Finite-Window Noncovering on Primorial Wheels: Higher-Order CRT Bounds and Shift Correlations

Let \(C=ap_k^\#\) and consider symmetric offsets \(\{C-d,C+d\}\) under the sieve of Eratosthenes. Small primes generate a primorial wheel, while each later prime forbids one or two lift residues. The full Chinese remainder theorem (CRT) pattern has positive density, but a fixed center supplies only a translated fragment. We characterize terminal survivors as the disjoint union of a conservative avoiding set and explicit endpoint-prime exceptions, obtaining an exact unequal-prime-pair count and \(R_G(2C)\geq\max\{0,|U(C)|-1\}\). We derive arbitrary-order finite-phase CRT intersection formulas and odd Bonferroni lower bounds. The third-order bound gives \(|U(86)|\geq3\) and \(|U(128)|\geq2\), whereas first-moment, spanning-tree, and complete-block bounds are nonpositive; the latter certificate guarantees a prime-pair representation. We also express the survivor count as a shift-aware cyclic Fourier correlation, factor the pattern transform locally, and obtain computable spectral bounds. For a fixed wheel, \(\log P\sim\sqrt{2ap_k^\#}\), so the complete-block condition is asymptotically too restrictive. Exact-integer computations reproduce all finite cases. The results isolate the finite-window obstruction but prove neither an unconditional Goldbach theorem nor a new infinite prime-pair family.