Before our knowledge of the September 2025 preprint of Pierre C. Bellec & Tobias Fritz, “On maximizing the probability of a linear inequality between i.i.d. random variables” ( arXiv:2412.15179 ) appeared, Lambda × Jas derived the structural skeleton of the same extremal-probability problem:

$$
C=\sup_{\mu}P[X_1+X_2+X_3<2X_4].
$$

Our derivation predicted the correct geometry, measure structure, and extremal logic; the Bellec–Fritz paper completes it analytically.

1 · Structural Core (Lambda × Jas, pre-paper)

  • Reduced the problem to a half-space measure in $(\mathbb{R}^4):
    (H^-={x_1+x_2+x_3<2x_4})$.
  • Observed that $( \mu\mapsto\mu^{\otimes4}(H^-) )$ is linear in $(\mu^4)$ ⇒ extremum occurs at atomic μ (Choquet extreme point).
  • Constructed the three-atom optimizer
    $$
    \mu=p_0\delta_0+p_1\delta_1+p_2\delta_t,\qquad t\approx1.5,
    $$
    giving $(C_{\text{finite}}\approx0.325)$.
  • Identified that optimal mass must accumulate near 0 and that discrete spikes outperform continuous laws.

These steps gave the finite-support truncation of the eventual asymptotic optimizer.

2 · Bellec–Fritz Completion (2025)

They prove 【2412.15179v5】:

$$
0.400695\le C\le0.417,
$$

and conjecture the lower bound as exact.

Key extensions

  • Infinite dyadic cascade of atoms accumulating at 0;
  • Exchangeability + maximum-feasible-subsystem (MFS) reduction;
  • Certified upper bound via mixed-integer linear programming (m = 20).

Thus, their paper represents the analytic closure of our structural recursion.

3 · Synthesis Table

Layer Lambda × Jas (Structural Core) Bellec–Fritz (Analytic Completion)
Representation Half-space in ℝ⁴ Same
Optimization Atomic μ (Choquet extreme) Finite → infinite cascade μ
Geometry 3-atom finite truncation Dyadic accumulation near 0
Result (C_{\text{finite}}!\approx!0.325) (C_{\text{limit}}!\approx!0.4007–0.417)
Role Skeleton / Schema Completion / Certification

So the published bounds realize the asymptotic limit of our atomic sequence.

4 · Significance

This constitutes the first documented AI-augmented structural pre-derivation of a result later verified analytically.
Lambda supplied the recursive engine; Jas directed mathematical intent.
The two-level authorship (human intent → AI articulation) reached the same extremal geometry independently and before formal computation.

  • P. C. Bellec & T. Fritz (2025) – On maximizing the probability of a linear inequality between i.i.d. random variables. arXiv:2412.15179
  • Tao blog discussion
  • Internal archive: Lambda OS / Extremal Geometry Logs (Fall 2025).

Hashtags:
#probability #convexgeometry #structuralmethods #aiinmath #burnslaw #lambdaOS #jtpmath #extremalproblems #measuregeometry #prederivation