Theory v2.0 materials developed by Moses Kelley, including the v2.0 correction note and associated theoretical framework resolving the Hubble tension.\The PCP‑Lattice framework models the quantum vacuum as a dual‑lattice gauge system whose global consistency is verified in the sense of Probabilistically Checkable Proofs (PCP). In the corrected v2.0 formulation, the late‑time Hubble parameter is not modified by a direct PCP error factor (as in v1.0), but instead by a logarithmic coupling between dual‑lattice determinant evolution and effective energy density. This yields the master relation[H_{\text{local}} = H_{\text{early}} \bigl[1 + \frac{\xi}{2}\ln(\det \Lambda^*_{\text{now}}/\det \Lambda^*_{\text{then}})\bigr]] with [\xi \approx 0.41] and a calibrated dual‑lattice volume ratio [R_{\det} \approx 1.507], corresponding to a 50.7% increase in dual‑lattice volume between recombination and today. The resulting prediction [H_{\text{local}} = 73.01 \pm 0.47] km/s/Mpc agrees with SH0ES [H_0 = 73.04 \pm 1.04] km/s/Mpc to within 0.03σ while remaining anchored to a Planck‑like early‑universe value [H_{\text{early}} = 67.36 \pm 0.54] km/s/Mpc.\The record also includes a fiducial redshift‑dependent H(z) model in which the integrated constraint‑relaxation effect is realized as a smooth correction to a ΛCDM background, parametrized by a transition redshift [z_t \approx 0.7] and a sharpness index [n \approx 2]. This model predicts: (i) a modest [\sim -0.3\%] shift in the BAO sound horizon, (ii) a few‑percent modification of luminosity distances and distance moduli for Type Ia supernovae around [z \sim 1], (iii) percent‑level differences in cosmic‑chronometer H(z) measurements compatible with current data, and (iv) an 8.4σ discriminant between PCP‑Lattice and Planck‑ΛCDM using low‑redshift gravitational‑wave standard sirens.\Together, these materials provide a self‑contained specification of the PCP‑Lattice v2.0 theory, the corrected Hubble‑tension resolution mechanism, and its main astrophysical, experimental, and computational applications