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

AbstractWe develop a finite cosine-sum profile whose coordinates unify prime counts, prime differences,and Goldbach sums. Exact discrete orthogonality identifies its Fourier average with the primedifference count plus half the ordered Goldbach count. This motivates the Original UnifiedMaster Conjecture, one asymptotic formula uniform in the shift. It recovers the Prime NumberTheorem at the central coordinate, gives the Hardy–Littlewood asymptotic at every fixed positiveshift, and gives the expected Goldbach asymptotic at the endpoint, hence representations of allsufficiently 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 |k| ≤ n + 2. Let C2 be the twin-prime constant. For every fixed α > 6/C2 ≈ 9.09, onlyOA(n(log 2n)−A) coordinates fail, for every A > 0; at α = 6/C2, only o(n) fail. For 3 < α < 6/C2,the failure proportion has an explicit positive limit: it equals 1 up to α = 2/C2 and is then givenby an integral against the Erdős–Wintner law of the singular factor, vanishing continuously at6/C2. At equality, the decay rate and eventual success at every coordinate remain open. Theresult is unconditional but ineffective, with no effective starting point claimed for either analyticinput. The result concerns the detector inequality, not the truth of the Original Unified MasterConjecture.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 prime-pairconjecture nor Goldbach’s conjecture for every even integer follows. On the extended range [6, 4n],we also conjecture that the primes below 2n miss asymptotically (CA/C2)(log 2n)2even targets,where CA is Artin’s constant. We further record obstructions to several direct approaches