Investigate central-mass CDF validation to avoid far-tail false failures

[tyrneh]

Summary

CDF validation currently checks the full derived probability grid, including extreme extrapolated tails. In live-data examples, tiny finite-difference or fitting issues at far-tail strikes can fail the whole probability materialization even when the economically relevant central distribution is well behaved.

One possible improvement: validate the central 99% of the implied distribution for hard failures, while treating far-tail violations as diagnostics or warnings unless they are severe.

Motivation

A live GME README example failed on the furthest fitted expiry (2027-01-15) during CDF materialization:

CalculationError: Failed to compute CDF: Direct CDF is materially below zero.

The fitted vol slice itself succeeded. The failure came from a tiny raw-CDF breach at the far-left boundary:

raw CDF minimum: -1.69e-05
current lower-bound threshold: -1.00e-05
failing strike: 0.01
observed strikes: 3 to 65
post-IV survival domain: 8 to 50
resolved probability domain: 0.01 to 1560

The issue disappeared with a denser fixed native grid:

auto / 2500 points: fails
4000 points: passes
8000 points: passes
12000 points: passes

This suggests the failure is a numerical far-tail artifact caused by a very wide extrapolated domain and finite-difference sensitivity, not a failure of the central fitted distribution.

Proposed direction

Investigate changing direct-CDF validation so that:

• hard failures focus on the central probability mass, e.g. the central 99% distribution;
• extreme-tail CDF bound violations are still recorded in metadata and warning diagnostics;
• severe tail violations still fail, especially if they imply material negative probability mass or large violations near observed strikes;
• validation distinguishes between observed/fit-supported regions and extrapolated tail regions.

Design questions

• How should the central 99% region be identified when the raw CDF is the object under validation?
• Should the threshold be configurable, e.g. central_mass=0.99?
• Should central-mass validation use raw CDF, minimally cleaned CDF, PDF-integrated CDF, or a two-pass approach?
• What tail diagnostics should be exposed so users can still audit model risk?
• Should the validation region be based on probability mass, observed strike support, or both?

Why this matters

The current behavior is conservative, but can make live examples brittle. From a user perspective, a tiny issue at a strike like 0.01, far outside observed quotes and economically irrelevant for most queries, should probably not block querying or plotting the central distribution.