[Sad Catapella N2][Scientific correctness] Tail second moments and pooled standard deviation can overflow despite finite true spread #33

Baseline and classification

Normalization Batch 2, commit f784c95e3049483f87466150ffc5764375fa631d. Confirmed numerical overflow path, with isolated NumPy squared-value probe. Full library runtime not tested.

Evidence

• _TailSums.add, lines 691–696 accumulates np.sum(values * values) in float64. A finite float64 pixel of magnitude > sqrt(float64 max) has an infinite square. _table_tail uses analogous squared values.
• PooledHistogram.std, lines 157–164 squares deviations directly for no-tail data; _std_with_tails, lines 343–357 subtracts large raw second moments, and max(...,0) can hide cancellation, while infinite second moments yield nonfinite results.
• WeightedHistogram.std, lines 281–288 shares these paths. z_score uses the returned standard deviation; its parameter constructor refuses nonfinite scale, despite a finite true standard deviation.
• Tail moment test uses int32 values (~2^31), not large finite float64 magnitudes.

Minimal counterexamples

• Two finite pixels [-1e200, +1e200] have population standard deviation 1e200 (finite), but their squared deviations overflow float64, causing PooledHistogram.std to return infinity or a nonfinite result.
• With a robust histogram whose extreme finite 1e200 values enter the tail, _TailSums.add stores sum_of_squares=inf, so pooled tail-based variance cannot be reconstructed reliably from saved schema-v4 moments, even though true std can be finite. An isolated NumPy probe confirmed (np.array([1e200])**2) == inf.

Impact

Valid floating-point measurements or upstream transformed arrays can yield an invalid z-score/weighted z-score despite mathematically finite dispersion; for tail-derived moments the persisted representation may already have lost information. This is a representation/algorithm issue, not just an output formatter.

Fix and tests

For direct/ordinary bins, use a scaled/centered variance computation that avoids overflow and catastrophic cancellation. For robust tails, introduce a stable mergeable moment representation (e.g., centered mean/variance with scale and count, or suitable normalized moments) and a versioned data/contract migration if necessary; do not silently reinterpret existing schema-v4 inf second moments. Test ±1e200, constant 1e200, 1e200±1e185, weighted conditions, mixed blocks, 32/64-bit floats and CSV round-trip, with high-precision reference and explicit diagnostic for truly unsupported magnitudes. Preserve ordinary-range numeric parity where required. Benchmark stable implementation before any Numba use.