Prime-Marked Reciprocal Dispersion for Binary Goldbach: Exact Heath-Brown Reassembly, Local Four-Form Covariance, High-Frequency Fourier Analysis, and the Outstanding Signed Long-Modulus Estimate (PIHR-Gold-45)

We present a self-contained formulation of the PIHR–Goldbach programme through Gold-45, centred on prime-marked reciprocal dispersion. The method couples an exact six-fold Heath-Brown identity with a two-prime correlation in which an external marked prime remains visible. We establish the relevant convolution reassembly, the finite local covariance identity for four affine forms, an exact packet-centring commutator, a high-frequency Fourier-energy decomposition, an elementary reciprocal-incidence estimate, and a Weil-bound argument for smoothly weighted reciprocal modulus sums. In the exponent configuration of interest, the latter argument yields a power-saving bound for genuinely separable, smoothly weighted packages. We show why the same deduction does not extend without proof to arbitrary long Möbius or Heath-Brown modulus weights. A quantitative conditional completion criterion is given for a signed long-modulus estimate together with the remaining Chen–Goldbach ledger. The fixed-shift Hardy–Littlewood–Chowla result and recent progress on Kloosterman bilinear forms motivate the approach but do not supply the missing fixed-target estimate. The document is a conditional research programme, not a proof of binary Goldbach.