A Master Conjecture Unifying the Hardy–Littlewood Prime Pair Conjecture and the Strong Goldbach Conjecture: Partial Results

We develop a finite cosine-sum profile whose coordinates include prime differences and Goldbach sums. Exact discrete orthogonality identifies its Fourier average with the prime-differencecount plus half the ordered Goldbach count. This motivates the Original Unified Master Conjecture, one asymptotic formula uniform over 1 ≤ k ≤ n. Its coordinates include every fixedpositive shift and the Goldbach endpoint. It gives the Hardy–Littlewood asymptotic at everyfixed positive shift and the expected Goldbach asymptotic at the endpoint, hence representationsof all sufficiently large even integers.Normalizing the reduced-residue part of the same average by the coordinate-dependent shapeof the conjectured main term gives a profile Yn,k(P), where P is a detector prime. For P the leastprime greater than αn, we prove a sharp transition for Yn,k(P) > 1 across the symmetric workingrange 1 ≤ |k| ≤ n + 2. Let C2 be the twin-prime constant. For every fixed α > 6/C2 ≈ 9.09,only OA(n(log 2n)−A) coordinates fail, for every A > 0; at α = 6/C2, only o(n) fail. For3 < α < 6/C2, the failure proportion has an explicit positive limit: it equals 1 up to α = 2/C2and is then given by an integral against the Erdős–Wintner law of the singular factor, vanishingcontinuously at 6/C2. At equality, the decay rate and eventual success at every coordinateremain open. The result is unconditional but ineffective, with no effective starting point claimedfor either analytic input. The result concerns the detector inequality, not the truth of theOriginal Unified Master Conjecture.For the least prime above 24n, exact computations verify Yn,k(P) > 1 throughout the rangefor 8 ≤ 2n ≤ 105. A bilateral corollary extracts quantitative prime-difference bounds at one endof the profile and, at the other, Goldbach bounds giving representations at every nonexceptionaltarget in a band near 2n. Exceptional coordinates remain, so neither the fixed-shift primepair conjecture nor Goldbach’s conjecture for every even integer follows. We further recordobstructions to several direct approaches.