussian SSM

Verdict: ✗ FAIL

Issue: 2000–2005 baseline requires ACLED/GDELT mark data that is absent before 2020.

BM2-T10: Orthogonalized-Background Marked Hawkes (Factor-PCA)

Verdict: ⚠️ FLAG

Issue: Annual survey data are sparse; DW-NOMINATE has gaps; specification is incomplete.

BM2-T11: Loss-Averse Hawkes: Reference-Dependent Background

Verdict: ✓ PASS

Issue: Honest about ACLED pre-2020 gap. Explicit two-regime framing (pre-2020 = survey-only; post-2020 = event-inclusive). No disqualifying defects.

BM2-T12: Cointegrated Background, Error-Correcting Hawkes

Verdict: ⚠️ FLAG

Issue: 2020 anchor unresolved; wage-composition problem (low-wage workers hit harder) is inherited from the background specification.

BM2-T13: Suppressor-Gated Hawkes: Threshold-Autoregressive

Verdict: ⚠️ FLAG

Issue: No disqualifying defect. Three orthogonalized suppressors (external threat, full employment, diffuse shock) are identified but unverified.

BM2-T14: Hierarchical Bayesian Marked Hawkes (Partial-Pooling)

Verdict: ⚠️ FLAG

Issue: Hindsight bias in calibration; pre-2020 episodes inherit parameter estimates from the full 1967–2023 sample.

BM2-T15: Wavelet-Multiscale Hawkes Spectral Dual

Verdict: ✗ FAIL

Issue: Calibration by assertion; no sample calculations provided.

BM2-T16: Copula-Coupled Hawkes (Transfer-Entropy Channel Weights)

Verdict: ⚠️ FLAG

Issue: E8 (2008–09) falsification: tail copula over-predicts financial-crisis contagion, producing CSI=55 when observed is 25.

Base Model 3: Multiplicative Threshold-Gated Product (SETAR on Exogenous Gates)

Core bet: Civic conflict is a product of conjunctive factors, several acting as near-binary gates. When all gates are open AND grievance × trigger product is large, intensity reaches 70–93; any closed gate forces it to 8–30.

Empirical basis: E12, E8, E7, E1, E5, E13 — full range of gated vs. ungated; challenged by distributional channels without strong gates (E6, E9, E11).

Base formula:

$$\text{CSI}t = \text{Baseline} + \text{CoreStress}t \cdot G{\text{econ}}(z{\text{econ},t}) \cdot G_{\text{attrib}}(z_{\text{attrib},t}) \cdot G_{\text{release}}(z_{\text{release},t})$$

where each gate $G_k(z_k)$ is a logistic transition function:

$$G_k(z_k) = \frac{1}{1 + \exp(-\gamma_k(z_k - z_{k,*}))}$$

The core stress is the product of grievance and trigger:

$$\text{CoreStress}_t = \left(\text{GrievanceStock}_t\right) \times \left(\text{TriggerMagnitude}_t\right)$$

BM3-T1 through BM3-T16

(16 variations with progressively more gates and more complex functional forms; see methodology page for verdicts and details)

Summary and Cross-Formula Patterns

The 7 passing formulas share three properties:

1. Estimated rather than imposed kernels (BM2-T7, BM2-T11): Let the data determine lag shapes and coupling strengths rather than asserting them from theory.

2. Honest about data gaps (BM2-T11): Explicitly two-regime (pre-/post-2020) rather than claiming one formula covers both.

3. Bayesian hierarchical structure (BM3-T14, BM7-T14): With only 6–9 effective calibration episodes, credible intervals are the only statistically defensible output, not false-precision point estimates.

4. Citation-locked conservatism (BM9-T1): Zero-weight any channel without a published effect size.

5. Regime-specific rather than global multipliers (BM10-T5): Annual-cadence gains are static between calibration points; treat them as fixed background, not fast drivers.

The 44 failing formulas fail on one of five dimensions:

• Data availability (18 FAILs): Formula’s dominant signal does not exist for the calibration period.
• Unspecified randomness (12 FAILs): Templates without operationalizations; no way to actually compute.
• Incompatible structure (6 FAILs): E.g., nonlinear functions in linear frameworks.
• Circular validation (5 FAILs): Uses ACLED/GDELT as both input and ground truth.
• Degenerate components (3 FAILs): Zero-sum weights, undefined thresholds, or weights summing to zero.

For detailed specification of each of the 170 formulas, cross-reference the methodology page where the 15 base models are fully documented.