cascade-heat14-heat11-stationary-ar2-fastpaths

Heat14 starts from Heat11. It preserves Heat11's family weights, curriculum, observation model, integrated-SV correction, and AR(1) stationary-SD correction. It adds one mathematical update and four narrow algorithmic fast paths.

Mathematical and algorithmic AR(2) update

Heat11 used a zero-initialized AR(2), generated 512 discarded burn-in points, and treated sampled sigma as innovation SD. Heat14 uses the PACF parameterization

a1 = p1*(1-p2), a2 = p2.

Yule-Walker gives

rho(1) = p1

and

sigma_eps^2 / sigma_x^2 = (1-p1^2)*(1-p2^2).

Heat14 therefore uses

sigma_eps = sigma_x*sqrt((1-p1^2)*(1-p2^2))

and samples the two presample states directly from the exact stationary Gaussian law:

x[-2] = sigma_x*z0

x[-1] = sigma_x*(p1*z0 + sqrt(1-p1^2)*z1).

A Numba recurrence then emits x[0:L]. This removes all 512 discarded points, removes finite-burn bias, preserves the configured PACF, and makes sigma the stationary process SD. AR(2) has 9.5% final and 12% curriculum-start mass.

References:

• Wharton ARMA covariance notes
• Burn-in selection in stationary simulation

Throughput fast paths

1. ou_stochastic_vol replaces two Python loops of per-row SciPy lfilter calls with the existing compiled batched AR(1) recurrence. Initial conditions and equations are unchanged.
2. physical_sensors does not generate a spectral GP for pressure rows because the pressure branch completely replaces the base containing that GP. The branch law is unchanged; only unused work is removed.
3. seasonal_counts samples Gamma shape parameters only for rows selected for Gamma-Poisson overdispersion. The Poisson call remains batched.
4. regime_shift samples its three sparse Bernoulli jump fields by drawing an exact Binomial event count and a uniform subset of positions, rather than allocating three dense uniform matrices. This is distributionally identical to independent Bernoulli events and is already used elsewhere in Heat11. No fastmath or parallel Numba mode is used. This avoids floating-point reassociation and thread-scheduling overhead for modest family batches.

Implementation guidance follows the Numba performance recommendations: compile stateful recurrences and avoid unnecessary temporary arrays. NumPy's random Generator API remains unchanged.

Scope and validation

Fixed-seed corpus bytes change because unused random draws are removed and AR(2) is intentionally recalibrated. Non-AR(2) family distributions are intended to remain unchanged. The relevant gates are:

• warmed, alternating Heat11/Heat14 throughput benchmarks;
• exact recurrence comparison for the OU fast path;
• AR(2) stationary SD and ACF checks across parameter extremes;
• unchanged config apart from name and description;
• cascade verify in the pinned worker image.