A Computational Audit of a Candidate Counterexample to Sun's (2,4,6,8) Conjecture

A long-standing computationally tested conjecture about representations by mixed-degree binomial coefficients may have reached an unexpected turning point. Sun's (2,4,6,8) conjecture (OEIS A306477) asserts that every positive integer n greater than or equal to 4 can be represented as: n = C(w,2) + C(x,4) + C(y,6) + C(z,8), with w >= 2, x >= 4, y >= 6, and z >= 8. The conjecture had previously been computationally verified up to 2 x 10^12. This study reports an independently audited candidate counterexample at: n* = 896,315,812,331,399, approximately 448 times beyond the previous verification range. The result is not based on a single program or an incomplete search. Using 12 independently implemented computational kernels in C++ and Python/Numba, together with strict algebraic bounds that make the relevant search space finite, we exhaustively obtain: R(n*) = 0. An independent external checker reaches the same conclusion. A particularly useful local control is provided by the neighboring integers. The immediately preceding integer satisfies R(n*-1) = 1, with an explicit representation: (w, x, y, z) = (33,663,667; 9,433; 16; 9). The integer n* itself has no representation, while n*+1 is again representable. Thus the candidate occurs at a sharply localized gap rather than simply at the edge of an incompletely explored search region. But the computational result raises a deeper question: Why does this particular integer fail? For fixed x, y, and z, representability reduces to a square-value condition. Define: D = 8 x (n* - C(x,4) - C(y,6) - C(z,8)) + 1. A representation exists precisely when D is an appropriate odd square. Across the complete finite search domain, no admissible triple produces the required square. Interestingly, the minimum observed discriminant gap is only 8. This suggests that the phenomenon is not explained by a simple modular obstruction. Instead, it may reflect a more subtle interaction between mixed-degree additive structure and square values. The study therefore has two connected objectives. First, it audits and formalizes the computational evidence for the first candidate counterexample, including the current status of Lean 4 verification. Second, it begins a targeted search for a possible second counterexample. So far, 5,030,500 candidate evaluations have been performed, including complete enumeration of the interval n* +/- 50,000 and a continuous search near the 10^15 scale. No second counterexample has been identified so far. The central question is therefore no longer simply whether the conjecture survives computation. The deeper questions are: Why does the exceptional integer n* fail to be represented? Do other exceptional integers exist? Can we develop a theory explaining the sparse "holes" in the mixed-degree additive set generated by C(x,4), C(y,6), and C(z,8)? The current evidence is deliberately separated into five levels: THEOREM, COMP_VERIF, FORMALIZATION, NUMERICAL, and OPEN. The candidate is supported by exhaustive computation and independent computational replication. However, a publicly reproducible end-to-end Lean certificate and an analytic explanation of the non-representability remain open. This work is therefore intended not merely as a report of a large computation, but as a starting point for a broader investigation into exceptional holes, representation density, and square-value avoidance in mixed-degree binomial coefficient sequences.