Exact and Certified Tail Computation for Walsh Interaction-Order Energies

Paper-3 develops an exact and reproducible method for computing difficult tail probabilities of Walsh interaction-order energies on the Boolean cube. This work is part of a three-paper progression: Paper-1 — Exact Permutation Moments: derives exact finite-sample mean, variance, and covariance formulas for Walsh interaction-order energies under permutation. Paper-2 — Exact Distribution Laws: extends the framework to structured exact distributions, Krawtchouk-based reductions, computational-complexity boundaries, and continuation-horizon classification. Paper-3 — Exact Certified Tail Computation: focuses on computing and certifying exact tail probabilities for a structured finite \(n=5\) Walsh/Clebsch problem without constructing the entire exact distribution. The central challenge in Paper-3 is that, even when exact permutation moments are known, evaluating a single extreme tail can still involve an enormous exact state space. The paper develops a sequence of exact reductions that converts this problem into a structured deterministic counting problem. What Paper-3 does: reduces Walsh interaction-order energies through exact quotient identities; uses the 16-vertex Clebsch graph representation to reconstruct \(E_2\) and \(E_4\); derives exact conditional moments through sixth order; shows that the generic exact support contains\(16!/1920 = 10,897,286,400\) atoms; constructs an exact sign-coordinate hierarchy; uses deterministic pruning certificates to eliminate whole branches; incorporates exact \(S_5\) orbit multiplicities; applies final-pair meet-in-the-middle reductions; produces certified exact tail probabilities for three prespecified thresholds. Main exact results for the locked \(n=5\) fixture: \(p = 0.1704015660\) \(p = 0.03839474972\) \(p = 0.01302539172\) These values come from exact integer counts over the finite permutation space rather than Monte Carlo estimation. How the three papers connect:Paper-1 establishes the exact finite-sample permutation-moment foundation. Paper-2 extends that foundation to structured exact distribution laws and classification results. Paper-3 then addresses the practical single-tail problem by developing certified exact counting methods for a fixed structured case. Why it is useful:The method provides a transparent route for computing rare-event probabilities when building the complete exact probability distribution would be unnecessarily large. The mathematical reduction, deterministic counting, exact certificates, and computational reproduction can each be independently checked. Reproducibility:The computational implementations for Papers 1, 2, and 3 are brought together in QDL Research Suite v1.1.0, DOI 10.5281/zenodo.22969308. Scope:The Paper-3 result is exact for the stated finite \(n=5\) Walsh/Clebsch problem. It does not claim a universal arbitrary-\(n\) exact PMF/CDF algorithm, a new physical mechanism, causation, future-state prediction, or retrocausality. In simple terms:Paper-1 explains the exact permutation moments, Paper-2 develops structured exact distributions, and Paper-3 shows how a very large exact tail problem can be compressed into a smaller certified counting problem so that difficult probabilities can be calculated exactly and independently verified.